WALSH FUNCTIONS
AND THEIR
APPLICATIONS
K. G. BEAUCHAMP
Director of Computer Services,
University of Lancaster,
Lancaster, England
1975
ACADEMIC PRESS
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Preface
The basis for development in many areas of electrical engineering is a system
of sine and cosine functions. This is due in no small part to the desirable
properties of frequency domain representation of a large class of functions
encountered in the theoretical and practical aspects of engineering design.
Examples are found extensively in communication and the analysis of
stochastic problems where the complete and orthogonal properties of such a
system lend themselves to particularly attractive solutions to the identifica-
tion problem. With the application of digital techniques and semiconductor
technology to these areas has come awareness of other more general
complete systems of orthogonal functions, which although not possessing
some of the desirable properties of sine-cosine functions in linear time-
invariant networks, have other advantages rendering their use more directly
applicable to these applications.
The Walsh and Haar sets of functions form the most important of these.
They are characterised by assuming only two states, thus matching the
behaviour of digital logic, and yet possess many of the attractive manipula-
tive properties of the sine-cosine series.
Historically the Haar series was described first by the Hungarian
mathematician, Alfred Haar in 1910 1 . He proposed a set of orthogonal
functions, taking essentially only two values, such that the formal expansion
of a given continuous function in the new functions converges uniformly to
the given function. This was a property not possessed by any orthogonal set
known up to that time and his proposals served to emphasise the unifying
theories on orthogonal series which had been developed at Gottingen by
Schmidt et al. at the turn of the century 2 .
The Walsh functions were defined in 1923 by the American mathemati-
cian J. L. Walsh 3 . These functions also formed a complete orthogonal set
and, although taking only the values +1 and -1, were found to have many
properties similar to the trigonometric series. Nearly at the same time
(in 1922), but independant of Walsh, the German mathematician,
H. Rademacher, presented yet another set of two-level orthogonal
functions, which were found later to form an incomplete but true subset to
the Walsh functions 4 .
These three function sets form the basis of a new direction in communica-
tion and processing which will be described in the following pages. All these
VI
PREFACE
functions form enumerably infinite sets of periodic orthogonal square-wave
functions which are characterised by having piecewise constant value
between many infinite jump discontinuities. We shall, however, for the most
part be concerned with finite and discrete sets of such functions.
In his original paper, Walsh gave a recursive definition of the Walsh
functions that orders the functions according to the average number of zero
crossings for the function in the' orthogonality interval. This order was also
used by the Polish mathematician, S. Kaczmarz 5 , in his important
mathematical work on the series and, because of this, has been referred to as
the Walsh-Kaczmarz order. More recently H. F. Harmuth has proposed the
term “sequency order” for this, which has gained widespread acceptance 6 .
In 1931, an entirely different definition of the Walsh functions was
described by R. E. A. C. Paley 7 , another American mathematician. His
definition is based on finite products of Rademacher functions and the order
obtained was quite different from that of Walsh. It is related to a binary
decomposition of the index to the function so that the order is referred to as a
binary-ordered or “natural” series. A relationship between the two series
was given by F. Pichler 8 .
A much earlier approach to the Walsh function definition is through the
application of certain orthogonal matrices, containing only the entries +1
and -1. Some work on such matrices had been carried out by the British
mathematician, J. J. Sylvester in 1867 9 . This was generalized by the French
mathematician, M. J. Hadamard in 1893, who established a class of matrices
called after him 10 . A special set of these matrices can be shown to be directly
related to the Walsh series through a Kronecker product operator on a basic
Hadamard matrix. The Walsh functions obtained in this way represent a
third order, known as Kronecker-ordering which is related quite simply to
natural-ordering through a binary bit-inversion of their numerical position
in that order.
The papers cited above laid a firm foundation for the mathematical
properties of the Walsh and related functions which was virtually complete
by the 1930’s. However, publications which referred to engineering and
other applications did not begin to make their appearance until thirty years
later when the semi-conductor and the digital computer came into use. The
first developments were concerned with communications problems and here
much of the credit is due to H. F. Harmuth who introduced the Walsh
function into telecommunication engineering 11,12 . In the last decade, a
number of experimental communications systems, involving sequency mul-
tiplex equipment, have been developed in several European countries, the
U.S.A. and Canada.
Amongst the many areas of development, several of which will be
described later, are those in which computer processing of data, particularly
two-dimensional or real-time data has been carried out. Reference can be
PREFACE
vii
made to the pioneer work of Pratt 13 , Andrews 14 and many others, together
with applications in the field of television transmission by Enomoto 15 ,
Taki 16 , Wintz 17 and others. The principal advantage for the Walsh or Harr
transform in this work is the reduction obtained in the calculation
speed-storage space product when this is compared with equivalent Fourier
methods. This makes it particularly attractive when real-time processing is
carried out and/or large amounts of data are to be handled.
The widespread interest in practical applications has stimulated further
contributions to the mathematical theory especially in terms of the use of
digital computational methods. Of particular interest is the logical differen-
tial calculus of Gibbs 18,19 . Whereas the cosinusoidal functions often repre-
sent the characteristic solution to certain linear differential equations, so the
Walsh functions can be shown to represent solutions to what has become
known as the logical differential equation. Applications of the Gibbs
derivative are found in mathematical logic 20 , approximational theory 21 ,
statistics 22 and linear system theory 23 .
The continued development of Walsh function theory and applications
poses a number of problems for which solutions are not yet fully apparent.
There are some doubts as to the interpretative value of Walsh analysis
applied to certain Fourier-based signals, e.g. seismic events. The more
recent areas of application in the fields of electromagnetic radiation and
radar must take into account the considerable technical development that
will become necessary to exploit adequately the capabilities of the functions.
Finally, the cost-effectiveness of replacing an existing system, such as a
multiplexed communication system, with one based on the new functions
must also include consideration of the capital investment costs in related
equipment as well as technical performance of the replacing system.
Nevertheless the overall progress in understanding and applying the
Walsh and related functions over the last two decades has been remarkable
and more than sufficient to ensure a place alongside the earlier well-
established developments based on the sine-cosine relationships.
It is the intention with this book to present a broad appraisal of the theory
and use of Walsh and related functions together with a necessarily limited
survey of the current range of applications. The first four chapters are
tutorial in character and attempt to summarize the basic relationships for the
new range of functions. Comparisons are made between these functions and
the sine-cosine functions used in Fourier analysis. Domain transforms are
developed and their properties described. From this mathematical basis the
derivation of fast transform algorithms is given and programs discussed for
implementation on the digital computer.
Chapters 5 and 6 are concerned with the general principles of sequency
analysis and filtering which form the basis of very many applications for the
functions. Concepts equivalent to power spectral density are developed and
PREFACE
viii
applied to specific random processes. Again comparison is made to conven-
tional analysis using Fourier techniques. A difficulty in the application of
Walsh theory to correlation and convolution is the absence of a shift
theorem similar to that found in Fourier theory. An equivalent operation to
correlation and convolution is available and may be applied, providing the
time variable is considered to behave somewhat differently. This affects the
way in which filtering of discrete sampled data is carried out and is discussed
with reference to the classical process of filtering originally described by
Wiener 24 .
The final chapters, 7 and 8, are concerned with applications. The rapidity
of progress and diversity of application for the theory over a relatively short
period has been outstanding. During this time the use of Walsh and related
functions has proceeded from interesting variants on well-established
methods to unique, and in many cases cost-effective, solutions to specific
problems. Much of this work is reported at yearly symposia held in
Washington and Hatfield, which is referred to repeatedly throughout the
text.
Few works of this kind, aimed at a broad overview of a subject, can
represent solely the authors’ experience and this book is no exception. I
would like to acknowledge with grateful thanks the assistance given to me by
very many people, including Professor Cappellini of the Consiglio Nazionale
Delle Ricerche, Drs Kennett and Gubbins of the University of Cambridge,
Dr. Schollar of the Rheinishes Landesmuseum, Dr. Decker of Spectral
Imaging Corporation, Dr. Ahmed of Kansas State University, Dr. Gibbs of
the National Physical Laboratory, Mr. Walker of the British Broadcasting
Corporation and particularly Professor Harmuth of the Catholic Univer-
sity of America for his generous contributions and to Miss M. E. Williamson
of Cranfield Institute of Technology who was responsible for the digital
programming work described.
Acknowledgement is also extended to the Institution of Electrical
Engineers, the National Electronics Conference and the World Organisa-
tion of General Systems and Cybernetics for permission to include some of
the author’s earlier published work.
Finally I would like to express my appreciation to Brenda Latus for her
painstaking preparation of the final manuscript.
References
1. Haar, A. (1910). Zur Theorie der orthogonalen Funktionensysteme, Math.
Annal. 69, 331-371.
2. Schmidt, F. (1905). Zur Theorie des linearen und nichtlinearen Integral-
gleichungen. Math. Annal. 63, 433-476.
3. Walsh, J. L. (1923). A closed set of orthogonal functions. Amer. J. Math. 45,
5-24.
PREFACE
IX
4. Rademacher, H. (1922). Einige Satze von allgemeinen Orthogonalfunktionen.
Math. Anna! 87, 112-138.
5. Kaczmarz, S. (1929). Uber ein Orthogonalsystem. Comptes-Rendus du I.
Congres des mathematiciens des pays Slaves. Warsaw, 189-192.
6. Harmuth, H. F. (1964). Die Orthogonalteilung als Verallgemeinerung der Zeit
und Frequenzteilung. Archiv. Elektr. Ubertragung , 18, 43-50.
7. Paley, R. E. A. C. (1932). A remarkable series of orthogonal functions. Proc.
Lond. Math. Soc. 34, 241-279.
8. Pichler, F. (1967). Das System de sal und cal Funktionen als Erweiterung des
Systems der Walsh-Funktionen und die Theorie der sal und cal Fourier
Transformation. Ph.D. Thesis, University of Innsbruck, Austria.
9. Sylvester, J. J. (1867). Thoughts on inverse orthogonal matrices, simultaneous
sign-successions and tesselated pavements in two or more colours, with applica-
tions to Newton’s rule, ornamental tile-work, and the theory of numbers. Phil.
Mag. 34, 4, 461-475.
10. Hadamard, M. J. (1893). Resolution d’une question relative aux determinants.
Bull. Sci. Math. A17, 240-246.
1 1 . Harmuth, H. F. (1960). On the transmission of information by orthogonal time
functions. Trans. A.I.E.E. Comm, and Electronics 79, 248-255.
12. Harmuth, H. F. (1963). Tragersystem fur die Nachrichtentechnik. W. German
Patent 1-191-416, H50289 (U.S. Patent 3,470,324).
13. Pratt, W. K. (1969), Hadamard transform image coding. Proc. I.E.E.E. 57,
58-68.
14. Andrews, H. C. (1970). “Computer Techniques in Image Processing”.
Academic Press, New York and London.
15. Enomoto, H. and Shibata, K. (1970). Orthogonal transform coding system for
television. J. Inst. TV. Eng. Japan. 24, 2, 98-108.
16. Taki, Y. and Hatori, M. (1966). P.C.M. communication system using Hadamard
transformation. Electron. Comm. Japan 49, 11, 247-267.
17. Wintz, P. A. (1972). Transform picture coding. Proc. I.E.E.E. 60, 7, 809.
18. Gibbs, J. E. (1969). Walsh functions as solutions of a logical differential
equation. National Physical Laboratory, Teddington, England, DES Report,
No. 1.
19. Gibbs, J. E. (1970). Sine waves and Walsh waves in physics. 1970 Proceedings:
Aplications of Walsh functions, Washington D.C., AD 707431.
20. Liedl, P. (1970). Harmonische Analysis bei Aussagenkalkuelen. Math. Logik
13, 158-167.
2 1 . Butzer, P. L. and Wagner, H. J. ( 1 972). Walsh-Fourier series and the concept of
a derivative. “Applicable Analysis”, Vol. 1, pp. 29-46. Gordon and Breach,
London.
22. Pearl, J. (1971). Applications of the Walsh transform to statistical analysis.
Proceedings: 4th Hawaii Int. Conf. on System Science, 406-407.
23. Pichler, F. (1970). Some aspects of a theory of correlation with respect to Walsh
harmonic analysis. Maryland University, U.S. A. Report No. AD 714 596.
24. Wiener, N. (1949). The Extrapolation, Interpolation and Smoothing of Sta-
tionary Data. M.I.T. Press, Cambridge and New York.
K. G. Beauchamp
February 1975
Contents
PREFACE v
Chapter 1. ORTHOGONAL FUNCTIONS 1
IA Preamble 1
IR Sine-cosine functions 3
IC Incomplete function sets 5
ID Walsh functions 7
IE Haar functions 9
IF Other orthogonal functions 10
References 11
Chapter 2. THE WALSH FUNCTION SERIES 12
II A Definition of the Walsh series 12
IIB Function ordering 17
IIC Walsh function derivation 20
IID Hardware function generation 26
HE Relationship between WAL and PAL series 31
IIF Waveformsynthesisusing Walsh and Fourier series .... 33
IIG Digital sampling 36
IIH Modulo-2 arithmetic 38
References 38
Chapter 3. WALSH TRANSFORM ATION 40
IIIA Definition of the Walsh transform 40
IIIB Comparison with the discrete Fourier transform 41
IIIC Effects of circular time shift 42
HID Behaviour of transform products 46
HIE Walsh transformation of a sinusoid 47
IIIF Conversion between discrete Walsh and Fourier transforma-
tion 49
IIIG Summary of Walsh transform characteristics 49
IIIH The fast Walsh transform 52
IIII The R transform 59
IIIJ The generalised transform 60
IIIK Transform programming 63
XI
xii CONTENTS
IIIL Two dimensional transformation 63
HIM Hardware transformation 64
References 70
Chapter 4. THE HAAR FUNCTION 72
IVA Introduction 72
IVB Haar function definition 72
IVC Relationship between the Walsh and Haar functions .... 76
IVD The discrete Haar transform 78
IVE The fast Haar transform 79
IVF Two dimensional Haar transformation 80
IVG The Haar power spectrum 82
References 83
Chapter 5. SPECTRAL DECOMPOSITION 85
VA Walsh spectral analysis 85
VB Correlation and convolution 93
VC Applicability of the Weiner-Khintchine theory 96
VD The sequency spectrum via the autocorrelation function ... 96
VE The periodogram approach 98
VF Comparisons between Walsh and Fourier spectra 99
VG The odd-harmonic sequency spectrum 104
VH Short-term spectral analysis 109
References 113
Chapter 6. SEQUENCY FILTERING 115
VIA Sequency filtering 115
VIB Analog sequency filters 116
VIC Generalised Wiener filtering 117
VID Sequency-based vector filtering 119
VIE Frequency-based scalar filtering 123
VIF Filtering anon-stationary signal 126
VIG Two-dimensional filtering 132
References 136
Chapter 7. APPLICATIONS IN COMMUNICATIONS 140
VII A General 140
VIIB Communications applications 140
VIIC Multiplexing 141
VIID Coding systems 147
VIIE Image transmission 148
VIIF Electromagnetic radiation 155
VIIG Radar systems 158
References 163
Chapter 8. APPLICATIONS IN SIGNAL PROCESSING 166
VIII A Signal processing applications 166
VIIIB Spectroscopy 167
VIIIC Pattern recognition and image-processing ' 171
VIIID Acoustic image filtering 178
CONTENTS
VIIIE Speech processing 180
VIIIF Medical signal processing 182
VIIIG Non-linear applications 185
References 189
Appendix I
A Signal processing computer programs 192
B Program summary 192
C Fast Walsh transform routines FWT and FFWT 193
D Fast Walsh transform routines FHT and FRT 195
E Fast Haar transform routines HAAR, HAARIN and
HNORM 197
F Walsh power spectral density program PSDW 199
References 205
Appendix II. TABLES FOR MODULO-2 ADDITION R©S (with
R m ax = S m ax=125) 206
AUTHOR INDEX 223
SUBJECT INDEX 223
Chapter 1
Orthogonal Functions
IA Preamble
Representation of a time series by the superposition of members of a set of
simple functions, which are easy to generate and define is a useful attribute
that has many uses. It is important, for example, to be able to represent
waveforms used for communication in this way since generalisations can
then be made from the values of the set of functions required permitting
equipment design characteristics to be evaluated.
Only orthogonal sets of functions can be made to synthesise completely
any time function to a required degree of accuracy. Further, the characteris-
tics of an orthogonal set are such that identification of a particular member
of the set contained in a given time function can be made using quite simple
mathematical operations on the function.
We can consider a signal time function, /(t), defined over a time interval
(0, T) as being represented by an orthogonal series, S n (t).
Thus
m=i c n s n (t ) (i.D
r=0
where C n is a number indicative of the magnitude of the series constituents.
The series S n (t) (n = 0, 1, 2 . . .) is said to be orthogonal with weight K
over the interval 0 ^ t ^ T if
T ^ ^ x ^ / x , (K if n = m ^
K ■ S n (t)S m (t) dt = \ (1.2)
J 0 10 it ni^m
1
2
ORTHOGONAL FUNCTIONS
when n and m have integer values and K is a non-negative constant (or fixed
function) which does not depend on the indices m and n. If the constant K is
one then the set is normalised and called an Orthonormal set of functions. A
non-normalised set can always be converted into an orthonormal set.
Since only a finite number of terms, N, is possible for a practical realisation
of the series given by equation (1.1), it is necessary to choose the coefficients
C n in order to minimise the mean-square approximation error.
M.S.E.
T [f(t)-Y c n s n (t)Ydt
which is realised by making
f(t)S n (t) dt
(1.3)
(1.4)
It is desirable that this error monotonically decreases to zero as N becomes
very large. This is the case for a complete orthogonal function series such as
the sine-cosine or Walsh functions.
A complete orthonormal function series S n (t) is always a closed series.
That is, if there exists no quadratically integrable function, /(r), where
Ocf f 2 (t)dt<oo (1.5)
and for which the equality
f(t),S H (t)dt = 0 (1.6)
Jb
is satisfied for all values of n.*
More generally, a function series, whether normal and orthogonal or not,
is said to be complete or closed if no function exists which is orthogonal to
every other function of the series unless the integral of the square of the
function is itself zero 1 . A necessary and sufficient condition for completeness
is that Parseval’s equation holds good for every function whose square is
summable in the interval of orthogonality.
Incomplete orthogonal function series do not converge and therefore
cannot represent exactly any given time function, although they may have
other properties of equal importance. For example the output of a low-
frequency filter can be comprised of an incomplete orthogonal series of
sin x/x functions. Another example of an incomplete series is one comprised
of Rademacher functions. These are a set of simple rectangular functions
This expresses the Riesz-Fischer theorem (1907).
IB
SINE-COSINE FUNCTIONS
3
which, as we shall see later, play an important part in the generation of other
function series.
Using an orthogonal time series representation, the signal can be expres-
sed as a limited set of coefficients or spectral numbers. Quite considerable
reductions can be made in the number of coefficients needed for complete
representation, in this way, without losing the identity of the signal. It is also
possible to use the orthogonal property of a data series to identify specific
components of the series as will be described later.
In summary we may state that any set of functions which are capable of
being integrated and of integrable square modulus may be used to form a
closed set of linear combinations of the functions which will be normal and
orthogonal. Also these combinations will be found to be obtainable from a
limited number of constituent functions. The circular functions are the most
well-known of these and their orthogonality will be considered in the next
section.
IB Sine-cosine functions
Let us consider a series of sine-cosine functions, the first eight members of
which are shown in Fig. 1.1. The significance of orthogonality may be seen if
we take the products of pairs of such functions over a limited time interval,
o
T
Fig. 1.1. A set of sine-cosine functions.
4
ORTHOGONAL FUNCTIONS
O^t^T. Thus if we let
S n (t) = V2 cos 2irnt or sin 2i mt
S m (t) -42 cos 27rmt or V 2 sin 27rmt (1.7)
then from equation (1.2)
2 cos 2irmt cos 2imX dt
= (cos (m + u)27rt + cos (m - n)2irt) dt (1.8)
= 0 if m^n and both m and n are integers since the average value of a
cosinusoidal waveform over an integer number of periods is zero.
Similarly,
2 sin 2rrmt sin 2 7 mt dt
2 sin 2'nmt cos 2rrnt dt
2 cos limit sin lirnt dt
However, for m - n then
2 sin 2 2 7 mt dt- J 2 cos 2 2 mt dt- 2
over the interval (0, T).
Since any time series can be expressed as a summation of sinusoidal
components (Fourier series) viz.
Clo 00
f(t) = -r+ I {a k cos (k(o 0 t) + b k sin (k<o 0 t)} (1.10)
2 k = 1
then multiplication by cos k(o 0 t or sin k(o 0 t and integrating over the period
27r/o)o will enable the Fourier coefficients a k and b k to be extracted.
This follows from the above where it is seen that the process of integration
will reduce all the other sine-cosine products to zero. The orthogonal feature
of sine-cosine representation, therefore, solves the problem of identifying a
particular sinusoidal component from a composite waveform comprising of
the summation of many elements and hence is the key to spectral decompos-
ition, filtering and similar operations.
also equal 0 if m^n
(1.9)
1C
INCOMPLETE FUNCTION SETS
5
1C Incomplete function sets
A function set is complete if the mean-square error of the signal representa-
tion, f(t) converges to zero with increasing number of terms viz.
lim C n S n (t)] 2 dt = 0 (1.11)
N -*°° J 0 n= 0
The mean-square error
M.S.E. = f T [f(t)-Y C n S n (t)] 2 dt (1.12)
depends on the system of functions chosen for the linear approximation to
the data series or waveform. If the shape of this waveform is similar to that of
the functions used then the M.S.E. will be small (see Section IIF2).
An alternative definition for completeness of a function set is that of
Parseval’s theorem or completeness theorem which will be given later. A
physical meaning for this theorem applicable to a complete series is to state
that the energy contained within the series is the same whether expressed in
the time or transformed domain.
One example of an incomplete series is the set of orthogonal block pulses
shown in Fig. 1.2. The condition of equation (1.2) obviously obtains here
Fig. 1.2. A set of orthogonal block pulses.
6
ORTHOGONAL FUNCTIONS
since only one of the signals is allowed to differ from zero at any one time,
although the system is not complete.
The definitions of completeness, normality and orthogonality may be
applied to all functions over a semi-infinite interval (0, oo) or the complete
infinite interval (- 00 , + 00 ) as well as to functions defined over a finite
interval (4- T/2, —T/2) or (0, T). The circular functions can apply to any of
these cases, but the Walsh, Haar and certain other functions are limited to
finite intervals only.
The limitation in the case of Walsh and other functions confers the
advantage that a time-limited signal composed of a limited number of
orthogonal functions will occupy a finite section of the transformed domain.
With the circular functions, on the other hand, a finite time signal occupies
an infinite frequency function in the transformed domain. This has a
relevance in signal reconstruction for a given accuracy and in the analysis of
non-stationary waveforms.
IC1 Rademacher functions
An important, incomplete but orthogonal function set are the
Rademacher functions 2 . These represent a series of rectangular pulses or
square-waves having unit mark-space ratio. The first six of these are shown
in Fig. 1.3. The first function, R(0, t), is equal to one for the entire interval
0 <t<T,
0 ~i R(a,)
R(l»t)
R(2,t)
R(3,t)
R (4,t)
R(5, t)
0
T
Fig. 1.3. A set of Rademacher functions.
ID
WALSH FUNCTIONS
7
The next and subsequent functions are square-waves having odd sym-
metry. The incompleteness of the series can be demonstrated if we consider
the summation of a number of Rademacher functions. This composite
waveform will also have odd symmetry about the centre and similar, even
symmetry functions required for completeness cannot be developed.
Rademacher functions have two arguments n and t such that R (n, t) has
2 n_1 periods of square-wave over a normalised time base The
amplitudes of the functions are +1 and -1. They can be derived from
sinusoidal functions which have identical zero crossing positions. Thus,
R(rc, t) = sign [sin (2 n irt)] (1.13)
and may be obtained from a sinusoidal waveform of appropriate frequency
by amplification followed by hard limiting. They are important principally
since other complete series, such as the Walsh series, can be derived from
them.
ID Walsh functions
Walsh functions 3 form an ordered set of rectangular waveforms taking
only two amplitude values +1 and —1 and are another example of an
orthonormal set of functions. Unlike the Rademacher functions the Walsh
rectangular waveforms do not have unit mark-space ratio. They are d efined
over a limited time interval, T, known as the time-base, which requires to be
known if quantitative values are to be assigned to a function. Like the
sine-cosine functions, two arguments are required for complete definition, a
time period, t , (usually normalised to the time-base as t/ T) and an ordering
number, n, related to frequency in a way which is described later. The
function is written
WAL(tM) (1.14)
and for most purposes a set of such functions is ordered in ascending value of
the number of zero crossings found within the time-base. Figure 1.4 shows
the first 32 of these with the ordering arranged in this way.
The orthogonality of the discrete Walsh series can be proved in the
following way. First an expression for the discrete Walsh function having
N = 2 P terms will be stated in terms of a continued product 4 as
WAL(n p _i, n p - 2 . . . n 0 ; t P - 1 , t p - 2 . . . t 0 )
=ff (-l)"'- 1 -^'^ (i.i5)
r= 0
where n and t are the arguments of the function expressed in binary
notation.
8
ORTHOGONAL FUNCTIONS
WAL(3I,T)
WAL(26,T)
WAL(25,T)
WAL (24,T)
SAL(I6,T)
CAL(I5,T)
SAL (15, T)
CAL (14, T)
SAL (I4,T)
CAL (13, T)
SAL (I3,T)
CAL (12, T)
WAL ( 23, T)
WAL(22,T )
WAL (21, T)
WAL (20, T)
WAL(I9,T)
WAL (18, T )
WAL (17, T)
WAL (I6,T)
WAL (15, T)
WA L (I4,T)
WAL (I3,T)
WAL (12, T)
WAL (II, T)
WAL ( 10 ,T )
WAL (9,T)
WAL ( 8,T)
WAL (7,T)
SAL(I2,T)
CAL (II ,T)
SAL(II ,T)
CAL (10, T)
SAL(IO,T)
CAL (9,T)
SAL (9,T )
CAL (8,T )
SAL (8,T)
CAL (7,T)
SAL (7,T)
CAL (6,T)
SAL (6,T)
CAL (5,T )
SAL (5,T)
CAL (4,T)
SAL (4,T)
WAL (6,T)
WAL (5,T)
WAL (4,T )
WAL (3,T)
WAL (2,T )
WAL (I ,T)
WAL (0 ,T)
CAL (3,T)
SAL (3,T)
CAL (2,T)
SAL (2,T)
CAL ( l,T )
SAL(I.T)
CAL (0,T )
Fig. 1.4. A set of Walsh functions arranged in sequency order.
IE
HAAR FUNCTIONS
9
The sum of the products of any two discrete Walsh functions is given as the
binary summation
11 i
I I ... I WAL(m p _i, mp -2 . . . m 0 ; t p - u t p - 2 ...t 0 )
tp— 1 *p— 2 r O=0
x WAL(n p _i, n p -2 . . . n 0 ; t P - u t P - 2 ■ ■ ■ to) (1.16)
Substituting equation (1.15) in (1.16) gives for the binary product-sum
i i ■ ■ ■ i n (-i) ( " p - i -' +m - i -^ +i>
t p - 1 t p -2 t 0 =0 r = 0
p-1 1
= HY, (— 1 y n p-l-r +m p-l-rX tr+t r+0
r=0 t r =Q
= P ff {1 + (-l) ( "o->-' +m -- i - ) } (1.17)
r = 0
Now if each n t = m h remembering that only two values are possible, zero
or one, then equation (1.17) becomes
n (i+i)=2 p =iv
r=0
If at least one n t ^ m t then at least one term in the product given by
equation (1.16) is zero giving a zero product. In terms of decimal indices we
have for the product of two Walsh terms
f N for n = m
X WAL(m, t) WAL(n, t) = \ (1.18)
t=0 10 forn^m
Hence the Walsh functions can be seen to form an orthonormal set which
can be normalised by division by N to form an orthogonal system.
IE Haar functions
These also form a complete orthonormal function set of rectangular
waveforms proposed originally by Haar 5 . The functions have several impor-
tant properties, including the ability to represent a given function with few
constituent terms to a high degree of accuracy.
They have three possible states 0, + A, and -A where ± A is a function of
a/2. Thus, unlike the Walsh functions, the amplitude of the functions vary
with their place in the series.
They may be represented over the interval 0 ^ t ^ 1 as
HAR(n, t)
(1.19)
10
ORTHOGONAL FUNCTIONS
where n also identifies the function in terms of zero crossings. For complete
definition it is also necessary to give some information concerning the
amplitude of the function since this is no longer confined to + 1 and -1 as
with the Walsh function series. An alternative representation giving zero
crossing order equality with the Walsh series is given later in Chapter 4.
The first eight Haar functions are shown in Fig. 1.5. The functions are
orthogonal and orthonormal and obey the condition for orthogonality
[ HAR(m, t) HAR(«, t) dt = torn = m (1.20)
J 0 10 for n ^ m
o
T
Fig. 1.5. A set of Haar functions.
IF Other orthogohal functions
A number of other function sets are known which are found to be orthogonal
and hence can be used for series representation.
The Karhunen-Loeve series is one of these and can be used in much the
same way as the Walsh and Haar function for signal filtering 6 and other
purposes. Unfortunately the Karhunen-Loeve transform required in the
application of the series can only be derived from the covariance matrix of
the data series where the eigenvectors of the covariance matrix are used.
REFERENCES
11
The implications of this in terms of practical application of the series are
discussed later in Section (VIIIE).
Additionally certain polynomials can be made orthogonal by multiplica-
tion by a weighting factor. These orthogonal polynomials consist of a series,
/„( x) (n = 0, 1, 2 ... ), where n is the degree of the polynomial. This class
contains many special functions commonly encountered in practical applica-
tions, e.g. Chebychev, Hermite, Laguerre, Jacobi, Gegenbauer and
Legendre polynomials 7,8 .
None of these contain the essential simplicity of the Walsh and Haar
functions where members of each of these two classes assume a single value
having either a positive or a negative sign which has the effect of reducing
multiplicative operations on the series to an appropriate sequence of sign
changes. This simplicity gives rise to favourable repercussions in calculation
and in digital hardware, which will be referred to time and again in the
following pages.
References
1. Chen Kien-Kwong. (1957). “Summation of the Fourier Series of Orthogonal
Functions”. Science Press, Peking.
2. Rademacher, H. (1922). Einige Satze von allgemeinen Orthogonalfunktionen.
Math. Annal. 87 , 122-38.
3. Walsh, J. L. (1923). A closed set of orthogonal functions. Ann. Journ. Math. 55 ,
5-24.
4. Pratt, W. K., Kane, J. and Andrews, H. C. (1969). Hadamard transform image
coding, Proc. I.E.E.E. 57 , 1 , 58-68.
5. Haar, A. (1910). Zur Theorie der orthogonalen Funktionensysteme Math.
Annal. 69, 331-71.
6. Pratt, W. K. (1971). Generalised Wiener filtering techniques. Proceedings: 1971
U.M.C. Two-dimensional digital signal processing conference.
7. Davis, H. F. (1963). “Fourier Series and Orthogonal Functions”. Allyn and
Bacon, Inc., Boston.
8. Lebedev, N. N. (1965). “Special Functions and Their Applications”. Prentice-
Hall, New Jersey.
Chapter 2
The Walsh Function series
IIA Definition of the Walsh series
A definition of the Walsh function WAL(rc, t ) was given in Section ID and a
series of such functions shown in Fig. 1.4.
An alternative notation to that given in equation (1.14) has been intro-
duced by Harmuth 1 to classify the Walsh functions in terms of even and odd
waveform symmetry, viz.
WAL(2lc, t) = CAL(k, t)
WAL(2fc-l,t) = SAL(fc, t )
N
k = 1,2...- (2.1)
which defines two further Walsh series having close similarities with the
cosine and sine series. The notation is also used in Fig. 1.4 as well as the
WAL(n, t) notation described earlier.
As indicated in this diagram the normalised Walsh functions are sym-
metrical about their mid or zero time point. Defining the range of the
function as — then the functions are either directly symmetrical
(CAL functions) or inversely symmetrical (SAL functions). In this latter case
the one’s found in the left-hand side are mirrored by zero’s in the right-hand
side and vice-versa. This enables a symmetry relationship to be stated as
WAL(n, t) = WAL(t, n) (2.2)
12
II A
DEFINITION OF THE WALSH SERIES
13
As discussed later, the practical importance of this is that the transform
and its inverse represent the same mathematical operation thus simplifying
the derivation and application of the transform.
The ordering shown in Fig. 1 .4 is known as Sequency order. Sequency is a
term, also proposed by Harmuth, to describe a periodic repetition rate which
is independent of waveform. It is defined as, “one half of the average
number of zero crossings per unit time interval”. From this we see that
frequency can be regarded as a special measure of sequency applicable to
sinusoidal waveforms only. The number of zero crossings ( Zps ) in the
half-open interval (— \ ^ t ^ |) is 2 k so that k represents sequency in the CAL
or SAL ordering.
The similarities between the circular and Walsh functions are also seen in
the expressions for the function series. Using the Fourier series expansion
we can express a time series, /(f), as the sum of a series of sine-cosine
functions each multiplied by a coefficient giving the value of the function for
that series viz.
where
f{t) =—+ X ( a k cos ( k(D 0 t) + b k sin ( kco 0 t ))
^ k = 1
flo 1 rT
2
a k
41 **
4I>
dt
cos kco 0 t dt
TJ
fit ) sin k(o Q tdt
(2.3)
(2.4)
The coefficients a k and b k represent the peak amplitudes of the spectral
components of f(t). A set of these coefficients can form a further series, /(k),
which expresses f(t) in the frequency domain.
We can also express a time series, /(f), in a similar way in terms of the sum
of a series of Walsh functions, viz.
where
f(t) = a 0 WAL(0, t) + Y a n WAL(n, t)
n = 1
Oo
2
U fit) WAL(0, t) dt
l Jo
~\ fit) WAL(n, t) dt
1 Jq J
(2.5)
( 2 . 6 )
14
THE WALSH FUNCTION SERIES
or from equation (2.1) using the sum of two series for CAL and SAL terms
having N / 2 — 1 and N/2 values respectively,
and
T f(t)CAh(j,t)dt
1 J 0
1
di = T J
f(t) SAL (i, t) dt
(2.7)
Using the SAL and CAL forms we can obtain an expression for the Walsh
series similar to that given for the sine-cosine series in equation (2.3), viz.
N/2 N/2— 1
f(t) = a 0 WAL(0, 0 + Z I (a,SAL(i, t) + ftyCAL(/, t)) (2.8)
i=l 7=1
The two new series of a t and bj coefficients taken together express f(t) in the
sequency domain. Note that WAL(0, t) = CAL(0, t) so that, in this expres-
sion, there is one less CAL term than the SAL terms in the summation.
The derivation of these coefficient series is referred to as decomposition
into the spectral components of /(t), although these components are now no
longer sinusoidal in form.
We may note that although it is possible to combine the sine and cosine
elements into a single complex variable, exp(jk(o 0 t), (where / = V— 1),
expressing the same frequency; this is not possible with Walsh functions due
to the absence of a similar shift theorem to that found in circular function
theory. Consequently the two separate series, that are developed from the
SAL and CAL functions of the Walsh series, are needed to express fully the
sequency behaviour of f(t).
Synthesis of a complex waveform using the principle of superposition
from a linear set of functions is obtained using the Walsh series in an
analogous manner to Fourier synthesis. A stepped equivalent waveform is
obtained which approximates the original waveform more closely as the
number of superimposed series are increased. As an example Fig. 2.1 shows
a very simple approximation of a sinusoidal waveform from the three
principle Walsh series, each having an appropriate amplitude. The number
of terms required for a given mean-square-error is dependant on the
characteristics of the reconstructed waveform in relation to those of the
constituent series. Consideration is given to this in Section IIF when
equivalent Fourier and Walsh syntheses are discussed.
IIA1 Relationship between the Fourier and Walsh series
From equations (2.4) and (2.6) it can be seen that for a normalised series,
since WAL(0, t)= 1 inside the open interval 0^ 1 then, the expressions
IIA
DEFINITION OF THE WALSH SERIES
15
Fig. 2.1. Synthesis of a sinusoidal signal from a limited number of Walsh series.
for a 0 / 2 shown for the Fourier and Walsh series are equivalent and give the
mean value of the function. AJso, if we take equation (2.8) to its limit by
extending the summation to infinity then the Fourier and Walsh series
representations are identical. Hence, we can substitute the extended version
of equation (2.8) in the expression for the cosine coefficient given in
equation (2.4) to obtain
a k= 7 f, “ 7 + Z Z a t SAL(/, t) + bj CAL(/, t)l cos kco 0 tdt (2.9)
1 Jo L Z i = 1 ;=1 J
Reversing the order of integration
2 °o ~ ' T
a k = tf 0 +— Z SAL(i, 0 cos ko) 0 tdt
1 i=i L J 0
2 °° r r T
+~ S ft, CAL(y, /) cos k(o 0 tdt
1 /= 1 LJo
( 2 . 10 )
16
THE WALSH FUNCTION SERIES
The terms in brackets represent the Fourier coefficients for SAL(/, t) and
CAL (/, t) respectively. (This can be seen if we substitute SAL(z, t ) and
CAL (/, t) for f(t) in the cosine equation of (2.4).) Writing a k (SAL) and
a k (CAL) for these we have
a k - ^o+~ X X I (SAL) + b } a k (CAL) (2.11)
Similarly we can derive for b k
b, = a 0 +^l f I a,b k (SAL) + b,b k (CAL) 1 (2.12)
1 j=\ i= i L J
To find the Fourier series in terms of the Walsh series the terms for a k and-
b k are substituted in equation (2.3) to give
/(0 = “ e +x { tfo+~ X X [^«fc(SAL) + fc 7 a k (CAL)] cos ka) 0 t
2 k = i l L 1 j= i ,=i
+ a 0 +2 Z f [aA(SAL) + ^ t (CAL)] sin fc&> 0 f] (2.13)
1 y= l , = i J J
Since both Walsh and Fourier series are orthogonal the terms containing
a A (SAL) and h y h k (CAL) will vanish so that equation (2.13) simplifies to
fit) = 7 “+“; XXX [aA(SAL) sin k^ 0 t + h y a k (CAL) cos (2.14)
2 l ,=i j=i
which approximates to a limited and normalised sampled form
9 N N N
f(t)=^+i e 1 1
^ iV k = 1 j = i y = i
x dib k (SAL) sin ^ 2 7r/c^ + fr,a k (CAL) cos 2 7rfc^
M, y = 0, 1...N-1 (2.15)
Equation (2.15) enables the Fourier series for f(t) to be obtained from the
Walsh series using a table of Fourier coefficients for the Walsh functions,
SAL(i, t ) and CAL(y, t). This method of derivation for the Fourier coeffi-
cients is used in hardware generation by Siemens and Kitai 20 who give
examples of coefficient tables for common types of waveforms.
IIA2 Relationship between CAL(/c, f) and SAL(k, f)
Although a simple relationship exists between cos (2 irkt) and sin (lirkt), the
relationship between CAL(/c, t) and SAL (fc, t) is rather complicated. This is
MB
FUNCTION ORDERING
17
a consequence of the absence of a simple shift theorem which is found in the
circular functions. A relationship has been stated by Gibbs 2 and modified by
Tam and Goulet 3 to give the expression
CAL(/c, t + t 0 ) = SAUK t) (2.16)
where
t 0 = (—l) q+1 • 2 _(r+2) (2.17)
q and r are expressed as factors in an expression for k viz.
k = 2 r (2q + l) r,q = 0,1,2... (2.18)
A table for k in terms of q and r is required in order to obtain suitable
values for equation (2.17). This is shown below for k = 1 to 8.
k r q
1 0 0
2 1 0
3 0 1
4 2 0
5 0 2
6 1 1
7 0 3
8 3 0
MB Function ordering
It is necessary now to consider the ordering of Walsh functions in some
detail. Two major ordering conventions are in common use and unless the
convention used is clearly defined confusion can arise when results and
algorithms from different sources are compared.
What we would like is an ordering based on the number of zero crossings
for the function which is related to our practical experience with other
orthogonal functions (e.g. sinusoidal waveforms) and yet which is easy to
define for analytical and computational purposes. Neither of the two forms
of ordering described below simultaneously satisfies both requirements so
that each plays a part in the application of Walsh theory. The two forms are:
(a) Sequency order (ordered form, Walsh order, Walsh- Kaczmarz order).
This was Walsh’s original order for his function, WAL(n, t ), and he arranged
the components in ascending order of zero crossing (Fig. 1.4). It is directly
18
THE WALSH FUNCTION SERIES
related to frequency where we find that Fourier components are also
arranged in increasing harmonic number (zero crossings divided by two).
The advantage of this order is that derivation of the alternate CAL and SAL
functions shown resembles that of orthonormal Cosine and Sine functions in
Fourier analysis and hence permits suitable comparisons to be made.
(b) Natural order (normal order, binary order, dyadic order, Paley order).
This is the order obtained by generation from successive Rademacher
functions. It was first used by Paley 4 and will be referred to as the Paley-
ordered function, PAL(n, t). This ordering has certain analytical and com-
putational advantages noted by Fine, Gibbs, Ahmed, Yuen and others. In
particular, Gibbs has shown that the Paley-ordered functions may be
defined as the eigenfunctions of a logical differential operator and that this
definition is of value in the mathematical development of the theory. The
relationship between Natural and Sequency order is considered in Section
HE.
Natural order is used in theoretical mathematical work, image transmis-
sion, and for computational efficiency. Sequency order is favoured for
communications and signal processing work such as spectral analysis and
filtering.
Figure 2.2 shows the first 32 Walsh functions arranged in natural order.
This may be compared with the sequency order shown in Fig. 1.4 where
quite considerable changes in relative positions for the various functions are
seen.
As will be shown later certain derivations of the transformed coefficient
series will result in PAL or WAL series arranged in bit-reversed order. Thus,
if we consider the binary equivalents to an ascending series 0, 1, 2, 3, 4, 5 as
000, 001, 010, Oil, 100, 101; then reversing the order of the binary digits
gives, 000, 100, 010, 110, 001, 101 which is expressed in decimal order as
0, 4, 2, 6, 1, 5 respectively. This is the bit-reversed order for the first six
integer values.
Rearrangement into the required sequency or natural order for a given
set, N, is made through suitable routing in the case of hardware derivation or
a bit-reversal software routine which may precede or follow the transform
routine.
IIB1 Phase of the ordered set
The diagrams given in Figs 1.4 and 2.2 are arranged to emphasise the phase
similarity with an ordered set of sine-cosine functions and will be referred to
as Harmuth phasing.
However, if the series is derived directly from the Hadamard matrix or
from Rademacher products, as described later, then the functions will be
phased such that they all start at a +1 level and this will be referred to as
20
THE WALSH FUNCTION SERIES
positive phasing. It involves a reversal of sign for some of the functions
shown in Figs 1.4 and 2.2.
IIC Walsh function derivation
The function series can be obtained in several different ways, each of which
has its own particular advantages. The methods considered in this chapter
are:
(i) by means of a difference equation 1
(ii) from products of Rademacher functions 5
(iii) through the Hadamard matrix 6
(iv) by the use of Boolean synthesis 7
The difference method gives the function directly in sequency order. The
other methods give results in natural order or bit-reversed natural order. A
method of converting these to sequency order has been described by Lackey
and Meltzer 8 and will be considered later.
A further derivation due to Swick 9 should also be mentioned because of its
mathematical elegance. This takes the symmetry relationship, given in
equation (2.2) as a starting point and develops a complete set of functions
from this consideration alone.
All these derivations are, of course, mathematical processes for which
computational algorithms can be developed and the series produced using
the digital computer or obtained directly using digital logic. However, due to
the simplicity of the fast Walsh transform algorithm and the speed with
which this can be implemented on the digital machine, the generation of a
function or series of functions can also be obtained simply by transforming a
unit input sample at the appropriate position in the input vector using the
fast Walsh transform. This will become apparent when signal flow diagrams
for this algorithm are studied (section IIIH2).
IIC1 From difference equations
This assumes that the normalised time-base is referenced to its centre, i.e.
A given Walsh function is defined from its .preceding harmonic
function so that, commencing with a definition of WAL(0, 0) = 1 within the
time-base and 0 outside the time-base, then the entire set of Walsh functions
can be obtained by an iterative process.
The difference equation* is given as
WAL(2 j + q, 0) = (-l) u/21+q WAL(j, 2(0 +i)) + (-l) y+<? WAL(/, 2 (0-|))]
q = 0 or 1
7 = 0, 1,2... (2,19)
* [y/2] means the largest integer smaller or equal to j/2. Thus, for j = 1 then [j/2] = 0.
lie
WALSH FUNCTION DERIVATION
21
A somewhat easier notation is obtained if we reference time to the
commencement of the generated function to avoid negative values of 0.
Thus, if we replace 0 by t and define this as O^t^l then, as before,
WAL(0, t) = 1 within this time-base and 0 outside the time-base, so that the
difference equation can be restated as
WAL(2/ + q, t) = (-l)“ /2]+ ’[WAL(/, 2t) + (-\) i+q WAL( j, 2 (*-$))]
q = 0 or 1
7 = 0 , 1 , 2 ... ( 2 . 20 )
which for N equally spaced discrete points (where N = 2 P and n =
0, 1, 2 . . . (TV— 1)) can be written
WAL(2; + q,n) = (-1)°' /21+ "
x [WAL(/, 2n) + (—l) i+q WAL (/, 2 (n - |))
( 2 . 21 )
Commencing with the known WAL(0, n) - 1 within the time-base (i.e.
j = 0, q = 1) for n ^ N/2 its value for WAL(/, 2n) will be 1 and for n > N/2
the function falls outside the time-base and will be 0. Similarly with
WAL(y, 2{n—N/2)) for n<N/2 the function again falls outside the time-
base and will become 0, whilst for n > N/2 the value is 1. The sign of these
functions will be modified by the factors (-l) u/21+q and (— l) 0+q) in accordance
with equation (2.21).
This is summarised to give the required function, WAL(1, n) in Table 2.1.
Note that values of n < 0 orn > 1 must result in any Walsh function having a
value of 0. From this result WAL(2, n ) can be obtained in a similar manner
and the process repeated to obtain further functions in the series.
n
WAL(y, 2n) = A
WAL(/,2(n —
(_l)« + i> = B = C
N/2))
(_1 = D
WAL(1, n)
= D(A + BQ
0
1
-1
0
-1
-1
1
1
-1
0
-1
-1
2
1
-1
0
-1
-1
3
1
-1
0
-1
-1
N/2 — 1
1
-1
0
-1
-1
N/2
0
, -1
1
-1
1
N— 1
0
-1
1
-1
1
Table 2.1 . Derivation of the Walsh function by the difference method.
22
THE WALSH FUNCTION SERIES
The operation of this difference equation may be considered as equivalent
to compressing the previous Walsh function WAL(;, 2 n) into the left-hand
part of the time-base, by selection of alternate points and, after left
adjustment adding to these on the right-hand side a similar valued set of
points but all having an opposite sign.
IIC2 From Rademacher functions
Rademacher functions were introduced in Section (IC1) and are illustrated
in Fig. 1.3. Although they form an incomplete series having odd symmetry it
is possible to form functions from them which will exhibit either odd or even
symmetry. Hence, a complete series can be developed from the incomplete
set of functions. In particular a complete set of Walsh functions in natural
order can be obtained from selected Rademacher function products. This is
considered in the following, commencing with the generation of the Walsh
series in natural order.
The product series for the Rademacher functions is expressed as
m
PAL(n, t)=Ubi R(i, t) (2.22)
i = l
where n is expressed as a binary number
n = b m 2 m + b m - i2 m " 1 + . . . b x V + b 0 2° (2.23)
and b t - 0 or 1.
Thus, to find PAL(13, /), we can write
PAL(13, r) = R(4, r) R(3, t) R(l, l)
since binary 13 is 1 101 and the 4,3 and 1 refer to ones found in the binary bit
positions.
This will give a natural-ordered series having positive phasing. If the open
interval for the set of Rademacher functions is defined as then the
functions will have the inverse sign to that shown in Fig. 1.3. Their products
will then give rise to the Harmuth phasing for the natural-ordered series.
The products actually refer to Rademacher functions represented as a
string of - 1 and + 1 ’s. If we adopt the convention -1 = 0 and +1 = 1 then the
products become sums and we write
PAL(n, t)= £ bi R(i, t) (2.24)
i = 1
But the summations are expressed as Modulo-2 addition i.e.: binary sums
without carry, and obey the rules
0 + 0 = 0 , 0+1 = 1 ,
1+0=1 and 1 + 1 = 0
lie
WALSH FUNCTION DERIVATION
23
(Modulo-2 arithmetic plays an important part in Walsh theory and its special
characteristics are referred to again in section IIH).
A recursive method of generating a sequency-ordered set is described by
Harmuth 1 . A simpler definition is given by Lackey 8 who notes that the
sequency order (n) is related to the natural order by means of the Gray code
given in Table 2.2. To find the value of bit position i of the sequency number
n expressed in the Gray code we need to add bit ( i ) to bit (i® 1) of the
original binary number (this addition is also Modulo-2). Thus defining n as a
string of binary bits
n = (b m b m - 1 . . . bo )2
Decimal
Code
Decimal
Code
0
0000
8
1100
1
0001
9
1101
2
0011
10
1111
3
0010
11
1110
4
0110
12
1010
5
0111
13
1011
6
0101
14
1001
7
0100
15
1000
Table 2.2. Gray single-digit change code.
and expressing this in the Gray code, we write
n = (g m g m -i ... go)
where
g i = b 1 ®b i+1 (2.25)
and © represents Modulo-2 addition.
Hence, to find WAL(9, t) we first express 9 in binary code as 1001 and
then rearrange this in Gray code as 1101. The second bit position gives
bi = 0, so that we can write directly from equation (2.22)
WAL(9, t) = R(4, t ) R(3, t ) R(l, t )
which is the same result as found earlier for PAL(13, t ). A graphical
illustration of this function product derivation is given in Fig. 2.3. This
~i_ x mrx uuuuuum =imnjL
R(l,t) R(3,t) B(4,t) WAL ( 9,t )
Fig. 2.3. Derivation of WAL(9, t) from three Rademacher functions.
24
THE WALSH FUNCTION SERIES
shows clearly the role of the function R(l, t) in the sign inversion occurring
at the centre of the generated function, WAL(9, t ).
Since the Rademacher functions can be obtained from a limited set of sine
or cosine functions then it is also possible to derive the Walsh series by taking
the sign of a product expansion of sines and cosines. A method for this is
described by Ross and Kelly 10 who give a general expression for WAL(n, t )
in terms of a binary representation for n (equation 2.23), viz.
WAL(n, i) = sign
m
(sin 2Trt)' , ° II (cos 2 k irt) bk
(2.26)
IIC3 From Hadamard matrices
The Hadamard matrix is a square array whose coefficients comprise only +1
and —1 and where its rows (and columns) are orthogonal to one another. In a
symmetrical Hadamard matrix it is possible to interchange rows and col-
umns or to change the sign of every element in a row without affecting these
orthogonal properties. This makes it possible to obtain a symmetrical
Hadamard matrix whose first row and first column contain only plus ones.
The matrix obtained in this way is known as the “normal form” for the
Hadamard matrix. The lowest order Hadamard matrix is of the order two,
viz.
H 2 =
"1
.1
(2.27)
Higher order matrices, restricted to having powers of two, can be obtained
from the recursive relationship
Hv — H.y/2 ® 11; (2.28)
where ® denotes the direct or Kronecker product and N is a power of two.
The Kronecker product means replacing each element in the matrix (in
this case, H n/2 ) by the matrix H 2 .
Thus, for
h., = h 2 @h 3
we replace each of the l’s and the — l’s of the matrix given above in H 2
(equation 2.27) by the complete matrix of H 2 or its inverse, thus
H4 =
1
1
1
1
1
-1
1
-1
1
1
-1
-1
1
-1
-1
1
(2.29)
lie
WALSH FUNCTION DERIVATION
25
Furthermore, if we now replace each element in the H 4 matrix by an H 2
matrix we obtain an Hs matrix. Replacing each row of this matrix by its
equivalent naturally-ordered Walsh function we can form a series of func-
tions which will indicate the ordering obtained through this derivation.
Therefore, for a series consisting of eight terms
H 8 = H4®H 2
1 1
1 -1
1 1
1 -1
1 1
1 -1
1 1
1 -1
1 1
1 -1
-1 -1
-1 1
1 1
1 -1
-1 -1
-1 1
1 1
1 -1
1 1
1 -1
-1 -1
-1 1
-1 -1
-1 1
1
1
-1
-1
-1
-1
1
1
1
-1
-1
1
-1
1
1
-1
PAL(0, t)
PAL(4, t)
PAL(2, t)
PAL(6, t)
PAL(1, t)
PAL(5, t)
PAL(3, t)
PAL(7, t)
(2.30)
The relationship between these Hadamard matrices and a sampled set of
Walsh functions is now clear. They simply express a Walsh function series
having positive phasing and arranged in bit-reversed natural order. This
ordering is sometimes referred to in the literature as Kronecker ordering or,
more recently, Lexicographic ordering.
IIC4 From Boolean synthesis
For this we use the expression developed by Gibbs 7 which defines the
discrete Walsh function, WAL(i, t), as a form of the continued product
definition (equation 1.14)
WAL(i, t ) = -l x "- |(VH ^ |,, ‘ (2.31)
where , , t t
0 S i 3= IV — 1 and 0 2 1 2 1
Defining the number of terms, N, as 2 P we can define i in terms of binary bits
as
i = (i p , ip-i . . . ii, ii) 2
Similarly we can define t as a binary expansion, viz.
t — (ti, t 2 , . . . 4)2
26
THE WALSH FUNCTION SERIES
Since the function WAL(z, t ) is bounded by p terms beyond which i k = 0 then
only the first p binary bits of the t expansion appear in equation (2.31), so
that by substituting a Boolean 0 for 1 and a Boolean 1 for -1 we can express
WAL (/, t) as a N - bit string, viz.
WAL(f, T)=t (i k ® i k+ 0 T p _ k+1 (2.32)
k = 1
where T is an integer constructed by taking the first p binary digits of t.
Since the Walsh function is symmetric (equation 2.2) we can also write
WAL(T, i)= t (T k ®T k +i)i p - k+1 (2.33)
k = 1
A complete set of N Walsh functions can thus be represented by a N by N
Boolean matrix using expressions (2.32) or (2.33).
The expression within the brackets represents a binary to Gray code
conversion defined in equation (2.25). Thus equations (2.32) and (2.33) can
be re-written as
WAL(/, T) = I gLT p _ fc+1 (2.34)
fc=i
and
WAL(T, i) = t gIi P - M (2.35)
i
where g % is the k th bit of the Gray code,
g(T) = (g;-gi_i.--gi)i (2.36)
This form of derivation is useful in considering hardware function genera-
tion (Section IID) since the process of Modulo-2 addition corresponds to a
Boolean exclusive-OR operation whilst multiplication corresponds to a
Boolean AND operation. Thus direct implementation of the above
mathematical expressions can be made using logical circuits.
IID Hardware function generation
In the previous section the derivation of a Walsh series was considered in
terms of mathematical formulation. For computational purposes the
generation of Walsh series is almost always carried out with the fast Walsh
transform, described in the next chapter. This we can term “software”
generation of the series. Generation for real-time purposes, such as com-
munication equipment or on-line signal processing, requires specially-
designed hardware to produce a continuous function series to whatever level
IID
HARDWARE FUNCTION GENERATION
27
is required by the application. Such “hardware” function generators are
described in this section.
Two classifications of hardware generators are possible. The first generate
fixed sets of Walsh functions, WAL(n, t), where only the sequency range of
the entire array is controlled externally. These array generators find a use in
multiplexing and signal processing. The second classification is where the
sequency order, n , and/or time interval, t , are controlled externally. These
are known as programmable generators. A further sub-classification of
programmable generators can be defined; namely serial programmable
generators where the time interval is fixed and the sequency order is
controlled, and parallel programmable generators where the sequency
range is fixed and the time interval is controlled.
IID1 Array generators
These were the first to be used for applications work. They are developed
from the multiplicative property of the Walsh function (Section HID) given
as
WAL(n, t) WAL(m, t) = WAL(n ® m, t) (2.37)
This is an easy design to mechanise since it consists simply of a p-bit binary
counter and 2p-p — l exclusive-OR gates to produce an array of 2 p
functions.
An example of Harmuth’s array generator 11 is shown in Fig. 2.4. The
binary counters C x to C 4 produce the four Rademacher functions,
WAL(1, t), WAL(3, t), WAL(7, t) and WAL(15, t ) commencing with
R(5, t). From these functions, using suitable multiplicative exclusive-OR
gates, may be generated a complete set of the first 16 Walsh functions. The
logical values of the functions generated are 0 and 1 corresponding to the +1
and -1 required by the Walsh function definition.
This generator is not well-suited for high switching speeds due to the carry
propagation that occurs through the four counter stages. A synchronous
counter having parallel triggering can remove this source of error and a
design of this type has also been described by Harmuth.
Further difficulties in the practical application of these generators are the
spurious pulses which can arise from Modulo-2 addition of two unsynchron-
ised signals and also the variable time delays that can occur due to transmis-
sion of the counter outputs through different numbers of gates. An improve-
ment has been suggested by Boesswetter 12 , who uses a bi-stable latch to hold
the outputs of each Walsh series constant until the arrival of the next clock
pulse. An alternative design due to Besslich 13 operates at a constant gate
delay irrespective of the number of stages used. This is in contrast with
28 THE WALSH FUNCTION SERIES
Binary counters
Harmuth’s original design which includes an additional delay for every
doubling of range for the generated series.
IID2 Programmable generators
The mathematical basis for hardware generation was given in Section IIC4.
A fully programmable generator to produce the formulation given by
equation (2.32) is shown in Fig. 2.5. Here the p pairs of i and k parameters
are input to p - 1 two-input exclusive-OR gates and p AND gates together
with a p input exclusive-OR adder required to carry out the Modulo-2
addition and to form the output Walsh function.
To produce a serial programmable generator it is only necessary to
replace the k parameter inputs by a p-bit binary counter. A description of
such a generator is given by Durrani and Nightingale 14 .
IID
HARDWARE FUNCTION GENERATION
29
Fig. 2.5. A fully-programmable Walsh function generator.
A number of parallel programmable generators have been developed
where the order i, is externally controlled. Examples are the generators of
Lee 15 and Peterson 16 which mechanise equation (2.33) by the use of a series
of linear shift registers without reference to preceding members of the set.
Peterson’s method, like that of Swick referred to earlier (Section IIC), is
based on symmetry considerations and is better suited to hardware construc-
tion since only a single one-directional shift register is needed.
Where N is large, the most effective designs, from the point of view of
cost-effectiveness, are those in which equation (2.35) is implemented by
using a Gray code counter. Yuen 17 has shown that a string of pulses
representing sign changes of a Walsh function can be generated by combin-
ing the p output bits of a Gray code counter using an OR gate. The required
Walsh function can then be reconstructed by triggering a bi-stable circuit
from this string of pulses. A further advantage of the method is that the
generator is unlikely to produce spurious outputs since only one non-zero
pulse is produced by the p-bit Gray code incrementation. Thus, there is little
chance of two slightly unsynchronised pulses being combined to produce
spurious outputs.
Generation of Gray code increments can be made using a binary counter,
as shown by Lebert 18 , so that a complete parallel generator of this type may
be realised using standard integrated circuit components. A design for p = 4
30
THE WALSH FUNCTION SERIES
Fig. 2.6. Yuen’s programmed generator.
due to Yuen 19 is shown in Fig. 2.6. This has a symmetric structure and is,
therefore, capable of expansion to form a larger series. Due also to the
logical operation of the generator, it is capable of high speed operation since
the outputs of the counter and register pass only two gates before arriving at
another clocked component.
HE RELATIONSHIP BETWEEN WAL AND PAL SERIES 31
HE Relationship between WAL and PAL series
This relationship was referred to in Section IIC2 and may be formalised as
PAL(g(fc), t) = WAL (k, t ) (2.38)
where g(fc) is a function based on the Gray Code.
If we consider the binary equivalent of order number, k, as
k = (k p ,k p - 1 ...kik o) 2
then g(fc) can be formed from
g i (k) = k i ®k i+1 (2.39)
where © = Modulo-2 addition.
Natural order
Sequency order
PAL(0, t)
CAL(0, t)
WAL(0, l)
PAL(1, t)
SAL(1, t)
WAL(1, t)
PAL(2, t)
SAL(2, t)
WAL(3, f)
PAL(3, t)
CAL(1, t)
W AL(2, t)
PAL(4, f)
SAL(4, t)
WAL(7, t)
PAL(5, t)
CAL(3, t)
WAL(6, t)
PAL(6, t)
CAL(2, t)
WAL(4, ()
PAL(7, t)
SAL(3,t)
WAL(5, t)
PAL(8, ()
SAL(8, t)
WAL(15, t)
PAL(9, t)
CAL(7, f)
WAL(14, t)
PAL(10, t)
CAL(6, t)
WAL(12, t)
PAL(11, t)
SAL(7, t)
WAL(13, t)
PAL(12, t)
CAL(4, t)
WAL(8, t)
PAL(13, t)
SAL(5, t)
WAL(9, t)
PAL(14, t)
SAL(6, ()
WAL(11, f)
PAL(15, t)
CAL(5, t)
WAL(10, ()
PAL(16, t)
SAL(16, t)
WAL(31, f)
PAL(17, t)
CAL(15, t)
WAL(30, f)
PAL(18, t)
CAL(14, t)
WAL(28, t)
PAL(19, t)
SAL(15, t)
WAL(29, t)
PAL(20, t)
CAL(12, t)
WAL(24, t)
PAL(21, f)
SAL(13, t)
WAL(25, t)
PAL(22, t)
SAL(14, t)
WAL(27, t)
PAL(23, t)
CAL(13, t)
WAL(26, t)
PAL(24, f)
CAL(8, t)
WAL(16, t)
PAL(25, t)
SAL(9, t)
WAL(17, t)
PAL(26, t)
SAL(10, t)
WAL(19, t)
PAL(27, t)
CAL(9, t)
WAL(18, t)
PAL(28, t)
SAL(12, t)
WAL(23, t)
PAL(29, t)
CAL(11, t)
WAL(22, t)
PAL(30, t)
CAL(10, t)
WAL(20, t)
PAL(31, t)
SAL(11, t)
WAL(21, t)
Table 2.3. Conversion for natural linear progression.
32
THE WALSH FUNCTION SERIES
Thus, for fc = 610 = 01 10 2 = fc 3 , k 2 , k u k 0
then digit gi(fc) = 0© 1 = 1
g 2 (k) = 101 = 0
g 3 (k)= 100=1
so that g(k) = 101 2 = 5io and we can write
PAL(5, t) = WAL(6, 0
Conversion tables are given in Tables 2.3 and 2.4 for the first 32 WAL and
PAL functions.
Sequency order
Natural order
WAL(0, t)
CAL(0, 0
PAL(0, t)
WAL(1, t)
SAL(1, r)
PAL(1, t)
WAL(2, t)
CAL(1, t)
PAL(3, t)
WAL(3, t)
SAL(2, f)
PAL(2, f)
WAL(4, t)
CAL(2, t)
PAL(6, t)
WAL(5, t)
SAL(3, ()
PAL(7, t)
WAL(6, r)
CAL(3, t)
PAL(5, r)
WAL(7, f)
SAL(4, t)
PAL(4, *)
WAL(8 7 1)
CAL(4 7 1)
PAL(12 7 f)
WAL(9, r)
SAL(5, t)
PAL(13, t)
WAL(10, t)
CAL(5, f)
PAL(15, r)
WAL(11, t)
SAL(6, t)
PAL(14, f)
WAL(12, f)
CAL(6, f)
PAL(10, f)
WAL(13, 1)
SAL(7, 1)
PAL(11, t)
WAL(14, <)
CAL(7, t)
PAL(9, t)
WAL(15, t)
SAL(8, f)
PAL(8, /)
WAL(16, f)
CAL(8, r)
PAL(24, r)
WAL(17, f)
SAL(9, f)
PAL(25, t)
WAL(18 7 t)
CAL(9, t)
PAL(27, t)
WAL(19, f)
SAL(10, t)
PAL(26, /)
WAL(20, t)
CAL(10, f)
PAL(30, ()
WAL(21 7 0
SAL(ll 7 f)
PAL(31, t)
WAL(22, r)
CAL(ll,r)
PAL(29, t)
WAL(23, t)
SAL(12, r)
PAL(28, t)
WAL(24, f)
CAL(12, /)
PAL(20, t)
WAL(25, t)
SAL(13, r)
PAL(21, t)
WAL(26, f)
CAL(13, f)
PAL(23, t)
WAL(27, f)
SAL(14, t)
PAL(22, t)
WAL(28, f)
CAL(14, f)
PAL(18, t)
WAL(29, t)
SAL(15, t)
PAL(19, t)
WAL(30, f)
CAL(15, t)
PAL(17, t)
WAL(31, f)
SAL(16, f)
PAL(16, i)
Table 2.4. Conversion for sequency linear progression.
IIF WAVEFORM SYNTHESIS USING WALSH AND FOURIER SERIES 33
IIF Waveform synthesis using Walsh and Fourier series
The advantages of synthesis of a complex waveform from Walsh rather than
Fourier series have been described by Siemens and Kitai 20 . They show that
for many commonly encountered waveforms very small errors are found
using only 32 Walsh coefficients to yield Fourier coefficients up to approxi-
mately the tenth harmonic. They also show that the method is particularly
valuable when used as a conversion procedure for a hardware Fourier
analyser based on the Walsh functions. Such a purely digital system requires
a very much smaller averaging time than its analog counterpart permitting
resolution of signals having extremely long periods (several days). The
comparative properties of Walsh and Fourier series for the synthesis of a
complex waveform may be demonstrated by carrying out the following
procedure 21 ’ 22
The waveform is transformed and normalised to give unit value for the
largest frequency or sequency coefficient. A threshold criteria, R , is chosen
to be less than one and all coefficients found to have value <R reduced to
zero. This enables a limited number of coefficients to be made available from
which reconstruction of the original series is carried out. The accuracy of
synthesis, using these coefficients can then be compared for the Walsh and
Fourier function series.
Typical examples of this approach are shown in Fig. 2.7 and 2.8. The first
diagram shows the effect of transformation, thresholding and reconstruction
for a section of a continuous seismic waveform. Approximately twice the
number of Walsh terms are required to give a similar accuracy as may be
obtained in the Fourier case.
Fig. 2.7. Synthesis of a continuous seismic waveform.
34
THE WALSH FUNCTION SERIES
4 samples pulse
Original
Walsh 24 terms
Fourier 18 terms
Fourier 44 terms
Fig. 2.8. Synthesis of a rectangular waveform.
The second example shows the transformation and reconstruction of a
rectangular waveform. This matches the form of the Walsh function and
results in efficient reconstruction for considerably less Walsh terms than
Fourier terms.
It will be recognised from these examples that a continuous type of
waveform favours the Fourier transform whereas a rectangular waveform,
or more precisely a discontinuous waveform, is reconstructed more easily
using the Walsh transform. This would also be true for a comparison
between the Fourier and Haar transformations. In order to investigate their
synthesis properties quantitatively, two specific waveforms may be chosen to
represent these two types of signal. A continuous function
y, = exp (Xi - x ?) - exp (-*,) (2.40)
which is representative of a critically damped structure found in physical
systems, and a step function
= 0 for 0 ^ Xi < x n
*
for x n <x^ x m
1=1 for x m <Xi
(2.41)
where
and
x m >x n
i = 0,l, 2... N
IIF WAVEFORM SYNTHESIS USING WALSH AND FOURIER SERIES 35
The effects of the level of quantisation and sampling interval for these
synthesised waveforms is given in the next section.
IIF1 Effects of quantisation level and sampling interval
Providing the sampling interval chosen is several times smaller than the
Nyquist interval, reconstruction using the Fourier transform results in a
linear interpolated version of the original waveform. This is to be expected
from the linear properties of the F.F.T. and has been noted elsewhere 23 .
Using the Walsh transform a limitation is found where the order of the
highest Walsh function determines the quantisation level of the recon-
structed signal. Let us take, for example, a known case for a threshold level
of 2% where only 24 non-zero Walsh coefficients are determined and the
highest coefficient found is WAL(128, t). Here N = 1024 and the minimum
number of reconstructed data points found to represent any quantised level
will be 1024/128 = 8, hence limiting to a known finite value the realisable
accuracy of representation.
No such limitation is found with the Fourier transform since a limited
time-base is not present and a closer approximation to the original function
can be obtained.
The effect of varying the quantisation level for a continuous and a step
waveform is shown in Tables 2.5 and 2.6. In general, as the permitted
number of levels for the waveform are increased, fewer coefficients are
required to represent the waveform in the sequency or frequency domain.
No. Levels No. terms for 2% threshold
Fourier
Walsh
4
24
43
8
11
25
37
9
23
74
10
25
Table 2.5. Effect of quantisation level on a continuous function.
No. Levels
No. terms for 2% threshold
Fourier
Walsh
12
34
24
25
34
24
50
33
24
Table 2.6. Effect of quantisation level on a step function.
36
THE WALSH FUNCTION SERIES
As expected the number of coefficients required to reconstruct the continu-
ous waveform is greater in the Walsh case and the ratio between Fourier and
Walsh terms appears to remain constant with change of quantisation level.
The converse is true with the step function which seems relatively insensitive
to changes in level.
IIF2 Error considerations
Where the signal can be related to a physical system having a spring-mass-
damped form, synthesis of the waveform from a given number of sinusoidal
functions is realistic. This is the case for many physical experiments and here
the use of Fourier, rather than Walsh, transformation is likely to give
reduced errors in any form of subsequent analysis of the signal. A
rectangular-based signal is generally resultant of technology generated
systems (e.g. a communication coding system) and some analysis advantages
are obtained if the Walsh or Haar transform is used, particularly if the signal
can be binary-related to the time-base of the series. A possible criteria for
the efficiency of reconstruction permitting the selection of the Walsh, Haar
or Fourier transform is to be found in a mean-square error technique. Thus,
if the original sampled signal, x h were processed with a given threshold level,
as described earlier, and the reconstructed version from a limited number of
coefficients designated as y„ then the M.S.E. coefficient can be stated as
M.S.E. = T £ (* ( _ y.) 2 (2.42)
This is not necessarily a valid criteria for two-dimensional data as discussed
later in Chapter 7.
Risch and Brubaker have investigated the M.S.E. arising from signal
reconstruction of a finite discrete signal using a sin x/x function, a Walsh
function and a zero-order hold representation 24 .
They show that the sin x/x reconstruction, which is related to a Fourier
series, gives the least error when TV is small. As the number of sampled
points is increased, reconstruction using the Walsh function will give less
error for TV > 32. At all times the error with Walsh reconstruction is less than
with zero-order hold.
IIG Digital Sampling
The Sampling Theorem for sine-cosine functions states that for a band-
limited and sampled signal containing frequencies up to f n Hz, a sampling
rate of 2f n =f s (or sampling period of h = l/2f n = l/fs) is the minimum rate
necessary to recover completely the original signal from the sampled
IIG
DIGITAL SAMPLING
37
version. A similar theorem is applicable to the Walsh series but the
minimum sampling rate may not be 2 Z for a sequency-limited signal
containing sequences no higher than Zps.
The relevance of sampling theory has been considered by Kak 25 who has
shown that the minimum sampling rate should be f s = 2 k+1 where the
sequency bandwidth is expressed as a power of 2, i.e. Z = 2 k (or sampling
period of h = l/2 k+1 ).
Thus, if Z = 3 as found in (SAL(3, t)), then we must put Z = 4.0 (the next
nearest power of 2), giving f s = 2 2+1 = 8 rather than f s = 2Z = 6 which would
be expected from the sampling theorem for circular functions. Note that for
Z = 4.0 as found in (S AL(4, t)) it is permissible to use the same sampling rate
of f s = 2 2+1 = 8.
This is illustrated in Fig. 2.9 which shows the impossibility of reconstitut-
ing SAL(3, t) from only six samples, compared with a minimum value of 8
needed for either SAL(3, t) or SAL(4, t).
Aliasing, which is a consequence of the sampling theorem, is equally
applicable to the Walsh and Haar functions even though the functions are
SAL (3,t )
i i i i i i i
fe = 6
A possible
reconstruction
of SAL(3,t)
with f s = 6
SAL ( 4,t)
Fig. 2.9. Reconstruction of SAL(3, t) from a limited number of sample values.
38
THE WALSH FUNCTION SERIES
both time and sequency-limited. It is necessary, however, to consider the
power-of-two representation of the sequency bandwidth, as defined above,
when determining the cut-off sequency for the band-limiting low-pass
sequency filter required.
IIH Modulo-2 arithmetic
This was briefly introduced in Section IIC2 when the rules for Modulo-2
addition were given;
0©0 = 0 001=1 100=1 101=0
Here the sign © means addition in this special sense of requiring no carry
over to the next digit. In hardware terms this will be recognised as the
operation carried out by the half-adder in binary computers.
As an example of this type of arithmetic, consider the multiplication of
function WAL(13, t) with WAL(10, t). It will be seen later (Section HID)
that this results in a further Walsh function WAL(130 10, t) which requires
Modulo-2 addition for its determination. Using binary numbers for 13 and
10 we have
1010
01110
0100 = 4 10
so that the result will be WAL(4, t). This addition Modulo-2 is also seen to
be associative.
For certain purposes it is necessary to extend the definition of addition
Modulo-2 to negative numbers so that we write
(— a)®(— b) = a®b
(-a)@(+b) = -(a®b)
(-6)®(+a) = ~(a®b)
A table showing the behaviour of number products in Modulo-2 addition is
given in Appendix II.
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I.E.E.E. Trans. Comp. C21, 1451-2.
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39
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22. Beauchamp, K. G. (1973). The use of Walsh functions in the computer proces-
sing of discrete data. 6th IMEKO Congress, Dresden, D.D.R.
23. Cooley, J. W., Lewis, A. W. and Welch, P. D. (1969). The fast Fourier transform
and its applications. I.E.E.E. Trans. Ed. E12, 1, 27. ^ (?o
24. Risch, P. R. and Brubaker, T. A. (1973). Evaluation of data reconstruction using
Walsh functions. Elect. Letters 9, 21, 489-90.
25. Kak, S. C. (1970). Sampling theorem in Walsh-Fourier analysis. Elect. Letters 6,
14.
Chapter 3
Walsh Transformation
IIIA Definition of the Walsh transform
The definition for the Walsh functions given in Section IIA may be restated
by saying that every function f(t) which is integrable (in the Lebesque sense)
is capable of being represented by a Walsh series defined over the open
interval (0, 1) as
x(r) = flo + fliWAL(l, r) + a 2 WAL(2, r)+ . . . (3.1)
where the coefficients are given by
Clk
fit) WAL(fc, t) dt
From this we are able to define a transform pair,
f(t) = I F(k) WAL(fc, t)
F(k) = [ fit) WAL(fc, t) dt
Jo
(3.2)
(3.3)
(3.4)
This definition applies to a continuous function limited in time over the
interval O^t^l. For numerical use it is convenient to consider a discrete
series of N terms set up by sampling the continuous functions at N equally
spaced points over the open interval (0, 1). In order that the properties of the
continuous and discrete systems should correspond then we must make N
equal to a power of 2, i.e. N = 2 P .
40
NIB
COMPARISON WITH THE DISCRETE FOURIER TRANSFORM
41
The integration shown in equation (3.4) may then be replaced by summa-
tion, using the trapezium rule on N sampling points, x h and we can write the
finite discrete Walsh transform pair as
Xn=^- T "Zx.WALfai)
i—o
n = 0, 1, 2 . . . N—l (3.5)
and
x, = Y X n WAL(n, i)
n= 0
i = 0,l,2...N-l (3.6)
Similar transforms, X c (k ) and X s (k) can be obtained for a time series, x *
using Harmuth’s CAL and SAL functions viz.
X c (fc) = -J- Yx,CAL(fc,/) (3.7)
A i=0
and
x s (fc) = ^ T X, SAL(/c, 0
™ i = 0
As with their functions the transforms are linear so that if
then
Xi<r>X n and y,- Y n
(3.8)
aXi + by t <r> aX n + b Y n
where a and b are real constants and <-> denotes a transform operator.
Also the transform is symmetrical (equation 2.2). Since WAL(n, i) is
symmetrical about the mid-point of the sequence, i = 0, 1, 2 . . . N— 1 when
n is even, and anti-symmetric when n is odd, then it also follows from
equation (2.1) that a sequence x t will have a transform composed only of
even-order Walsh function coefficients (CAL function) if it is symmetric
about its mid-point and be composed only of odd-order (SAL function)
coefficients if the series is inversely symmetric.
I1IB Comparison with the discrete Fourier transform
The transform and its inverse given by equations (3.5) and (3.6) may be
obtained by matrix multiplication using the digital computer. Since the
42
WALSH TRANSFORMATION
matrices are symmetrical for the Walsh transform (unlike the Fourier
transform) then both transform and inverse transform are identical, except
for a scaling factor, 1/N.
If we compare equation (3.5) with the corresponding discrete Fourier
transform
= I x, exp (-jlvif/N)
/ = 0, 1, 2 . . . N— 1 (3.9)
we note that whilst WAL(n, i) is real and limited to values ± 1 the kernal,
K = exp {—jlirif/ N), is complex and can assume N different values for each
coefficient. As a direct consequence of this the Walsh transform proves
considerably easier and faster to calculate using digital methods.
It may also be noted that since the sine and cosine functions cannot be
represented exactly by a finite number of bits then a source of truncation
noise is introduced by the discrete Fourier transform which involves
repeated multiplication by a complex number. The Walsh transform, on the
other hand, involves only addition and subtraction and precise representa-
tion is possible, so that the transform is not a noisy one.
IIIC Effects of circular time shift
The discrete Fourier transform is invariant to the phase of the input signal so
that the same spectral decomposition can be obtained independently of the
phase or circular time-shift of the input signal. This is not the case for the
discrete Walsh transform.
Walsh transform signals conform to ParsevaPs theorem where the energy
in the time and sequency domains are shown to be equivalent. Expressing
this energy in terms of the sum of the squared coefficients for a discrete time
series, x i9 and its transformed coefficients, X„, a special and simplified case of
ParsevaPs theorem may be expressed as
ilW ! = Tft) 2 (3.10)
j =0 n=0
If the transformed coefficients, X n , are expressed terms of CAL and
SAL transformed coefficients (see Section VE), we can also write
1 N—l N/ 2-1 N/ 2-1
- z xl = XK0,t)+ l (Xl(k, t))+ I (X?(M)) + X?(JV/2,f))
1 > i—O k = 1 k = 1
k = l,2 . . . (N/2 — 1)
i = 0, 1, 2 . . . (N-l)
0=st=s;l
(3.11)
MIC
EFFECTS OF CIRCULAR TIME SHIFT
43
where X c (k , t) and X s (k , t) are the CAL and SAL function coefficients for x h
If now Xi is circularly-shifted forming
yi = Xi+ p (3.12)
then, as noted by Whelchel and Guinn 1 ,
Yl(K t)+Yi(K t)*X?AK t) + X 2 s (k, t) (3.13)
However, Pichler 2 has shown that if the time-shift is obtained through a
dyadic translation,
Zi = x mp (3.14)
(where © indicates Modulo-2 addition for the binary representations of /
and p ) then
Z 2 (/c, t) + Z 2 s (h t) = X 2 c (K f)+Xj(fc, 0 (3.15)
and the sequency spectrum is invariant under dyadic time-shift of the input
signal. This also indicates that a relationship must exist between the
sequency values obtained, namely
Z(k,t) = X(k@p,t) (3.16)
and is shown in the following example.
0 0.663
0.063
0
0
-0.263
0.025 0
0 -0.052
-0.006
0
0
-0.126
0.013 0
0 -0.013
-0.002
0
0
0.006
0 0
0 -0.025
-0.002
0
0
-0.062
0.006 0
Table 3.1 (a). Walsh transform coefficients for
a simple sine waveform. N = 32.
0
-0.063
0.663
0
0
0.025
0.263
0
0
0.006
-0.052
0
0
0.013
0.126
0
0
0.001
-0.013
0
0
0
-0.006
0
0
0.002
-0.025
0
0
0.006
0.062
0
Table 3.1 (b). Walsh transform coefficients for the sine waveform shifted by 90°.
Table 3.1 shows the discrete Walsh transform of a single cycle of a
sampled sinusoidal waveform having 32 values, which is compared with a
transform of the same waveform circularly-shifted through it] 2 radians. If
we take the shift index, p, as p = log 2 8 = 3, then we obtain the following
index values for the sequency coefficients k and fc®p:
44
WALSH TRANSFORMATION
90° shift ( k ) added binary values of k and p 0° shift (fc@p)
1
0001
2
0010
5
0101
6
0110
10
1010
13
1101
14
1110
18
10010
26
11010
30
11110
0010
2
0001
1
0110
6
0101
5
1001
9
1110
14
1101
13
10001
17
11001
25
11101
29
+ 00011
Taking respective values from Table 3.1 we obtain,
90° shift 0° shift
k value fc0p value
1
-0.063
2
0.063
2
0.633
1
0.633
5
0.025
6
0.025
6
0.263
5
-0.263
10
-0.052
9
-0.052
13
0.013
14
0.013
14
0.126
13
-0.126
18
-0.013
17
-0.013
26
-0.025
25
-0.025
30
0.062
29
-0.062
Squaring these values will give the expected invariance.
Figure 3.1 shows the effect of circular phase-shift on a sampled sinusoidal
signal. Two features of interest may be seen from this diagram. The value of
the sequency coefficients changes quite considerably over a 2tt radian phase
shift and a change of sign also occurs. It may further be noted that
complimentary changes in value occur for related pairs of CAL and SAL
coefficients. As a consequence the effect of circular phase shift is of less
importance where the sum of the squares of pairs of transformed coefficients
of the same sequency are taken, as with the power spectrum derivation
(Chapter 5 ), but will account for minor variations between spectra unless the
time histories are obtained commencing from the same time instant or
adjusted for zero phase shift.
IMG
EFFECTS OF CIRCULAR TIME SHIFT
45
Order of (A) Sequency — ►
Fig. 3.1. Walsh transforms of a sinusoidal waveform with circular time shift. N = 32.
Finally we may note that phase shift is unimportant in many circumstances
where a fixed reference can be taken for the signal to be analysed. Examples
are transient signals (shock transients, seismic disturbances etc.) or repeti-
tive waveforms in which the start of the signal can be readily defined (e.g.
E.C.G. signals).
46
WALSH TRANSFORMATION
HID Behaviour of transform products
The behaviour of transform products for Walsh functions are determined
from an addition relationship 3
WAL(n, t) WAL(m, t) = WAL(n©m, t) (3.17)
as may be seen from the following.
From equation (1.15) we have
WAL(n, t)=U (-1 )vi-^ +1 ) (3.18)
r—0
and
WAL(m, t) = n (3.19)
r= 0
where n , m, and t are expressed in p binary terms and AT = 2 P .
The product of the two functions will be
WAL(n, t) WAL (m, f)
p-i
r = 0
r= 0
= WAL[(n©m), i]
since the addition of binary terms of the same index must be carried out by
Modulo-2 addition.
Using equation (2.1) the following set of relationships can also be
obtained
CAL (k, t) CAL (p, t) = CAL (k®p, t)
SAL(fc, t) CAL(p, t) = SAL(p®(k - 1) + 1, t)
CAL(k, t) SAL(p, f) = SAL(fc®(p- 1)4-1, f) [ (3.21)
SAL(fc, 0 SAL(p, t) = CAL((fc - 1)® (p - 1), 0
CAL®, r) = WAL(0, 0
which correspond to the set of circular function relationships
2 cos kt cos pt = cos (k -p)t + cos (k +p)t
2 sin kt cos pt = sin (k— p)t4-sin (fc 4-p)t
2 cos kt sin pt = — sin (fc — p)t + sin (k-\-p)t
2 sin kt sin pt — cos (fc — p)t — cos (k+p)t
(3.22)
WALSH TRANSFORMATION OF A SINUSOID
47
HIE
with the important difference that a shift theorem for Walsh functions does
not exist, so that whilst the product of two Fourier transforms can be
transformed to obtain a convolution of the two original time series, a similar
result is not obtained with the Walsh transform. This is considered further in
Chapter 5 when convolution and correlation using Walsh functions are
considered.
If the addition relationship of equation (3.17) is combined with the
symmetry relationship given in equation (2.2) then we have the interesting
result
WAL (k © p, t) = WAL(fe, t) WAL (p, t)
= WAL(t, k) WAL(t, p)
= WAL(f, k®p) (3.23)
This can provide the means for generation of a series of Walsh functions
from the symmetry relationship alone, as noted in Section IIC.
HIE Walsh transformation of a sinusoid
A knowledge of the sequency content of a set of sinusoidal waveforms is
useful when comparing the sequency and frequency content of complex
waveforms. As discussed in the previous chapter, any function of time, g(t ),
can be represented by an expansion of Walsh functions over an interval
(0, T) as
g(t)= I W n WAL(n,t/T) (3.24)
n—O
where
w " = t1 8(t) WAL(n > t/T) dt < 3 - 25 )
For a sinusoidal function the order of the series expansion, n, is large so
that a high harmonic content, in the sequency sense, is obtained 4 . This is
similar to the extensive harmonic frequency content of a rectangular
waveform. The behaviour of the sequency content of a sinusoidal series may
be seen from Table 3.2. This shows a limited series of 32 Walsh transform
coefficients taken for a set of sinusoidal waveforms of normalised fre-
quency, F= 1 to 32 Hz. A broad sequency spectrum is obtained, whose
maximum is at the normalised coefficient value n = 2F (Hz) and which is
surrounded by many side-lobes. These side-lobes vanish when the input
frequency becomes equal to the Rademacher functions, which are periodic
over the time interval (e.g. WAL(1, t ), WAL(7, t) etc.).
WAL(n, t) CAUk, t) SAL (k, t)
48
WALSH TRANSFORMATION
cMmM"‘n©c-~ooON©’— icMm^in©
(Mm^in©!^-- 1 ^©©
cm m '3- in
oooooooooooooooooooooooooooooooo
o
m
O’— iino©<nTt©ococooosor-'-oo<nooosoo\oo*— imoor^oo
men t-h^h I I © r- cm cm m h in
I II I II enm
ooomsooooooovocMOOOOOOOOsoooooo^mooo
sc cm m m in h oo
| | hh mm
O’-HCMOOincMOOOoooOr-toooooNmoosO’^toooNinoomaNO
mrf t*- o in so cm cm soon so cm Tf m ©m
^ || I r-l N CM m r— I T— (
I I I
OOOOOOOmr-OOOOOOOOOOOOOOcMCMOOOOOOO
cm co ©in
< — h m m-
I
0’-HCM©©»nso©©moo©©c^ooo©ooso©OT-HCM©©’— iNooooNooo
mm t*" cm h oo m- r- oo hid M-m in on
I t-h T-H^H II H CM m • CM
I I I
©©omr-©©©o©©^cM©©©©o©©m©©o©o©M-inooo
so’— i in oc cm cm on 0
r- 1 ’—i CM CM M - | ’— i
©’-iso©©mr'©©>noo©©t-cM©©’— i'n©©ONoo©©som©©oO’— i©
mso ’—i cm som »nm os© c^-so ’— i m moo
I - m - M- I - II I
o^t^oomsooomfMOOt^oooaNooomONOOr-^ooosDfMO
m x r-i m ©oo in m cm © m ^ m cm r-
I II Y ^ -m ^ I
ooom^ooooooH r -©©©©©©’— icmoooooocnooooo
so© m o\ ©m m- m
cm ’— i ^ r-i m t- i
I I
©T-<in©©m’-i©©m^©©t^i>©©in©©©©T-H©o©'^©©t^t^©
m CM t^O -H in 00 CM© © ^ Tt © tH©
r-i m H Tf I H r-i | CM r-i |
I I
OOOOOOOinoCOQOOOOOOOOOOOOMOOOOOOOO
CM CM in ©
’—I © I CM
o^^oomONooooooHHOO^moommoomi-oooTto
m r-i r> o in cm "d- tj- h- i o cm in th©
cm *n | m | i— i | ’—i
I I
000min000©00©m000000inM000000M©000
© m cm © | in t-h cm
© | CM | | t-4
OH\ooomtoom(NOOsohOOomoooinoOH\ooommo
mm t-h© I >n i cm t-h I | cm I©
© | CM ' | 't-h I 1 |
o,_icMmr!-in©t^ooaNOHCMm'*tin©t'-ooONO’-iiMm'' 3 -‘n©t-'OOONOr-!
t-h h t-h t-h r— ( t-h t-h t-h t-h ’-h cm CM CM CM CM CM CM CM CM CM m m
Table 3.2. Walsh transformation coefficients for a set of sine waveforms of normalised frequency, F= 1 to 16 Hz (WAL(0, t ) to WAL(31, t )).
IMG
SUMMARY OF WALSH TRANSFORM CHARACTERISTICS
49
IIIF Conversion between discrete Walsh and Fourier transformation
From equations (3.6) and (3.9) we can express a sampled time series, x h of
size N, in terms of its Fourier transform series, X f , or its Walsh transform
series, X n , viz.
x i= X X f exp (jlrrif/N) = £ X n WAL(n, i) (3.26)
/= 0 n— 0
where /, i, n = 0, 1 . . . (N— 1).
Hence conversion from Fourier to Walsh transformation is
Xn=^"l x t WAL(n, i)
N i = 0
= ^ I* X f (Y WAL (n, i ) exp (j2mf/N)) (3.27)
and conversion from Walsh to Fourier transformation is
Xf = -k I X, exp (-jlmf/N)
lV j=0
= Z* Z* WAL(n, i) exp (- j2mf/N)) (3.28)
The limitation here is, of course, that N be sufficiently large so that an
accurate representation of the sampled time series is possible from a limited
number of terms. This was discussed in Section IIF where the number of
terms required was shown to be dependent on the shape of the original
continuous time function, f(t).
DIG Summary of Walsh transform characteristics
The characteristics of the Walsh function and its transform described in the
preceding text are compared with the Fourier function and summarised in
Table 3.3. The relationships given refer to a discrete time series where the
number of terms N is expressed as a power of 2.
If we consider the discrete Walsh transform, given by equation (3.5), as a
Walsh function matrix obtained by sampling the continuous function of
equation (3.4) then the relationship
W • W -1 = W” 1 • W = IV • I
always applies.
(3.29)
50
WALSH TRANSFORMATION
Walsh inverse x t = Y. X n WAL(n, i) Fourier inverse
transform " =0 transform
IIIG
SUMMARY OF WALSH TRANSFORM CHARACTERISTICS
51
52
WALSH TRANSFORMATION
Here W and W" 1 are the direct and transposed Walsh function matrices,
and / is an identity matrix (i.e. an N by N unit matrix). This derives from a
general property of the Hadamard matrix (discussed in Chapter 2) which is
related simply to the Walsh matrix by means of a permutation of the rows for
a symmetrical matrix. The relationship given in equation (3.29) is funda-
mental to the derivation of the fast Walsh transform which is examined in the
next section.
IIIH The fast Walsh transform
The basis for efficient implementation of the transformations discussed here
is the high degree of redundancy present in the transform matrix. If this
redundancy can be removed by using matrix factorisation then the efficiency
of transformation will be improved. Such a technique was described by
Good 5 and has resulted in the development of a fast Fourier transform
(F.F.T.) 6 . Similar computation algorithms are available for many other
orthogonal transformations including the Walsh and Haar transforms.
Computation of the discrete Walsh transform given in equation (3.5)
requires N 2 mathematical operations, where an operation is either an
addition or a subtraction. Using matrix factorisation techniques an
algorithm may be found to enable the transformation to be carried out using
only Nlog 2 N operations. This is known as the fast Walsh transform
(F.W.T.).
A procedure for obtaining a fast Walsh transform algorithm is described
in this section. This follows the well-known Cooley-Tukey algorithm used
for fast Fourier transformation 6 and has similar limitations. In particular the
value of N must be a power of two which will be seen later to be an essential
condition for the factorisation method used in the algorithm. It is worth
noting that certain fast Fourier transform programs may be converted to the
Walsh transform by simply setting all the trigonometric values to ± 1 and
removing the complex part of the transformation (since the Walsh transfor-
mation is a real one). The ordering of the Walsh functions obtained with this
method will, however, correspond to natural order or bit-reversed natural
order.
IIIH1 Sequency-ordered transforms
A derivation of a fast Walsh transform algorithm having sequency-ordered
coefficients is most conveniently obtained from the continued product
representation given by Pratt et aV . This was defined earlier for a series of
IIIH
THE FAST WALSH TRANSFORM
53
N = 2 P terms as
WAL(n, i)=U (-1 )v-w ( w +1 >
r=0
i, n = 0, 1, 2 . . . N— 1
r = 0, 1, 2 . . . p (3.30)
Here /, n are expressed in terms of their binary digits, i r and n r , e.g.
i = (i p - 1, ip — 2 • • • ii, io)2 (3.31)
The algorithm is developed by substitution of equation (3.30) into equa-
tion (3.5) and factorising the calculation into p separate stages. Carrying out
this substitution we obtain a product-sum expression for the discrete Walsh
transform as,
x„ = V Xl WAL(n, i) = n i (3.32)
i=0 r= 0 i r =0
(neglecting scaling by N).
Here x, is expressed as x (Ip _ 1 ... io) and X n is expressed as X(„ p _ lmmmno) where i p
and Yi p are the binary bits of i and n with r = 0, 1, 2 . . . p.
The calculation of the fast Walsh transform is carried out in a series of
stages, one stage for each power of 2 for N. The first calculation stage is to
derive a partial transformation series, A„, from the input series, x h by placing
r = 0 in equation (3.32) giving
A(n p - U i P -, .../,)= I (3.33)
i 0 =0
To see how this is calculated the case of N= 16 can be studied. We take
the first adjacent pair of data samples and look at the i x bit in Four values
of
(— 1 )" p_l(l ° +, i) for i 0 = 0 or 1
rip-i = 0 or 1
are obtained.
These indicate whether to add or subtract the two adjacent values of x t
and x i+1 (where i and i + 1 are decimal values). From these sums and
differences the two intermediate transformed values are obtained as
and either
Xi+X
i + 1
Xi x,+i or X/+1 X,-
54
WALSH TRANSFORMATION
depending on the sign for (— l)v- i(, ° +ii) worked out earlier. This process is
continued for the remaining consecutive pairs of x, coefficients.
Further stages of calculation proceed serially by using the results of the
preceding stage as input for the next stage. This may be expressed in the
general expression for the intermediate transformation as
■A r {jlp— 1? . . . Tlp- r , Ip-1) Ip— 2 • • • fr)
= I (-1) ^^Ar-Mp-l . . . ttp-r+i, ip-1 . . . ir-l) (3.34)
i r - 1=0
with the values of A i to A p retained in temporary storage during the course
of the calculation.
Finally we need to normalise the result through division by N, viz.
v
n p -i,n p - z ,...n 0 ) '
,1 A
p(n p -i,n p - 2 ,...no)
(3.35)
Thus it is seen that the complete transform may be obtained in N log 2 N
addition and subtraction operations rather than N 2 operations demanded by
the direct method of calculation for equation (3.5).
A summary of the mathematical operations required in the evaluation of
several orthogonal transformations is given in Table 3.4. These compare the
number and type of operations needed to calculate the transform for the
discrete and fast forms of the algorithm. A fast transform algorithm does not
exist for the Karhunen-Loeve transform.
Transform
Mathematical operations
Discrete Fourier
N 2 complex multiply-additions
Fast Fourier
iVlog 2 N complex multiply-additions
Discrete Walsh
N 2 addition/subtractions
Fast Walsh
iVlog 2 N additions/subtractions
Discrete Haar
N 2 additions/subtractions
Fast Haar
2 (N- 1) additions/subtractions
Discrete Karhunen-Loeve
N 2 multiply-additions
Table 3.4. Summary of transformation operations.
IIIH2 Signal flow diagrams
Computation for the fast transform algorithms can be described con-
veniently by means of the signal flow diagram. This consists of a series of
nodes, each representing a variable which is itself expressed as the sum of
other variables originating from the left of the diagram, with the nodes
connected together by means of straight lines. The weighting of these
IIIH
THE FAST WALSH TRANSFORM
55
B3
additions is indicated by a number appearing at the side of an indicating
arrow shown on these connecting lines. Thus from Fig. 3.2, the variable A7
is derived from variables originating at nodes B3 and C 3, with the latter
weighted by W 2 , so that we can write
A7 = B3+W 2 • C3 (3.36)
Figure 3.3 shows a signal flow diagram for a sequency-ordered Walsh
transform 8 which corresponds to the mathematical development given in
Section IIIH1. The solid lines in the diagram indicate that the calculated
value is to be carried forward to the addition at the next node with the same
sign, and a dotted line indicates multiplication by -1 before addition takes
place.
IIIH3 'In-place” algorithms
The final calculation stage, illustrated in the signal flow diagram of Fig. 3.3, is
interesting since it shows a series of identical self-contained operations
carried out on two values only. This is reproduced in Fig. 3.4 and has been
referred to as a “butterfly” diagram. Here the output value D 1 is obtained
from a linear combination of C x and C 2 , whilst D 2 is obtained from the
combination Ci and C 2 with the latter modified by a sign inversion.
These butterflies have the inherent advantage that in each of the transfor-
mation steps once a pair of data locations have been read then they can be
overwritten by the calculated pair of output values. This is the key to
“in-place” algorithms in which memory storage for intermediate stage
calculations is not needed since the calculated values can be placed back into
memory locations occupied previously by the initial data values. A signal
flow diagram for an “in-place” algorithm is shown in Fig. 3.5. This gives
results in bit-reversed order so that an auxiliary sorting sequence is required
to achieve binary reversal and hence linear sequency order. A study of this
flow diagram will show that calculation can proceed as a series of simple
“butterfly” calculations requiring no intermediate storage locations.
Input signal samples
(time history )
Output transformed samples
( x N sequency history)
Fig. 3.3. A signal flow diagram for a sequency-ordered discrete Walsh transform (F.W.T.)
Fig. 3.4. A “Butterfly” of a signal flow diagram.
mi
THE R TRANSFORM
59
III I The R transform
The sequency-ordered flow diagram of Fig. 3.3 is arranged to commence
with addition/subtraction of adjacent pairs for values of x h This corresponds
to Walsh’s original algorithm. An alternative version in which non-adjacent
points are selected has been described by Ulman 9 . A flow diagram for this
version is shown in Fig. 3.6. This also produces a sequency-order series.
A form of Ulman’s fast Walsh transform, known as the R -transform ( R n ),
can be obtained which is invariant to the cyclic shift of the input data (see
section IIIC). Here the subtractive terms obtained during the stage-by-stage
calculation of the transform are replaced by their absolute values. However,
unlike other transforms discussed above the original signal can no longer be
recovered by means of a second transformation.
Fig. 3.7. Comparative examples of the R-transform and the Walsh transform.
A comparative example of the R transform and the Walsh transform is
shown in Fig. 3.7. Whilst the former is effective in providing a transform
which is unaffected by cyclic phase shift of the input data, rather less
information is available concerning the sequency distribution of the trans-
formed coefficients. In particular a lack of detail is apparent in the upper
sequency range. The penalty for this cyclic invariance is seen as a restricted
60
WALSH TRANSFORMATION
dynamic range for the spectral values and increased emphasis of the lower
sequency components. Both of these are undesirable attributes limiting the
useful applications for this particular transform.
IIIJ The generalised transform
The similarities in mathematical form of the Fourier, Walsh and other
domain transformations has suggested the development of a generalised
transform which can express any of these. Caspari has developed such a
transform 10 which has led to the implementation of a fast transform
algorithm by Ahmed and Rao 11 which includes the Walsh and Fourier
transforms as special cases. These implementations are accomplished
through a Kroneker product of a set of sparse matrices, as discussed earlier,
or by matrix factorisation. The computational algorithm describes an identi-
cal sequence of operations for all the possible transforms with only the
multiplying factors changing. This is shown in terms of a signal flow diagram
in Fig. 3.8 for N= 16. The multiplying factors are shown in Table 3.5 for the
Walsh transform, the complex Walsh transform 12 and the Fourier transform.
Multiplier
F.W.T.
C.W.T.
F.F.T.
a x
1
-y
0-2
-1
)
i
1
1
K 2
a 4
-1
-1
K 10
0 5
1
1
K 6
a*
-1
-1
K 1 *
a 7
1
1
K
08
-1
-1
K 9
a 9
1
1
K 5
010
-1
-1
K 13
011
1
1
K 3
012
-1
-1
K 11
013
1
1
K 7
014
-1
-1
K 15
i =
V— I K = exp (-/ 2 Trifj 1 6)
Table 3.5. Multipliers for signal flow graphs, N = 16.
It can be shown that for N = 2 P there are p = log 2 N possible discrete
orthogonal transforms, each having a different set of multiplying constants.
In the case of the Walsh transform derived from this general transform
(known as the BIFORE transform, B n ) the order of the coefficients is in
62
WALSH TRANSFORMATION
bit-reversed natural order which relates to a dyadic rather than a real time
base 13 . (The significance of a dyadic time-base will be considered later). It
may be observed that the original Cooley-Tukey F.F.T. algorithm corres-
ponds to the generalised flow diagram of Fig. 3.8 so that, as noted earlier, it is
possible with certain computer programs to obtain a fast Walsh transform
program simply by replacing the trigonometric multiplying factors by ±1
and removing the complex part of the operation. It is also worth noting here
that, whereas the inverse Fourier transform requires the replacement of the
exponential multiplying factor, K = exp (—j2rrif/N), by its complex conju-
gate, K* = exp (j2irif/N), in Fig. 3.8 (neglecting the scaling of one or both
transforms which will become necessary), the Walsh transform forms its own
inverse and a separate transform for this is not required. Manz has also
described a particular algorithm of this type in which the Fourier, Walsh and
other transforms can be implemented as alternatives in the calling routine 14
(see also Section IVE).
Transform algorithms derived from the generalised equation require that
either the input data vector be arranged in bit-reversed order of placing or
that a similar rearrangement is carried out on the output vector. This
bit-reversed natural order for the coefficients is not a particularly convenient
order for most signal processing and communications purposes due to
SUBROUTINE WALSH (N,X,Y)
N2 = N/2
DIMENSION X(N), Y(N2)
M = ALOG2 (FLOAT(N))
Z = -1.0
DO 4 1,M
N1 = 2**(M-J+1)
J1 =2**(J-1)
DO 3 L= 1, J1
IS = (L-1)* N1+1
II = 0
W = Z
DO 1 I = IS, IS + N1-1, 2
A = X(l)
X (IS + I1) = A + X(l+1)
II = 11+1
Y(I1) = (X(I+1)-A)*W
w=w*z
1 CONTINUE
CALL FMOVE (Y(1),X(IS+N1/2),N1/2) t
3 CONTINUE
4 CONTINUE
RETURN
END
Fig. 3.9. A fast Walsh transform computer program.
t An ICL function used to make every element of an array equal to a constant value or to make
two arrays equal.
MIL
TWO DIMENSIONAL TRANSFORMATION
63
difficulties in calculation and interpretation. For this reason the sequency-
ordered algorithms described previously are to be preferred.
IIIK Transform programming
The simplicity of programming the fast Walsh or Haar transforms in a high
level language, such as Fortran, is shown by the subroutine for calculating
the Walsh transform given in Fig. 3.9. The efficiency of the routine is,
however, dependent on the way the repetitive arithmetic operations are
carried out by the high-level compiler. Programming in machine-code will
always prove faster, particularly for the bit-reversal routines required for the
“in-place” fast transforms. Further details of computer transformation
programs are given in Appendix I.
The speed advantage over the fast Fourier transform is dependent on the
particular algorithm chosen and the way this is calculated in the machine.
Some comparative figures are given below for Fortran implementation on an
ICL 1903T computer 15 . The transformed data series is 1024 samples in
length for each transformation.
Transform Time(s) Data storage (IV = 1024)
Fourier
9.48
4K
Walsh
1.60
3K
Walsh-Ulman
1.25
4K
Walsh “in-place”
2.20
2K
Haar
0.29
4K
IIIL Two-dimensional transformation
The two-dimensional finite Walsh transform of a two-dimensional array, Xu,
of N 2 points is given by the expression
X+. = Y Y Y WAL(n, i) WAL(m, j) (3.37)
IS j=o j = o
and the inverse transformation by
Xi,, = I* Y X,„ WAL(n, i) WAL (m, /) (3.38)
m =0 n =0
using the symmetry of the discrete Walsh transform.
Transformation may be carried out in two steps. First the transformation
for the variable, i , is performed, viz.
X^J = N £ x y WA L(rc,i)
(3.39)
64
WALSH TRANSFORMATION
This is equivalent to a one-dimensional transformation along each row of
the array. Then a second one-dimensional transform is taken along each
column of the transformed array for the variable, 7, viz.
X m , n = Y X^j WAL (m, 7) (3.40)
1=0
This two-stage operation is used in the digital filtering procedures to be
described in Chapter 6.
The zero sequency term for equation (3.37) is a measure of the average
value for the summation of terms in the data matrix. Thus we can write
^0,0= E E x itj (3.41)
i = 0 ;=0
If Xij represents a positive real function then the maximum possible value for
X 0 ,o is N 2 A where A is the maximum value of the function. All the Walsh
domain samples other than X 0 , 0 will vary between ±N 2 A/2 thus establish-
ing a bound for all other Walsh domain samples.
As with the single-dimensional transform Parseval’s relationship can be
applied giving
N—l N- 1 1 N—l N—l
1 1 = I I |X_| 2 (3.42)
This has important implications in bandwidth reduction since if a few of
the Walsh domain samples are of a large magnitude then it follows that the
remainder will be of a small magnitude. These small magnitude samples may
be discarded to achieve a reduced bandwidth for the data sample. This is the
method of threshold filtering discussed in Chapter 6.
HIM Hardware transformation
For certain computation purposes, such as real-time digital filtering, image
transmission and pattern recognition, there is considerable interest in
hardware transformation which can outclass digital computer (software)
transformation in terms of speed of operation 16 . The fast Walsh transform
algorithm was shown earlier to provide some speed advantages over other
forms of transformation when programmed for the digital computer and that
these advantages arise from the simpler form of the calculation since only
additions and subtractions are involved. Similar advantages are obtained
when the transformation is carried out using logical circuits 17 .
Three methods of implementation are available: analog, digital and
hybrid. Analog methods provide the speed, efficiency and flexibility
required in those applications concerned with continuous systems such as
him
HARDWARE TRANSFORMATION
65
biomedical engineering. Digital systems, using a sampled input series, can
result in fairly simple solid-state designs which are well adapted to conven-
tional logic and integrated circuits. An example of this type of realisation is
described in Section VIIE in connection with television transmission techni-
ques. It is also possible to use analog transformation methods using sampled
input data 18 . Finally the two methods can be combined to give a hybrid
system which uses logic control of an analog recursive operation to reduce
effectively the number of operational amplifiers required 19 .
HIM 1 Analog transformation
The transformation of single and two-dimensional sampled analog informa-
tion has been described by Harmuth using combinations of operational
amplifiers to carry out the processes of addition and subtraction.
The most straight-forward way for single dimensional data is the direct
implementation shown in Fig. 3.10 for N= 16. The disadvantages of this
circuit are that each input terminal must supply current to feed 16 resistors
and each operational amplifier must compensate for this current feed.
in i,1 i,2 i,3 i,4 i,5 1,6 i,7 1,8 i,9 i,10 i,11 i,12 i,13 i,14 1 ,15
♦
a-
"«=h
Jfl:
W-
■“1 ■
y
y
-s
ffl:
y
°i
y :
-p ■
-p
y
p
■
P
p ■
p
P ■
y
1
'S
S
s ■
■-<=>1
S
p
P •
-p
P
p
P •
f o, ■
;p
1_
S
S
-t=h
"S ■
■p
°l
o, ■
°i ■
■
:P _
°1
4
p
1
-S
s
S
"“I '
■p
■°i
P
p
p
P
-p
1
"“i
■<=h ■
p
■«=>i
■0|
-CD-]
■°l '
-p ■
.p
- c - h
<=>1
is
I
--«=h
‘S ■
-<=h
■•=>1
°1
Pl
°L
p
°i
-p
-p
-p
:P
1
"S
-c=h ■
■°l '
"S
-S
'°1
.-p .
.PL
.p.
.P.
_Pl
.P
.P
1
--C3-! ■
”0|
■°1
1°1 ■
■°1
-p
:PJ
:P
ip
°1
S
■°1
S
■■=>1
■p
'P
P
"P
P
-p
.P
1
-£=h
S ■
P
p
" C> 1
-p
p
+p
pi
p
1
"°1
S
<=^
S
-p
-p
P
-p
p
p
p
1
<=h
"S
p
-p
■p
P
-p
p
p
-p
p
:S
1
S
s
■°1
-S
P
-p
p
p
-p
p
ip
:P
-O]
1
S
-p
-p
P
p
■p
ip
p
-p
P
P
:S
1
S
S
1
P
■s
p
P
P
.P.
:Pl
jp]
.Pq
.Pj
:P
:p
.S
1
p
4
■“i
1
P
P
■°i
■°1
:P.
-p
P
p
:S
r
(
i,
,i
2
3,
, i
4
,i
5,
,i
<
6,
i
•
7,
,i
8^
,i
9,
,i
<
10,
,i
i
11
,i
(
12
,i
13
i
14
!
15
6Ai
7
Fig. 3.10. Direct analog hardware transformation.
A more efficient circuit which avoids these problems is suggested by
Harmuth 18 . This is based on the fast Walsh transform algorithm which is
shown in Fig. 3.11. Here two types of operational amplifier are used, one to
produce the sum of two imput voltages and the other to produce the
difference of two imput voltages. These operations are all that are needed to
implement the algorithm, independently of the size of the sampled and
66
WALSH TRANSFORMATION
Fig. 3.11. Hardware transformation based on the fast Walsh transform.
transformed input signal. Each input terminal now feeds only two summing
resistors and each amplifier need only compensate for the current flowing
through the two resistors. It is necessary, however, to use N log 2 N opera-
tional amplifiers with this circuit. Intermediate solutions are possible in
which the amplifiers sum 2 m input voltages with increased input loading. In
one design by Harmuth, used for sonar imaging, he has found it economical
to sum four voltage in this way (see Chapter 8).
An alternative design, due to Carl and Swartwood, exploits the recursive
nature of the fast transform algorithm 19 . If each summing function is used
log 2 N times through a feedback loop, the number of amplifiers is again
reduced to N with each amplifier receiving two inputs. A schematic diagram
of the arrangement is shown in Fig. 3.12. A bank of sample switches carries
out the process of serial-to-parallel conversion and provides a set of N
inputs simultaneously to the transformation logic. These input values are
sampled simultaneously by a set of sample-and-hold amplifiers, SH„, and
presented to the operational amplifiers, A n . The sums and differences of the
HARDWARE TRANSFORMATION
67
HIM
Gate control
Fig. 3.12. A Hybrid transform system.
sampled input are fed back through a gated set of sample-and-hold ampli-
fiers, FB n , to the relevant inputs, SH n . The process is repeated N— 1 times
to obtain the final transformed values at the output terminals. Control of the
gating and sample-and-hold amplifiers is accomplished by a cyclic sequence
of control pulses having appropriate duration and timing.
Note that the pairs of sample-and-hold amplifiers and sum/difference
amplifiers, shown in Fig. 3.12, together perform the function of the
“butterfly” stage operation already described in Section IIIH.
68
WALSH TRANSFORMATION
N 2 - picture
input samples
Fjhj = B
transformed
samples
Fig. 3.13. Two-dimensional real-time transformation.
The complete transformation carries out the conversion in natural order.
Sequency ordering may be obtained by arranging the samples obtained at
the output of the serial-to-parallel conversion logic into bit-reversed order
of their position in the output register prior to parallel transformation.
MM2 Digital transformation
A digital transform circuit employing recursive techniques is described by
Elliott and Shum 20 . This is similar to the sampled analog system described in
the preceding section. A data ordering register drives a series of “butterfly”
logic units and carries out a re-cycling of the results from the output of the
addition units to the inputs of the register. After log 2 N recursions the output
is available at the output register in sequency order.
A simpler solution is adopted where the order of the transform is small
(e.g. with AT =$64). The transform circuit is constructed from a series of
log 2 N identical stages, the results of each intermediate vector transforma-
tion being transferred as input to the rest. The process follows closely the
working of the fast Walsh transform algorithm described in Section IIIH
with the output at each of the intermediate stages being obtained in a serial
manner (unlike the parallel operation implied in Fig. 3.3). The transforma-
tion circuit of Walker and Clarke 21 is of this type and is described later in
Chapter 7.
Inverse transformation to carry out the function
x(t) = Y X n • WAL(n, t) (3.43)
n=0
can be performed by similar hardware realisation of the fast transform
algorithm. An alternative arrangement, which has the merit of extreme
simplicity, is to multiply each Walsh coefficient separately with a programm-
able Walsh generator. The products X n , WAL(n, t) are then summed in a
binary adder. This is carried out for each transformed sample value. In a
him
HARDWARE TRANSFORMATION
69
design due to Brown and Elliott 22 the logical output is converted to analog
form to produce a continuously transformed output signal. Updating of the
input coefficients and Walsh generator values are controlled by means of a
small digital computer.
Two-dimensional transformation for image transmission operates on N 2
sample values. In order to achieve a reasonable transmission speed some
form of parallel transformation of the vector series is desirable. Where N is
small, it is possible to duplicate the transform hardware so that N separate
transform systems, each operating on N samples, can be operated simul-
taneously. This gives rise to some redundancy in hardware components and,
furthermore, has the disadvantage that the transform operation cannot be
commenced until after the entire picture has been scanned and digitised.
The decomposition into rows and columns given in Section IIIL also
requires the complete matrix x ih to be available.
An alternative decomposition theorem has been suggested by Alexan-
dras 23 which is based on the matrix representation for the two-dimensional
transformation given by equation (3.37), viz.
B = H • A • H (3.44)
where H represents an N by N Walsh matrix and A is the N by N set of data
values, Xij. If h, = (ha, h i2 , . . . h iN ) is the ith row vector of the matrix H and we
define a column vector, F,-, to represent the product vector
/li
h
Fj = : = H A, (3.45)
/m
then the transformation of A will be given by the sum of the products F,- • h,
as
B = I F t h, (3.46)
i = 1
Thus, the process of two-dimensional transformation can proceed by imple-
menting three distinct steps:
(1) Each picture line which has been scanned and digitised is multiplied
by the Walsh vector component to form a partial transformation, F*.
(2) The products are retained in an accumulating array of N elements
until N such products have been stored (i.e. one scan line has been
processed). The vector products F, • h, are then obtained and stored.
(3) Finally the elements of F f • h, are added to give the transformation of
the picture when all the picture lines have been scanned.
70
WALSH TRANSFORMATION
The important feature to note about this transformation is that only one
line is operated on at a time, so that immediately step (1) is carried out and
the partial transformed vector, F,-, becomes available, it can be copied into a
store for step (2) leaving the store for step (1) vacant to hold the results of the
next line calculation. The delay in processing is thus reduced from N 2
elements to N elements, constituting one line of the picture.
A machine for carrying out parallel transformation of pictures in this way
has been described by Alexandridis and Klinger 24 . This is shown in Fig. 3.13.
The digitised picture elements are input to the parallel transformer one
column (scanning line) at a time. Each column is thus processed separately
and all the samples from a column are processed simultaneously in parallel.
A Walsh matrix is generated or stored and applied as input to steps PI and
P2 of the machine. The Walsh matrix and picture element lines are put into
the PI unit one at a time. Each line produces one component of the vector F*.
The P2 unit carries out the operation F, • h, for / = 1 , 2 , ... IV and outputs its
results to the summation unit, S, to produce the transformed picture.
In operation, overlapping of transmission and computation can take place
since, while one line of data is placed in storage, calculations on the samples
of the previous line are carried out to obtain a partial transformation. The
accumulations of these partial transformations represents the complete
transform of the picture.
Further discussions on hardware transformation is given in Chapters 7
and 8.
References
1. Whelchel, J. E. and Guinn, D. F. (1968). The fast Fourier-Hadamard transform
and its use in signal representation and classification. Tech. Report. PRC68-1 1.
Melpar Inc. Falls Church, Va 22046.
2. Pichler, F. (1970). Some aspects of a theory of correlation with respect to Walsh
harmonic analysis. University Maryland Report, R-70-11, College Park, Mary-
land.
3. Pichler, F. (1967). Das system der sal und cal Funktionen als Erweiterung des
Systems der Walsh Funktionen und die Theorie der sal und cal Fourier Transfor-
mation. Ph.D. Thesis, University of Innsbruck, Austria.
4. Boesswetter, C. (1970). Sequency analysis and synthesis of signals. Nachrichten
Zeitung 6, 313-19.
5. Good, I. J. (1958). The interaction algorithm and practical Fourier analysis. I.
Royal Statist. Soc. ( London ) B20, 361.
6. Cooley, J. W. and Tukey, J. W. (1965). An algorithm for the machine calculation
of complex Fourier series. Math. Comp. 19 , 297.
7. Pratt, W. K., Kane, J. and Andrews, H. C. (1970). Hadamard transform image
coding. Proc. I.E.E.E., 57, 1, 58-68.
8. Beauchamp, K. G. (1972). The Walsh Transform — a new tool for control
engineers. Kybernetes 2, 113-25.
REFERENCES
71
9. Ulman, L. J. (1970). Computation of the Hadamard transform and the R-
transform in ordered form. I.E.E.E. Trans. Comp. C19, 359-60.
10. Caspari, K. (1970). Generalised spectrum analysis. 1970 Proceedings: Applica-
tions of Walsh Functions, Washington D.C., AD 707431.
11. Ahmed, N. and Rao, K. R. (1971). The generalised transform. 1971 Proceed-
ings: Applications of Walsh Functions, Washington, D.C., AD 727000.
12. Ahmed, N. and Rao, K. R. (1970). The complex Bifore transform. Elect. Letters
6, 8, 256-8.
13. Gibbs, J. E. and Millard, M. L. (1969). Walsh functions as solutions of a logical
differential equation. National Physical Laboratory, Report No. 1.
14. Manz, J. W. (1972). A sequency-ordered fast Walsh transform. I.E.E.E. Trans.
Audio and Electroacoustics AV-20 3, 204-5.
15. Beauchamp, K. G., Kent, P., Torode, S. E., Hulley, E. and Williamson, H. E.
(1974). The BOON system — a comprehensive technique for time-series
analysis. 1974 Proceedings: COMPSTAT symposium, University of Vienna,
437-46.
16. Harmuth, H. F. (1970). Survey of analog sequency filters based on Walsh
functions. 1970 Proceedings: Applications of Walsh Functions, Washington
D.C., AD 707431.
17. Wishner. H. D. (1972). Designing a special-purpose digital image processor.
Computer Design 11, 71-6.
18. Harmuth, H. F. (1974). Two dimensional spatial hardware filters for acoustic
imaging. 1974 Proceedings: Applications of Walsh Functions, Washington,
D.C.
19. Carl, J. W. and Swartwood, R. V. (1973). A hybrid Walsh transform computer.
I.E.E.E. Trans. Comp. C22, 669-72.
20. Elliott, A. R. and Shum, Y. Y. (1972). A parallel array hardware implementa-
tion of the fast Hadamard and Walsh transforms. 1972 Proceedings: Applica-
tions of Walsh Functions, Washington D.C., AD 744650.
21. Walker, R. (1974). Hadamard transformation — a real-time transformer for
broadcast standard P.C.M. television. B.B.C. Research Department Report,
BC RD 1974/7.
22. Brown, W. O. and Elliott, A. R. (1972). A digital instrument for the inverse
Walsh transform. 1972 Proceedings: Applications of Walsh Functions,
Washington D.C., AD 744650.
23. Alexandridis, N. A. (1971). Walsh-Hadamard transformations in image proces-
sing. University of California, L.A. Eng. Report 7108.
24. Alexandridis, N. A. and Klinger, A. (1972). Real-time Walsh-Hadamard
transformation. I.E.E.E. Trans. Comp. C21, 288-92.
Chapter 4
The Haar Function
IVA Introduction
The Haar function set forms a complete set of orthogonal rectangular
functions similar in several respects to the Walsh functions. They were
established rather earlier than the Walsh functions by the Hungarian
methematician, Alfred Haar in a paper published in 1910 1 . He described a
set of orthogonal functions, each taking essentially only two values, and yet
providing an expansion of a given continuous function, using these new
functions which could be made to converge uniformly and rapidly. This was
a property not obtained by any other set of non-sinusoidal orthogonal
functions at that time.
Little practical use was made of these functions for over half a century
until the 1960’s when they were seen to provide some computational
advantages in certain areas of communication 2 , image coding 3,4 and digital
filtering 5 . This interest prompted a review of the mathematical properties of
the Haar series and notable papers have been published by Uljanov 6,7 and
McLaughlin 8 .
IVB Haar function definition
The Haar functions form an orthogonal and orthonormal system of periodic
square waves. The amplitude values of these square waves do not have
uniform value, as with Walsh waveforms, but assume a limited set of values,
0, ± 1 , ± V2 ± 2, ± 2>/2, ± 4 etc. They may be expressed in a similar manner
72
IVB
HAAR FUNCTION DEFINITION
73
to the Walsh functions as
HAR(n, t) (4.1)
If we consider the time base to be defined as 0^ t ^ 1 then, following the
simplified definition suggested by Kremer 9 , we can write
HAR(0, t ) = 1
for
o
/A
***.
/A
HAR(1, t) =
■{-1
for
for
0
l
: s.
for
O
/A
A
•Nf*
HAR(2, t) = !
-42
for
k^t< 2
. 0
for
5*Sf=Sl
0
for
0=Sf<5
HAR(3, t) = <
42
for
t*t<l
-42
for
HAR(2 P + n, t) =
p = 1,2.
f 44 p for n/2 p ^t<(n+k)/2 p
-42 p for (n+i)/2 p ^t<(n + l}j2 p
0 elsewhere
n = 0, 1 . . . 2 P — 1
This allows a sequential numbering system analogous to that adopted by
Walsh for his function series.
The first eight Haar functions are shown in Fig. 4.1. The first two functions
are identical to WAL(0, t ) and WAL(1, t). The next function, HAR(2, t ), is
74
THE HAAR FUNCTION
t'lr
[i]
[-1]
/2
- s /2
1
/2
~/2
2
-2
2
-2
2
-2
HAR (0,t)
HAR(l.t)
HAR (2,t)
HAR (3,t)
HAR (4,t)
HAR (5,t)
HAR (6,t)
HAR(7,t)
Fig. 4.1. The first eight Haar functions
simply HAR(1, t) squeezed into the left-hand half of the time base and
modified in amplitude to ± yfl. The next function HAR(3, t) is identical but
squeezed into the right-hand half of the time-base. Subsequent pairs of
functions are similarly squeezed and shifted having amplitudes ± 1 multi-
plied by powers of V2. In general all members of the same function subset
(such as HAR(2, t) and HAR(3, t), or HAR(4, t), HAR(5, t), HAR(6, t)
and HAR(7, t), etc.) are obtained by a lateral shift of the first member along
the time axis by an amount proportional to its length. Since the functions
behave very much as the series of block pulses found in sequential multiplex-
ing systems, it is not possible to apply the term sequency used for the Walsh
IVB
HAAR FUNCTION DEFINITION
75
series. By adopting a different definition for the series, namely
HAR(0, t) = 1 for j
HARO', j, f) = V? for
= -J2‘ for
>
= 0 elsewhere
i = 0,1,2... j = 1...2'
(4.3)
then the Haar functions can be referred to by order, y, and degree, i, as well
as time, t. The degree, i, then denotes a subset having the same number of
zero crossings in a given width, 1/2*, thus providing a form of comparison
with frequency and sequency terminology. The order, y, gives the position of
the function within this subset. All members of the subset with the same
degree are obtained by shifting the first member along the axis by an amount
proportional to its order.
From Fig. 4.1 the essential characteristic of the Haar function is seen as a
constant value everywhere except in one sub-interval where a double step
occurs. This type of function is also found in a single line scan for certain
types of images and has led to the suggestion that the Haar function may be
useful in edge detection as part of a pattern recognition technique 10 .
From the definition given in equation (4.2) it can be seen that the Haar
functions are orthogonal, thus
HAR(m, t) HAR(n, t)
1 for n — m
0 for n^m
(4.4)
The proof of completeness of the series is given by Haar 1 .
A given continuous function, f(t ), within the interval and
repeated periodically outside this interval can be synthesised from a Haar
series by
where
f(t)= I C n HAR(n, t)
n= 0
(4.5)
C„ =
f(t) HAR(n, f) dt
(4.6)
76
THE HAAR FUNCTION
As with other complete series, ParsevaPs equivalence holds and we can write
f f(t)dt=lc 2 n (4.7)
Jt=0 n =0
There is no theorem analogous to the shift or addition theorems found
with Fourier and Walsh series respectively.
IVB1 Convergence of the Haar series
The convergence features of the expansion in Haar functions are superior to
the Walsh functions, as noted by Alexits 11 , since for some continuous
functions the Walsh expansion can actually diverge at a given point. This
cannot occur in the case of Haar expansions.
Any finite approximation to a function, /(f), using the Haar series,
will take the form of a step function having 2 P equal length steps. Some
examples are given in Fig. 4.2. The effect of additional terms is simple and
intuitive, unlike the effect of adding Fourier or Walsh terms to a finite
approximation.
If we consider the value of a partial sum at each step size we find that this is
simply the mean value of f(t) in the interval covered by the step. This is the
condition for the best step function approximation of f(t) in terms of the
mean-square-error and accounts for the comparatively small number of
terms necessary to synthesise a waveform from the Haar series 12 . A related
property is that the expansion coefficients are proportional to the difference
in the mean value of f(t) over the adjacent subintervals.
IVC Relationship between the Walsh and Haar functions
This has been derived by Kremer 9 in terms of a set of block pulse functions
which are similar to the positive excursions of the Haar functions shown in
Fig. 4.1. They are defined as
(1 for n/2 p ^t^(n + l)/2 p
q(2 p ,n;t)=
(0 elsewhere
p = 0, 1 . . . n = 0, 1 . . . 2 P — 1 (4.8)
The functions are rectangular, having values of 0 and + 1, and are defined in
term of degree, p, which is inversely related to the pulse width period, and an
order, n , which gives the position of the pulse width within a given degree (or
function subset) along the time axis.
IVC
RELATIONSHIP BETWEEN THE WALSH AND HAAR FUNCTIONS
77
The reason for using these set of pulses is that they can act as a link
between the Haar and Walsh function sets. The block pulses defined in
equation (4.8) can easily be described in terms of limited and linear
combinations of Walsh functions.
Thus,
q(l, 0; t) = WAL(0, t) = 1
q( 2 P , n ; f) = ^ l‘ WAL(n, k/ 2”) WAL (fc, t )
^ k =0
(4.9)
Furthermore, by expressing the Haar functions as combinations of posi-
tive and negative block pulses, using the definition of equation (4.8), we can
78
THE HAAR FUNCTION
write
HAR(0, t) = q(l,0;t)
HAR(1, t) = q(2, 0; t)-q(2, 1; t)
HAR(2, t) = J2[q(4, 0; t)-q( 4, 1; ()]
HAR(3, t) = V2[q( 4, 2; t)-q( 4, 3; f)]
(4.10)
HAR(2 P + n, t) = J2lq(2 p+1 , 2 n ; () - q(2 p+1 , 2n + l;t)] J
Equations (4.9) and (4.10) can be combined to obtain a general relationship
between the Haar and Walsh functions, viz.
1 (2P +1 -1)
HAR(2 P + n,t) = — -j= I [WAL(2n, fc/2 p+1 )
2V2 P k=o
- WAL(2 n + 1, fc/2 p+1 )]WAL(fc, t) (4.11)
where p and n are equivalent to the degree i and order j given in equation
(4.3).
Kremer also gives a method of simultaneously describing the first N = 2 P
Haar functions by means of an N dimensional mapping matrix, enabling a
suitable conversion algorithm to be obtained for the digital computer.
IVD The discrete Haar transform
From equations (4.5) and (4.6) the discrete Haar transform and its inverse
can be stated as
X, HAR(n, i/N)
N j= 0
(4.12)
x,= I* X„HAR (n,i/N)
(4.13)
n =0
i,n = 0, 1...N-1
IVD
THE DISCRETE HAAR TRANSFORM
79
Written in matrix form, equations (4.12) and (4.13) become
Hx
(4.14)
and
x = H _1 • X (4.15)
where H and H " 1 are the direct and transposed Haar function matrices.
Unlike the Walsh transform the matrix is not symmetric, so that separate
transform operations are required for transformation and inverse transfor-
mation.
It can be shown 9 that the transform matrix can also be written as the
product of a diagonal matrix containing weighting factors consisting of
multiples of and a matrix consisting only of 0, +1 and — 1.
IVE The fast Haar transform
If the transformation given in equation (4.12) is carried out directly then N 2
additions will become necessary. This can be reduced to p • N where N = 2 P
if only the non-zero values are considered.
A considerable improvement in computational efficiency is obtained if a
factorisation algorithm similar to that used for the fast Fourier and Walsh
transforms is employed. A flow diagram for a 16 point Haar transform is
shown in Fig. 4.3. The solid lines show addition and the dotted lines
subtraction at the nodal points. Multiplication of the sum/differences by 1 or
V2 is indicated in the diagram. It will be seen that at each step in the
calculation (other than with the first) half the points require no further
calculation. (The multiplications can all be delayed until the transformation
is complete.) Thus, the total number of additions or subtractions is
N N
N+-+-+...+2 = 2(N-l) (4.16)
Tranformation time is therefore linearly proportional to the number of
terms, N, in contrast to Walsh or Fourier where it is proportional to N times
the logarithm of N to the base of 2 (see Table 3.3). It may also be noted that
the average number of operations per sample is independent of transform
size for the Haar transformation. The Haar algorithm gives the fastest linear
transformation presently available and due to its simplicity, it is particularly
valuable for small computers having no floating-point hardware. Since the
matrix for the Haar transform is not symmetrical a separate inverse trans-
form is required. A flow diagram for this is shown in Fig. 4.4.
82
THE HAAR FUNCTION
Matrix relationships between the Walsh and Haar transforms have been
developed by Fino 13 who has shown that a fast Walsh transform algorithm
can be obtained directly from a fast Haar transform algorithm by means of a
recursive operation. This could have some computational advantage where
both transformations are needed, such as in the slant transform described in
Section VIIE.
Although the computational algorithm for the Haar transform, shown in
Fig. 4.3, gives the fastest transformation, it may not be the most desirable
where the transformation is accomplished by means of processor hardware.
Ahmed 14 has described a Cooley-Tukey type of algorithm which enables a
single fast processor to be constructed so that fast Fourier, Walsh or Haar
transforms can be computed by simple modifications to the basic logic. A
Wiener filtering application is described for such a processor.
1VF Two-dimensional Haar transformation
The two-dimensional Haar transform of an array x of N 2 points may be
stated from equation (4.12) as
X m , n =±Y Y *y HAR(n, i/N) HAR(m, j/N) (4.17)
N j= 0 j=o
Transformation is carried out through decomposition in the same way as
with the Walsh transformation (equations 3.39 and 3.40) using a partial
transformation
Knj = Y X y HAR(n, i/N ) (4.18)
i= 0
followed by a second single-dimensional transformation
X m , n = Y X m ,j HAR(m, j/N) (4.19)
j=0
This process is essentially an operation on a single-dimensional data series
constructed from successive subsets of two-dimensional data.
A better approach to processing data which is intrinsically two-
dimensional is to use a two-dimensional set of functions having analogous
properties to the Haar series. A Haar-like series has been proposed by
Shore 15 which comprises three types of orthonormal two-dimensional func-
tions having similar characteristics to the Haar series. When a function on
the unit square, f(x, y), is expanded in terms of these functions, the Nth
partial sum, P N (x, y), (see section IVB1) is a step function of 2 2N square
steps, each covering an area of 1 /2 2N . The value of P N {x, y ) at any step is the
IVF
TWO-DIMENSIONAL HAAR TRANSFORMATION
83
mean value of /( x, y) over the area covered by the step. Also, the expansion
coefficients are proportional to the difference in the mean value of /( x, y)
over adjacent sub-areas of the unit square. As with the Haar single-
dimensional series, this gives the optimum convergence conditions for step
function approximation and leads to an efficient computational algorithm.
The desired sequence is defined by Shore in terms of the three functions
shown in Fig. 4.5. The first of these has a saddle-shape and is given by
Sl(x, y) = HAR y (n, /) HAR x (n, i ) (4.20)
The second has a horizontal shape and is given by
Hi I(jc, y) = HAR y (n, i) HAR x (n, i) (4.21)
The third has a vertical shape and is given by
Vn(x, y) = HAR y (n, i) HAR x (n, i) (4.22)
The Nth partial sum which defines the transformation is then given as
P„(jc,y) = Co+ x I z [dnSn(x, y) + bnH%x, y) + c n V%x, y)]
n = 1 i = 1 ;'= 1
(4.23)
where C 0 is a constant term. The coefficients a% b i] n and cl constitute the
two-dimensional transformation.
An analog method of determining the set of coefficients for a given scene
depicted on a film transparency would be to project a beam of light through a
Fig. 4.5. A series of Haar-like two-dimensional functions.
84
THE HAAR FUNCTION
series of masks each having transmitting areas only in the positive regions
shown in Fig. 4.5. Shore gives a version of the modified fast Haar transform
for a digitally sampled scene transformed in accordance with equation
(4.23).
This two-dimensional transform is appropriate for image transmission
where large areas of the scene are constant or slowly changing with time. It
could also be useful for edge detection since coefficients bl are sensitive to
horizontal edges whilst cl are sensitive to vertical edges of the object
outline 10 .
IVG The Haar power spectrum
The periodogram definition for the Haar spectrum such as will be described
later for the Walsh series (equation 5. 16) is difficult to use with the definition
given by equation (4.2) since the equivalent periodicity of a number of
adjacent functions can be identical. If we define the effective sequency of the
Haar function series as “one half the average number of zero crossings per
unit time interval” then it is seen that the Haar functions fall into discrete
groups, each member of a group having the same effective sequency as other
members of the same group (Table 4.1). This was referred to earlier as the
definition by order and degree, given by equation (4.3). Using this definition
a power spectrum can be defined which can be considered as analogous to
the sequency or frequency energy spectrum.
The procedure is to take the sum of the normalised value of the squares of
the line spectra that fall within each grouping and to use these values, plotted
on a suitable expanded time scale (Fig. 4.6) to show the energy contained in
J i i i I
2° 2 1 2 2 2 3 Zps
Sequency groups
Fig. 4.6. Haar line spectra.
the Haar transformation. This has some similarities with Ahmed’s odd-
harmonic sequency spectrum (Section VG) and shares with it the disadvan-
tage of providing a small number of spectral values for a given number of
data points. However, the computational advantages will in many cases
outweigh the sparsity of calculated values and the technique has been
IVG
THE HAAR POWER SPECTRUM
85
applied successfully by Thomas 16 , to detect and identify bursts of energy
widely distributed over frequency and time, in a way which cannot easily be
carried out by other forms of analysis.
Sequency group
HAR(0, t)
1
HAR(1, t)
2
HAR(2, t), HAR(3, t)
3
HAR(4, t), HAR(5, t) j
l 4
HAR(6, 0, HAR(7, t) J
r 4
HAR(8, t), HAR(9, t)
HAR(10, t), HAR(11, f) |
HAR(12, f), HAR(13, t)
HAR(14, 1), HAR(15, t)
y 5
Table 4.1. Haar sequency groupings for the first 1 6 Haar functions.
References
1. Haar, A. (1910). Zur theorie der orthogonalen Functionensysteme. Math .
Annal. 69, 331-71.
2. Hammond, J. L. and Johnson, R. S. (1962). A review of orthogonal square-wave
functions and their application to linear networks. J. Franklin Inst. 273, 21 1-25.
3. Andrews, K. C., Pratt, W. K. and Caspari, K. (1970). “Computer Techniques in
Image Processing”. Academic Press, New York and London.
4. Gerardin, L. A. and Flanent, J. (1969). Geometrical pattern feature extraction
by projection of Haar orthogonal basis. International Joint Conference on
Artificial Intelligence.
5. Gubbins, D., Scoll^r, I. and Wisskirchen, P. (1971). Two-dimensional digital
filtering with Haar and Walsh transforms. Annales de Geophysique 27, 2,
85-104.
6. Uljanov, P. L. (1964). On Haar series. Mat. Sb. 63, 105, 356-91.
7. Uljanov, P. L. (1967). On some properties of Haar series. Mat. Zametki 1,
17-24.
8. McLaughlin, J. R. (1969). Haar series. Trans. Am. Math. Soc. 137, 153-76.
9. Kremer, H. (1971). Algorithms for the Haar functions and the fast Haar
transform. Symposium: Theory and Applications of Walsh functions, Hatfield
Polytechnic, England.
10. Rosenfield, A. and Thurston, M. (1971). Edge and curve detection for visual
scene analysis. I.E.E.E. Trans. Computers C20, 562-9.
11. Alexits, G. (1961). “Convergence Problems of Orthogonal Series”. Pergamon
Press, New York and London.
12. Shore, J. E. (1973). On the application of Haar functions. I.E.E.E. Trans.
Communications COM 21, 3, 209-16.
86
THE HAAR FUNCTION
13. Fino, B. J. (1972). Relations between Haar and Walsh-Hadamard transforms.
Proc. I.E.E.E. 60 , 5 , 647-8.
14. Ahmed, N., Natarajan, T. and Rao, K. R. (1973). Some considerations of the
Haar and modified Walsh-Hadamard transforms. 1973 Proceedings: Applica-
tions of Walsh Functions, Washington D.C., AD 763000.
15. Shore, J. E. (1973). A two-dimensional Haar-like transform. NRL Report 7472,
AD 755433.
16. Thomas, D. W. (1973). Burst detection using the Haar spectrum. 1973 Proceed-
ings: Theory and Applications of Walsh functions, Hatfield Polytechnic, Eng-
land.
Chapter 5
Spectral Decomposition
VA Walsh spectral analysis
Spectral analysis in terms of sequency rather than frequency was first
suggested in a paper by Polyak and Schneider 1 where the Walsh function is
applied to the spectral analysis of discontinuous and transient waveforms. A
form of spectral analysis based on the Walsh series was developed by Gibbs
and Millard 2 who produced a theory for Walsh functions analogous to that of
Wiener-Khintchine leading to a definition of the power spectrum through
the classical route. The relationships between the Walsh and Fourier spec-
trum derivation in terms of matrix manipulation have been discussed further
by Pichler 3 and described in terms of digital computer usage by Robinson 4
and Yuen 5 .
An important feature of the definition of power spectra using Walsh
functions is that it is possible for the power spectrum to be sequency-limited
although the corresponding time functions are time-limited. This is in
contrast to the behaviour of the Fourier transform in a power spectrum
definition where a time-limited function cannot have a frequency-limited
power spectrum. This has relevance to the application of Walsh functions in
the analysis of non-stationary data as noted by Gibbs 2 .
Apart from the more normal methods of spectral analysis advantage has
been taken of the special properties of natural Walsh ordering to define
other forms of power spectrum, such as those proposed by Ohnsorg 6 and
developed by Ahmed and Rao 7 " 10 , to give a highly compressed spectral
representation.
87
88
SPECTRAL DECOMPOSITION
A review of the characteristics of these various forms of Walsh spectral
analysis has been given elsewhere 11 and is repeated here in an expanded
form.
The sequency spectral decomposition of a smoothly-varying waveform,
such as a sinusoid, was shown earlier to have a more complex spectrum than
is obtained with Fourier analysis. This is illustrated in Fig. 5.1, which shows
the pattern of sequency power distribution for a wide band of sinusoidal
waveforms. For those frequencies which are not binary multiples of the
sequency time-base, considerable energy is found outside the sequency
which has zero-crossing equality with that of the sinusoidal waveform. A
similar complexity is obtained by the Fourier analysis of a rectangular
waveform. Figure 5.2 gives a comparison between the power spectra
obtained in the two cases.
Walsh sequency no. (Zps)
H Peak power 0 >30% ^>3% H Finite power present
Fig. 5.1. Sequency power distribution for a set of sinusoidal waveforms, of frequency 1 to
32 Hz over a Walsh sequency spectrum extending from 1 to 32 Zps.
VA
WALSH SPECTRAL ANALYSIS
89
Corresponding
Fourier
analysis
15 Hz
(b) Fourier power spectrum of
Corresponding
Walsh analysis
t
15 Zps
Fig. 5.2. Comparison between Walsh and Fourier spectra of sinusoidal and rectangular
waveforms.
A second feature of Walsh spectral decomposition is the effect of circular
time-shift noted earlier (Section IIIC). As seen in Fig. 3.2, corresponding
SAL and CAL functions of the same sequency will vary quite considerably in
amplitude, but in a reciprocal manner. This suggests that the summation of
squares of the CAL and SAL function coefficients, such as occur in one
definition of the power spectrum, will reduce this effect. Figure 5.3 gives the
power spectrum of a random process calculated in this way for various values
of circular time-shift. Comparatively small changes in spectral shape are
seen in this example. The general case can be formulated from a statement of
VA
WALSH SPECTRAL ANALYSIS
91
the energy equivalence in the time and sequency domains as discussed
earlier.
It should be noted that Ulman’s R transform (Section III I) is, of course,
invariant with phase shift and its coefficients may be squared and normalised
to yield a phase-invariant energy spectrum. An example of the R energy
spectrum for a single sinusoidal waveform is shown in Fig. 5.4 where it is
compared with a similarly-derived Walsh energy spectrum for the two cases,
x = sin (ot and x = sin [o> + (tt/2)]i. The shaded area gives the error bounds
for the Walsh spectrum arising from the added 7t/2 phase shift of the
applied waveform. As noted earlier, the phase invariance of the R trans-
form is obtained at the cost of greatly reduced detail in the higher sequency
region, making the method of little value for other than strongly periodic
data.
The lack of an equiValent shift theorem for the Walsh function when this is
considered in terms of spectral analysis means that the derivation of the
power spectrum via the autocorrelation function, which represents the
“classical method” of Fourier spectral analysis, is not directly possible. It has
been shown by Gibbs 12 that a dyadic equivalent of the Weiner-Khintchine
transformation theory is valid providing the time-shift is obtained from
Modulo-2 addition. This gives a direct evaluation of the sequency power
spectrum from the Walsh transformation of the dyadic autocorrelation
function. This may be used for spectral analysis of real data providing a
relationship between arithmetic autocorrelation and dyadic autocorrelation
can be found. A technique for deriving the power spectrum in this way is
discussed later.
As with Fourier analysis three alternative classes of operation can be
carried out to derive the Walsh power spectrum.
(1) An indirect method via the dyadic autocorrelation function (equiv-
alent Wiener-Khintchine method).
(2) Direct evaluation via the squared value of the Walsh transform
(equivalent periodogram method).
(3) Narrow-band Walsh filtering.
Only method (1) can be invariant with circular time-shift. The spectral
co-ordinates in all cases are based on the general concept of sequency
instead of frequency which is relevant only to the sinusoidal case.
The additional relationships required for a discussion of spectral decom-
position for the Walsh function are summarised in Table 5.1. These relate
to a discrete time series, x h of length N and complete the comparison
between Walsh and Fourier characteristics, commenced in Table 3.2. Some
of these were discussed or derived in earlier chapters; others will be
considered here.
92
SPECTRAL DECOMPOSITION
a« + a*
t + ^ +
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8 §+ 8
‘P^ I 15
1 I ■* I
.V w
w-^*i -c
°< a, 5,‘S,
g g.g C
o § 'S •£
^ ^ ^ ^
>?
Convolution x t ®yi+>X k • Y fc Convolution jc, * y , ^X„ •
product <h> = Transform product
relation relation
VA
WALSH SPECTRAL ANALYSIS
93
c
o
13
o
3
<
O
<u
JC
*3 03 S
O > o
tS <D f
.5 £ o
X! & £
■ w (S £
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n ©*
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Jwl
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94
SPECTRAL DECOMPOSITION
Normalised sequency
Fig. 5.4. Comparison between the R and Walsh spectra for a phase-shifted waveform.
VB
CORRELATION AND CONVOLUTION
95
VB Correlation and Convolution
This is where the behaviour of the Walsh function is so different from the
sine-cosine functions requiring a different view of a series of time lag values
to that experienced in Fourier analysis. Before we are able to understand the
first of the methods of spectral analysis defined above, namely derivation
through the autocorrelation function, the special properties of correlation in
the Walsh domain must first be considered.
Let us define two time series, x t and y h which have transformed values in
the Walsh domain as X k and Y k respectively. The convolution of the two
time series can be expressed as
Z(r) = T N £ Xl y r - i =X k *Y k (5.1)
rv ,-=o
where r indicates the incremental lag value.
Replacing jc f and y T _/ with their transformed values and letting Y k = Y,
Z(r) =-J- Y I* X, WAL(fc, Off Y,WAL(/, r-i)l
N ,_o U_ 0 JL,-o J
= Y Y X^P- Y WAL (k, 0 WAL(Z, T- i)l (5.2)
/c =0 1 = 0 LiV j =0 J
The expression in brackets is the convolution of the discrete Walsh func-
tions.
If we carry out a similar convolution using the Fourier transform then the
corresponding time delay function, Y(r — t) exp (-jlTrin/ N ) can be decom-
posed into sine-cosine products by the use of the shift theorem to permit the
simple relationship
x(t)*y(t)<*X(f)-Y(f) (5.3)
to be applied, where <-> indicates a transform operator. Thus, convolution of
x (t) and y(t) becomes equivalent to the inverse transform of the product of
their Fourier transforms, X(f) and Y(f).
No such simple relationship between a Walsh function and a delayed
version of the same function exists, so that a direct equivalence between the
products of the Walsh transforms and the convolution of the time domain
representations cannot be used. However by making use of the Walsh
addition theorem (equation 3.17) we find that this can play a similar part to
the shift theorem in Fourier convolution.
Thus, for two series,
Xi *^>X k
y^Y k
(5.4)
96
SPECTRAL DECOMPOSITION
we now define dyadic convolution as
z(r) =-^ Y * • y T *. = x k ®Y k
N i =0
where © represents the operation of dyadic convolution, and
Z(r)4z x\Y Y k WAL(fc, T®i)
iV i=0 Lfc=o
Using equation (3.17)
Z(t) = T Y £* X, WAL(fc, i) WAL (k, r)
iV k=0 i=0
= YXk- y t WAL(fe, T)
k = 0
This results in an equivalent relationship to equation (5.3), viz.
Xi®yi±*X k • Y k
(5.5)
(5.6)
(5.7)
(5.8)
Hence two different sets of relationships exist side by side for Fourier and
Walsh series. Both express a form of convolution theory but, whereas the
Fourier version implies arithmetic addition for the recursive time-shift, the
Walsh version requires the substitution of dyadic or Modulo-2 addition.
Using these expressions we are now in a position to compare discrete
correlation in the Fourier and the Walsh case.
Discrete autocorrelation in real time may be defined as
R f (t)=— I XjX i+T (5.9)
X j= 0
where /= 0, 1, 2 ... m and m « N. Here m is the total correlation lag.
Discrete autocorrelation in dyadic time is given as
l N- 1
R w ( t )=— I XiXiOr (5.10)
The difference between the two expressions lies in the addition behaviour
of the incremental time lag. In the real time domain this addition is
arithmetic and in the Walsh domain this is Modulo-2 addition. Similar
remarks apply to cross-correlation of two discrete time series where a
translation matrix is also required.
Dyadic convolution and correlation are identical because addition and
subtraction, Modulo-2 are identical operations. Neither can be evaluated
rapidly through the product of the Walsh transforms, as with Fourier
VB
CORRELATION AND CONVOLUTION
97
analysis, for reasons stated earlier. Kennett 13 has described computer
algorithms to determine these functions by the summation of the products of
the Walsh transforms (equation 5.2) but their usefulness is limited since they
require even more operations than direct evaluation using equation (5.1).
VB1 The dyadic time scale
It was shown in the previous section how the process of correlation for Walsh
series is dependent on the time addition being carried out using Modulo-2
addition rather than arithmetic or linear lag addition. The characteristics of
Modulo-2 addition are seen from the tables given in Appendix B. This
process imparts a peculiar behaviour to time in what has become known as
the dyadic domain. Dyadic time is compared with real (linear) time in Fig.
5.5. Here consecutive data samples for a process are shown equally spaced
(that is in real time) so that, whereas in arithmetic correlation the lag
increases uniformly with time, in the dyadic (Walsh) case the time lag varies
in the manner shown. Here consecutive data samples are taken at time
instants which progress in a series of jumps unequal in length over both
forward and backward time intervals.
It is difficult to associate this with our normal experience of correlation
and this is what makes correlation results obtained using the Walsh series of
so little direct value. The results of this type of correlation, obtained in
dyadic time, are of use, however, in that conversion to arithmetic correlation
can be made through a matrix translation. This has certain advantages in
spectral evaluation which are discussed later in Section VD.
Time
Real
Fig. 5.5. Linear and dyadic time.
98
SPECTRAL DECOMPOSITION
VC Applicability of the Wiener-Khintchine theory
The convolution theorem, expressed by equation (5.8), is closely related to
that of Wiener-Khintchine which states that the Fourier transform of the
autocorrelation function for a time series is equal to its power spectrum, thus
Px(f)=\ Rx(t) exp (— /27t/t) dr (5.11)
J—oo
The converse also holds so that we have the general relationship
R x (t)<-> P x (f) (5.12)
Gibbs 2 has proved that a similar relationship is applicable between the
Walsh transform of a function and its dyadic autocorrelation. Thus, we can
write
R w (r)^P x (k) (5.13)
where P x (k) is the discrete Walsh power spectrum expressed in terms of a
sequency series and R w (r) is the dyadic autocorrelation. This has become
known as the logical derivation of the Wiener-Khintchine theorem.
VD The sequency spectrum via the autocorrelation function
A major advantage of deriving the Walsh power spectrum in this way is that
the spectra obtained will be phase-invariant. To obtain the spectrum by this
route an additional calculation step is involved. This is shown in Fig. 5.6. The
arithmetic autocorrelation is obtained first and the result transformed
through a matrix operation to yield the dyadic autocorrelation of the
function. This may then be transformed, using the fast Walsh transform, to
obtain the power spectrum through equation (5.13). The additional step
required is the correlation matrix operation needed to convert from the
arithmetic to the dyadic correlation function. Whilst the Walsh transforma-
tion is obtained quite rapidly the correlation matrix operation involves N 2
additions and if N is large this represents a serious lengthening of the
calculations involved.
The translation between the correlation functions of these two systems
has been shown by Pichler to take the form of two linear matrices,
T a -i, — D/v * T/sr and T l _a - T^ 1 • Djv 1 (5.14)
where T A - L refers to translation from arithmetic to logical correlation and
T^_ a to translation from logical to arithmetic correlation. D N is a N by N
diagonal matrix whose elements are simply related to the binary representa-
tion for the numbers 0, 1,2...N— l.T N is also an N by N matrix generated
recursively.
VD THE SEQUENCY SPECTRUM VIA THE AUTOCORRELATED FUNCTION
99
Real time
Dyadic time
Translation
matrices
1 Fo urier
Walsh
^transform
transform
Power
Power
spectral
spectral
density
density
Fourier
Walsh
domain
domain
Fig. 5.6. Relationship between Fourier and Walsh spectrum in real and dyadic time.
A recursive algorithm using a “shuffling” matrix is defined by Robinson 4
and will permit the required matrix and its inverse to be developed for a
known size of the series, N, commencing with IT 1 = 1.
Gibbs and Pichler 14 have succeeded in combining the two steps into one
matrix multiplication but at a cost of N 2 multiplications and additions which
is even slower than carrying out the translation and fast transform sepa-
rately.
A more economical approach is carried out by Yuen 15 who has established
a relationship between the Walsh power spectrum and the autocorrelation
function without the intermediate calculation via the logical correlation
function. To do this a square matrix, consisting of N by N values, operates
on a column vector made up of sampled values of the auto-correlation
function. This square matrix is related to a Paley-ordered matrix and its rows
obey certain recursive relations. Because of these relationships a measure of
redundancy is found so that the matrix may be factorised into a product of
sparse matrices and multiplication carried out in a manner similar to the fast
Walsh transform. Yuen 5 has shown that the power spectral calculation can
be carried out in 3 N log 2 N additions plus N log 2 N divisions by 2. This latter
simply represents a series of shifts to the right in a holding register.
This seems about the best approach to power spectrum derivation through
the Wiener-Khintchine method that is currently available. For very many
engineering applications, however, where some phase-related variation of
100
SPECTRAL DECOMPOSITION
the result can be tolerated, the periodogram method is to be preferred due to
its simplicity of calculation.
VE The Periodogram approach
The power spectrum coefficients can also be determined in a manner
analogous to the periodogram used in Fourier power spectral analysis, viz.
P x (k) — Re (X fc ) 2 + Im (X k ) 2 (5.15)
where Re (X k ) is the real (sinusoidal) component of the complex Fourier
transform and Im (X k ) is the imaginary (cosine) component. Thus, for the
Walsh spectrum we would calculate
p(o)=xm
P{k) = X 2 c (k)+X 2 s {k) >
P(N/2) = X 2 s (N/2)
fc = 1, 2 . . . (N/2 — 1)
(5.16)
giving ( N/2 ) + 1 spectral points.
It should be noted that whilst equation (5.16) can represent energy and
hence conforms to Parseval’s theorem , VP x (/c) does n ot give a sequency
amplitude spectrum as obtained with Vcos 2 art + sin 2 cot in circular function
theory since an addition theorem similar to that found with sine and cosine
would be needed.
The realisation of equation (5.16) will result in a change in the energy
structure dependant on the cyclic phase of the time series data. This was
noted earlier and an illustration of its effect given in Fig. 5.3.
VE1 Degrees of freedom
By analogy with Fourier spectral analysis we can regard the Walsh power
spectral coefficients (other than the first and last) as being derived from a
sequency bandwidth equivalent to two degrees of freedom.
If we now average the coefficients contained in a number of adjacent
bands (making sure that we average over an even number to include pairs of
the same sequency CAL and SAL coefficients) then we can write for 2D
degrees of freedom
P D (k) = ^ Z [X 2 c (k+L-l) + X 2 s (k + L-l)]
giving [(N/2) — l]l/£> spectral points.
(5.17)
VF COMPARISONS BETWEEN WALSH AND FOURIER SPECTRA 1 01
VF Comparisons between Walsh and Fourier spectra
A choice of Fourier or Walsh spectral estimation for a given purpose will be
found to be strongly dependent on the signal characteristics, as indicated
earlier in Section IIIC in connection with transform theory. Some examples
will now be given to illustrate this dependence.
Figure 5.7 shows a short-term transient signal obtained from a shock-
excited mechanical structure. A comparison between the Walsh and Fourier
power spectrum obtained is shown in Fig. 5.8. We can see from this that
close similarities exist in the main region of spectra power. However, a
significant region of higher sequency power is present in the Walsh case,
well-removed from the main region found in both representations. This is a
characteristic of certain types of shock phenomena which has been com-
mented upon by Both and Burman 16 and Kennett 17 , who suggest that this
second region may indicate a useful characteristic for the signal. The
underlying reasons for this are associated with the origin of the particular
Fig. 5.7. A shock transient waveform.
VF
COMPARISONS BETWEEN WALSH AND FOURIER SPECTRA
103
signal being analysed. Since this is obtained from harmonic motion of a
mechanical structure and hence defined by means of a linear differential
equation, we can expect it to be represented by an exponential series, i.e. a
sum of sinusoidal and cosinusoidal terms. We saw in Fig. 5.1 that a
smoothly-varying sinusoidal signal will give rise to just these additional
energy regions of higher-sequency terms when analysed in this way.
A similar region of high-order frequency power coefficients is found with
the Fourier analysis of a rectangular synthesised signal, such as the compara-
tive spectra of the pulse-coded modulation (P.C.M.) waveform shown in the
second example, Fig. 5.9. The Walsh spectral representation of the P.C.M.
signal shows a precise sequency-limited bandwidth, due to the finite number
of terms required to synthesise a binary-coded signal having a length which
is related to the Walsh time-base. In the Fourier case a theoretically
unlimited series of harmonic coefficients is produced by the spectral
analysis.
An example which exploits this difference in representation is shown in
Fig. 5.10. This concerns the differences in the Walsh power spectra obtained
for two rectangular types of signal. The first signal (a) is derived from a given
morse code message of 15 characters which is digitally sampled into 512
samples having either of two values, 0 or 1. The mark/space ratio of the
encoded morse is maintained at a precise value, such as would be obtained
with machine-generated code. The second signal ( b ) represents the same
message but this time the Mark/space ratio is varied slightly in a random
manner, as would be experienced, for example, if the message were sent
using a hand-key. Comparison between the two spectra shows very clearly
the essential difference between the two sets of code. A regular sequency-
limited spectra is obtained in the machine-sent case, showing that
discrimination between the two cases can easily be obtained in this
domain whereas the difference would be difficult to detect in the original
time domain.
These examples indicate clearly the respective roles of Walsh and Fourier
spectral analysis for discontinuous and smooth-varying signals respectively.
Where the signal is derived from a sinusoidally-based waveform, such as
would be obtained from a spring-mass-damped system (mechanical struc-
ture), then Fourier analysis is relevant. Where the signal contains sharp
discontinuities and a limited number of levels such that it may be synthesised
from a combination of rectangular waveforms, then Walsh analysis is
appropriate. However, for some special purposes (e.g. on-line applications),
the speed with which the Walsh transform may be obtained by digital
computation becomes more important in the analysis of sinusoidally-based
signals despite the increased complexity in the spectrum so obtained.
Fig. 5.9. Comparison between the sequency and frequency spectrum of a P.C.M. waveform
signal, coded in NRZ-M code.
104
106
SPECTRAL DECOMPOSITION
VG The odd-harmonic sequency spectrum
This has been developed using the naturally-ordered BIFORE transform,
(Section IIIJ) and results in a highly compressed spectrum of k + 1 points
which is invariant to the cyclic shift of the input waveform. As with the
periodogram spectrum, the resultant CAL and SAL functions are squared
and summed. However, in this case they are grouped as the squared values
of odd harmonics of a fundamental sequency before being added together.
This is shown in Table 5.2. The bit-reversed natural order for the BIFORE
transform coefficients simplifies this grouping.
This grouping of coefficient values may be compared with the determina-
tion of the Haar spectrum (Section IVG) where the groupings are concerned
with finding identical average zero-crossing values (Table 4.1). In both cases
the number of spectral coefficients is reduced considerably when compared
with those obtained by the application of equations (5.15) and (5.16). The
comparison between the two forms of coefficient grouping is commented
upon by Ahmed etal. 18 who also point to similarities in the calculation of the
two transforms.
Power spectral
BT coefficient Sequency coefficient coefficient
BT(0, t)
CAL(0, t)
BT(1, t)
SAL(8, t)
BT(2, t)
SAL(4, t) + 1
BT(3, t)
CAL(4, t) J
BT(4, t)
SAL(2, t) 1
BT(5, t)
CAL(6, t)+ l
BT(6, t)
CAL(2, t) (
BT(7, t)
SAL(6, t) J
BT(8, t)
SAL(1, t ) '
BT(9, t)
CAL(7, t)
BT(10, t)
CAL(3, t)
BT(11, t)
SAL(5, t)
BT(12, t)
CAL(1, t) +
BT(13, t)
SAL(7, t)
BT(14, t)
SAL(3, t)
BT(15, t)
CAL(5, t) J
where P 0 = zero sequency coefficient
Pi = odd-harmonics of 1st sequency
P 2 = odd-harmonics of 2nd sequency
P 3 = 4th sequency component
P 4 = 8th sequency component
Table 5.2. Derivation of odd-harmonic sequency power spectrum for N= 16.
VG
THE ODD-HARMONIC SEQUENCY SPECTRUM
107
The k + 1 components of the odd-harmonic sequency spectrum are
defined as
P(0) = B 2 o
P(P)= Y Bl
n = 2'“ 1
(5.18)
i = l,2 . . . p
n = 0, 1...N N = 2 P
where B n is the BIFORE transformed value.
It is not necessary to evaluate all the transform coefficients, JB n , before
carrying out the summation and a simplified signal flow diagram for the
complete spectrum, due to Ohnsorg 6 , is shown in Fig. 5.11. It will be seen
that the sparse nature of this computational matrix permits considerable
economy in the calculation of the spectral coefficients. Algorithms for
BIFORE spectral calculations have been described by Ahmed et al 19 .
In common with the HAAR power spectrum, the BIFORE spectrum
suffers from the disadvantage of producing only p spectral coefficients for a
data series set of 2 P points. As a consequence, it is not able to characterise
the sampled data set particularly well. Despite this, its economy in computer
calculation time is attractive and has resulted in a number of applications.
Examples are found in systems analysis 20 , image coding 21 , coding of vocoder
speech signals 22 and in speech synthesis 23 .
VG1 Sequency octave analysis
By making the assumption that the information in each odd-harmonic
sequency grouping extends to the highest sequency (N/ 2), it is possible to
derive a form of sequency octave analysis based on these results. Table 5.3
indicates the error arising from this assumption which is acceptable for all
but the highest harmonic groupings. Referring to Fig. 5.12, we can sum the
power in a number of octaves spanned by a given odd-harmonic sequence
and equate the average power per octave with the average power per
spectral coefficient.
Thus, for N= 1024
B + C + D + E + F+G + H+I + J P 1
9 “512
C+D+E + F+G + H+I+J P 2
256
8
etc.
108
SPECTRAL DECOMPOSITION
Fig. 5.11. Flow diagram for the BIFORE power spectrum.
VG
THE ODD-HARMONIC SEQUENCY SPECTRUM
109
Power
Odd harmonic
sequency groupings
Octave
% error
Po
0
0
Pi
1,3, 5, 7... 511
0.2
P 2
2, 6, 10, 14... 510
0.4
Ps
4,12, 20, 28... 508
0.8
P 4
8, 24, 40 . . . 504
1.6
Ps
16, 48, 80 . . . 496
2.3
Pe
32, 96, 160 .. . 480
6.7
Pn
64, 192, 320 .. . 448
14.3
Ps
128,384
33.4
P 9
256
—
P\o
512
—
Table 5.3. Sequency groupings for N— 1024, k = 10, with octave error percentage.
P
8
1
P 7
1
r 6 r
i
p 5 r
r i r
p o
Sequency
P 9
A
B
C
D
E
F
G
H
1
J
0 I 2 4 8 16 32 64 128 256 512
Fig. 5.12. Sequency octave analysis.
110
SPECTRAL DECOMPOSITION
Rearranging,
B + C+D + E + F+G + H+I+J = 9Pj/512
C+D+E+F+G + H+I+J = 8P 2 /256
D + E + F + G + H + 1 + J = 7 P 3 / 128
E + F+G + H+I+J = 6PJ64
G + H+I+J = 5P 5 /32
H+I + J = 3P 7 /8
I+J = 2P S / 4
J = P 9 /2
K = P io
from which
A=P 0
B = (9/512)Pi -(8/256)P 2
C = (8/256)P 2 -(7/128)P 3
D = (7/128)P 3 — (6/64)P 4
E = (6/64)P 4 — (5/32)P 5
P = (5/32)P 5 -(4/16)P 6
G = (4/16)P 6 — (3/8)P 7
H = ( 3/8)P 7 -(2/4)P 8
/ = (2/4)P 8 — (1/2)P 9
J = (1/2)P 9
K = P 10
Hence, generally for p sequency octaves (2 P = N)
v ' 2 P -' 2 F ~‘~ 1
i = 1,2 . . . p -1
P 0 (out) = P 0
Pp(oct) = P p
(5.19)
VG
THE ODD-HARMONIC SEQUENCY SPECTRUM
111
Since these represent average powers the total power in each octave is given
by the product of P*( oct) and the octave frequency band, (f 2 —fi). Thus,
Oi = Pi(oct)i (5.20)
(for a spectrum normalised to 1/T).
For random data where the harmonic components are randomly distri-
buted (i.e. not consisting entirely of odd-harmonic components as found
with a symmetrical ramp or half-sine wave), some general idea of octave
spectral decomposition can be obtained. A computationally rapid evalua-
tion of octave sequency content for a random signal can be obtained, but
considerable error results for data not falling into the limited class described
above.
VH Short-term spectral analysis
The spectral analysis of non-stationary data presents the problem of
representing a three-dimensional view of the spectral content of a time
series (Section VIF). It is necessary to show the energy content of the data
simultaneously in terms of both time and frequency. Several methods are
available to do this. One is to assume that the data series is stationary, or
nearly so, over a short time sequence and to divide the series into a
consecutive number of shorter series, each of which can be analysed as a
stationary series and a series of two-dimensional representations given one
for each of the short series. An example is given in Fig. 5.13. Here the
spectra of each of the short series are shown, one behind the other, using a
technique of “hidden line suppression” so that the changes in the spectral
content of the entire series can be seen along the dotted time axis.
An alternative method of analysis is to apply a running transform which
acts on a short segment of the signal and is progressively shifted along the
time series to give a running power spectrum along the record length. This
technique is known as short-term spectral analysis 24 which is shown dia-
grammatically in Fig. 5.14.
A short-length discrete transform operator is slid progressively along the
signal record and the resulting short-length power spectra over a time
window equal to the length of the operator are displayed as a function of
time at the centre point of the operator. The resolution of the analysis is
proportional to the length of the transform operator whilst the accurate
localisation in time requires a short operator series. Some compromise is
necessary in the practical case in order to secure acceptable results.
The use of a Walsh transform operator has been considered by
Gulamhusein 25 and Kennett 17 who compare the results obtained with
Fourier methods. Kennett has found that the use of a triangular “window”
VH
SHORT-TERM SPECTRAL ANALYSIS
113
Transform
operator
time
Power
spectra
i
f l j 2 *3
time
FIG. 5.14. Short-term spectral analysis.
for the sliding transform sequence is desirable and its use corresponds to the
effect obtained with a cosine taper window in the Fourier case to reduce
discontinuity errors. An example of the display method due to Kennett, is
shown in Fig. 5.15. This shows the short-term sequency time plot of a seismic
disturbance recorded at a seismograph array station compared with a similar
display using Fourier spectral analysis. The sequency time-plot is seen to be
more complex than its frequency counterpart which is due to the additional
sidebands generated by the Walsh function. The position of these sidebands
can provide useful information as an aid to earthquake pattern recognition.
Further examples of this are given in Kennett’s paper.
Gulamhusein has developed a somewhat similar method which defines a
short-time Walsh energy spectrum as the square of the Walsh transform
weighted by a dyadic function, h(t® A), which corresponds to the impulse
response of a linear dyadic invariant system, viz.
E(n, t ) =
[/(A)WAL(n, A)]/i(t©A)
(5.21)
The weighting function, h(t® A), is chosen such that its product with the
signal f(X)h(t®\) is capable of transformation by the naturally-ordered
Walsh transform.
(a)
Fig. 5.15. Sequency time plot of a seismic disturbance: (a) Seismic time record, (b) Frequency-
time plot, (c) Sequency-time plot.
114
VH
SHORT-TERM SPECTRAL ANALYSIS
115
This method can be shown to be equivalent to the well-known technique
of Fano 26 who carries out a running power spectrum in which the signal is
weighted such that earlier contributions contribute very little to the calcu-
lated spectrum. As the time of the sampled signal is advanced so the
contribution of these earlier samples decreases and prominence is given to
those samples near the current sampled value. Either of these methods is
applicable to on-line spectral analysis of a continuous function of time.
References
1. Polyak, B. T. and Schreider, Y. A. (1962). The application of Walsh functions in
approximate calculations. Voprosy Theor. Matem. Mashin Coll. II, 174-90.
2. Gibbs, J. E. and Millard, M. J. (1969). Walsh functions as solutions of a logical
differential equation. National Physical Laboratory report No. 1, N.P.L., Eng-
land.
3. Pichler, F. R. (1970). Some aspects of a theory of correlation with respect to
Walsh harmonic analysis. Report R-70-11, Department Electrical Eng., Mary-
land University., AD 714596.
4. Robinson, G. S. (1972). Discrete Walsh and Fourier power spectra. 1972
Proceedings: Applications of Walsh Functions, Washington D.C., AD 744650.
5. Yuen, C. K. (1972). The computation of Walsh power spectrum. Tech. Report
No. 72. Basser Dept, of Comp. Sciences, University of Sydney.
6. Ohnsorg, F. R. (1971). Spectral modes of the Walsh-Hadamard transform. 1971
Proceedings: Applications of Walsh Functions, Washington D.C., AD 727000.
7. Ahmed, N. (1971). The generalised transform. 1971 Proceedings: Applications
of Walsh Functions, Washington D.C., AD 727000.
8. Ahmed, N. and Rao, K. R. (1971). Fast complex BIFORE transform by matrix
pardoning. I.E.E.E. Trans. Computers C20, 707-10.
9. Ahmed, N. and Rao, K. R. (1970). Complex BIFORE transform. Elect. Letters
6, 256-8.
10. Ahmed, N. et al. (1972). On an analogy between the Fourier and Walsh-
Hadamard transforms. Proceedings: N.E.C. Chicago 27, 383-7.
11. Beauchamp, K. G. (1972). The Walsh power spectrum. Proceedings: N.E.C.
Chicago 27, 377-82.
12. Gibbs, J. E. (1970). Walsh spectroscopy, a form of spectral analysis well-suited
to binary digital representation. Unpubl. National Physical Laboratory report,
N.P.L. England.
13. Kennett, B. L. N. (1971). Introduction to the finite Walsh transform and the
theory of the fast Walsh transform. 1971 Proceedings: Theory and Applications
of Walsh Functions, Hatfield Polytechnic, England.
14. Gibbs, J. E. and Pichler, F. R: (1971). Comments on the transformation of
Fourier power spectra into Walsh power spectra. I.E.E.E. Trans. Electromag-
netic Compat. EMC13, 3, 51-4.
15. Yuen, C. K. (1973). A fast algorithm for computing Walsh power spectrum.
1973 Proceedings: Applications of Walsh Functions, Washington D.C., AD
763000.
116
SPECTRAL DECOMPOSITION
16. Both, M. and Burman, S. (1972). Walsh spectroscopy of Rayleigh waves caused
by underground explosions. 1972 Proceedings: Applications of Walsh Func-
tions, Washington D.C., AD 744650.
17. Kennett, B. L. N. (1974). Short-term spectral analysis and sequency filtering of
seismic data. NATO Advanced Study Institute, Sandefjord, “Exploitation of
Seismograph Networks” (Ed. K. G. Beauchamp). Noordhoff Int. Pub. Co.,
Leiden, Netherlands.
18. Ahmed, N., Natarajan, T. and Rao, L. R. (1973). Some considerations of the
Haar and modified Walsh-Hadamard transform. 1973 Proceedings: Applica-
tions of Walsh Functions, Washington, D.C., AD 763000.
19. Ahmed, N., Abdussattar, A. L. and Rao, K. R. (1972). Efficient computation of
the Walsh-Hadamard transform spectral modes. 1972 Proceedings: Applica-
tions of Walsh Functions, Washington D.C., AD 744650.
20. Ahmed, N. and Rao, K. R. (1970). Spectral analysis of linear digital systems
using BIFORE. Elect Letters 6, 43-4.
21. Pratt, W. K., Kane, J. and Andrews, H. C. (1969). Hadamard transform image
coding. Proc. I.E.E.E. 57, 58-68.
22. Crowther, W. R. and Rader, C. M. (1966). Efficient coding of Vocoder channel
signals using linear transformation. Proc. I.E.E.E. 54, 1594-5.
23. Campanella, S. J. and Robinson, G. S. (1970). Analog sequency composition of
voice signals. 1970 Proceedings: Applications of Walsh Functions, Washington
D.C., AD 707431.
24. Beauchamp, K. G. (1973). “Signal Processing”, Chapter 11. George Allen and
Unwin, London and John Wiley, New York.
25. Galamhusein, M. N. and Fallside, F. (1973). Short-term spectral and autocorre-
lation analysis in the Walsh domain. I.E.E.E. Trans. Inf. Theory IT-19, 5,
615-23.
26. Fano, R. M. (1950). Short-term autocorrelation functions and power spectra. /.
Acoust. Soc. Amer. 22 , 546-50.
Chapter 6
Seqtiency Filtering
VIA Sequency filtering
Numerous filtering techniques have been developed for use with Walsh and
Haar functions. Early work was concerned with analog circuits in which the
essential ingredients are integrators, switches and sample/hold amplifiers.
Harmuth 1,2 gives several examples of these. Golden 3 has described a reso-
nant LC filter consisting of inductors, capacitors and a switch driven by a
sampled Walsh function. Later developments of the resonant filter are the
digital designs of Nagle 4 and others using logical hardware.
Matched analog filters have been described 5 in which the coefficients of a
Walsh function generator are modified by the characteristics of the signal
and the desired filter response before summation, to form a synthesised
version of the filtered signal. A design of transversal analog filter has been
constructed which involves a resistive Walsh matrix in place of a delay
network 6 .
Digital filter development has used the fast Walsh transformation either
as a means of simplifying the hardware logic requirements 7,8 or to produce
efficient software filters 9-11 . The lines of development for software filters
have been constrained by the difficulty in carrying out convolution with the
Walsh series and this has resulted in the use of generalised Wiener tech-
niques 12 . Two-dimensional filtering techniques have been applied to recent
work in image analysis and transmission 13 , and this has led to the construc-
tion of fast matrix hardware specifically for real-time applications 14 .
The basis for a number of these development will be discussed here as a
preliminary to a description of applications given in later chapters.
117
118
SEQUENCY FILTERING
VIB Analog sequency filters
Let us consider the definition of a periodic function, f(t), given by equation
(3.3) in terms of its transformed value, F(fc), given in equation (3.4). To filter
the function in the sequency domain, it is necessary to modify each coeffi-
cient value, F(fc), by a transfer characteristic, H(k), and to transform the
modified value at some arbitrary time, t'(k), later to produce an output from
the filter
y(t)= X F(k) • H(k) WAL(fc, t-t') (6.1)
fc =0
The simplest realisable system is obtained when t'(k ) = 1 for all values of k
and when
y (t) = X F(k) • H(k) WAL (fc, t - 1) (6.2)
fc =0
Harmuth 1 was the first to propose an analog sequency low-pass filter in
which H(k) = 0 for k > 1 and H(k) = a constant for k = 1, thus giving an
output
y(t) = F(0) • H( 0) WAL(0, t- 1) (6.3)
Since WAL(0, * — 1) = 1 for all values of t then
y(t)=\ 1 fit) dt (6.4)
Jo
with unit delay, i.e. the output represents the average value of the input at
each step value.
This may be mechanised by the simple combination of an integrator and a
sample-and-hold amplifier shown in Fig. 6.1. The output from such a
low-pass filter will be a stepped version of the input signal. Some further
smoothing of the output may be necessary to recover a continuous
waveform.
Control Control
Fig. 6.1. Analog low-pass sequency filtering.
VIB
ANALOG SEQUENCY FILTERS
119
Sequency band-pass filtering is possible using this basic design by employ-
ing the addition relationship of equation (3.17)
WAL(fc, t) WAL (p, t) = WAL(fc © p, t) (6.5)
If k = p then the product gives a Walsh function of zero order and if k = 0
then the sequency order of the product is unchanged. Hence the product of
an input signal, f(t), and a Walsh function, WAL(fc, t), will contain a zero
order component (d.c. level) of identical amplitude to a selected Walsh
component, WAL (fc, f), contained in the input signal, f(t). It will also
contain other, higher sequency components, which may be removed by a
low-pass sequency filter. It is then only necessary to multiply the output by a
unit-delayed Walsh function, WAL(fc, f+1), to recover this selected
sequency element. A form of band-pass filter is realised which may be
produced by combining the three steps given above, which is shown dia-
grammatically in Fig. 6.2.
Control
Fig. 6.2. Analog band-pass sequency filtering.
An f xtension of this method is given by Vandivere 6 and Lee 5 who have
implemented a general-purpose filter in which a programmable Walsh
function generator drives a number of such band-pass filters in parallel so
that their outputs may be summed to give the desired filtered output.
VIC Generalised Wiener filtering
Since arithmetic convolution cannot be applied directly to Walsh digital
filtering, development has been carried out somewhat differently to that
found with non-recursive Fourier filtering. In general the classical technique
of Wiener filtering has been applied 15 which includes Fourier non-recursive
filtering as a special case.
Figure 6.3 shows the generalised one-dimensional Wiener filtering sys-
tem. A signal vector, x(t) is assumed to consist of additive zero-mean signal,
120
SEQUENCY FILTERING
x(t
Filter
matrix
G
Fig. 6.3. Generalised Wiener filtering.
s(t ), and noise, n(t), components which are assumed to be uncorrelated with
each other, A unitary transformation operation utilising an N by N matrix,
A, is performed on x(t) to yield
X(f) = Ax(t) = As(t)+An(t) = S (f) + N(f) (6.6)
The resultant vector, X(/), is multiplied by an N by N filter matrix, G, and
inversely transformed to produce a filtered output
y (t) = A -1 • G • A • x(t) (6.7)
This is the familiar transform-modify-inverse transform method of filtering.
If G is chosen correctly then the required filtered output will consist of the
signal component, s(t), plus a much reduced noise component, n(t). Ideally
the filter matrix G is chosen in conjunction with the transformation matrix,
A, to minimise the mean-square error between the required signal, s(t ), and
its estimated value, y(t).
Pratt 16 has shown that discrete Wiener filtering may be implemented by
any unitary transformation, including Fourier, Walsh, Haar and Karhunen-
Loeve, for the same mean-square error. Fourier non-recursive filtering is a
special case of Wiener filtering where the filter matrix is a vector represent-
ing a conjugate symmetric set of filter weights derived from sampling the
required frequency response 17 .
Where the Walsh transformation matrix is used two possible types of filter
matrices are applicable. The filter matrix may be vector (Le. contain
non-zero components along the diagonal only) as with Fourier non-
recursive filtering, or it may be scalar (containing non-zero components off
the diagonal.)
VIC
GENERALISED WIENER FILTERING
121
Vector filtering using a Walsh transformation can be extremely fast and
may be carried out using 2 N log 2 N additions/subtractions plus N real
multiplications. However, very few practical filter requirements in terms of
actual frequency-derived specifications result in a simple diagonal filter
matrix. The situation is different for a sequency-derived specification and
this will be discussed later in Section VID.
Scalar filtering is the general case and although the process of transforma-
tion may be fast, multiplication by the filter matrix can demand up to N 2
multiplications. A reduction in computation time is obtained if many of the
off-diagonal coefficients can be made zero and it is possible to design a
sub-optimal filter of this type from known properties of a Fourier filter which
is economical in both design and computation time 18 .
A particular form of sequency-filtering, applicable to the analysis of
non-stationary signals, is discussed in Section VIF. This produces a series of
filtered signals from a single input signal, where each output series gives the
sequency content for a single sequency coefficient at each sampled value
along the time axis. The filter matrix in this case is the Walsh function
relevant to the sequency coefficient for that particular output series.
These various forms of Walsh filtering will now be described, commencing
with the simplest of these, namely, sequency-based vector filtering.
VID Sequency-based vector filtering
A simple technique of Walsh filtering is referred to as sequency-limited
filtering. Here a column vector is employed in place of the diagonal filter
matrix. The coefficients of the vector are limited in value to 0 or 1 . In the case
of a low-pass filter for example, all the sequency coefficients above a certain
value are set to zero through multiplication by the zero coefficients of the
vector before re transformation takes place. This is an extremely efficient
process and can result in a perfect “brick-wall” filter.
Analog signals can be handled by such a filter through conversion to a
stepwise approximation to the signal. The signal is integrated over the step
length and held at its final value by a sample-and-hold amplifier (Fig. 6.1).
Since the stepped approximation obtained cannot contain any Walsh func-
tion components of higher sequency than the step rate, then an ideal
(sequency) low-pass filter results.
An example is given in Fig. 6.4 where a low sequency signal (a) has added
to it a second signal (b) comprised of elements of higher sequency. Carrying
out the Wiener process using a series of l’s in the top of the filter vector and
zero elsewhere results in the original low-sequency signal being reconsti-
tuted exactly. The process is ideally suited to filtering of rectangular data to
which is added a noise component, such as would occur for example, in
122
SEQUENCY FILTERING
Fig. 6.4. Example of sequency-limited filtering.
digital transmission systems. The applicability of Wiener vector filtering to
sampled continuous signals can result in a stepped representation of the
filtered signal where a Walsh or Haar transformation is used. An example is
given in Fig. 6.5 which compares low-pass filtering of a noisy seismic
transient using Fourier, Walsh and Haar transformations. A similar form of
distortion can occur with two-dimensional image filtering where a super-
position of a chequer-board pattern occurs. The effect may be minimised to
negligible proportions if a sufficiently high order for the transformation
matrix is chosen.
VID1 Matched filtering
An extension of this method based on the known properties of the
original signal can result in an efficient matched-filtering technique which
may be used to recover a known signal immersed in noise. Here the form of
the signal is known so that some distortion of the recovered signal is
acceptable.
The relationship between the transformed values of the matching signal
and the set of limited-value column vector weights is obtained by first
defining a threshold level in the magnitude of the transformed series. A
corresponding series of weights is formed in which all values in excess of this
threshold value are made 1 and all those below it are made 0. Because of this
method of derivation the procedure has been referred to as threshold
filtering and has been applied successfully to the recovery of geological
data 19 and the enhancement of video images 20 .
If we consider the synthesis of a function by means of the summation of a
number of Walsh functions it becomes apparent that the threshold level
must be set so as to allow sufficient of these functions to be summed to form
an acceptable reconstruction of the filtered signal. Thus, for smoothly-
varying matching signals we would expect a low threshold level to minimise
distortion in the reconstructed signal. Where only a few terms are required
VID SEQUENCY-BASED VECTOR FILTERING 123
Fig. 6.5. Low-pass filtering using Fourier, Walsh and Haar transformation.
to adequately synthesise the signal, such as pulse or discontinuous functions,
then a high threshold level can be used (see Section IIF).
A flow diagram for a matched filtering program is given in Fig. 6.6. The
threshold level, L, is defined as a percentage of the largest sequency
coefficient value found in the Walsh transform of the matching signal. L is
124
SEQUENCY FILTERING
Enter N values of Enter N values of
signal matched signal
Fig. 6.6. Flow diagram for Walsh threshold filtering.
provided as a parameter to the program together with the number of
samples, N, for the signal and the matching signal.
It is interesting to carry out matched filtering on a series of pulses having
various sample widths to which increasing amounts of random noise is
added. The effect of raising the threshold level will be found to stretch the
pulse recovered from the signal. This is due to the limited number of
non-zero sequency coefficients which change their state over an increasing
larger series of samples. The stretching obtained is inversely proportional to
the width of the pulse, being greatest for narrow pulses 1 sampling unit in
VID
SEQUENCY-BASED VECTOR FILTERING
125
width. The accuracy of reconstruction for the pulse is thus dependant on the
number of samples used to define it (pulse width) and to the threshold level.
For the purpose of identification of a pulse immersed in noise we may be
less interested in its shape than in the signal/noise ratio of the resultant
filtered signal, particularly where the original signal has zero or a negative
value of signal/noise ratio.
Some quantitative results of the matched filtering process acting on a
sampled series representing a single pulse immersed in noise are shown in
Table 6.1. These results show very little variation for pulse widths varying
from 1-8 sample values. For the higher threshold levels only the presence or
absence of the pulse is determined. All information concerning the shape of
the pulse or its amplitude relative to other matched pulses will be lost.
After filtering
S/N ratio (d.b.)
Signal/Noise
ratio (d.b.)
20%
Threshold level
30% 50%
70%
6
19
22
23
33
0
12
15
17
37
-3
9
12
13
34
-6
9
12
10
22
Table 6. 1 . Signal/noise ratio for the threshold filtering.
Some appreciation of the improvement obtained is shown in Figs. 6.7 and
6.8. The first figure shows a series of pulses obscured by random noise with
matched filtering carried out on the noisy signal. At a given level of L = 33%
the location of the pulses are clearly determined although some distortion of
their relative height and width is seen. The second figure shows the effect of
varying the threshold level on a given signal. It is noted that Walsh matched
filtering is most effective in the identification of pulse signals and produces
equivalent results to those obtained by Brown 21 , who used direct correlation
methods.
VIE Frequency-based scalar filtering
The difficulty of implementation for scalar Walsh filtering lies in the deter-
mination of the matrix weights required for effective frequency domain
filtering. This was indicated in Fig. 5.1 which shows that to pass only a
limited-frequency band the filtering coefficients required for Walsh filtering
are numerous and extend over a wide sequency bandwidth. However, a
126
SEQUENCY FILTERING
(a)
Original
signal
large amount of filter design information for filtered sampled data via the
Fourier transform is available and, somewhat naturally, attempts have been
made to utilise this through a relationship between the Fourier and Walsh-
Wiener filtering equations.
Kahveci and Hall 18 have shown that if we express the two forms of filtering
in matrix terms (equation 6.7) then we can obtain this relationship. We can
write the output column vector, y t (r), for Fourier filtering as
y 1 (t) = F- 1 -G 1 -F-x(f) (6.8)
where F and F _1 are the direct and inverse Fourier transform, and G I is the
set of filter weights. Walsh filtering can be written in the same way as
y 2 (t) = W- 1 G 2 W-x(f) (6.9)
where W and W” 1 are the direct and inverse Walsh transforms, and G 2 is a
second set of filter weights appropriate to Walsh filtering. If we assume
VIE
FREQUENCY-BASED SCALAR FILTERING
127
Matched filtered TL=20%
Fig. 6.8. Matched filtering — variation of threshold level. TL = Threshold level.
similar outputs for the two filtering operations, then y t (t) — y 2 (t) and we can
write
F GiF • x(t) = W G 2 W • x(t) (6.10)
from which
G 2 -W F^GjFW 1 (6.11)
The filter weights, Gi, may be derived using well-tried methods. Using the
fast Fourier and Walsh transforms, the required filter weights, G 2 , necessary
128
SEQUENCY FILTERING
for Walsh vector filtering are obtained. Whilst this process of filter weight
derivation will be slower than simple determination of Gi for Fourier
filtering, the actual productive process of filtering using the Walsh transform
can be considerably faster. This would be important for complex or repeti-
tive filtering operations such as two-dimensional image filtering, or on-line
television applications.
In the case of a single filtering operation on N samples a maximum of
2 N log 2 N additions/subtractions plus N 2 real products would be required.
The calculation involving the N 2 real products will generally be the domi-
nant factor in determining the speed of the filtering operation and may be
reduced by a process of selective computation, originally suggested by Pratt.
In this case only the non-zero elements are subject to matrix multiplication,
thus reducing the number of mathematical operations needed. An improve-
ment of up to 75% reduction in computational time has been claimed for
image enhancement applications.
VIF Filtering a non-stationary signal
A signal that is found to exhibit amplitude characteristics that are a function
of frequency as well as time is termed a non-stationary signal 15 . Most signals
derived from physical systems are of this type, as are much of the data
acquired from time-sampled economic situations. If the variation with
frequency during the period of measurement or acquisition is slight then the
signal obtained can be considered as stationary, or very nearly so, and the
techniques discussed earlier are applicable.
A particular advantage of using Walsh functions for non-stationary
signals is that CAL (fc, t) and SAL(fc, t ) vanish outside the time interval,
— The Walsh transform, X n , also reduces to zero outside the
sequency intervals, -(k + l)^Seq.^ +(k 4- 1). Consequently, any time
function which may be represented as a superposition of a finite number of
Walsh functions is both time and sequency limited. Hence it will occupy only
a finite area of a time-sequency domain, rather than the infinite area it would
occupy in the time-frequency domain.
Tests for non-stationarity can include variance checks which indicate the
extent of time/sequency variation, permitting determination of partition
length for short-term analysis 22 . With the partition method, the signal is
divided into a number of short series over which stationarity can be assumed.
Spectral analysis can be carried out for each series and the results plotted
separately on the same time (or sequency) axis as a form of three-di-
mensional display. A continuous form of sequency analysis of this type has
become known as parallel sequency filtering which must be distinguished
from the sequency vector filtering already described in Section VID.
VIF
FILTERING A NON-STATIONARY SIGNAL
129
VIF1 Parallel sequency filtering
Parallel sequency filtering has similarities with non-stationary spectral
analysis using a set of narrow-band analog or digital filters. It is in fact, a
form of spectral analysis in the sequency domain and gives information
about the sequency content along the signal length. This can be regarded in
the conventional way by considering each filtering operation as resulting
from a limited impulse-response function. Alternatively, it can be regarded
as an application of the Walsh orthogonal property to the unfiltered signal,
where it is used to select the required Walsh function coefficient and the
process repeated for M different Walsh functions.
The impulse response function is formed from a limited version of the
Walsh function
H(n , t) = WAL(n, t)G(t)
where
G(t) = 1 for 0 ^t<T
= 0, elsewhere.
This is used in a convolution expression
y(n, t) = [ f(T)H(n, t-T)dr
J t—T
( 6 . 12 )
(6.13)
With a sequency-limited function (i.e. low-pass filtered and sampled), this
convolution integral becomes
y(n, t)= X WAL(n, i)x(t-ih) (6.14)
i= 0
where h = T/N = sampling interval.
This is realised on the continuous basis to form a transversal filter bank,
having M sequency history outputs (Fig. 6.9). A limited sample set, consist-
ing of the first M samples from the total set N, is transformed to provide M
transformed samples, each forming a separate stored or recorded channel
output. The set of input samples is then advanced by one, admitting a further
sample from N and neglecting the oldest (i.e. first) sample. Transformation
of this second set is carried out to provide the next parallel set of M
transformed samples at the channel outputs. This process is repeated until
the entire set N has been transformed. The result is to give a series of M
channel outputs each representing the continuous value of one Walsh
transform coefficient along the signal length of N values.
130
SEQUENCY FILTERING
L. P filter and sample/hold
The form of the output obtained is given in Fig. 6.10 which shows parallel
sequency filtering of a sampled seismic signal. Only the first 16 Walsh
coefficients are used.
An analog form of this filter has been implemented by Gethoffer 23 for the
analysis of voice signals. A schematic diagram of his method is shown in Fig.
6.11. The input analog signal is sequency-limited and applied to a tapped
delay line. The Walsh transformation is carried out in a resistor network
which performs the functions of addition and subtraction of the delayed
pre-filtered signals. The delay line will require N - 1 delay elements, each
having a delay of T 0 /N seconds where T 0 refers to the time-base for the
transformation and N is the order of the Walsh system.
The interconnections required in the case shown (AT = 4) are obvious
when the pattern is compared with the first four of the sequency-ordered
series given in Fig. 1.4.
An application of parallel sequency filtering for trend prediction is
described below as an alternative to least-squares calculation using extrac-
tion and extrapolation of polynomial trends which is presently used 24,25 .
Since many economic time series are discontinuous in form and free from
the dependence on sinusoidal generation procedures, it is reasonable to
consider orthogonal trends other than polynomial ones.
If we take, for example, the airline ticket statistics used by Box and
Jenkins 24 then it is found that fewer components are needed to reconstruct
this data from sequency decomposition than with comparative frequency
decomposition (Fig. 6.12). This suggests that trend determination using
sequency filtering can usefully be employed with only a few of the principle
trends being considered. A simple process designed to carry this out is
VI F FILTERING A NON-STATIONARY SIGNAL 131
Fig. 6.10. Parallel sequency filtering of a seismic event.
illustrated in Fig. 6.13. The sampled data is first subjected to sequency
filtering, as described earlier, with M limited to a value of 16. Each of the 16
outputs is plotted separately and studied to determine any apparent
sequency trend. In this particular case only six outputs showed any signifi-
cant trend which, in each case, could be represented by a simple linear
equation. The last 16 values of the data were taken and Walsh transformed.
Ten of the coefficients were left unaltered and the remaining six modified by
one unit in accordance with the trends previously determined. Repeated
operation on the data by succeeding extrapolated trend values will produce a
new set of sequency coefficients each modified by the observed trends. The
132
SEQUENCY FILTERING
Delay line
x ( t )
(Sequency low
pass filter
n:
J
i
] f
i
44
1 ^.
1— T
] r
i-4
M
' T
T
r
Walsh transform
Fig. 6.11. Analog sequency filtering.
- y (0, t )
-yd,t)
~y (2,t)
-y(3,t)
process is repeated until the next expected cyclic point is reached (yearly
cycles in this case). The complete data set is then transformed back into the
time domain to obtain the first projected cycle of values. A second following
set of 16 values can be treated in the same way to obtain a further projected
set of values.
One result of this technique applied to the airline ticket data is shown in
Fig. 6.14. The forecast data are shown by the dotted lines which may be
compared with the realised extended time series shown by the full lines. An
improvement which could be carried out on this reconstruction would be to
subject sample sets earlier than X to X-16 to the process in order to derive
further projected sets. These may then be averaged to obtain the final result.
VIF2 Power sequency filtering
Some redundancy is present with the technique shown in Fig. 6.9, since pairs
of CAL and SAL filtered signals for the same sequency will contain identical
information apart from a sign change. A power spectral analysis version of
the sequency filter may be developed in which the squares of the SAL and
CAL filtered signals are summed, using equation (5.16), to give (M/2) + 1
VI F
FILTERING A NON-STATIONARY SIGNAL
133
Data
value
0
Manual
inspection
0 /
T = trend found
N = no trend found
sample
N = 10
T * I Ot trend normalised
to unity
Fig. 6.13. Forecasting analysis procedure.
134
SEQUENCY FILTERING
Fig. 6.14. Airline ticket data reconstruction.
filtered signals. Some results are shown in Fig. 6.15 for the same non-
stationary transient shown analysed in Fig. 6.10. Use of power sequency
filtering also produces a simpler display pattern since the values obtained
can take only a positive sign. Gethoffer 23 gives some examples of this
technique applied to short-term power sequency filtering of biomedical
E.E.G. signals.
VIG Two-dimensional filtering
Transform filtering of sampled image data can demand substantial comput-
ing resources, due to the size of the data matrix required for good resolution.
VIG
TWO-DIMENSIONAL FILTERING
135
.•w/^ ^ J\AA/\aaa/nAAA/\a^^
If the filtering is to be carried out in real time, there is the added problem of
achieving the high transmission rate required. For both of these reasons
considerable developments have taken place using the fast Walsh and Haar
transforms.
In general terms, the process of transform filtering implies spectral
decomposition of the sampled two-dimensional data into a domain which
permits either linear or non-linear processing operations to be carried out
before re-transformation back into the original domain for recovery of the
filtered signal or image. The transformation process was described in
Sections IIIL and IVF and is one step in the General Wiener filtering process
discussed earlier.
Linear filtering is defined as a linear combination of the entire trans-
formed coefficients to produce a modified transform of AT 2 points and is
136
SEQUENCY FILTERING
given by
X m '„' = I I X m , n G(m', m, n', n) (6.15)
m =0 n=0
where G(m\ m, n\ n) is a filter weighting function.
It is convenient to consider the filter weighting function as the product of
two matrices, viz.
G(m', m, n\ n ) = G m (m', m) • G (n\ n) (6.16)
where the filter matrices, [G m ] or [G n ], can take the form of a diagonal (and
therefore vector) matrix or the more complex case of a scalar matrix.
Unfortunately the reconstructed data matrix cannot be obtained by simple
convolution of the original data with the Walsh transform of [G m ] or [G n ], as
is possible with the Fourier transform, and no fast processing algorithms
have been found for the computation. However, if the matrix filter coeffi-
cients can be expressed by zeros and ones then the process resolves itself into
the case of selection for the coefficients to be retained.
There are a number of filtering operations that do not require the
convolution property. One such example is where the data matrix consists of
additive signal and noise components. An optimum filter can be designed,
through a two-dimensional transformation of a matrix based on the known
covariance matrices of the signal and noise 26 , to give a minimum mean-
square error for the filtered signal. This may not lead to the best subjective
reconstructed image, as has been pointed out by Hart 27 , although it will be
mathematically correct.
Optimum filter design will generally result in filter coefficients which have
levels other than zero or one. It is possible to simplify the practical
application of such a filter by replacing those filter coefficients having very
small values, close to zero, with zero values. Pratt has claimed that, under
certain conditions, up to 90% of the filtering multiplications can be avoided
in this way 28 .
Non-linear filtering involves carrying out a non-linear operation on each
of the transformed samples. Two types of non-linear operation that have
been employed in the area of image processing are logarithmic operation
and a power-law operation
K • X m ,„ log {|X m
IXJ
V —
IXJ
(6.17)
(6.18)
where K and k are constants.
VIG
TWO-DIMENSIONAL FILTERING
137
A logarithmic operation will attenuate transform domain samples by a
factor proportional to their magnitude and can give an “edge” enhancement
to a processed picture. It is related to a similar Fourier non-linear operation
known as the Cepstrum which has been used successfully to remove cyclic
background noise 29 .
A power-law operation tends to emphasise the difference between low
amplitude and high amplitude samples and can also be used for image
enhancement. A number of examples of picture filtering using these tech-
niques is given by Pratt 30 .
As can be seen from equation (6.15), the filtering operation consists of
selectively modifying the frequency, sequency or degree, depending on the
type of transform used, according to the filter weights given by the transfer
function of the filter, G m ,„. Two-dimensional filtering differs only in the
necessity to carry out a two-pass transformation and product-formation of
what is essentially single-dimensional data.
When we consider filtering in one dimension, we generally desire to refer
to the characteristics of the transfer function as low-pass, high-pass, band-
pass or band-stop, depending on its relative effects on the frequency
spectrum of the transformed data. Ideally, we would like the filter charac-
teristic to have the unit value inside the pass-band and zero outside of it. This
is impermissable using the Fourier transform due to the oscillatory effects
imposed on the filtered signal (Gibbs phenomenon), but is perfectly practical
using the Walsh or Haar functions which are themselves discontinuous in
form. Hence, the tapering and other processes found necessary with Fourier
design filtering 31 are not required, which permits consequent simplification
in the transfer function design.
With discrete Fourier filtering, the possible information content for the
data is defined for wavelengths ranging from half the total period of the data
series to twice the sampling interval (Nyquist criteria). Information at
shorter wavelengths will add to produce an aliased error content within the
defined data range. This is also generally true for the Walsh and Haar
transformed data, with the provision given earlier in Section IIG.
As indicated previously, although the number of transformed values in all
three transforms are equal to the number of discrete input data samples, the
values of the transform coefficients are constant for Haar functions of equal
degree which is not the case with the Fourier and Walsh transformations.
This means that the transfer function has only log 2 N different values and, as
a consequence, Haar transform filtering can be very much faster than Walsh
filtering although, due to the limited number of unique transformed values,
the resolution of the filtered data will be poorer. Some examples of two-
dimensional filtering using all three transforms will be discussed later in
Chapter 8.
138
SEQUENCY FILTERING
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London.
26. Pratt, W. K. (1971). Fast computational techniques for generalized two-
dimensional Wiener filtering. Proceedings: Two-dimensional digital signal pro-
cessing conference, University of Missouri, Columbia.
27. Hart, C. G., Durrani, T. S. and Stafford, E. M. (1974). Digital signal processing
for image deconvolution and enhancement. “Proceedings: NATO Advanced
Study Institute on New Directions in Signal Processing and Communications
(Ed. J. K. Skwirzynski). Nordhoff Int. Pub. Co., Leiden.
28. Pratt, W. K. (1972). Walsh functions in image processing and two-dimensional
filtering. 1972 Proceedings: Applications of Walsh Functions, Washington
D.C., AD 744650.
29. Thomas, D. W. and Wilkins, B. R. (1970). Determination of engine firing rate
from the acoustic waveform. Electr. Letters 6 , 7 , 193-6.
30. Pratt, W. K. (1971). Linear and non-linear filtering in the Walsh domain. 1971
Proceedings: Applications of Walsh Functions, Washington D.C., AD 727000.
31. Beauchamp, K. G. (1973). “Signal Processing”. George Allen and Unwin,
London and John Wiley, New York.
Chapter 7
Applications in Communications
VIIA General
The literature on applications for Walsh and related functions is extensive.
Bramhill 1 lists over 400 references to applications alone, excluding those
concerned with theoretical and mathematical development. These refer-
ences cover the subjects of Signal Processing, Transform Spectroscopy,
Image Coding and Transmission, Statistical Analysis, Voice Processing and
Vocoding, Switching Functions and Logic Circuitry, Filtering, Multiplexing,
Electromagnetic Waves, Optical Devices and Mathematical Modelling. To
this already impressive list can be added Radar Processing, Seismology,
Holography, Pattern Recognition, Data Compression and Chemical
Analysis 2 . A brief outline of some of the most important of these develop-
ments is given in this and the succeeding chapter. Much of this work is
reported in the proceedings of symposia on the Applications of Walsh
Functions held annually in Washington D.C. and sponsored by the I.E.E.E.
and other bodies. Full details of these are given in the bibliography included
at the end of this chapter.
VIIB Communications applications
The previous chapters have indicated that the Walsh function series can be
applied to many of those areas where sinusoidal techniques have previously
dominated. This is particularly so in the design of digital equipment for
communication and computer applications, where the two-levels form the
function matches binary logic and binary computer algorithmic techniques.
140
VilB
COMMUNICATIONS APPLICATIONS
141
Additionally, there are certain characteristics of the Walsh function which
have no counterpart in Fourier-based series and which permit new ideas in
hardware and software development.
Historically, the mathematical foundations for later communications
developments were laid by Paley and Wiener 3 , Fine 4 and Pichler 5 . The
treatment in these early works was concerned with a continuous time-
sequency representation in much the same way as the older theory is based
on continuous time-frequency representation. Practical applications of
Walsh function theory in communications have shown that, in most cases, it
is sufficient to use a finite set of discrete Walsh functions. The theory
becomes simpler and only the elementary results of linear algebra are
necessary. The finite theory is described in the works of Gibbs 6 and Pichler 7 ,
to which the reader is referred.
The impetus for much of the application of Walsh and related functions to
communications can be attributed to the outstanding achievements of
Harmuth and his team 8 . Especial reference should also be made to the work
of Taki 9 and particularly Hiibner who was one of the first to demonstrate
some of the practical advantages of the new techniques in a real working
system 10 . These advantages include efficient multiplexing hardware, reduc-
tion of transmission bandwidth and lowering of error rates. A discussion of
these and other benefits of the application of Walsh theory to communica-
tion is given below, commencing with the most successful of them all, namely
digital multiplexing.
VIIC Multiplexing
Multiplexing refers to those techniques that enable simultaneous transmis-
sion of many independent signals over a common communications channel.
All forms of multiplexing are based upon systems of orthogonal functions
used as communication carrier signals. The two systems in general use are
known as frequency-division multiplex (F.D.M.) and time-division multi-
plex (T.D.M.).
Using F.D.M. , each signal is allocated to a different part of the frequency
spectrum and demultiplexing at the receiving end of the system is accom-
plished by frequency-selective filters. In the case of T.D.M., each signal is
allocated to a different time-slot for transmission and demultiplexing is
obtained by time-dependent gating circuits.
Figure 7.1 shows a general multiplexing system. The n signals to be
transmitted are each modified by a unique carrier signal such that separa-
tion, i.e. demultiplexing of all the signals, can be carried out at the receiving
end by combining the multiplexed signal separately with each of the unique
142
APPLICATSONS IN COMMUNICATIONS
carrier signals. In the case of F.D.M. the carrier consists of sine-cosine
signals and in the case of T.D.M. block pulses are used.
Referring to Fig. 7.1, the signal sent over the single transmission line is the
sum of the products of each of the input channels, S k , and its associated
carrier, C k , both expressed as functions of time, viz.
S(t)= t s k (t) • c k {t) (7.1)
k = 1
At the receiving end the composite signal, S(t), is distributed to a series of
multipliers each of which forms the product of S(t) with one of the replica
Summer
Recovered
channel
output Sk
Replica
carrier
C'k
Fig. 7. 1 . A general multiplexing system.
carriers, C k . The products are filtered to remove the higher frequency
cross-products and the original channel signals, S k , are recovered. Unfortu-
nately the signal at a given output terminal is the sum of the desired filtered
output
\ T m dt (7.2)
I J 0
where f a (t) is the carrier signal associated with input channel, S a , 1/T is the
filter cut-off frequency and a series of cross-modulation products of which
f a (t)- f b (t) dt
(7.3)
represents that due to the carrier signal, f b (t), with which is associated input
channel, S b .
A necessary condition under which expression (7.3) reduces to zero is
when the n carriers form a set of orthogonal functions, so that for a ^ b
\fa(t)-f b (t) = 0 (7.4)
VIIC
MULTIPLEXING
143
As shown earlier this property enables the identification of a particular
carrier signal to be made from the summation of a number of signals, each
conveyed by means of a unique carrier signal. It is only necessary to multiply
and average the multiplexed signal with the carrier signal appropriate to the
required channel in order to identify and separate this carrier and hence the
required channel signal. The process is, of course, one of multiple correla-
tion, in which the required signal is extracted by means of autocorrelation,
giving a peak amplitude for the identified signal 11 .
Apart from the sinusoidal and block functions, there are many other
different sets of orthogonal functions which may be used in a multiplex
system. Systems using Legendre polynomials 12 , Hermite polynomials 13 ,
trigonometric products 14 and Rademacher functions 15 have all been pro-
posed or used. However, the only set of functions which are as efficient as
the sinusoidal functions, in terms of bandwidth utilisation, are the Walsh
functions.
Early use of Walsh functions for this purpose simply replaced the sine-
cosine carrier signal of a F.D.M. system by a continuous Walsh function
signal (Fig. 7.2)*. A separate function is applied to each of the transmission
channels. To avoid distortion due to the limitations in representation of
Mulhplexed
signal
WAL(n.t)
Fig. 7.2. Analog Walsh carrier system.
* A continuous Walsh function repeats the pattern, shown in Fig. 1.4 at each multiple of the
time-base interval which can be considered as the period of a complex waveform.
144
APPLICATIONS IN COMMUNICATIONS
analog signals by discrete samples, the analog samples must first be passed
through an aliasing filter (F). Samples are then taken at uniform intervals
and constrained by means of a sample-and-hold amplifier (H) to result in a
step-shaped signal which can then be transmitted, without distortion, over
the sequency multiplex system. To achieve this the stepped signals are
applied to a series of multipliers (M), to which a series of contiguous carrier
sequencies are also applied. The output of the multipliers are linearly added
before transmission in S.
Demultiplexing using a similar set of synchronised carriers relies on the
orthogonal property of the Walsh series and, as with the conventional
system using sine-cosine functions, separates out the mixed signals by a
process of autocorrelation. The advantages of this system over frequency
multiplexing are due, in part, to the single-sideband features of the addition
theorem for the Walsh functions, noted in equation (3.17), which permits
sideband filters to be omitted and also eases implementation using inte-
grated circuit technology. A working system designed for 256 voice channels
has been described by Hiibner of the West German Post Office 10 .
The main problems with these analog multiplexing systems is that of
cross-talk, caused by difficulties in achieving accurate synchronisation,
together with the realisation of sufficiently linear analog multipiers. Syn-
chronisation of the transmitting and receiving systems has been considered
by Harmuth 15 who has shown that for numbers of channels that are a power
of two the synchronisation problem is simplified if Rademacher functions
are used, the orthogonality of which is invariant with a time-shift.
Despite improvements that have been made other difficulties remain. An
almost insuperable one is the widespread development of conventional
F.D.M. equipment causing a justifiable reluctance on the part of communi-
cations authorities to change for even quite considerable technical gains.
The position is different, however, if we consider Walsh multiplexing of
binary signals. The transmission of binary data streams for communication
and computer purposes is beginning to impose its own requirements, for
which the equipment in service is as yet limited in quantity.
In order to study the operation of a digital system we may consider the
system shown in Fig. 7.3. Here the data signals to be transmitted, a k and
b k (k = 0, ±1, ±2 . . .), are quantised signals existing over a finite time inter-
val, 0^ t ^ T. The carrier signals are Walsh-coded sequencies, / f (t) and / ; (t),
having values ±1 and are mutually orthogonal to each other.
The pairs of data and carrier signals are multiplied and linearly added for
transmission as shown. It is assumed that the coded sequencies, fi(t) and / y (f),
are stored or recovered at the receiving end. The combined and transmitted
signal is multiplexed separately with each of these coded sequencies and the
VIIC
MULTIPLEXING
145
result averaged to give an effective cross-correlated signal represented, in
the particular case of data signal, a k , as
At = ? i m dt r bk ■ m m dt (7 - 5)
Now since the carrier signals fi(t) and fj(t) are orthogonal, the second term in
the equation becomes zero (from equation 7.4) and we can write
A fc =T[ a k • ff(t) dt (7.6)
1 J 0
but fi(t) = ±1 so that A fc = a k and the desired signal is recovered.
This system, although simple in concept, suffers from several disadvan-
tages. The transmitted signal is a multi-level signal, having a variable peak-
to-average power relationship and is, therefore, susceptible to corruption
by noise. The system also requires very linear multipliers, a disadvantage
found with the analog system described earlier. These difficulties have led to
the development of systems in which the multiplexed transmitted signal is
also a binary signal having only two levels. This gives a peak power which is
equal to the average power and hence there is optimum immunity from
noise. A further advantage is that it is easier to design simple digital
multipliers for multi-level operation to a much higher degree of accuracy
than can be achieved with analog multipliers.
Gordon and Barrett 16 have described a digital multiplexing technique of
this kind in which only the sign of the multiplexed signal is transmitted. This
146
APPLICATIONS IN COMMUNICATIONS
is possible since with binary signalling it is only necessary to determine the
sign of the correlation coefficient for the transmitted information, relative to
a single channel, for this to be unambiguously recovered.
The binary data to be multiplexed are each amplitude modulated with one
of a set of binary Walsh codes acting as the transmission carrier signals. The
summated signal is passed through a hard-limiting device which transmits
only the sign of the majority logical value of the modulated channel signal in
each time slot. For this reason the system is known as Majority Logic
multiplexing. Demultiplexing is carried out by finding the correlation coeffi-
cient of the transmitted signal with each of the replica Walsh codes gener-
ated at the receiver. The number of channels which may be used with this
system depends on the existence of a matrix, the rows of which show a
sign-invariant correlation coefficient after being modulated, summed and
threshold-limited. Such a matrix can be formed from the Walsh functions,
WAL(1, t) to WAL(7, t). Unfortunately the use of a larger matrix than this
does not provide the unambiguous signal transmission obtained with this
definition. A way of extending the number of channels through concatena-
tion has been described by Gordon and Barrett 17 . Here the output of one
multiplexor is taken to be the input to the second and so on. Thus, each
successive demultiplexor is subject to a lower error rate and itself reduces
this rate.
With suitably chosen codes for the carriers, the system offers a trade-off
between number of channels in use and automatic error correction. This
results from the fact that the magnitude of the correlation coefficient for a
given channel increases inversely with the number of channels mixed so that
the correct sign is maintained even in the presence of errors in the multi-
plexed signal. This is illustrated in Fig. 7.4 which shows the effect of adding
an error bit to the multiplexed signal and its elimination during the process
of recovery. In this example three out of a possible seven channels are
shown. Although some error correction capability can be provided when the
system is not fully loaded, it is still necessary for the multiplexor to be able to
determine which channels are active. Durst 18 has analysed the allocation of
channel power in such a system by plotting the user channel bit error
probability against the channel signalling error rate. He concluded that in a
seven channel system deterministic errors, causing ambiguous performance,
begin to occur when four or five channels are active. Because of the way that
the available channel power is redistributed in the system, it is possible that
demodulation of a quiescent channel will yield a measure of output signal.
Hence, the need to be able to monitor the channel activity and act upon this
information to suppress the channel output. This difficulty can be overcome
if one of the multiplexor channels is reserved for this purpose. The majority
logic multiplexor may be constructed entirely from digital integrated circuits
VIIC
MULTIPLEXING
147
Input
data
+ 1-1 -I
and may, in fact, be designed using “read-only” memory and a shift register
as noted by Durst.
VIID Coding systems
A special class of binary group codes, known as the Reed-Muller codes 19 ,
have been in use for some two decades and are orthogonal codes based on
the Walsh function. Their principle use is in multiple error correction. More
recently efforts have been made to adapt the coding used to the statistical
properties of the transmitted signal. It has been found, for instance, that in
order to obtain the best performance for a given transmission, it is necessary
to represent some data samples more accurately than others. This implies a
variable word length for the samples transmitted. By allocating word lengths
according to the expected variation in the amplitude of the Walsh trans-
formed coefficients, it has been found that reduced quantisation error is
realised compared with a P.C.M. system having constant word length equal
to the average word length of the transform coding system . 20
A further development is described by Redinbo 21 who considers the
possibility of using Walsh spectral analysis of the state of the communication
148
APPLICATIONS IN COMMUNICATIONS
channel to permit the development of an automatic coding system. Trans-
mission of information using redundant coding elements is a well-known
method of combating the statistical effects of the transmission system. In the
spectral analysis system, Redinbo describes a system of minimum mean-
square error codes using the results of Walsh analysis to optimise the
operation of the binary error-detecting codes used. These codes are based
directly on the statistics of the communication channel which is continually
analysed by the use of the Walsh transform.
Since the Walsh transforms of the channel statistics determine the coding
rules, the equipment for real-time processing to determine these rules need
not be complex. For example, a flat sequency spectrum is one indication of
the optimisation of such channels. It is suggested that these adaptive
methods could enable sustained optimum, or near optimum, transmission to
be achieved, even on channels where the statistics are slowly varying.
VUE Image transmission
An area of Walsh and Haar application that has shown consistent progress
over the past decade is that of image data processing. The reason is not too
far away to seek. The amount of data is large and bit transmission rate is
high. Many of the more desirable techniques, such as image transformation,
if carried out by Fourier or other methods, demand too high an overhead in
terms of equipment or processing time. Television transmission is a typical
example of this.
Much interest is being shown by television authorities in the transmission
of television pictures through pulse code modulation due to the extremely
good immunity to noise that such a system confers. The band- width required
for conventional linear P.C.M. systems is unfortunately rather high. As an
example, for the 625 line, 50 frame PAL system used throughout Western
Europe, a transmission rate of 106.4 Mbits/s has been quoted using a grey
level resolution of 8 bits 22 .
One of the techniques used to overcome this disadvantage relies on an
orthogonal transformation of the picture elements. This is identical to the
transform filtering methods described in the previous chapter. In this case
the transformation is made to affect a reduction of redundant coefficients by
filtering, suppression of certain coefficients and quantisation of others. This
allows the converted version to be encoded for transmission using fewer bits
per word than the original signal. An inverse transformation may then be
employed to recover the signal for display. An analog system of this type has
been described by Kraus 23 and employs the hard-wired resistor matrix
referred to earlier in Section IIIM1. A disadvantage of analog transforma-
VIIE
IMAGE TRANSMISSION
149
tion, as discussed previously, is the amount of equipment required and the
high order of accuracy demanded to obtain a reasonable resolution.
Transformation techniques applied to video transmission have generally
been digital, that is apart from the actual process of picture reconstruction.
The principle is shown in Fig. 7.5. Following quantisation and digitising of
the image on a line-by-line basis, the data is transformed and filtered in such
Fig. 7.5. Transform image coding.
a way as to reduce the total number of samples available for transmission.
P.C.M. coding is carried out on the reduced number of samples. Decoding
and inverse transformation of this produces a digital signal for conversion
into the analog form required by the display tube.
Shibata and Enomoto 24 have shown that if the Walsh transform, or the
slant transform derived from it, is used for precoding the signal in this way
before transmission then quite considerable reductions in band-width
become possible.
The Slant transform 25,26 is an interesting transform which attempts to
match its structure with that of typical structures found in a television signal.
The orthogonal vector series for this representation are chosen in the
following way. The first series vector is given identical components, i.e. it is a
d.c. vector. This matches those long intervals in each line when the signal is
constant. The second vector has a uniform quantised ramp and is chosen to
match those regions where the signal exhibits a fairly uniform rate of change
from low to high density. Further vectors are divided into two groups. The
first group resembles quantised sawteeth of increasing frequency. The
second group is chosen to maintain the row vectors orthogonal to each other,
by making the waveform resemble Walsh and Haar vectors. Figure 7.6
shows a set of 16 of these slant basis waveforms.
Using a 16 by 16 picture element mosaic and slant transform coding, Chen
and Pratt 26 claim a 12 : 1 reduction of domestic colour TV image band-
150
APPLICATIONS IN COMMUNICATIONS
Wave no.
5
6 n
7
8 OJlTUTi 1
8 4_ruiTLnj
10 muuu 1
- J bfiniuinj
'2 iT-^rP-nr
13
14
15 if^r^T-n^
Fig. 7.6. A set of Slant basis waveforms.
width. This is currently the most effective coding structure employed for this
purpose.
The problems of two-dimensional filtering of television images, particu-
larly colour television signals, do not lend themselves to simple digital
filtering solutions mainly because of the complexity of the signal and the
need to process this in real time. Pratt 27 has carried out some work using
Wiener filtering with Walsh and other non-sinusoidal functions, showing
that the complexity of the hardware required can be reduced considerably
for this application and that only a small coding error of smaller than 1%
mean-square error is obtained for small coding block sizes. This compares
very favourably with other methods (e.g. Fourier) which demand more
complex coding equipment.
Of the orthogonal transformations that can be employed for such
filtering, the Walsh transformation is pre-eminent due to the ease with which
VIIE
IMAGE TRANSMISSION
151
it can be implemented using real-time digital logic 22,28 . The only logic
elements needed to build such a hardware transform are adder/subtractors,
shift registers and a two-level logic switch. By using a fast Walsh transform
algorithm, the amount of logic required is not only minimised but may be
built from a very simple logic system. A single logic cell of such a system is
shown in Fig. 7.7.
T( x) = I T( x) = 0
Fig. 7.7. A logical fast Walsh transform “cell”.
This is due to Walker 29 and has been used experimentally in a real-time
television system to achieve band- width compression. The processing is
carried out in three stages, each stage being a combination of a unique
storage block and a common arithmetic block of which Fig. 7.7 forms a
single cell. In this circuit, the shift register has two functions. Initially it is
used to delay the input to the stage, so that the words forming the first half of
the block are available at the inputs to the adder and subtractor simultane-
ously with the words in the second half. During the second half of the process
the adder and subtractor simultaneously produce the sums and differences
of the elements word by word. The data selector, T(p-r)a, switches the
sums to the stage output and T{p — r)b switches the differences to the shift
register input. When all the sums have been clocked out, switch T(p-r)a
changes to direct the differences stored in the shift register to the output. At
the same time T(p-r)b loads the shift register with data from the next
block.
Table 7.1 indicates how the process is carried out in terms of the contents
of the storage units at intermediate steps in the process. This may be
compared with the matrix method of fast Walsh transformation described in
Chapter 3. It may be seen from Table 7.1 that the control signals required
are simply a set of Rademacher functions, suitably phased with respect to
152
APPLICATIONS IN COMMUNICATIONS
Clock
pulse
No.
Input
samples
Output
of 1st
stage
Output of
2nd stage
Output of
3rd stage
1
A
2
B
3
C
4
D
5
E
A + E
6
F
B + F
7
G
C + G
(A + E) + (C + G)
8
H
D + H
(B + F) + (D + H)
[(A + E) + (C+G)] + [(B + F) + (D + H)]
9
A-E
(A + E)- (C+G)
[(A + E) + (C+G)]-[(B + F) + (D + H)]
10
B-F
(B + F) — (D + H)
[(A + E)-(C + G)] + [(B + F) — (D + H)]
11
C-G
(A — E) + (C— G)
[(A + E)-(C + G)]— [(B + F) — (D + H)]
12
D-H
(B — F) + (D — H)
[(A-E) + (C — G)]+[(B — F) + (D — H)]
13
(A-E) — (C— G)
[(A-E) + (C-G)]-[(B-F) + (D-H)]
14
(B-F)-(D-H)
[(A — E)-(C-G)]+[(B — F) — (D-H)]
15
[(A-E) — (C-G)] - [(B-F)- (D-H)]
Table 7.1. Contents of a hardware transformer at intermediate stages in the transformation.
each other and to the input sample block. These functions can be generated
by means of a 3-stage binary counter.
The principle may be extended to produce a larger transformer as a series
of stages. Each stage in the transformer is constructed of two distinct parts,
the arithmetic and data selection logic (which is identical in every stage), and
the storage array (which is of a different length in each stage). To transform
N words each of m bits requires p transformer stages, where p = log 2 N.
Each stage contains a shift register, an adder, a subtractor and two logic
switches. In the r’th stage, the shift register contains 2 (p_r) words each of
(m + r) bits, the adder and the subtractor each accept inputs of (m + r— 1)
bits and produce outputs of (m + r) bits, whilst the logic switches control
(m + r) bits. Thus, the processing part is seen to increase approximately
linearly with the number of stages whilst the storage required doubles with
each additional stage.
Walker’s real-time transform is of the order 32 by 32. The signal input
consists of sets of 32 samples taken consecutively from a line of the television
raster. The P.C.M. input word length is 8 bits and the transform domain
word length is 13 bits. During inverse transformation the output word length
is limited to 8 bits by the digital-to-analog converter. The speed of hardware
processing using the Walsh transformation permits video filtering to be
undertaken in one dimension, i.e. line-by-line. The advantages claimed for
this approach are compatibility with existing picture scanning equipment
and reduction in the amount of high speed memory required. Murray 30 gives
some examples of this.
VIIE
IMAGE TRANSMISSION
153
Bit reduction following transformation is usually achieved by limiting the
dynamic range of the transformed signal through omission of some of the
higher sequency digits. It can be shown that in a linear P.C.M. system
involving transformation, the theoretical number of bits required to repre-
sent each sample coefficient is
B = m + log 2 N (7.7)
where m is the number of bits in the P.C.M. input word and N is the order of
the transformation.
The number of bits necessary to represent the transform may be reduced
considerably without noticeably affecting the picture quality. It is possible to
do this in two ways. First, because television signals rarely utilise the full
range of possible coefficient values, some of the most significant bits from the
coefficient values may be removed and second, the least significant bits may
be removed since the signal theory contained therein is negligibly small.
A further reduction is possible by making use of a non-linear quantising
step prior to transmission. Linear quantisation has a deleterious effect on
image quality due to the large quantisation errors found with high sequen-
cies. A desirable quantisation function is one which has a high variance at the
origin in the Walsh domain and which decreases towards the higher
sequency end of the range. A Gaussian function possesses these properties
and has been used by Pratt to improve the subjective quality of several
scenes 31 . Unfortunately, due to slightly different variance requirements for
different scenes, the optimum quantisation function is difficult to define
although the Gaussian function is considerably better than the linear
function for this purpose. Ohira et al 32 has described a working system
where the type of non-linear quantising characteristics is selected heuristi-
cally from examination of a number of typical picture scenarios.
A form of coding which has analogies with Redinbo’s optimum coding
techniques (Section VIID) is that of Adaptive Coding. This is a technique,
pioneered by Wintz 28 and others, in which the filtering coefficients applied to
the transformed signal are made to adapt to the local picture structure by
allowing a number of alternative modes of operation. To do this the picture
is first divided into a number of sub-picture elements containing a small
number of samples. Each sub-picture element is transformed and all those
elements retained which exceed a predetermined threshold value. Only the
retained coefficients are coded for transmission together with information
giving the position of the subset of coefficients retained. The threshold level
can be adapted to the average brightness, which improves the subjective
quality of the picture, as well as reducing the number of bits required for
each picture element from 8, required for conventional P.C.M. transmis-
sion, to 1, in the case of adaptive coding 28 .
154
APPLICATIONS IN COMMUNICATIONS
The previous discussion has been concerned primarily with exploiting the
statistical redundancy in image transmission which exists by virtue of the
dynamic constraints on the process. A second form of redundancy, namely
perceptual redundancy, can also be exploited by means of transform coding.
It is well known that in a television P.C.M. system, isolated or burst errors
occur during transmission and result in localised intense noise “spikes”
which are subjectively irritating. This also occurs with digital recording of
certain forms of data, such as seismograph signals. On the other hand
low-level noise distributed over the entire band-width spectrum is usually
acceptable. An example of this is quantising noise which is imperceptible
providing at least 256 levels are used 11 . The objectionable feature of impulse
noise is, therefore, its localised extent rather than the energy it contains. One
way of improving the subjective quality of the processed or transmitted
image is to arrange for the errors to be spread over a sufficiently large region.
This is carried out automatically by the type of transformation —
transmission — inverse transformation described above. By correct choice of
transform size, N, the amplitude of most of the spread errors will be found to
fall below the quantisation noise level so that it becomes possible to
eliminate completely the spike errors that occur in transmission.
Subjective assessments of picture quality, using these methods of image
processing, form the only currently reliable way of comparing results,
particularly for colour images. Attempts have been made to find some
quantitative performance measure of image quality and only limited success
has been achieved. The most widely used method is to utilise the mean-
square error between an original image, /( x, y), and a processed image,
g(x, y), viz.
error = -L £ I [/(x, y)-g(x, y)] 2 (7.8)
iV x=0 y =0
Some improvement over this has been suggested by Sakrisin and Algazi, 33 to
take into account the impulse response of the human eye. The modified
error is then given by.
error = — 5 £ £ [/(*, y) * H(x, y)-g(x, y) * h(x, y)] 2 (7.9)
A JC=0 y =0
where H(x, y) is the estimated response function of the human eye and *
indicates a convolution process. Neither of these error expressions has been
found completely adequate to compare processed images since the results
obtained do not always agree with a concensus subjective result.
VIIF
ELECTROMAGNETIC RADIATION
155
VIIF Electromagnetic radiation
The form of electromagnetic waves used in communications has invariably
been sinusoidal. This need not necessarily be so since, from D’Alambert’s
solution of the one-dimensional wave equation, it can be shown that the
electromagnetic field will transmit waves of any function /( x - ct) or g(x +
ct).
Whilst it is theoretically possible to produce such general waves by the
superposition of sinusoidal waveforms of different frequencies having the
correct amplitude and relative phase values the practical difficulties atten-
dant on maintaining these values to the accuracy required at radio frequen-
cies is almost insuperable. The problems are associated with the way in
which Fourier series converge at a discontinuity to give rise to the well-
known Gibbs phenomenon. Walsh series, on the other hand, being them-
selves discontinuous functions, converge readily and thus lend themselves to
this type of synthesis.
There are a number of reasons why the radiation of non-sinusoidal
electromagnetic waves are considered desirable. They can be used effec-
tively for target discrimination, particularly as discrimination between con-
ducting and non-conducting targets then becomes possible. Some of the
resolution problems caused by multi-path transmission are eased. It is also
convenient technically to develop transmitting equipment from high-
powered semi-conductor switching devices which are ill-suited to support
sinusoidal generation.
The theoretical work supporting a study of Walsh electromagnetic radia-
tion owes much to the pioneering efforts of Harmuth 8 who has also contrib-
uted to the design of experimental transmission and reception equipment
(Lally et a/. 34 ). A transmitter for electromagnetic Walsh radiation is also
described by Fralick 35 who used horn aerials for both transmission and
reception. The more difficult problem of the reception of Walsh waves is
discussed by Frank 36 and follows the earlier work of Harmuth noted above.
Harmuth 37 has stated several basic differences between sinusoidal and
Walsh function electromagnetic radiation which could possibly be exploited.
These are:
(a) The technology for implementation is different. Pure sinusoidal
waveforms are relatively difficult to generate. Walsh waveforms require
only suitable switch matrices operating in the nano-second region —
currently achievable with solid-state devices.
(b) The differentiation of a sinusoidal function yields a shifted sine
function (actually a cosine function) of the same frequency, whilst the
differentiation of a Walsh function yields a differently shaped function.
This is illustrated by Fig. 7.8. Here a Walsh function (a) is integrated to
156
APPLICATIONS IN COMMUNICATIONS
Fig. 7.8. Differentiation of sine waveforms and Walsh functions.
give waveform (b) and differentiated to give (c). A sum of the Walsh
function and its derivative could clearly be separated due to the different
shapes, whereas this would be impossible for sinusoidal functions.
(c) The summation of sinusoidal functions having arbitrary amplitudes
and phases but equal frequency yields a sinusoidal function of the same
frequency. Walsh functions are summed differently (see Fig. 7.1 1) so that
interference effects do not behave in the same way.
(d) The Doppler effect can transform a sinusoidal function into another
for any velocity ratio, v/c. With a Walsh function a threshold ratio of
l»/c|^3 .5 is necessary before a transformation occurs to another Walsh
function of the same system.
(e) Sinusoidal waves exhibit polarity symmetry; that is a reversal in
amplitude produces the same effect as a time shift. Walsh waves do not
have polarity symmetry.
One of the results of (c) is that quadrupole radiation in free space appears to
be easily possible with Walsh waves, whereas sinusoidal waves give rise to
predominantly dipole radiation. Thus it is theoretically possible to radiate
more power, using the Walsh quadrupole mode, for an aerial of a given size.
In addition the interference effects of quadrupole radiation should yield a
better resolution than that with dipole radiation.
Practical implementation of these several possibilities have yet to yield
acceptable results and for this reason the generation and use of Walsh
electromagnetic waves is one of the weaker areas of development.
Interesting and new possibilities exist however and include Harmuth’s
solution for the problem of identifying radar reflections from conducting and
non-conducting targets. This uses the property (e) since it is found that a
conducting reflector or scatterer reverses the amplitude of the electric field
strength of a wave, whereas a non-conducting reflector or scatterer does not.
Discrimination between the two cases is, therefore, possible with Walsh
waves in a way that is impossible using sinusoidal electromagnetic radiation.
ELECTROMAGNETIC RADIATION
157
VIIF
The two possible forms of Hertzian radiating dipoles are (1) the electric
dipole requiring high potentials and low energising currents and (2) the
magnetic dipole requiring low applied potentials but high energising cur-
rents. The latter is better suited to semi-conductor technology and has been
implemented in the experimental equipment described in the literature.
One of these, due to Lally 34 , is shown in Fig. 7.9. Four Hertzian magnetic
dipoles are energised through switching transistors to which the driving
voltages WAL(n, t ) and -WAL(n, t ) are applied. By including a separate
power transistor or switching dipole within each loop the radiating power is
confined to the loops, thus minimising feeder line losses through radiation.
Two or three-dimensional arrays follow the same principle. Using these the
array more clearly simulates a point source and is, therefore, suitable for
location at the focal point of a parabolic reflector. Stacked magnetic dipoles
of this type are not suitable for very short switching times due to transmis-
sion delay effects around the loops. For switching times below 1 nS, Har-
muth has suggested a helical form of radiation in which the direction of helix
rotation is made to correspond to the +1 or -1 of the required energising
Walsh function 37 .
Reception of radiated Walsh waves is a more difficult problem. As with
earlier receivers used for sinusoidal transmitted waves the selectivity is
limited to the characteristics of the detecting circuit. A Walsh wave, having a
period T, is extracted from a noisy background by means of a circuit which
resonates with periodic waves of this period. The principle is shown in Fig.
158
APPLICATIONS IN COMMUNICATIONS
7.10. The received signal, consisting of pulses of energy, enters a summing
amplifier with which is associated a delay line of the correct length such that
only those pulses fed back with the proper time delay will be additive. Pulses
of the wrong time delay will tend to interfere with each other and fail to
accumulate.
Harmuth has described an experimental receiver based on this principle,
operating in the region 1 MHz to 1 GHz and constructed of non-dispersive
coaxial transmission components 37 which has established the feasibility of
the technique. Further developments are proposed to take advantage of the
theoretically improved Doppler resolution by using several line-stretchers
and feedback amplifiers in parallel. This has obvious applications to side-
looking radar considered in the next section.
VIIG Radar Systems
It has been pointed out by Lackey 38 that the resolution of point targets can
be enhanced if the target area is illuminated with a Walsh wave rather than a
sinusoidal signal. The effect of a secondary target, close to the required point
target, may be seen during the entire period of the transmitted Walsh signal
which is not the case for a single period of a sinusoidal signal.
This is shown in Fig. 7.11. Here (a) and (b) show the signals reflected from
two target areas, T x and T 2 , where a sinusoidal radar pulse train is used. The
summation signal, (c), which is the signal observed by the radar receiver,
shows very little sign of the existence of a second target, except for a slight
discontinuity of the beginning and end of the pulse train. The duration of this
VUG
RADAR SYSTEMS
159
T,
Fig. 7.11. Resolution of a point target.
discontinuity is slight, (2(d 2 — djc) and, since a typical radar pulse contains
only about 1000 carrier cycles, the relative energy in this period will be
extremely small.
If a Walsh pulse waveform (shown here in dipole form) is reflected from
the two targets, the reflected signals, shown as (d) and (e), will sum to give (f ).
The difference between a reflection from two separate targets in terms of
their summation is no longer a small perturbation of the reflected signal but a
major change in the summed and reflected waveform. This appears to
indicate an improvement in resolution attainable with Walsh radiated
waves, although in practice the attainment of sufficient bandwidth to achieve
these idealistic waveforms may present considerable difficulty.
160
APPLICATIONS IN COMMUNICATIONS
Harmuth 39 has compared the received patterns of amplitude modulated
radar transmission for sinusoidal and Walsh carriers. These transmissions
consist of a block pulse modulation of the carrier permitting a short train of
sinusoids or pulses to be radiated. Thus, in the case of a sinusoidal carrier,
the radiated signal has the form
mT mT
cos 2 irt/T between — (7.10)
and for the Walsh carrier
tnT mT
d WAL(2, t/T)/dt between (7.11)
where T is the period of the carrier and m is the number of cycles contained
within the period of the block pulse modulation.
In order to detect the time differences (and hence distance travelled)
between the transmitted and reflected signals, the auto-correlation of the
received amplitude modulated carrier is used. Autocorrelations are shown
in Fig. 7.12 for sinusoidal and Walsh carriers. The important difference
between the two functions is that in the Walsh case fairly long sections of
zero value are found, whereas the sinusoidal case produces a continuous
curve. This is significant in the case shown diagrammatically in Fig. 7.11,
where two reflectors are considered. The propagation time difference
needed for detection differs in the two cases. Using a sinusoidal carrier, a
propagation time of about t p = mT is required and in fact there is little
Fig. 7.12. Normalised auto-correlation for (a) an amplitude modulated sinusoidal carrier and
(b) an amplitude modulated Walsh carrier.
VIIG
RADAR SYSTEMS
161
advantage in using the full-line signal so that envelope detection (shown
dotted) may be used. The Walsh carrier allows discrimination with a
propagation time difference between the intervals, T^t p ^ 7/2- At,
7/2 + At t p T— At etc. If At is short compared with 7 then a considerable
theoretical improvement in range resolution can be obtained. Harmuth has
also shown that this advantage can be combined with that of more sophisti-
cated modulating codes, such as the Barker code, to obtain a radiation
characteristic involving smaller sidelobes.
An alternative approach using Walsh sequences for radar signal genera-
tion is to use these to synthesise a modulating waveform having optimum
characteristics. Some analytical work has been carried out by Rihaczek 40 on
these optimum waveforms. The choice of the optimum radar waveform is
found to be greatly dependant on the exact nature of the target and type of
discrimination required. The formulation of the modulating waveform
required demands a generation process which is flexible and easy to mechan-
ise. For reasons given in Section 7.6, the Walsh sequence has particular
advantages in this prospect.
The basic system configuration for an optimum radar transmitter is shown
in Fig. 7.13. The Walsh function generator can be arranged to permit
Fig. 7.13. Optimum radar system configuration.
variation of the weighting coefficients and to select the function series
presented for summation. This results in the availability of a large class of
modulation waveforms. The hardware advantages of such a system over one
employing sinusoidal synthesis methods are particularly significant in digital
processing systems and in correlation receiver processing. A discussion of
fast digital Walsh transformation techniques for this purpose is given by
Griffiths 41 .
162
APPLICATIONS IN COMMUNICATIONS
The effectiveness of a radar system is a measure of the resolution
capabilities of the receiver. Resolution performance in the presence of noise
depends very much on the precise form of the modulation signal in terms of
its amplitude, a(t ), and phase, cf)(t), modulation waveforms. A measure of
the receiver resolution is the radar ambiguity function, F(t, cu d ), introduced
by Woodward 42 .
F(t, o) d )= J a(t)a(t-r)exp(j[(t)(t)-(l)(t-T)])exp[-j(o d t]dt
(7.12)
where r is the time delay (range) and co d the Doppler shift. An ideal
ambiguity function would have a large narrow peak at r = (o d = 0 and small
values everywhere else.
Whilst this is unattainable, the value of the ambiguity function remains a
measure of the suitability of a given modulation waveform. Hence, it is
important to determine a relationship between this function and a given
modulating waveform, in order to maximise the ambiguity function and
efficiency of the process. This has been carried out by Griffiths and Jacob-
son 43 who have shown that the ambiguity information for all N=2 P Walsh
functions is contained in a knowledge of the ambiguity for p Walsh basis
functions. Therefore, a knowledge of relatively few ambiguity relationships
for a Walsh basis series will enable a wide range of radar modulation
waveforms to be developed having specific ambiguity properties. The
ambiguity functions obtained for the higher order Walsh functions are
surprisingly good and rival the performance of pseudo-random Barker
codes, often used for this purpose 44 .
A modification of the synthesis procedure used to achieve the optimum
ambiguity function is also described in the paper. This describes a system
configuration for binary phase modulation in which a series of controlled
delay elements are included in each of the generated Walsh function
outputs. This leads to an extremely flexible system permitting the generation
of a wide class of periodic waveforms.
References
1. Bramhill, J. N. (1974). An annotated bibliography on Walsh and Walsh related
functions. Technical Memorandum TG 1198B, The John Hoskins University,
Applied Physics Laboratory, Baltimore, U.S.A.
2. Various, (1974). 1974 Proceedings: Applications of Walsh Functions, Washing-
ton D.C.
3. Paley, R. E. and Wiener, N. (1933). Characters of Abelian groups. Proc. Nat.
Acad. Sc. 19, 253-7.
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163
4. Fine, N. J. (1949). On the Walsh functions. Trans. Am. Math. Soc. 65, 372-414.
5. Pichler, F. (1967). Das System der SAL und CAL-Funktionen als erweiterung
des Systems der Walsh-funktionen und die Theorie der SAL und CAL-Fourier
Transformation. Ph. D. Thesis, University of Innsbruck, Austria.
6. Gibbs, J. E. (1970). Discrete complex Walsh functions. 1970 Proceedings:
Applications of Walsh Functions, Washington D.C., AD 707431.
7. Pichler, F. (1973). Walsh Functions: Introduction to the theory. 1973 Proceed-
ings: NATO Advanced Study Institute Loughborough. “Signal Processing”, p.
23-41 (Ed. J. W. R. Griffiths et al). Academic Press, London and New York.
8. Harmuth, H. F. (1972), “Transmission of information by orthogonal func-
tions”, 2nd Edn. Springer-Verlag, Berlin.
9. Taki, Y. and Hatori, M. (1966). P.C.M. Communication system using Hadamard
transformation. Elect. Comm. Japan. 49, 11, 247-67.
10. Hiibner, H. (1971). Analog and digital multiplexing by means of Walsh func-
tions. 1971 Proceedings: Applications of Walsh Functions, Washington D.C.,
AD 727000.
11. Beauchamp, K. G. (1973). “Signal Processing”. Allen and Unwin, London, and
John Wiley, New York.
12. Ballard, A. H. (1962). A new multiplex technique for telemetry. Proceedings:
National Telemetering Conference, U.S.A.
13. Karp, S. and Higuchi, P. K. (1963). An orthogonal multiplexed communication
system using modified Hermite polynomials. Proceedings: International Tele-
metering Conference, London.
14. Filipowski, R. F. (1967). Trignometric product waveforms as the basis of
orthogonal sets of signals. Proceedings: National Telemetry Conference, U.S.A.
15. Harmuth, H. F. (1969). Applications of Walsh functions in telecommunications.
I.E.E.E. Spectrum 6, 11, 82-91, November.
16. Gordon, J. A. and Barrett, R. (1971). Correlation recovered adaptive majority
multiplexing. Proc. I.E.E.E. 118, 314, 417-22.
17. Gordon, J. A. and Barrett, R. (1972). Group multiplexing by concatenation of
non-linear code division systems 1972 Proceedings: Applications of Walsh
Functions, Washington D.C., AD 744650.
18. Durst, D. (1972). Results of multiplexing experiments using Walsh functions.
1972 Proceedings: Applications of Walsh Functions, Washington D.C., AD
744650.
19. Reed, I. S. (1954). A class of multiple error-correcting codes and the decoding
scheme. I.R.E. Trans. Inf. Theory IT 4, 38-49.
20. Robinson, G. S. (1972). Quantization noise considerations in Walsh transform
image processing. 1972 Proceedings: Applications of Walsh Functions,
Washington D.C., AD 744650.
21. Redinbo, G. R. (1972). Linear mean-square error codes. 1972 Proceedings:
Applications of Walsh Functions, Washington D.C., AD 744650.
22. Clarke, C. K. P. and Walker, R. (1973). Slow-time analysis of television pictures.
1973 Proceedings: Theory and Applications of Walsh Functions, Hatfield
Polytechnic, England.
23. Kraus, U. (1972). A wired-in resistor circuit realization of the two-dimensional
Hadamard transformation of broadband television signals. 1972 Proceedings:
Applications of Walsh Functions, Washington D.C., AD 744650.
24. Shibata, K. and Enomoto, H. (1971). Orthogonal transform and coding system
for television signals. 1971 Proceedings: Applications of Walsh Functions,
Washington D.C., AD 727000.
164
APPLICATIONS IN COMMUNICATIONS
25. Pratt, W. K. and Welch, L. R. (1972). Slant transforms for image coding. 1972
Proceedings: Applications of Walsh Functions, Washington D.C., AD 744650.
26. Chen, W. H. and Pratt, W. K. (1973). Colour image coding with the slant
transform. 1973 Proceedings: Applications of Walsh Functions, Washington
D.C., AD 763000.
27. Pratt, W. K. (1972). Walsh functions in image processing and two-dimensional
filtering. 1972 Proceedings: Applications of Walsh Functions, Washington D.C.,
AD 744650.
28. Wintz, P. A. (1972). Transform picture coding. Proc. I.E.E.E. 60, 7, 809.
29. Walker, R. (1974). Hadamard transformation — a real-time transformer for
broadcast standard P.C.M. television. B.B.C. Research Department, Report
No. BC RD 1974/7.
30. Murray, C. G. (1972). Modified transforms in imagery analysis. 1972 Proceed-
ings: Applications of Walsh Functions, Washington D.C., AD 744650.
31. Pratt, W. K., Kane, J. and Andrews, H. C. (1969). Hadamard transform image
coding. Proc. I.E.E.E. 57, 1, 58-68.
32. Ohira, T. et al. (1973). Picture quality of Hadamard transform coding using
non-linear quantising for color television signals. 1973 Proceedings: Applica-
tions of Walsh Functions, Washington D.C., AD 763000.
33. Sakrisin, D. J. and Algazi, V. R. (1971). Comparison of line-by-line and
two-dimensional encoding of random images. I.E.E.E. Trans. Inf. Theory IT 17,
4, 386-398.
34. Lally, J. F., Hong, Y. K. and Harmuth, H. F. (1974). Experimental transmitter
and receiver for electromagnetic Walsh waves. 1974 Proceedings: Applications
of Walsh Functions, Washington D.C.
35. Fralick, S. (1972). Radiation of E.M. waves with Walsh function time variation.
Conference Record: 1972 International Conference on Communications 38-16
to 38-19 I.E.E.E. order no 72 CHO 622-I-COM.
36. Frank, T. (1971). Circuitry for the reception of Walsh waves. 1971 Proceedings:
Walsh Function Applications, Hatfield Polytechnic, England.
37. Harmuth, H. F. (1974). Electromagnetic Walsh waves: transmitters, receivers
and their applications. 1974 Proceedings: Applications of Walsh Functions,
Washington D.C.
38. Lackey, R. B. (1972). The wonderful world of Walsh functions. 1972 Proceed-
ings: Applications of Walsh Functions, Washington D.C., AD 744650.
39. Harmuth, H. F. (1974). Range-Doppler resolution of electromagnetic
Walsh waves in radar. I.E.E.E. Trans. Electromagnetic Compatibility (to be
published).
40. Rihaczek, A. W. (1971). Radar waveform selection — a simplified approach.
I.E.E.E. Trans. Aerospace and Electronic Systems AES 7, 6.
41. Griffiths, L. J. (1973). The extraction of target information from radar signals
which use Walsh function modulation formats. 1973 Proceedings: Applications
of Walsh Functions, Washington D.C., AD 763000.
42. Woodward, P. M. (1953). “Probability and Information Theory, with Applica-
tions to Radar”. McGraw-Hill, New York.
43. Griffiths, L. J. and Jacobson, L. A. (1974). The use of Walsh functions in the
design of optimum radar waveforms. 1974 Proceedings: Applications of Walsh
Functions, Washington D.C.
44. Barker, R. H. (1953). Group synchronizing of binary digital systems. “Com-
munication Theory”. W. Jackson, London.
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165
Symposia on the Applications of Walsh Functions are held annually in Washington
D.C. The proceedings are published and made available by the National Technical
Information Service, U.S. Department of Commerce, Springfield, VA 22151,
U.S.A.
Details of the proceedings now available
1970 Ed., C. A. Bass, AD 707431.
1971 Eds., R. W. Zeek and A. F. Showalter, AD 727000.
1972 Eds., R. W. Zeek and A. E. Showalter, AD 744650.
1973 Eds., R. W. Zeek and A. E. Showalter, AD 763000.
1974 (To be Published).
Chapter 8
Applications in Signal Processing
VIIIA Signal processing applications
The Walsh and Haar transform can be implemented fairly easily on the
digital computer and this fact has stimulated a search for a class of applica-
tions where these transforms can replace, or possibly augment, the tradi-
tional role of the Fourier transform. In spite of their similarity and the
common properties of the Walsh and Fourier transforms the transforms are
characteristics of two different topological groups and will not in general be
interchangeable. Whilst the Fourier basis, consisting of exponential func-
tions, constitute the natural representation for systems with translational
symmetry, the Walsh transform is the natural representation of systems with
dyadic symmetry.
In certain application areas these differences are of less consequence than
the significant reduction in computation time that may be realised, such as,
for instance, in the transformation of very long series or tot two-dimensional
data. Also, as discussed in earlier chapters, the identity of the Walsh function
with certain non-linear series can lead to simplication in spectral decomposi-
tion and more efficient filtering methods.
The special characteristics of dyadic symmetry are now beginning to be
exploited, not as a replacement for Fourier techniques, but for the unique
properties they have in matching certain types of processing problems. A
fundamental difference is the change in form obtainable upon differentia-
tion which enables SAL and CAL functions to be separated quite easily 1 .
Again the limited level representation of the Walsh and Haar functions
matches digital operations, thus enabling economic translation to logical
processing hardware.
166
VIIIA
SIGNAL PROCESSING APPLICATIONS
167
Some applications in the area of single-dimensional spectral analysis and
filtering have already been discussed in the relevant chapters. Further uses
of the functions, using signal processing methods, are described in this
chapter. These include examples of processing hardware for real-time
applications, computer processing of physiological signals and applications
in the area of non-linear stochastic problems.
VIIIB Spectroscopy
One of the earliest uses of the Walsh function was its application by
Gebbie 2 to the problem of on-line spectroscopy. He applied the Walsh
transform to the decoding of the output of a two-beam interferometer such
that a very rapid result was obtained. Whilst the original impetus for the
work was the expectation of better matching of the functions to digital
computer operations, his conclusion was that it could result in a spectro-
scopic system of high overall efficiency.
Some simplicity in spectroscopy equipment requirements has been
demonstrated by Despain and Vanasse 3 who used a series of Walsh-related
masks to code and decode the optical signals. The use of coded masks has the
effect of improving the signal/noise ratio of the measurement and, in the
case of the Walsh function, this advantage is obtained without loss of
resolution.
The most successful of the current applications to spectroscopy is the
development of an infra-red spectrometer using the techniques proposed by
Decker 4 and Harwit 5 . The difficulties with infra-red spectroscopy are
twofold. The low flux levels of infra-red sources, and the high attenuation of
the narrow exit slit required to separate out the spectral elements of a
dispersed light spectra. Attempts at improving the light collection
capabilities of the spectrometer and to increase the transmission of the
spectral detection device have led to the development of scanning inter-
ferometer devices. These are precise laboratory instruments, expensive to
produce and difficult to align. The Walsh transform spectrometer gives a
similar performance but is simpler in design, rugged and cheaper to produce.
A conventional dispersive element is used (fixed prism or grating) and the
light distributed over the entire spectra is passed through a multi-slit mask
coded in an orthogonal manner such that the slits in the mask correspond to
a binary 1 and the opaque portions to a binary 0 (Fig. 8.1). The total light
passing through the mask is collected on a single infra-red detector. The
orthogonal mask pattern is then changed and the process repeated. This is
carried out a number of times equal to the number of discrete slots in the
mask (i.e. binary 1 or 0 locations on the mask). The result is a series of output
values for the detector from which the spectral values can be obtained
through a mathematical transform.
168
APPLICATIONS IN SIGNAL PROCESSING
Fig. 8.1. A Hadamard Spectrometer — schematic diagram.
If we define n such measurements, each consisting of the summation of n
products of the dispersed spectral value output x h which determines whether
the slot is opaque or transparent, then the n summated values, I h can be
represented as a series of simultaneous equations, viz.
a 1 . 1 x 1 + a 1 . 2 x 2 + - • ' + a 1 . n x n = Ji 1
(I2-1X1 T (I2-2X2 + • • • + a 2n x n — 12 I
(8.1)
^n-1-^1 2X2^’ ’ ’ & n-nXn In J
which may be expressed in matrix form as
Mxy} = tt} (8.2)
To determine the value of the spectral series, x h it is only necessary to
multiply the summation vector, I„ by the inverse of the coefficient matrix,
viz.
{x,} = [ay] 1 . {I/}
( 8 . 3 )
VIIIB
SPECTPOSCOPY
169
This latter operation is carried out by a digital computer operating on the
values of I, previously recorded on some suitable media, e.g. punched paper
tape.
The choice of matrix value and hence coding of the transmission slit is
important if a best estimate to the true values of the power spectral densities
is to be obtained. Codes based on the Hadamard matrices provide this
optimum performance and in practice a cyclic code is used. One such code is
shown in equation (8.4) for n = 19.
[««]=
1011000010101111001
1101100001010111100
0110110000101011110
0011011000010101111
1001101100001010111
1100110110000101011
1110011011000010101
1111001101100001010
0111100110110000101
1011110011011000010
0101111001101100001
1010111100110110000
0101011110011011000
0010101111001101100
0001010111100110110
0000101011110011011
1000010101111001101
1100001010111100110
0110000101011110011
(8.4)
Here each row of code is simply the one above shifted by one element. The
advantage of these codes is that only one strip mask need be designed and
can be moved mechanically along one position for each determination of the
summation coefficient, I,.
170
APPLICATIONS IN SIGNAL PROCESSING
400 425 450 475 500 525 550 575 600 625
Wave number (cm -1 )
Fig. 8.2. Comparison between measured and calculated spectral transmission through the
Earth’s atmosphere.
Although this type of development enables a rugged device to be con-
structed overcoming many of the instrumental difficulties of the well-known
Michelson interferometer, its performance in terms of high energy transmis-
sion (luminosity) is generally inferior. An improvement can be made by
using a double multiplexing scheme associated with the dispersive grating 6 .
Such a system involves the modulation of the dispersed radiation by means
of a mechanical chopper or mask at both the entrance and exit slits. This
gives simultaneously the advantages of high luminosity and effective
polychromatic transmission and enables comparable results to the Michel-
son interferometer to be obtained with the advantages in simplified design
and operation outlined above.
A doubly-multiplexed instrument of this type has been constructed by
Phillips and Briotta for use in astronomical spectrum analysis 7 . One field of
measurement in which it has been successfully employed is the determina-
tion of the spectral transmission profile for Jupiter and the Earth. Figure 8.2,
reproduced from their paper, shows a comparison between the measured
and calculated spectral transmission profile of the earth’s atmosphere
obtained under field conditions. The agreement in position, width and depth
VIIIB
SPECTROSCOPY
171
of most of the spectral lines is good and compares favourably with that
obtained using earlier laboratory methods.
This information on infra-red spectrometers has been included here by
courtesy of Spectral Imaging Inc., U.S.A. who have supplied such equip-
ment to N.A.S.A. for reflection spectrometry of the surface of Mars and
other planets and also, on a more mundane level, for routine analytical
spectral chemistry in the laboratory.
VIIIC Pattern recognition and image-processing
In both of these applications, the motivation for using transforms other than
Fourier is either to reduce computation time for a given resolution or to
increase the resolution without incurring the penalty of drastically enhanced
computation time. Both Walsh and Haar transforms have been used effec-
tively to satisfy these requirements.
The general problems of pattern recognition using orthonormal represen-
tation of the pattern environment have been stated by Carl 8 who has
indicated some advantages in using Walsh functions to model the human
visual system. A fairly general approach towards pattern recognition is to
carry out a transformation of the pattern and its model and to cross-correlate
the transformed sets of values to determine the degree of recognition, rather
than to attempt cross-correlation of the original signals. Kabrinsky 9 has
pointed out that substantial savings in the computation time involved can be
obtained using this method, whilst Carl 10 and others have used the Walsh
transform as a means of reducing still further the complexity of the two-
dimensional processing to the level of coefficient additions and subtractions
only.
A form of spectral pattern recognition has been described by Kennett 11
who has employed sequency spectrograms for nonstationary data. Two-
dimensional plots of the processed data were obtained in which the sequency
power was plotted against time. The technique used is described in Chapter
6, where it was shown that information can be obtained which is additional
to that obtained by equivalent Fourier methods.
Image processing and enhancement using transform techniques exploits
the redundancy found in nearly all pictures. The two methods currently in
use are:
(a) To carry out threshold filtering of the picture data following transfor-
mation, either by considering the lower sequency or frequency compo-
nents of the transformed picture as important and worthy of retention
whilst considering that the higher values represent noise and, therefore,
these coefficients can be set to zero; or by truncating the transformed
signal coefficients below a constant threshold level to zero value.
172
APPLICATIONS IN SIGNAL PROCESSING
(b) To divide the picture area into a number of squares and to define a
minimum word length necessary to express the maximum amplitude value
found within each square. This represents a bandwidth compression
approach, referred to briefly in the previous chapter as adaptive coding.
In either of the threshold filtering methods the picture is reconstituted by
means of a further transformation, either directly or following some filter-
ing process. This latter method can take the form of enhancement of a
particular feature of the image, such as giving emphasis to the outlines of the
picture content. This implies non-linear forms of filtering which are found
easier to apply to a Walsh or Haar transformed data matrix.
The main objective in discrete filtering for image enhancement is to
improve the signal-to-noise level of the picture. In order to determine some
quantative values for this improvement, Kennedy has carried out Walsh
filtering of a given quantised and sampled image subjected to various
amounts of additive noise. Under certain conditions, he has found that the
addition of up to 50% of additive noise to a picture transmitted in sequency
form has little discernable effect on the reconstructed picture 12 .
The design of optimum filters for image enhancement has proceeded by
the development of various forms of matched filtering. Since it is very
inconvenient to have to know the position of the desired feature contained
within the data, alternative sub-optimum solutions have been proposed.
One of these, due to Treitel and Robinson 13 , is to derive the filter transform
coefficients not directly from the required signal but from another signal that
has the same power spectrum as the required signal. The only requirement is
that for real data the filter output must also be real, i.e. its transfer function
must be symmetric. Using a Fourier transformation the filter constants are
obtained by taking the modulus of the complex Fourier transform of a
matching image having the desired size, shape and orientation. It is not quite
so easy to do this with a Walsh transform since the result is not positionally
invariant. Instead, the Walsh power spectrum is taken which combines the
squares of CAL and SAL functions to give approximate independence from
the effects of phase shift which would be very apparent if the transform was
used directly (see Section VE). A similar spectrum is obtained from the sum
of the squares of the Haar transform but in this case it is necessary to average
the absolute value of all the coefficients derived from functions of the same
degree (see Section IVG).
Gubbins et al , 14 have applied this form of matched filtering to data
simulating magnetic measurements made on buried archaeological sites.
The structures encountered are usually of geometric shape and, therefore,
readily identifiable, but due to irregularities dn the upper soil layers, the
strengths of the magnetic anomalies is low and subject to poor signal/noise
ratios.
vine
PATTERN RECOGNITION AND IMAGE-PROCESSING
173
A dot density plot for a simulated buried Roman fort is shown in Fig. 8.3.
This is composed from a 10 000 point array having a maximum of five dots
near each point to indicate the amplitude of the magnetic anomoly. The
improvement obtained by Gubbins et a/., using matched Fourier filtering, is
shown in Fig. 8.4. This may be compared with the results obtained by Walsh
matched filtering in Fig. 8.5. The Walsh results are obtained in one-eighth of
the computational time of the Fourier calculations and give almost the same
information. The block structure in the display is due to the supression of
part of the higher sequency information. Haar filtering (Fig. 8.6) gives
results which are slightly inferior to the Walsh results but are obtained at a
small fraction of the computer time used for the latter and a still smaller
r
h
b
b
b
b
b
L
W <* $ A t ^
v , m.
, , .b.in "‘-S* :'•« ^ ■
$'■■■: ..r v /-• :
• ' *f-' % S -T’. „ # **V
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0 -999 100 0 -999 100 - 0 . - 0 . - 0 . - 0 . - 0 . - 0 . - 0 . - 0 . - 0 . - 0 .
n
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H
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Fig. 8.3. Simulated magnetic anomaly due to a buried Roman fort in soil.
174 APPLICATIONS IN SIGNAL PROCESSING
Fig. 8.4. Matched filtering of the data shown in Fig. 8.3 using the Fourier Transform.
fraction than Fourier filtering. These results are typical for the image
enhancement obtained when the filtering characteristics of the different
transforms are compared using either matched or threshold filtering. In both
cases a trade-off is obtained between computational time and subjective
resolution of the reconstructed image.
The second method, namely that of bandwidth compression is of value
chiefly in permitting efficient processing methods for large amounts of data.
This is relevant to situations involving telemetry transmission of video
images, such as for example, transmission from earth resource satellites and
planetary exploration. The procedure consists of dividing the transformed
image data into several equal squares and defining, for each, a minimum
VIIIC PATTERN RECOGNITION AND IMAGE-PROCESSING 175
0 1 2 3 4 5 6
n~ r ” r ~ r ~ r " r “ r ~]
0 -999 100 -<J 0 200 -0. -0. -0. -0. -0. -0. -0. -0. -0. -0.
Fig. 8.5. As Fig. 8.3 using the Walsh transform.
word length necessary to represent the maximum absolute amplitude of the
transformed coefficients found within the square. Each set of square coeffi-
cients needs to be preceded by a constant length word to define the variable
characteristic word length and, therefore, represents a slight but constant
overhead on the system.
If N = 2 P is the number of rows and columns for the image and L is the
number of luminosity levels, the maximum value for the transformed
coefficients will be JS^L. If q is the quantization value for these, then the
number of bits required to specify the word length used in a square will be 15
p s = log 2 [^log 2 (~ ; + 1 j
(8.5)
176
APPLICATIONS IN SIGNAL PROCESSING
M * & gp
^ '** C - ; B w
& * "&'&%• *>•«:
•5 M A, u *
*• #&*’ -fife, ’ ■ - J .ft
T«L 4> '
J"- -V- #*" .jKC
"-••• <V 21 Vi' a r-;#^.y •■*•:
¥ “■ -v- tagfe .M-
%k w & %* .,, mv " w
^ $' ;» « •" * suj.
■<> #’ ',. .*. ., - ,,# “W’
s . ^ » „ .»
^ j*v*ssr- *« '■«,.*?*,
ilst/ Sr faf "'K# jJIjP'.*
0 -999 300 -3 0 100 -0. -0. -0. -0. -0. -0. -0. -0. -0. -0.
Fig. 8.6. As Fig. 8.3 using the Haar transform.
where the logarithms are rounded to the maximum integer value. Figure 8.7
gives an example of the bandwidth compression capability of the Walsh
transform used in this way. This is due to the work of Professor Cappellini
and his colleagues who have carried out measurements of compression
obtained for the standard image shown in the diagram (obtained from the
Image Processing Institute of the University of Southern California) and on
images received from an earth resources satellite.
The original image used for Fig. 8.7 is constituted from 256 by 256 data
samples quantised to 256 luminosity levels (8 binary bits). The transformed
image data is divided into equal squares of 8 by 8 samples and a minimum
vine
PATTERN RECOGNITION AND IMAGE-PROCESSING
177
Hi
R£C. PICT. L m -i7 bits! simple
S?£C PICT. L m ‘ Q75 bits] sample
Fig. 8.7. Processing a standard image using area subcoding.
word length used to represent the square data (a bit number sufficient to
represent the maximum absolute amplitude value plus 1 sign bit). A constant
length word of 5 bits is required to specify the range of the encoded squares
and corresponds to an overhead of 0.08 bits/sample.
In the figure, the first illustration (a) shows the original image having a
mean word length of L m = 8 bits/sample. Two examples of reconstruction
are shown; (b) where the quantisation value for the transform coefficients is
q = 32 N (N = 256) and L m = 1.7 bits/sample and (c) where q = 64 N and
L m = 0.75 bits/sample. These results show that sub-area encoding using the
Walsh transform can give appreciable data compression ratios in excess of
5 : 1 with low image degradation.
178
APPLICATIONS IN SIGNAL PROCESSING
VIIID Acoustic image filtering
Image generation using the echo principle in sonar has been developed
extensively for underwater exploration and sounding. The method requires
the radiation of short pulses of energy for reflection at the target surface and
subsequent detection of the reflected signals at a scanning detector. The
resolution obtained by such a system is limited by the wavelength-to-
aperture ratio; a feature of all pulsed systems. Continuous illumination of
the target area by means of an unbroken sinusoidal (or other) waveform
could permit a resolution limited only by the signal-to-noise ratio which can
be very great.
Consideration of the reflected signal at the receiving plane shows that this
differs by a linear transformation from the signal present at the object plane.
Thus, a localised pulse-like signal, reflected at a given point on the object
plane, would be spread (transformed) over the entire area of the receiving
plane.
It is only necessary to carry out an inverse transformation of the signal
detected at the receiving plane for the reconstruction of the transmitted
“image” to become possible. The acoustic image filter described by Har-
muth 16 operates in this way. The principle is shown in Fig. 8.8.
Source of sound
waves
Fig. 8.8. Principle of acoustic imaging.
VIIID
ACOUSTIC IMAGE FILTERING
179
The object, O, is “illuminated” with continuous sound waves radiated
from source, 5. In general, these waves are sinusoidal in form although other
waveforms could be employed. The sound waves impinging on the object
are scattered and detected by a linear matrix of microphone transducers
(hydrophones), forming a reception plane, P. A signal scattered at a point on
O will arrive at each hydrophone subject to a different transit time delay.
Knowing these propagation delays, it would be possible to reconstitute the
scattered signal by superposition at a corresponding point on the reception
plane, P, in order to derive an “electronic” image of the object, at least as far
as its boundary location is concerned.
This procedure can be carried out without the use of delays by means of
two-dimensional spatial filters, of the type described earlier in Chapter 6,
which carry out the process of inverse transformation. The filter described
by Harmuth 17 is an analog device and, in order to reduce the loading on the
input source, a version of a fast transform algorithm is used which allows
only four voltages from the microphone matrix to be summed at any one
amplifier (see Section HIM). This has the result of introducing a quadrant
ambiguity in the location of a particular point in the source plane. For this
reason it is necessary to include an additional identifier, shown in Fig. 8.8,
which forms part of the sampling device required prior to transmission of the
transformed image matrix to the display system.
The classical limit for the resolution angle, 0 , that can be achieved with a
wavelength, A, and an aperture, A = 2(n — l)d, where n is the number of
microphones used and d their distance apart, is
0 — A/A (8.6)
It is possible in the case of radar systems to obtain a smaller resolution
angle, than that given by equation (8.6), by making use of the Doppler effect.
Whilst this is difficult to apply in the acoustic case, a similar improvement
may be obtained by increasing the number of microphones used and
combining their outputs through a transformation. The way in which this
may be carried out and quantitative values for the resolution improvement
obtained is also given by Harmuth 16 .
It should be pointed out that this work represents an early stage in
development and that imaging techniques by two-dimensional electric filters
currently exhibit a number of technological limitations. The resolution is *
limited to that of the television display and by the frequency of the
illuminating wave which cannot in practice be much higher than 1 MHz. On
the other hand, the resolution is bound by signal-to-noise ratio, rather than
the wavelength-to-aperture ratio of reflection sonar, and in this sense makes
much better use of the available information than present pulse reflection
techniques.
180
APPLICATIONS IN SIGNAL PROCESSING
VIIIE Speech processing
Walsh functions have been used in the processing of speech signals in several
different ways: as a method of reducing the bandwidth occupied by the
transmitted signals, as a tool for efficient speech synthesis and as a technique
for automatic speech recognition.
The earliest work was concerned with bandwidth compression. Important
contributions were made by Campanella and Robinson 18 and Boesswetter 19
which showed that advantage could be taken of Walsh coding to remove
some of the redundancy from speech transmission. Later work was success-
ful in reducing the transmission rate for Walsh coding to 48k bits/s,
compared with 56k bits/s for conventional P.C.M. coding in a similar
situation 20 . Further reductions were demonstrated using the hardware pro-
cessing system of Gethoffer 21 , who claims a bit rate of 22k bits/s with no
subjective loss of intellegibility.
Quantative comparisons have been made by Pratt and others between the
performance of different transforms used for bandwidth compression, based
on the measured mean-square error 22,23 . Their conclusions indicate that,
theoretically, the Karhunen-Loeve transformation gives the best reduction.
This transform was referred to earlier in Section IF. It is not generally a
separable transform and is defined in matrix terms from
[A] _1 C X [A] = diag[A f ] (8.7)
where [A] and [A] -1 are the matrix forms of the Karhunen-Loeve transform
and inverse transform respectively. C x represents the covariance matrix of
the data series and A, are the eigenvalues of C x arranged in a diagonal matrix
form. Although the Karhunen-Loeve transformation is optimum for this
case, it has (wo major shortcomings. First, as the size of the data series, N,
increases, so does the size of the covariance matrix required for computa-
tion. Second, there is no general fast algorithm to compute the Karhunen-
Loeve transformed coefficients, so that computation demands N 2 multiply
and add operations. For large values of N the computation can represent a
formidable task, particularly for real-time operation. Consequently, other
sub-optional transforms which do not have these shortcomings are used. In
many cases the Walsh transformation, representing the next favourable
case, would be selected in terms of its computation efficiency.
The basic problem with speech synthesis is to maximise the efficiency of
voice transmission whilst retaining acceptable distortion levels for the voice
characteristics. Early work was due to Sandy 24 who first envisaged the
possibilities of using a Walsh analysis for this purpose. Speech synthesis
techniques have developed in terms of either a reconstruction from the
power spectrum or by using only the dominant sequency coefficients.
VINE
SPEECH PROCESSING
181
Gethoffer 21 has constructed a hardware system to carry out synthesis by the
power spectrum method. He showed that the quantative energy distribution
of German speech in the Walsh spectrum differs for voiced and unvoiced
vowels. Consequently, he concludes that it should be possible to improve
bandwidth compression or synthesis by adopting different coding schemes
for the two voice parts. In order to adapt the time base of the coding Walsh
transform, it is necessary to extract the pitch of the signal with some
accuracy. Synthesis of speech and other forms of processing are then
possible from the pitch synchronous Walsh spectra produced from this
adaptive system.
The use of dominant sequency coefficients in signal reconstruction is
based on the assumption that the ear is insensitive to phase, so that the
minimisation of phase shift obtained with the power spectrum (see Section
VE) may not be necessary. This approach has been used by Shum and
Elliott 25 who have obtained fairly good reproduction of speech from a
selection of only four to eight dominant coefficients, out of a total of 64
comprising the sampling window.
Automatic speech recognition, or more particularly word recognition, is a
difficult field of research. At the present time given a restricted vocabulary,
such as the numbers 0 to 9, a good recognition is possible with up to 95%
success, where a single speaker is involved. This falls to around 50% if a
second speaker is used. The difficulties are due partly to the variable length
for spoken identical words and, more fundamentally, due to the non-
stationarity of speech waveforms. This latter problem was recognised by
Gethoffer 26 who devised means of continuous and parallel sequency filtering
and power spectrum analysis for investigation into vowel sounds. The analog
methods used for this non-stationary analysis have already been discussed in
Chapter 6. Due to the non-stationarity of speech waveforms, it is necessary
to define a fairly narrow window for the data over which stationarity can be
assumed. Early work used fixed window lengths and gave poor results.
Current work by Gethoffer and Elliott 21,25 analyse the speech waveform with
a window period that varies with the pitch period. This followed earlier work
by Gethoffer 26 where the Walsh time-base was adapted to the pitch period of
the vowel. An extensive discussion of these methods of synthesis and
recognition is given by Flanagan, to which the reader is referred 27 .
A correlation method of speech recognition is described by Clark et al. 28 .
This is based on the comparison between Walsh sequency components of
spoken words and those taken from a stored library of words. The Walsh
sequency components of successive time segments of spoken words are
obtained. These components are arranged in an amplitude sequency and
time matrix order and correlated sequentially against stored matrices
representing the transformed values of a library of words. Recognition is
182
APPLICATIONS IN SIGNAL PROCESSING
obtained by extracting the word giving the highest correlation coefficient. A
major difficulty lies in the variation of the duration of the spoken word
mentioned earlier. This has also been noted by Edwards and Seymour 29 who
compare Fourier and Walsh spectra of a restricted vocabulary of words.
They conclude that whilst the Walsh pattern contains more indications than
the Fourier pattern, this could be advantageous since small variations
between different utterances of the same word have less effect on the whole
pattern and should lead to more accurate identification.
VIIIF Medical signal processing
Transformation techniques used for bandwidth compression or identifica-
tion have been applied extensively in the analysis of physiological data 30,31
particularly in on-line situations such as patient monitoring. The application
of the Walsh or Haar transform to these problems has not been uniformly
successful due to the difficulties in applying the analysis to what is very often
a sinusoidally-based waveform and, in some cases, to the need for a
phase-invariant transformation. As a consequence, the most successful
results are found in the exploitation of the speed of calculation for these
transforms, compared with Fourier transform computations, and in the data
compression properties of Ahmed’s odd-harmonic spectrum (see Section
VG).
Automatic classification of Electrocardiograph (E.C.G.) data has formed
the subject of several papers, notably those of Milne et al 32 and Ahmed and
Rao 33 , whilst Thomas and Welch 34 have used the E.C.G. in the determina-
tion of heart rate. Morgan 35 has also carried out some work in the analysis of
aortic blood pressure.
The analysis of Electroencephlograph (E.E.G.) data has been amongst
the least rewarding of the applications of Walsh and related functions for
physiological work. A realistic appraisal of the current situation vis-a-vis
Fourier methods is given by Yeo and Smith 36 .
The principle use of E.E.G. analysis lies in clinical diagnosis of epilepsy
and in sleep research. Both of these studies involve the production of large
amounts of data so that automated techniques of analysis are widely used in
an attempt to reduce the magnitude of the interpretive task. The methods
provided invariably use spectral transformation where the Walsh transform
has a speed advantage over the Fourier transform. Yeo and Smith have
compared the sequency and frequency spectra derived from the analysis of
E.E.G. sleep processes. A better waveform discrimination is found with the
Fourier transform, which is to be expected, since the E.E.G. waveforms are
sinusoidal in content. The poorer discrimination to sinusoidal waveforms
found with the Walsh transform may be compensated by the reduced
VIIIF
MEDICAL SIGNAL PROCESSING
183
computation time and memory requirements, particularly if small-on-line
computers are used for the investigation.
Apart from their role in processing physiological data, the Walsh func-
tions have also been applied to the modelling of biological systems. An early
paper, due to Meltzer et al 37 , applied Walsh functions to define a two-
dimensional form, in his case morphological patterns. The study of mor-
phogenesis is concerned with the development of an organism characterised
by a succession of metabolic shapes having distinct morphological patterns.
The reduction of these shapes to a series of Walsh coefficients enables the
nature of the patterns and their classification to be examined with some
precision.
The relationship between the Boolean function and the Walsh function
has been used by Gann et al. 38 to determine a dynamic Boolean model of a
physical system, and was developed further through the use of a related
transformation in a later paper. The Walsh function has also been applied to
define a system transfer function for biological systems by Seif 39 , and as a
filtering system in the modelling of the action of the nervous system by
Boeswetter 40 .
VIIIFI E.C.G. and related analysis
The electrocardiogram (E.C.G.) gives a measure of the electrical activity of
the heart in terms of a continuous time-history. The most important
characteristic of a normal cardiac cycle is the segment shown in Fig. 8.9
R
Q
Fig. 8.9. The QRS cycle of cardiac activity.
which repeats itself, with minor variations, once per cycle. This is known as
the QRS cycle and corresponds to the electrical activity of the ventricles
during a single heart beat. Variations in the shape of the QRS waveform are
important in determining the onset of ventricular fibrillation, so that on-line
analysis and classification of this segment of the cardiac system forms an
important feature of automatic patient monitoring systems.
184
APPLICATSONS IN SIGNAL PROCESSING
Many physiological waveforms of this kind are suitable for Walsh analysis
since a consistent feature of the waveform can be recognised and used as a
reference point, thus overcoming the changing variance with phase of the
Walsh function (Aortic blood pressure variation is another example). Past
work in this form of analysis has emphasised a time domain approach
in which the cardiologist looks for certain identifiable characteristics
and compares these with the known and normal characteristics of the
patient.
A sequency or frequency approach to E.C.G. analysis has a number of
advantages, not least being the bandwidth reduction capability, which is
important where large quantities of data are to be analysed. A secondary
reason is the increase in high-frequency content which is found to accom-
pany many cases of function abnormality.
The odd harmonic power spectrum, described in equation (5.18), has
been used by Milne because of its powerful data reduction properties and its
invariance to shifts of the sampled E.C.G. waveform. In his analysis he
substitutes a two-dimensional transformation in place of B n , which is defined
as
[B-.J = [H N| ][x„.J[H N J (8.8)
where H Nl and H N2 are Hadamard matrices and x M2 is a Ni by N 2 data matrix.
Only two leads are attached to his subject so that the two-dimensional
transform, given by equation (8.8), can be^expressed as a one-dimensional
transform in terms of the sum and difference of the two channel sampled
values. The sampling of the E.C.G. commences with the start of the QRS
waveform and is completed 80 ms later after having recorded 32 samples.
Using two acquisition lead points, the resulting two-dimensional spectra
from 64 samples is reduced to 12 transformed values. A number of the
spectral points obtained were used in a recursive normal/abnormal classifi-
cation algorithm, achieving almost 90% success for canine subjects. An
analogous technique is reported by Ahmed 33 , using the Haar transform,
which gives similar data reduction properties to the Walsh odd-harmonic
spectrum. A 3 : 1 data compression reduction over the identity transform is
claimed for the Haar transform.
Morgan 41 has used the R transform (Section III I) to achieve a transform
for E.C.G. work which is insensitive to cyclic shifts of the input data. Some
results of the application of simple Walsh vector filtering are also given.
Here the transformation of the E.C.G. waveform, related to the location of
the QRS occurrence, is followed by selection of those components to be
retained and by re-transformation. The resultant time domain waveform
VI1IF
MEDICAL SIGNAL PROCESSING
185
gives information related to predictive analysis of heart abnormalities, such
as occur prior to ventricular fibrillation.
Using the E.C.G. signal as a basis for measurement, Thomas 34 has
developed a method of heart (pulse) rate determination based directly on
sequency calculation. Each time the large spike of the QRS cycle occurs, it
initiates a bistable circuit, the state of which is sampled at a rate > heart rate.
The sampled values are then transformed using a sequency ordered Walsh
transform. Under normal conditions a sampled square waveform
(Rademacher function) is generated, the sequency of which gives the heart
rate directly. Any variations in the triggering of the bistable affect the
mark-space ratio of the waveform generated and will be reflected in the
average zero crossing (sequency) measurement of the resulting waveform.
VIIIG Non-linear applications
An interesting area of application which makes use of the unique features of
the Walsh series is that of non-linear stochastic problems. In particular,
some success has been obtained in improving the efficiency of signal
detection for those transducers which are essentially non-linear in opera-
tion.
The special feature of the Walsh series found useful in this connection is
illustrated in Fig. 8.10. If we assume that a given waveform can be rep-
resented by a group of Walsh functions of a given order, the result will be a
stair-step approximation to the waveform. If this is now operated upon by a
single-valued non-linear transformation, then another stair-step function is
obtained having the same number of steps but with changed step heights.
Any further non-linear operations will have a similar result such that the
amplitudes of the individual steps will change but not their total number.
This has the effect of limiting the number of new sequency terms generated.
From equation (3.17), we see that the only intermodulation products
produced by the multiplication of two Walsh functions will be the Modulo-2
addition of each of the sequency terms. Therefore, if the input signal can be
represented by a finite number of Walsh functions, N, then only
(N— 1) • (IV— 2) • (TV— 3) ... 1
intermodulation products will be generated. The situation is different with
the Fourier representation of the input signal. Here a set of new harmonics
would be generated which will have combination frequencies equal to the
sum differences of all the possible harmonic values of the signal waveform
and the non-linear function.
186
APPLICATIONS IN SIGNAL PROCESSING
An example will illustrate the difference between these two types of
representation, both operated on by the same type of nonlinearity. If we
define a signal to consist of two Walsh functions, viz.
x(t) = WAL (r, t) + WAL(s, t) (8.9)
which are passed through a power-law device having an output of the form
y (t) = aijc(f) + a 2 x 2 (t) + a 3 x 3 (t) (8.10)
then y(t) is given as
y(t) = a 1 [WAL(r, f) + WAL(s, t)] + a 2 [WAL 2 (r, 0 + WAL 2 (s , t)
+ 2 WAL(r, f)WAL(s, *)] + a 3 [WAL 3 (r, 0 + WAL 2 (s, l)WAL(r, t)
+ 2 WAL 2 (r, f)WAL(s, t) + WAL(s, f)WAL 2 (r, t)
+ WAL 3 (s, f) + 2WAL(r, f)WAL 2 (s, *)] (8.11)
VIIIG
NON-LINEAR APPLICATIONS
187
Using the addition theorem for the Walsh transform (equation 3.17),
y(0 = fl 1 [WAL(r, f) + WAL(s, t)]
+ all WAL (r ©5, t) + 2 WAL(0, t)]
+ a 3 [WAL(r, t) + WAL(r, t) + 2 WAL (s, t)
+ WAL (s, f) + WAL(s, t) + 2 WAL(r, t)]
= 2a 2 WAL(0, 0 + (a, + 4a 3 )WAL(r, t)
+ (fli + 4a 3 )WAL(s, 0 + 2a 2 WAL(r©5, f) (8.12)
Thus, the output consists of the desired terms having changed amplitude
coefficients plus a d.c. term and an intermodulation product.
If we now define the input signal as consisting of two sinusoidal functions,
viz.
x(t) = sin A +sin B (8.13)
where
A =f(co. r.t)
B s.t)
then, if x(t) is passed through a device having the relationship given by
equation (8.10), the output signal would be
y(t) = a^sin A +sin I?] + a 2 [sin 2 A +sin 2 B + 2 sin A sin B]
+ a 3 [sin 3 A + sin 3 B + sin A sin 2 B + sin jB sin 2 A 4- 2 sin 2 A sin B
+ 2 sin 2 B sin A]
= ai[sin A + sin JB] + a 2 [i(l — cos 2A) + §(1 — cos 2 B)
+ cos ( A-B ) - cos (A+B)]
+ ^[1(2 sin A — sin 3 A — sin (— A))+|( 2 sin B — sin 3JB — sin (-2?))
+1(2 sin A - sin (A +2B)-sin (A-2B))
+1(2 sin B - sin (JB - 2 A) - sin (jB - 2A))] (8.14)
Here the output consists of the original signal plus its second and third
harmonics, a d.c. term and six intermodulation products. The position is
much more complex and careful bandpass filtering is necessary to extract the
required signal from the complex modulated output. In the general case for
Walsh functions only high or low-pass sequency filtering is needed.
Several workers have considered the effects of non-linearities, as
described above, on the Walsh modulation of a sinusoidal carrier signal.
188
APPLICATIONS IN SIGNAL PROCESSING
With any multi-channel carrier system of communication of this type
non-linearities within the system will give rise to cross-modulation products
(cross-talk), which can be serious. These products can be discriminated
against more easily using Walsh modulation compared with sinusoidal
modulation. However, in the case of Walsh functions, there is a danger that
the Modulo-2 addition of the addition theorem will result in Walsh functions
being generated which will coincide with one or more of the desired signals.
To avoid this, the use of a Rademacher subset of the Walsh functions has
been suggested by Frank 42 . Rademacher functions form an incomplete set
and have the property that their products yield, a Walsh function that cannot
be a Rademacher function. This system has the disadvantage in a practical
case that wider transmission bandwidth would be required. Harmuth 43
suggests other alternative methods of selection for the modulating signals to
minimise the cross-talk, without incurring this penalty.
Corrington 44 gives several examples related to the non-linear modulation
operations involved in phase-shift keying. He shows that the derivation of
the frequency spectrum, through the use of Walsh functions, considerably
eases the analytical problems involved. As a further example, we can
consider the application of Moss 45 who has investigated the use of a
Walsh series to modify a pseudo-random binary sequence used in the
estimation of the impulse response of a non-linear system (gas chromo-
tography).
The difficulty with detectors used in such a system is that the sampling
valve used can assume only an “on” or an “off” position and that the
detection operations for the gas samples are frequently very non-linear.
Thus, the estimation of linear impulse response would normally have a large
variance and the linearisation techniques, proposed prior to Moss’s work,
have been only partially successful in removing the effects of the square and
cubic terms. The linearisation scheme using a Walsh series employs only two
levels of input and is effective in such a situation.
The modified pseudo-random binary sequence is applied as an input to
the system, i.e. to operate the sampling valve. Operation of the non-linear
detector on the admitted gas samples results in the generation of an output
sequence, modified by the transfer function of the gas chromatographic
column. Measuring the cross-correlation between the output due to the
modified pseudo-random binary sequence and the sequence itself, a func-
tion results which contains contributions from the even-power terms of the
non-linearity only. Hence, as long as the only even power present is in the
second (a function of this particular system), the method is applicable with
equal success in situations where the output non-linearity contains an
arbitrary number of higher-order odd power terms.
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12. Kennedy, J. D. (1971). Walsh function imagery analysis. 1971 Proceedings:
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13. Treitel, S. and Robinson, E. A. (1969). Optimum digital filters for signal-to-
noise ratio enhancement. Geophysical Prospecting 17, 248-93.
14. Gubbins, D., Scollar, I. and Wisskirchen, P. (1971). Two-dimensional digital
filtering with Haar and Walsh transforms. Annales de Geophysique 27, 2,
85-104.
15. Cappellini, V., Lotti, F. and Stricchi, C. (1974). Some methods of image
compression by using the fast Walsh transform. Report CM-R. 179-10.48,
Consiglio Nazionale Delle Ricerche, Firenze, Italy.
16. Harmuth, H. F. et al. (1974). Two-dimensional spatial hardware filters for
acoustic imaging. 1974 Proceedings: Applications of Walsh Functions,
Washington D.C.
17. Harmuth, H. F. (1973). Applications of Walsh functions in communications:
State of the art. 1973 Proceedings: NATO Advanced Study Institute, Lough-
borough. “Signal Processing” (Ed. J. W. R. Griffiths). Academic Press, London
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18. Campanella, S. J. and Robinson, G. S. (1970). Digital sequency decomposition
of voice signals. 1970 Proceedings: Applications of Walsh Functions, Washing-
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APPLICATIONS IN SIGNAL PROCESSING
19. Boeswetter, C. (1970). Analog sequency analysis and synthesis of voice signals.
1970 Proceedings: Applications of Walsh Functions, Washington D.C.,
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20. Campanella, S. J. and Robinson, G. S. (1971). A comparison of Walsh and
Fourier transformations for application to speech. 1971 Proceedings: Applica-
tions of Walsh Functions, Washington D.C., AD 727000.
21. Gethoffer, H. (1972). Speech processing with Walsh functions. 1972 Proceed-
ings: Applications of Walsh Functions, Washington D.C., AD 744650.
22. Pratt, W. K. (1972). Walsh functions in image processing and two-dimensional
filtering. 1972 Proceedings: Applications of Walsh Functions, Washington D.C.,
AD 744650.
23. Campanella, S. J. and Robinson, G. S. (1971). A comparison of orthogonal
transformations for digital speech processing. I.E.E.E. Trans. Comm. Tech.
COM 19, 1, 1045-50.
24. Sandy, G. F. (1969) Speculations on possible applications of Walsh functions.
1969 Proceedings: Applications of Walsh Functions, Washington D.C.
25. Shum, Y. Y., Elliott, A. R. and Brown, W. O. (1973). Speech processing with
Walsh-Hadamard transforms. I.E.E.E. Trans. Audio Electroacoustics , AU21 ,
3, 174-9.
26. Gethoffer, H. (1971). Sequency analysis using correlation and convolution. 1971
Proceedings: Applications of Walsh Functions, Washington D.C., AD 727000.
27. Flanagan, J. L. (1972). “Speech Analysis, Synthesis and Perception”. Springer-
Verlag, Berlin.
28. Clark, M. T., Swanson, J. E. and Sanders, J. A. (1972). Word recognition by
means of Walsh transforms. 1972 Proceedings: Applications of Walsh Func-
tions, Washington D.C., AD 744650.
29. Edwards, I. M. and Seymour, J. (1973). Discrete Walsh functions and speech
recognition. 1973 Proceedings: Theory and Applications of Walsh Functions,
Hatfield Polytechnic, England.
30. Wartak, J. (1970). “Computers in electrocardiography”. C. C. Thomas Publica-
tion Inc., Illinois.
31. Start, L., Okjima, M. and Whipple, C. H. (1962). Computer pattern recognition
techniques. Commun. Assoc, for Computing Machinery 5, 10.
32. Milne, P. J., Ahmed, N., Gallagher, R. R. and Harris, S. G. (1972). An
application of Walsh functions to the monitoring of electrocardiograph signals.
1972 Proceedings: Applications of Walsh Functions, Washington D.C.,
AD 744650.
33. Ahmed, N. and Rao, K. R. (1974). Data compression using orthogonal trans-
forms. 1974 Proceedings: Applications of Walsh Functions, Washington D.C.
34. Thomas, C. W. and Welch, A. J. (1972). Heart rate representation using Walsh
functions. 1972 Proceedings: Applications of Walsh Functions, Washington
D.C, AD 744650.
35. Morgan, D. G. (1971). The use of Walsh functions in the analysis of
physiological signals. 1971 Proceedings: Theory and Applications of Walsh
Functions, Hatfield Polytechnic, England.
36. Yeo, W. C. and Smith, J. R. (1972). Walsh power spectra of human
electroencephalograms. 1972 Proceedings: Applications of Walsh Functions,
Washington D.C., AD 744650.
37. Meltzer, B., Searle, N. H. and Brown, R. (1967). Numerical specification of
biological form. Nature 216, 32-6.
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38. Gann, D. S., Seif, F. J. and Schoeffler, J. D. (1972). A quantized variable
approach to description of biological and medical systems. 1972 Proceedings:
Applications of Walsh Functions, Washington D.C., AD 744650.
39. Seif, F. J. and Gann, D. S. (1972). An orthogonal transform approach to the
description of biological and medical systems. 1972 Proceedings: Applications
of Walsh Functions, Washington, D.C., AD 744650.
40. Boeswetter, C. (1972). Modelling the compound action potential of the nerve.
1972 Proceedings: Applications of Walsh Functions, Washington D.C.,
AD 744650.
41. Morgan, D. G. (1971). A study of problems and solutions for a centralized
on-line computer facility. I.E.E. Conference: Computers for Analysis and
Control in Medical and Biological Research, University of Sheffield.
42. Frank, T. H. and Harmuth, H. F. (1971). Multiplexing of digital signals for
time-division channels by means of Walsh functions. 1971 Proceedings: Theory
and Applications of Walsh Functions, Hatfield Polytechnic, England.
43. Harmuth, H. F. (1969). Applications of Walsh functions in communications.
I.E.E.E. Spectrum 6, 11 , 82, 91.
44. Corrington, M. A. (1962). Advanced analytical and signal processing
techniques. Astra document AD 277942. Applied Research, R.C.A., New
Jersey.
45. Moss, G. C. (1971). The use of Walsh functions in identification of systems with
output nonlinearities. 1971 Proceedings: Theory and Applications of Walsh
functions, Hatfield Polytechnic, England.
Appendix I
A Signal processing computer programs
A number of programs for Walsh and Haar processing of a single-
dimensional data series referred to earlier in the text are described here. The
programs are all written in FORTRAN for the ICL 1900 series computer
and form part of a suite of signal processing programs developed at Cranfield
Institute of Technology 1-3 .
B Program summary
FWT A fast Walsh transform sub-routine giving the transformed coeffi-
cients in sequency order.
FFWT Similar to FWT but carries out computation “in-place” and
includes a bit-reversal operation on the output data to retain sequency
order for the transformed coefficients.
FHT A sequency-ordered fast Walsh transform which is slightly faster
than FWT but requires a larger working space.
FRT A modification to the sequency-ordered Walsh transform to give a
particular form of transform, known as the R-transform, which is cyclic
invariant.
HAAR A fast Haar transform sub-routine.
HAARIN A fast Haar inverse transformation sub-routine.
PSDW A Walsh power spectral density program based on the Periodo-
gram model.
c
WALSH TRANSFORM ROUTINES FWT AND FFWT
193
C Fast Walsh transform routines EWT and FFWT
The sub-routines carry out the finite discrete Walsh transform of a time
series, x h consisting of N samples into a sequency series, X n , also of N
samples, viz.
X n =Y xWAUnJ) (A.l)
i=0
where WAL(n, i) is a Walsh function of order n and time index i.
The ordering is carried out to give sequency order (i.e. related to the
ascending number of zero crossings for the function) and a normalisation of
the sequency series by divison by N is not included.
Two routines are given which perform the transformation in a fast
manner, analogous to the fast Fourier transform, where only Nlog 2 iV
mathematical operations are performed compared with N 2 operations if the
equation is evaluated directly. The sub-routine FWT is faster and requires
working storage space. The sub-routine FFWT carries out the computation
“in-place” but is slower since it needs to incorporate a bit-reversal operation
on the data.
A signal flow diagram for FWT was given in Fig. 3.4 and a similar diagram
for FFWT was given in Fig. 3.6. The arithmetic steps required for the
transformation of a time series for N=16 samples are shown in these
diagrams. Solid lines meeting at an intersection indicate addition, whereas
dotted lines indicate subtraction.
Cl Program details
The programs are written in FORTRAN for the ICL 1900 computer, using
all single precision variables. Approximate program length in statements is
24 for FWT and 49 for FFWT, corresponding to word lengths of 137 for
FWT and 24 1 for FFWT.
The data is input via a calling sequence. The results replace the input
sequence and are returned on exit from the routine. Working space required
for FWT is N/2 values, whereas FFWT carries out its calculations “in-place”
and requires no extra storage space.
C2 Calling sequence
CALL FWT (N,X, WORK)
or CALL FFWT (N,X)
where N is an integer constant or variable and must be a power of 2. X is the
name of a real array which on input contains the N data samples and on
194
APPENDIX I
output contains the transformed signal. WORK is an array dimensioned
N/2 and is used for temporary work space.
SUBROUTINE FWT(N,X,Y)
C THIS ROUTINES PERFORMS A FAST WALSH TRANSFORM ON AN INPUT SERIES X
C LEAVING THE TRANSFORMED RESULTS IN X, THE ARRAY Y IS USED FOR WORKING
C SPACE. X AND Y ARE DIMENSIONED N WHICH MUST BE A POWER OF 2
C THE RESULTS OF THIS WALSH TRANSFORM ARE IN SEQUENCY ORDER
DIMENSION X (N) ,Y(N)
N2=N/2
M=ALOG2 (FLOAT (N) )
DO 4 L=1,M
NY=0
NZ=2** (L-l)
NZI=2*NZ
NZN=N/NZI
DO 3 1=1, NZN
NX=NY+1
NY=NY+NZ
JS= (1-1) *NZI
JD=JS+NZI+1
DO 1 J=NX,NY
JS=JS+1
J2=J+N2
Y(JS)=X(J)+X(J2)
JD= JD- 1
Y ( JD) =ABS (X ( J) -X ( J2 ) )
1 CONTINUE
3 CONTINUE
CALL FMOVE (Y(l) ,X(1) ,N)
4 CONTINUE
RETURN
END
SUBROUTINE FFWT(N,X)
C THIS SUBROUTINES PERFORMS AN IN PLACE FAST WALSH TRANSFORM LEAVING THE
C TRANSFORMED VALUES IN SEQUENCY ORDER AFTER BIT-REVERSAL
DIMENSION X (N) ,INT (24)
M=ALOG2 (FLOAT (N) )
DO 10 1=1, M
10 INT (I) =2** (M-l)
DO 4 L=1,M
NZ=2** (L-l)
NZI=2*NZ
NZN=N/NZI
N.z2=NZ/2
c
WALSH TRANSFORM ROUTINES FWT AND FFWT
195
IF (NZ2 . EQ . 0) NZ2=NZ2+1
DO 3 1=1, NZN
JS=(I-1) *NZI
Z=1 . 0
DO 2 11=1,2
DO 1 J=1,NZ2
JS=JS+1
J2=JS+NZ
HOLD=X(JS) +Z*X(J2)
Z=-Z
X ( J2) =X(JS) +Z*X(J2)
X ( JS) =HOLD
Z=-Z
1 CONTINUE
IF (L.EQ. 1) GO TO 3
Z=-1.0
2 CONTINUE
3 CONTINUE
4 CONTINUE
C BIT-REVERSAL SECTION
C THE TRANSFORMED ARRAY IS REARRANGED INTO SEQUENCY ORDER
NW=0
DO 50 K = 1 , N
C CHOOSE CORRECT INDEX & SWITCH ELEMENTS IF NOT ALREADY SWITCHED
NW1=NW+1
IF (NW1-K) 55,55,60
60 HOLD=X (NW1 )
X(NW1)=X(K)
X (K) =HOLD
55 CONTINUE
C BUMP UP SERIES BY ONE
DO 70 1=1, M
II=I
IF (NW.LT.INT (I) ) GO TO 80
MW=NW/INT(I)
MWl=MW/2
IF (MW.GT.2*MW1) GO TO 70
GO TO 80
70 NW=NW- INT ( I )
80 NW=NW+INT (II)
50 CONTINUE
RETURN
END
D Fast Walsh transform routines FHT and FRT
The program FHT forms an alternative method to that of FWT described
previously and is slightly faster. A signal flow diagram was given in Fig. 3.7
which indicated the arithmetic steps required for the transformation of a
time series for N = 16 samples. Solid lines meeting at an intersection indicate
addition, whereas dotted lines indicate subtraction.
A disadvantage of the Walsh transform for some purposes is that it is not
invariant to circular time shifts of the series being transformed. The particu-
lar sequence of calculations, used in FHT, enables a slight modification to be
carried out in order to obtain a transform, known as the R-transform (see
Section III I) which is invariant to circular time shift. This modification
involves the replacement of the subtractive terms obtained during the
196
APPENDIX I
interim calculations by their absolute values. However, unlike the Walsh
transform, the R-transform does not permit the original series to be re-
covered by a second transformation, i.e. it is not its own inverse.
D.1 Program details
The programs are written in FORTRAN for the ICL 1900 computer, using
all single precision variables. Program length for the routines are 26
statements or 134 words.
The data input and output is made via a calling sequence. A working space
of N values is required.
D2 Calling sequence
CALL FHT (N,X,WORK)
or CALL FRT (N,X,WORK)
where N is an integer constant or variable and X and WORK are the names
of real arrays holding N samples. WORK is used for temporary working
space. X contains the input signal on entry to the routine and the trans-
formed series is left in X in sequency order on return from the routine.
SUBROUTINE FHT(N,X,Y)
C THIS ROUTINES PERFORMS A FAST WALSH TRANSFORM ON AN INPUT SERIES X
C LEAVING THE TRANSFORMED RESULTS IN X, THE ARRAY Y IS USED FOR WORKING
c SPACE. X AND Y ARE DIMENSIONED N WHICH MUST BE A POWER OF 2
c THE RESULTS OF THIS HADAMARD TRANSFORM ARE IN SEQUENCY ORDER
DIMENSION X (N) ,Y(N)
N2=N/2
N=ALOG2 (FLOAT (N) )
DO 4 L=1 , M
NY=0
NZ=2 ** (L-l)
NZI=2*NZ
NZN=N/NZI
DO 3 1=1, NZN
NX=NY+1
NY=NY+NZ
JS=(I-1) *NZI
JD=JS+NZI+1
DO 1 J=NX,NY
JS=JS+1
J2=J+N2
Y(JS)=X(J)+X(J2)
JD=JD-1
Y(JD)=X(J)-X(J2)
1 CONTINUE
3 CONTINUE
CALL FMOVE(Y(l) ,X(1) ,N)
4 CONTINUE
RETURN
END
197
D FAST WALSH TRANSFORM ROUTINES FHT AND FRT
SUBROUTINE FRT(N,X,Y)
C THIS ROUTINE PERFORMS A FAST R TRANSFORM ON AN INPUT SERIES X
C LEAVING THE TRANSFORMED RESULTS IN X, THE ARRAY Y IS USED FOR WORKING
C SPACE. X IS DIMENSIONED N AND Y N/2 , WHERE N MUST BE A POWER OF 2
C THE RESULTS OF THIS R TRANSFORM ARE IN SEQUENCY ORDER
N2=N/2
DIMENSION X (N) ,Y(N2)
M=ALOG2 (FLOAT (N) )
Z=-1.0
DO 4 J=l, M
NL=2** (M-J+l)
Jl=2** (J-l)
DO 3 L=1 f Jl
IS= (1-1) *N1+1
11=0
W=2
DO 1 I=IS,IS+N1-1,2
A=X(I)
X (IS+Il) =A+X (1+1)
11 = 11+1
Y (II) =(X(I+1) -A) *W
W=W*Z
1 CONTINUE
CALL FMOVE (Y ( 1 ) ,X(IS+Nl/2) ,Nl/2)
3 CONTINUE
4 CONTINUE
RETURN
END
E Fast Haar transform routines HAAR, HAARIN and HNORM
The subroutine HAAR performs a HAAR transform on a sequence of N
real numbers. A signal flow diagram was given in Fig. 4.3 which indicated the
arithmetic steps required for the transformation of a time series for N= 16
samples. Solid lines meeting at an intersection indicate addition, whereas
dotted lines indicate subtraction. The operation cannot be formed in place
and a working array of length N is required to hold intermediate results. The
subroutine HAARIN performs the inverse HAAR transform (Fig. 4.4) and
a working array is required as in HAAR. No scaling is done by the routines
HAAR and HAARIN and the subroutine HNORM should be used to apply
the scale factors to either the transform or the inverse transform.
El Program details
The programs are written in FORTRAN for the ICL 1900 computer, using
all single precision variables. Program lengths for the routines are 1 12 words
for HAAR, 106 for HAARIN and 92 for HNORM.
The data input is made via a calling sequence. A working space of N values
is required.
198
APPENDIX I
E2 Calling sequence
CALL HAAR (N,X,WORK)
CALL HAARIN (N,X,WORK)
CALL HNORM (N,X)
where N is an integer constant or variable and must be a power of 2. WORK
is a real array used for temporary working space. X contains the input signal
on entry to the routine and the transformed series is left in X on return from
the routine.
SUBROUTINE HAAR(N,X,Y)
DIMENSION X (N) , Y (N)
K-ALOG2 (FLOAT (N) )
DO 1 1=1, K
L=K+1-I
L2=2** (L-l)
CALL FMOVE(X(l) ,Y(1) ,2*L2)
DO 3 J=1,L2
L3=L2+J
JJ=2* J-l
X ( J) =Y ( JJ) +Y (JJ+1)
3 X (L3) =Y ( JJ) -Y (JJ+1)
1 CONTINUE
RETURN
END
SUBROUTINE HAARIN (N , X,Y)
DIMENSION X (N) ,Y (N)
K=ALOG2 (FLOAT (N) )
DO 1 1=1, K
L=2**(I-1)
CALL FMOVE (X(l) ,Y(1) ,2*L)
DO 2 J=1 , L
LJ=L+J
JJ=2*J
JJ1=JJ-1
X ( JJ) =Y ( J) — Y (LJ)
2 X ( JJl) =Y ( J) +Y (LJ)
1 CONTINUE
RETURN
END
SUBROUTINE HNORM (N, A)
DIMENSION A (N)
K=ALOG2 (FLOAT (N) )
A (1) =A (1) /2 . **K
A ( 2 ) =A ( 2 ) /2 . * *K
DO 1 11=2, K
1 = 11-1
WLK=1 . /2 . ** (K-I )
JMIN=2**I+1
JMAX=2**II
DO 2 J= JMIN , JMAX
2 A ( J) =A (J) *WLK
1 CONTINUE
RETURN
END
F WALSH POWER SPECTRAL DENSITY PROGRAM PSDW 199
F Walsh power spectral density program PSDW
Using this program, the power spectrum coefficients, P(k), are determined
in a manner analogous to the Periodogram used in Fourier power spectral
analysis, where the sum of the squares of the real and imaginary coefficients
are taken.
Thus, for the Walsh spectrum, we obtain
P(0) = X 2 c (0, t)
P(k) = X 2 c (k,t)+X 2 (k,t) (A.2)
P(N/2) = X 2 (N/2,t)
fc = 1, 2 . . . (N/2 — 1)
giving N/2 + 1 spectral points.
Here X s (k , t) and X c (fc, t) are the SAL and CAL transform coefficients
and are related to the Walsh transform coefficients by the function relation-
ships
CAL(/c, r) = WAL(2fc, t)
SAL (fc, f) = WAL(2fc — 1, t)
They are obtained from a fast Walsh transform routine FWT described
earlier.
A concept equivalent to degrees of freedom, which defines the analysis
bandwidth used, has been included in this program. Thus, the average value
of the coefficients contained in an even number of pairs of CAL and SAL
coefficients of the same sequency is obtained, so that for D degrees of
freedom
P D (k) = I (X2(*+L-l,f)+X?(*+L-l,f) (A.4)
giving [(N/2)- 1]1/D spectral points.
The resulting spectral points may be optionally smoothed by means of a
Hanning smoothing routine. A flow diagram for this program is given in Fig.
A.l. Some examples of the use of this program were given in Chapter 5.
FI Program details
The program is written in FORTRAN for the ICL 1900 series, using all
single precision variables. The core storage required is 22|k words.
The input medium is cards for parameters and cards or magnetic tape for
the input signal. The output medium is line printer for summary information
and magnetic tape for output values.
(A.3)
F
WALSH POWER SPECTRAL DENSITY PROGRAM PSDW
201
COMPSINWALSH 0 1024
PSDWTESTDATA 0 512
2
1024 2 O 1.0
O 1
O
(a)
NO. OF SAMPLES = 512
THE SAMPLING INTERVAL H = 0.200000E 02 SECONDS
MEAN POWER VALUE = 0.893616E 00
TIME-BASE FOR ANALYSIS = 0.102400E 05 SECONDS
SEQUENCY-BASE FOR ANALYSIS = 0.250000E-01 Z.P.S.
THERE ARE 10 DEGREES OF FREEDOM
THE EQUIVALENT ANALYSIS SEQUENCY BANDWIDTH IS 0.976563E-03 Z P S
THE NO. OF SPECTRAL VALUES OUTPUT =52
THE 0/P BLOCK SIZE = 256
THE DATA IS NOT SMOOTHED
(SEQUENCY IS DEFINED AS 1/2 (AVERAGE NO. OF ZERO CROSSINGS PER SECOND) Z.P.S.)
(b)
Fig. A.2. Summary Print Output for PSDW.
F2 Card Input Data
Card 1 Contains the file name of the input magnetic tape starting in
Column 1, its generation number starting on or after Column 18 and the
number of samples per data block on this tape (Format 2A8, 210).
Card 2 Contains the file name, generation and number of samples per
block on the output magnetic tape (Format 2A8, 210).
Card 3 NIN, where NIN = 0 to terminate the program,
= 1 for card input, = 2 for magnetic tape input.
Card 4 Contains N,M NSMOOTH,FS (Format 3I0,F0.0), where N =
number of data points (ideally a power of 2 ^ 8096) and M = number of
degrees of freedom and must be an even number. NSMOOTH = 0, Output
spectral values that are not smoothed, = 1, Output spectral values are
smoothed using a Hanning routine, and FS = Sampling frequency in Hertz.
IF NIN = 2
Card 5 NSKIP, NBLOK (210)
Where NSKIP = the number of blocks to be skipped on the input magnetic
tape and NBLOK = the number of data blocks to be read for this run. The
input data signal is then read from cards or magnetic tape depending upon
the value of NIN.
Repeat from Card 3 for each run. To terminate the program NIN = 0.
N.B. The card input is double buffered and so the last data card should be
blank. An example of input parameter format is given in Fig. A.2a.
202
APPENDIX I
F3 Output
There is a summary print out on the line printer for each run. An example of
this summary print is given in Fig. A. 2b.
The spectral coefficients, optionally smoothed, are output to magnetic
tape. The number of output points is controlled by the degrees of freedom M
requested for that run and will be equal to
N'-2
M
+ 1
(A.5)
where
N' = N if N is a power of 2 otherwise
N' = the next highest power of 2 > N.
If (N' — 2)/M is not a whole number the result will be rounded down.
MASTER PSDCWALSH
C THIS PROGRAM PRODUCES WALSH POWER SPECTRAL DENSITY COEFFICIENTS
DIMENSION FILI (2) ,FILO(2)
C THE ARRAY X CONTAINS THE INPUT SIGNAL (UP TO 8096 SAMPLES) READ
C FROM CARDS OR MAGNETIC TAPE
COMMON/DATA/X ( 8096 )
COMMON/MAGT/NOUT , NS I , NSO
COMMON /PAR/XMEAN , M , NSMOOTH / MN ,FS
C FILI = FILE NAME OF INPUT TAKE , IGENI = GENE RAT I ON NO. OF INPUT TAPE
C NS I = NO. OF SAMPLES /BLOCK ON INPUT TAPE ( .LE. 1024)
READ (1,5) FILI , IGENI ,NSI
C FILO = FILE NAME OF OUTPUT TAPE,IGENO= GENERATION NO. OF OUTPUT TAPE
C NSO = NO.OF SAMPLES /BLOCK ON OUTPUT TAPE ( .LE. 1024)
READ (1,5) FILO, IGENO, NSO
5 FORMAT (2A8, 2 10)
KFI ,KF0=1
C NIN = 1 FOR CARD INPUT OF SIGNAL X
C NIN = 2 FOR MAG. TAPE INPUT OF SIGNAL X
C NIN = 0 TO TERMINATE PROGRAM
9 READ (1,2) NIN
2 FORMAT (210)
IF (NIN o EQ. 0) GO TO 10
C N = NUMBER OF INPUT DATA POINTS .LE.809 (IDEALLY A POWER OF 2)
C M = NUMBER OF DEGREES OF FREEDOM AND MUST BE AN EVEN NUMBER
C NSMOOTH =0 NO SMOOTHING OF OUTPUT VALUES
C =1 SMOOTHING OF OUTPUT USING HANNING TECHNIQUE
READ (1,1) N,M, NSMOOTH, FS
I FORMAT ( 310 , FO. 0)
GO TO (0.3) ,NIN
CALL CARDIN (N)
GO TO 4
3 GO TO (0,1 1) , KFI
C OPEN INPUT TAPE AND LABEL IT WITH FILI
CALL FILE(3,FILI(1) , IGENI, 0)
KFI=2
II CALL MAGIN(N)
4 GO TO (0,6) ,KFO
C OPEN OUTPUT TAPE AND LABEL IT WITH FILO
CALL FILE (5, FILO(l) , IGENO, 4095)
KF0=2
C MAKE N EQUAL TO A POWER OF 2 IF IT WAS NOT SO ALREADY
L=AL0G2 (FLOAT (N) )
WALSH POWER SPECTRAL DENSITY PROGRAM PSDW
IF (2**L.EQ.N) GO TO 7
L=L+1
N=2**L
7 CALL FFWT (N,X)
CALL PSDCAL(N)
CALL MAGOUT(MN)
CALL PROUT(N)
GO TO 9
10 ENDFILE 5
REWIND 5
REWIND 3
STOP
END
SUBROUTINE CARDIN (N)
COMMON/DATA/X (8096)
READ (1,2) (X (I) ,I=1,N)
2 FORMAT (8F0.0)
RETURN
END
SUBROUTINE FFWT(N,X)
THIS SUBROUTINES PERFORMS AN IN PLACE FAST WALSH TRANSFORM LEAVING THE
TRANSFORMED VALUES IN SEQUENCY ORDER AFTER BIT-REVERSAL
DIMENSION X (N) , I NT (24)
M=AL0G2 (FLOAT (N) )
DO 10 1=1, M
10 INT (I) =2** (M-l)
DO 4 L=1 f M
NZ=2** (L-l)
NZI=2*NZ-
NZN=N/NZI
NZ2=NZ/2
IF (NZ2 . EQ . 0) NZ2=NZ2+1
DO 3 1=1, NZN
JS=(I-1) *NZI
Z = 1.0
DO 2 11=1 , 2
DO 1 J=1 ,NZ2
JS=JS+1
J2=JS+NZ
K0LD=X(JS)+Z*X(J2)
Z=-Z
X(J2)=X(JS)+Z*X(J2)
X ( JS ) =HOLD
Z=-Z
1 CONTINUE
IF (L.EQo 1) GO TO 3
Z=-1.0
2 CONTINUE
3 CONTINUE
4 CONTINUE
BIT-REVERSAL SECTION
THE TRANSFORMED ARRAY IS REARRANGED INTO SEQUENCY ORDER
NW=0
DO 50 K=1,N
CHOOSE CORRECT INDEX & SWITCH ELEMENTS IF NOT ALREADY SWITCHED
NW1=NW+1
IE (NW1-K) 55,55,60
HOLD=X (NWl)
X (NWl) =X (K)
X (K) =HOLD
CONTINUE
DO 70 1=1, M
204
APPENDIX I
IF (NW. LT. INT (I) ) GO TO 80
MW=NW/INT(I)
MWl=MW/2
IF (MW. GT 0 2*MW1) GO TO 70
GO TO 80
70 NW=NW-INT (I)
80 NW=NW+INT(II)
50 CONTINUE
RETURN
END
SUBROUTINE MAGIN (N)
COMMON/MAGT/NOUT , NS I ,NSO
COMMON /DATA/X (8096)
C NSKIP =NO.OF BLOCKS TO SKIP READING
C NBLOK =N0. OF BLOCKS IN DATA SIGNAL
READ (1,1) NSKIP , NBLOK
1 FORMAT (410)
5 IF (NSKIP . EQ. 0) GO TO 9
DO 6 1=1 , NSKIP
6 READ (3)
9 NS=1
DO 7 1=1, NBLOK
NF=NS+NSI-1
READ (3) (X (J) , J=NS ,NF)
V NS=NF+1
RETURN
END
SUBROUTINE PROUT(N)
COMMON/MAGT/NOUT , NS I , NSO
COMMON/DATA/X ( 809 6 )
COMMON/PAR/XMEAN , M , NSMOOTH , MN ,FS
H=1 . O/FS
WRITE (2 , 1) N ,H , XMEAN
T=FLOAT (N) /FS
S=FS/2 o 0
B=FLOAT (M) *FS /FLOAT (N)
WRITE (2,2) T,S ,M,B
WRITE (2,9) MN, NSO
9 FORMAT (5X, 35HTHE NO. OF SPECTRAL VALUES OUTPUT = , I6/5X, 21HTHE 0/P
1BL0CK SIZE = ,15)
L=NSM00TH+1
GO TO (3,4) ,L
3 WRITE (2,5)
GO TO 6
4 WRITE (2,7)
6 WRITE (2,8)
RETURN
1 FORMAT (1H1,4X,15HN0.0F SAMPLES= , I5/5X, 26HTHE SAMPLING INTERVAL H
1= , E12 . 6 , 8H SECONDS, /5X,19HMEAN POWER VALUE = ,E12.6)
2 FORMAT (5X, 2 5HTIME-BASE FOR ANALYSIS = ,E12.6,8H SECONDS , /5X , 29HSEQ
1UENCY-BASE 4 ANALYSIS = ,E12.6,7H Z . P . S . /5X , X , 9HTHERE ARE, 15, 19H
2DEGREES OF FREED0M/5X, 46HTHE EQUIVALENT ANALYSIS SEQUENCY BANDWIDT
3H IS ,E12.6,7H Z.P.S.)
5 FORMAT (5X,24HTHE DATA IS NOT SMOOTHED)
7 FORMAT ( 5X, 20HTHE DATA IS SMOOTHED)
8 FORMAT (5X, 7 9H (SEQUENCY IS DEFINED AS 1/2 (AVERAGE NO. OF ZERO CROSS
1INGS PER SECOND) = Z.P.S.))
END
F
WALSH POWER SPECTRAL DENSITY PROGRAM PSDW
205
SUBROUTINE MAGOUT(N)
COMMON/MAGT/NOUT,NSI ,NSO
COMMON/DATA/X (8096)
NFB=N/NSO
IF (NFB*NSOo EQoN) GO TO 2
DO 1 I=N+1 ,NSO* (NFB+1)
1 X (I)=0.0
NFB=NFB+1
2 NS=1
DO 3 1=1 , NFB
NF=NS-l+NSO
WRITE (5) (X ( J) ,J=NS,NF)
3 NS=NF+1
RETURN
END
SUBROUTINE PSDCAL(N)
COMMON/PAR/XMEAN ,M / NSMOOTH,MN ,FS
COMMON/DATA/X (8096)
AN=N
XMEAN= ( X ( 1 ) / AN ) **2
MN=(N-2)/M+l
MN2=M/2
D=1 . O/FLOAT ( M)
K=2
DO 1 1=1, MN-1
A=0. 0
DO 2 J=1,MN2
A=A+(X(K)/AN)**2+(X(K+1)/AN)**2
2 K=K+2
1 X (I) =A*D
X (MN) = (X (N) /AN) **2
IF (NSMOOTH.EQ.O) GO TO 3
SMOOTH THE OUTPUT USING HANNING TECHNIQUE
A=. 5* (X (1) +X (2) )
DO 4 1=2 ,MN-1
XA=X ( 1-1)
X (1-1) =A
4 A= .25* XA+ . 5 * X ( I ) + . 2 5 * X ( I +1 )
X (MN) =.5* (X (MN-1) +X (MN) )
X (MN-1 ) =A
3 RETURN
END
FINISH
References
1. Beauchamp, K. G., Pittem, S. E. and Williamson, M. E. (1972). Analysing
vibration and shock data. J. Soc. Environmental Engineers , Sept, and Dec.
2. Beauchamp, K. G., Pittem, S. E. and Williamson, M. E. (1973). Computing
facilities for the processing and analysis of random time series. Memo. No. 65,
Cranfield Institute of Technology, England.
3. Beauchamp, K. G. et al. (1974). The BOON system — a comprehensive technique
for time series analysis. 1974 Proceedings: COMPSTAT symposium, University
of Vienna, p. 437-46.
Appendix II
Tables for Modulo-2 addition R ® S (with R max = S max = 125).
206
Table of modulo-2 addition ( R © S)
R = 1 to 25 S = 1 to 42
APPENDIX II
207
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APPENDIX II
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220
APPENDIX II
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APPENDIX II
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Author Index
A
Ackroyd, M. H., 136
Ahmed, N., viii, 18, 60, 70, 80, 82, 83,
85, 104, 105, 113, 182, 184, 190
Alexandridis, N. A., 69, 70, 71
Alexits, G., 20, 39, 83
Algazi, V. R., 164
Andrews, H. C., vii, ix, 11, 70, 83, 114,
164
Carl, J. W., 65, 71, 171, 189
Caspari, K., 60, 70, 83
Chen, W. H., 149, 164
Clark, M. T., 181, 190
Clarke, C. K. P., 68, 163
Cooley, J. W., 39, 70
Corrington, M. A., 188, 191
Crowther, W. R., 114
D
B
Ballard, A. H., 163
Barker, R. H., 164
Barrett, R., 145, 146, 163
Beauchamp, K. G., 39, 70, 71, 1 13, 1 14,
136, 137, 163
Besslich, P. M., 27, 39
Boesswetter, C., 27, 39, 70, 180, 184,
190, 191
Both, M., 99, 114
Box, G. E., 128, 137
Bramhill, J. N., 140, 162
Briotta, D. A., 170, 189
Brown, C. G., 123, 136
Brown, R., 191
Brown, W. O., 68,71, 190
Brubaker, T. A., 36, 39
Burman, S., 99, 114
Butzer, P. L., ix
C
Campenella, S. J., 114, 136, 180, 189
Cappellini, V., viii, 176, 189
Davis, H. F., 11
Decker, J. A., 167, 189
Despain, A. M., 167, 189
Durrani, T. S., 28, 39, 137
Durst, D., 146, 163
E
Edwards, I. M., 182, 190
Elliott, A. R., 68,71, 181, 190
Enomato, H., vii, ix, 149, 163
F
Fano, R. M., 1 14
Filipowski, R. F., 163
Fine, N. J., 18, 141, 163
Fino, B. J., 83
Fino, T., 189
Flament, J., 83
Flanagan, J. L., 181, 190
Fralick, S., 155, 164
223
224
AUTHOR INDEX
Frank, T., 155, 164
Frank, T. H., 188, 191
G
Gallagher, R. R., 190
Gann, D. S„ 183, 191
Gebbie, H. A., 167, 189
Gerardin, L. A., 83
Gethoffer, H., 128, 132, 136, 180, 181,
190
Gibbs, J. E., vii, viii, ix, 17, 18, 19,
25, 38, 71, 85, 89, 96, 113, 141, 163
Golden, J. P., 115, 136
Good, I. J., 52, 70
Gordon, J. A.,' 145, 146, 163
Goulet, R. Y., 17, 38
Griffiths, L. J., 161, 162, 164
Gubbins, D., viii, 83, 136, 172, 189
Guinn, D. F., 43, 70, 136
Gulamhusein, M. N., 109, 111, 114
H
Haar, A., v, viii, 9, 11, 72, 75, 83
Hademard, M. J., vi, ix
Hall, A. L., 124, 136
Hammond, J. L., 83
Harmuth, H. F., vi, vii, ix, 12, 13, 20, 23,
27, 38, 39, 65,71, 115, 136, 144, 155,
160, 163, 164, 178, 179, 188, 189,
190
Harris, S. G., 191
Hart, C. G., 134, 137
Harwit, M., 167, 189
Hatori, M., ix, 163
Higuchi, P. K., 163
Hook, R. C., 136
Hong, Y. K., 164
Hiibner, H., 141, 144, 163
J
Jacobson, L. A., 162, 164
James, S. R, 136
Jenkins, G. M., 128, 137
Johnson, R. S., 83
K
Kabrinsky, M., 171, 189
Kaczmarz, S., vi, ix, 17
Kahveci, A. E., 124, 136
Kak, S. C., 37, 39
Kane, J., 11, 70, 114, 164
Karp, S., 163
Kelly, J. J., 24, 39
Kennedy, J. D., 189
Kennett, B. L. N., viii, 95, 99, 109, 113,
114, 171, 189
Kien-Kwang, C., 1 1
Kitai, R., 16, 33, 39
Klinger, A., 69, 71
Kraus, U., 148, 163
Kremer, H., 73, 76, 83
L
Lackey, R. B., 20, 23, 39, 158, 164
Lally, J. F., 155, 157, 164
Lebedev, N. N., 1 1
Lebert, F. J., 29, 39
Lee, J. S., 29, 39, 116
Lee, T. R., 136
Lewis, A. W., 39
Liedl, P., ix
Lotti, F., 189
M
Manz, J. W., 62, 71
McLaughlin, J. R., 72, 83
Meltzer, B., 183, 191
Meltzer, D., 20, 39
Millard, M. J., 85, 113
Milne, P. J., 182, 190
Morgan, D. G., 182, 184, 190
Moss, G. C., 189, 191
Murray, C. G., 152, 164
N
Nagle, H. T., 115, 136
Natarajan, T., 83, 114
Nightingale, J. M., 39
AUTHOR INDEX
225
O
Ohira, T., 153, 164
Ohnsorg, F. R., 85, 105, 113
Okjima, M., 190
P
Paley, R. E. A. C, vi, ix, 18, 39, 141, 162
Pearl, J., ix
Peterson, H. L., 29, 39
Phillips, P. G., 171, 189
Pichler, F., vi, ix, 43, 70, 85, 97, 113,
141, 163
Polyak, B. T., 85, 113
Pratt, W. K., vii, ix, 11, 52, 70, 83, 114,
118, 126, 134, 136, 137, 149, 153,
164, 180, 190
R
Rademacher, H., v, ix, 11
Rader, C. M., 114
Rao, K. R., 60, 70, 83, 85, 101, 113,
114, 190
Redinbo, G. R., 147, 153, 163
Reed, I. S., 163
Riesz, M., v
Rihaczek, A. W., 160, 164
Risch, P. R., 36, 39
Robinson, E. A., 172, 180, 189
Robinson, G. S., 85, 97, 113, 114, 136,
163, 190
Rosenfield, A., 83
Ross, I., 24, 39
Rushforth, C. K., 20, 39
S
Sakrisin, D. J., 154, 164
Sanders, J. A., 190
Sandy, G. F., 180, 190
Schmidt, F., v, viii
Schollar, I., viii, 83, 136, 172, 189
Schreider, Y. A., 85, 113
Searle, N. H., 191
Seif, F. J., 184, 191
Seymore, J., 182, 190
Shibata, K., ix, 149, 163
Shore, J.E., 80, 81,83
Shum, Y. Y., 68, 71, 181, 190
Siemens, K. H., 16, 33,39
Sloane, J. A., 189
Smith, J. R., 182, 190
Stafford, E. M., 137
Start, L., 190
Stricchi, C., 189
Swanson, J. E., 190
Swartwood, R. V., 66, 71
Swick, D. A., 20, 39
Sylvester, J. J., vi, ix
T
Taki, Y., vii, ix, 163
Tam, L. D. C., 17,38
Thomas, C. W., 185, 190
Thomas, D. W., 82, 84, 137
Thurston, M., 83
Treitel, S., 172, 189
Tukey, J. W., 70
U
Uljanov, P. L., 72, 83
Ulman, L. J., 59, 70, 89
Y
Vandivere, E. F., 117, 136
Vanasse, G. A., 167, 189
W
Wagner, H. J., ix
Walker, R., vii, 68, 71, 136, 151, 163
Walsh, J. L., v, viii, 11
Wartak, J., 190
Welch, A. J., 190
Welch, P. D., 39, 164
Whelchel, J. E., 43, 70, 136
Whipple, C. H., 190
226
AUTHOR INDEX
Wiener, N., viii, ix, 136, 141, 162 Woodward, P. M., 162, 164
Wilkins, B. R., 137
Williamson, M. E., viii, 136 Y
Wintz, P. A., vii, ix, 153, 164
Wishner, H. D., 71 Yeo, W. C., 182, 190
Wisskirchen, P., 83, 136, 189 Yuen, C. K., 18, 29, 30, 39, 85, 97, 113
Subject Index
A
Acoustic image filtering, 1 78
Adaptive coding, 147
Addition relationship, 46, 47, 89, 117
theorem, 76, 93, 98
Algorithm, 63, 95, 105
— Cooley-Tukey, 52, 62, 80
— Haar, 80
—“in place”, 55
Aliasing, 37
“error”, 135
Ambiguity function, 162
Amplifiers— sample and hold, 66, 144
— operational, 65
Amplitude modulation, 160
Analog matched filter, 115
multiplexing system, 144
multipliers, 144
sequency filter, 116
transformation, 64, 148
transversal filter, 115
AND — operation, 26
Aortic blood pressure, 182, 184
Applications, 140
Arguments, 7
Arithmetic autocorrelation, 89, 96
— to dyadic correlation, 96
Array generators, 27
Array — two dimensional, 63
Autocorrelation, 142, 144, 160
— arithmetic, 89
— dyadic, 89, 96
function, 96
Automatic error correction, 146
Average power, 105
B
Bandpass filter, 117, 135, 187
Bandstop filter, 135
Bandwidth compression, 172, 174, 180
reduction, 64
Barker code, 160, 162
BIFORE Walsh transform, 12, 60,
104
spectrum, 105
Binary counter, 151
notation, 7
— order series (see also Natural
order), vi, 18
product-sum, 9
summation, 9
-to-Gray-code conversion, 26
Walsh codes, 146
Bit inversion, vi
— reversed natural order, 20, 25, 52,
62, 63, 104
— reversed order, 18, 55, 63, 68
Block functions, 142
pulse modulation, 160
pulses, 74, 76, 142
series, 5
Boolean function, 183
synthesis, 19, 25
“Brick wall” filter, 119
Burst errors, 154
“Butterfly” diagram, 55
logic, 68
operation, 67
C
CAL function, 12, 14, 16, 41, 42, 87,
126, 166, 172
228
SUBJECT INDEX
coefficients, 44
Cardiac cycle, 183
Cepstrum, 135
Chebychev polynomial, 1 1
Chemical analysis, 140
Chequer-board distortion, 120
Circular functions, 3, 6, 37
function relationships, 46
phase shift, 44, 59
of R transform, 59
time shift, 42, 87
Code — Barker, 160, 162
— binary error detecting, 148
— cyclic, 169
— mean square error, 148
— Reed Muller, 147
Coding, 105
— adaptive, 153
— automatic, 148
error, 150
— optimum, 153
— P.C.M., 149
— sub-area, 177
— sub-optional, 119
system, 148
Coefficient table, 16
Communications, 18
applications, 140
Compiler, 63
Complete function series, 2
infinite interval, 6
Completeness, 6, 7, 75
— of Haar function series, 75
theorem, 5
Complex conjugate, 62
Fourier transform, 98
variable, 14
Walsh transform, 60
Compressed spectrum, 104
Computational efficiency, 180
Conducting targets, 155, 156
Continuous function series, 26, 40
Continued product, 7, 52
Continuous Walsh function, 143
Convergence — of Haar series, 76
— of step function, 81
Conversion tables — WAL and PAL
functions, 39
Convolution, viii, 46, 93, 127, 134, 154
— dyadic, 94
integral, 127
Cooley-Tukey algorithm, 52, 62, 80
Correlation, viii, 93, 127, 134, 154
coefficient, 146
— discrete, 94
function — logical, 96
lag, 94
matrix, 96
COS function, 18
Cost effectiveness, viii
Covariance matrix, 10, 180
Cranfield Institute of Technology, 192
Cross-correlation, 94, 145, 171, 188
Cross-modulation products, 142, 188
talk, 144, 188
Cyclic code, 169
D
D’Alambert’s solution to wave equa
tion, 155
Data compression, 140, 184
Data matrix, 134
Decomposition into spectral compo
nents, 4, 14, 80, 86
theorem, 69
Degrees of freedom, 98
Degree of a polynomial, 1 1
of Haar function, 75, 76, 82
Demultiplexing, 141, 144, 146
Deterministic error, 146
Diagonal matrix, 78, 180
Difference equation, 20, 21
Differentiation of a sinusoidal wave
form, 155
of a Walsh waveform, 155
Digital computer, 69
filtering, 64, 150
hardware, 11
multiplexing, 141
sampling, 36
-to-analog converter, 152
transformation, 68
transmission, 120
Digitising, 149
Dipole radiation, 157
Direct Fourier transform, 124
Discrete correlation, 94
Fourier transform, 41, 49, 124
SUBJECT INDEX
229
function series, 40
Haar transform, 78
Walsh series, 7
Walsh transform, 49, 52, 53
Fourier filtering, 135
Domain — dyadic, 95
— frequency, 13, 126
— sequency, 14, 127
— time, 5, 126
— transform, vii, 5, 6
—Walsh, 64, 93
Doppler-effect, 156
resolution, 158
shift, 162
Dot density plot, 173
Dyadic autocorrelation, 89, 96
convolution, 93
domain, 95
order (see also Natural order), 18
time, 95
time-base, 62
time-scale, 95
-to-arithmetic conversion, 96
translation, 43
symmetry, 166
Dynamic range, 59, 153
E
ECG (see Electrocardiograph),
analysis, 183, 184
signal, 45, 184
Echo principle, 178
Economic time series, 128
Edge detection, 75, 82
EEG (see Electroencephalograph)
analysis, 182
signals, 132
Eigenvalues, 180
Eigenvectors, 10
Electric radiating dipole, 157
Electrocardiograph, 182, 183
Electroencephalograph, 182
Electromagnetic radiation, vii, 140,
155
Error considerations, 36
correction, 146
detecting codes, 148
rate, 146
Even symmetry, 7, 12
Exclusive-OR operation, 26, 27
Exponential multiplying factor, 60
F
Fast Fourier transform, 35, 50, 52
Haar transform, 79, 133
processor, 80
R transform, 59, 89, 184
transform algorithms, vii, 62
Walsh transform, 20, 49, 52, 59, 64,
133, 151
FDM (see Frequency division multi-
plexing)
FFT (see Fast Fourier transform)
FHT (see Fast Haar transform)
Filter — analog sequency, 1 15
bandpass, 116
bank, 127
— “brick-wall”, 119
cut-off frequency, 142
— low pass, 116
— matched analog, 115
matrix, 117, 132
— optimum, 134
— resonant, 115
— sequency, 38, 116
— sideband, 144
— software (see digital filter)
— suboptimal, 119
— transversal, 115
weighting function, 134
weights, 118, 124
Filtering, vii, 4, 10, 18, 89, 140, 167
— acoustic image, 178
a non-stationary signal, 126
—digital, 64, 115, 150
—Fourier, 124, 135
— frequency domain, 123
— Haar transform, 135
— low pass, 120
— matched, 120, 121, 172
— non-recursive, 118
— parallel sequency, 127, 128
— power sequency, 130
—scalar, 119, 123
— sequency, 115, 126, 130
— sequency limited, 119
230
SUBJECT INDEX
—threshold, 64, 120, 122, 171
— transform, 133, 148
— two-dimensional, 120, 126, 133
— vector, 119, 120
—Walsh, 171
—Wiener, 80, 117, 133
Finite approximation, 76
discrete Walsh transform, 41
interval, 5
time signal, 6
FORTRAN implementation, 63
Fourier analysis, viii, 18, 86, 101
— based signals, vii
coefficients, 4, 16
convolution, 93
filter, 118
filtering, 123
methods, vii
series, 4, 13, 15, 155
spectral analysis, 101
spectrum, 99
synthesis, 14, 33
techniques, viii
terms, 34, 36, 76
to Walsh conversion, 49
transform, 42, 96, 124, 166, 182
— discrete, 42
— complex, 98
Frequency, 13, 14, 18
division multiplexing, 141, 144
domain, 13, 35
domain filtering, 123
multiplexing, 141
FRT (see Fast R transform)
Function — ambiguity, 160
— Boolean, 183
—CAL, 12, 16,41,42, 87, 126
— cosine, 18
— filter weighting, 134
generation, 26
— Haar, 10
horizontal-shaped, 81
— impulse response, 127
ordering, 17
—PAL, 18
— Rademacher, 2, 6, 7, 18, 19, 22, 47,
144, 151
— saddle-shaped, 81
—SAL, 12, 16, 37, 41, 42, 87, 126
— sequency limited* 127
— sine, 18
— transfer, 135
— vertical-shaped, 81
— WAL (see Walsh Function)
—Walsh, 14
— weighting, 111
FWT (see Fast Walsh transform)
G
Gas chromotography, 189
Guassian function, 153
Gegenbauer polynomial, 1 1
Generalised transform, 60
Wiener filtering, 115, 117
Gibbs derivative, vii
Gray code, 23, 26, 29
Grey level resolution, 148
H
Haar and Walsh function relationship,
76
function, 9, 10, 72
— definition, 72, 75
— degree of, 74, 75, 82
—order of, 73, 75, 82
— like series, 80
power spectrum, 82, 104, 105
series, 76
series — MSE, 76
transform, vii, 166, 171, 172, 184
— discrete, 79
— modified, 80
— two-dimensional, 80
Hademard matrix, 19, 24, 52, 169, 184
Half-adder, 38
Hardware function generation, 26
transformation, 64
HAR 'function (see Haar function)
Harmonic components, 109
frequency content, 47
function, 20
motion, 101
number, 18
Harmuth array generator, 27
phasing, 20, 22
Heart rate determination, 185
SUBJECT INDEX
231
Hermite polynomial, 11, 143
Hertzian radiation, 157
Hidden line suppression, 109
High-level compiler, 63
Holography, 140
I
ICL computer, 63, 192
Identity matrix, 52
Image analysis, 115
coding, 105, 140
enhancement, 126, 172, 174
filtering, 120, 126
— acoustic, 178
matrix, 179
processing, 154, 171
Processing Institute, 176
transmission, 18, 69, 82, 140, 148
Impulse response. 111, 127, 154, 189
Incomplete function sets, 5, 21
series, 5, 7
Infinite frequency function, 6
Infra-red spectrometer, 167
“In-place” algorithms, 55, 63
Institution of Electrical Engineers, viii
Integrated circuit technology, 144
Interferometer — infra-red, 167, 171
— Michelson, 171
Intermodulation products, 185, 187
Interval — complete, 6
— finite, 6
— semi-infinite, 6
Inverse transform, 12, 62, 68, 152
J
Jacobi polynomial, 11
K
Karhunen-Leove series, 10, 180
transform, 10, 54, 118, 180
Kernel (see also Complex conjugate),
42, 62
Kronecker ordering, vi, 24, 25
product, vi, 24, 60
L
Laguerre polynomial, 1 1
Lag value (see Correlation lag), 93
Lebesgue integration, 40
Legendre polynomial, 11, 143
Left adjustment, 22
Level of quantisation, 34
Lexicographic ordering, 25
Linear sequency order, 55
Logarithmic operation, 135
Logical differential calculus, vii
equation, vii
correlation function, 96
Logic circuitry, 140
Low-pass filter, 116, 119, 135
Luminosity level, 175
M
Magnetic dipole, 157
Majority logic multiplexing, 146
Mapping matrix, 78
Mark-space ratio, 6, 7, 101
Mask pattern-orthogonal, 167
Matched filter, 115, 172
filtering, 120, 123, 172
Mathematical modelling, 140
Matrix — covariance, 180
data, 134
— diagonal, 79, 134, 180
— filter, 134
— image, 179
— mapping, 78
multiplication, 126
relationship, 80
— scalar, 118, 134
transformation, 120
translation, 94
— vector, 118, 134
—Walsh, 70
Mean-square approximation error, 2
error, 5, 14, 36, 76, 118, 134, 148,
150, 154, 180
Medical signal processing, 182
Michelson interferometer, 171
Modelling of biological systems, 183
Modulo-2 addition, 22, 23, 27, 38, 43,
46, 89, 94, 95, 185, 188
arithmetic, 38
232
SUBJECT INDEX
Morphological patterns, 183
M.S.E. coefficient, 36
Multiplexing communication system,
vii
Multiplexing, 27, 140, 141
— analog, 144
— digital, 141
— frequency division, 141
hardware, 141
— majority logic, 146
— of binary signals, 144
— sequential, 74
— time division, 141
Multiplicative property, 27
Multiple error correction, 147
N
NASA, 171
National Electronics Conference, viii
Natural order, 18, 19, 22, 52, 68
Natural-ordered series, vi, 18
N-bit string, 26
Non-conducting targets, 155, 156
Non-linear applications, 185, 189
filtering, 134
quantising, 153
transformation, 185
Non-normalised set, 2
Non-recursive filtering, 117
Non-stationary analysis, 109, 127
signals, 119, 126, 181
waveforms, 6
Normalisation, 53
Normalised sequency,
series, 14
set, 9
time base, 20
Walsh function, 12
Normality, 6
Normal order (see Natural order)
Nyquist criteria, 135
interval, 35
O
Octave analysis — sequency, 105
spectral decomposition, 109
Odd-harmonic spectrum, 82, 104, 182,
184
Odd symmetry, 7, 12, 22
Open interval, 14
Operational amplifiers, 65
Optimum coding, 153
filter, 134
Order — binary, 18
— dyadic, 18
— Haar function, 75, 82
— harmonic, 18
— Harmuth, 18
— Kronecker, vi, 24, 25
— lexicographic, 25
— natural, 18, 68
— normal, 18
— Paley, 18
— sequency, vi, 13, 17, 19, 27, 68
—Walsh, 17
— Walsh-Kaczmarz, 17
Ordered form, 17
— phase of, 18
Orthogonal block pulses, 5
functions, v, 1, 5, 10, 72
interval, v, 2
mask pattern, 167
polynomial, 11
property, 3, 127
series, 1, 2, 3
— incomplete, 2
set, v, 1
system, 9
transformation, 54, 60, 148, 150
matrices, v
Orthogonality, 1, 4, 6, 7, 10
Orthonormal set, 2, 7, 9
P
Paley order, 18
Paley-ordered function, 18
PAL function, 18, 11, 31
system, 148
Parabolic reflector, 157
Parallel programmable generators, 27,
28
sequency filtering, 127, 128
transformation, 69
Parsevals equation, 2, 64
theorem, 5, 42, 75, 98
SUBJECT INDEX
233
Partial transformation, 69, 80
Partition method, 126
Patient monitoring, 182
Pattern recognition, 111, 140, 171
PCM (see Pulse-coded modulation)
Peak-to-average power ratio, 144
Perceptual redundancy, 154
Periodogram, 82, 89, 98, 114
Phase of the ordered set, 18
Phase-shift keying, 188
Physiological data, 182
Picture quality, 153
reconstruction, 148
Picture transformation, 69
Pitch synchronous spectra, 1 8 1
Polarity symmetry, 156
Polynomial, 1 1
Power law operation, 135, 186
sequency filtering, 130
spectral density,
spectrum, 44, 86, 172
coefficients, 98
— Haar, 82, 105
— odd harmonic, 184
— sequency limited, 85
Predictive analysis, 185
Programmable generators, 38, 68, 117
Pseudo-random binary sequence, 189
Pulse coded modulation, 101, 147, 148,
149, 153, 180
stretching, 122
width, 123
Q
QRS cycle, 183, 184, 185
Quadrupole radiation, 156
Quantisation, 149
error, 153
level, 34, 35
Quantised signals, 144
Quantising noise, 154
— non linear, 153
R
Radar, vii
ambiguity function, 162
processing, 140, 158
pulse, 159
transmission, 160, 161
Rademacher function, 2, 6, 7, 18, 19, 22,
47, 144, 151, 185
products, 20
“Read only” memory, 147
Reception of Walsh electromagnetic
radiation, 157
Reconstructed image, 174
waveform, 14
Rectangular waveform — harmonic con-
tent, 47
Recursive algorithm, 66
Reed-Muller codes, 147
Replica carriers, 142
Resonant filter, 115
Resolution angle, 179
of a receiver, 162
of a Walsh radiated wave, 158
Riesz-Fischer theorem, 2
“R” transform, 59, 89, 184
energy spectrum, 86
Running transform, 109
S
SAL coefficient, 44
function, 12, 14, 16, 36, 41, 42, 87,
126, 166, 172
Sample-and-hold amplifiers, 67, 116,
119,144
Sampling interval, 34, 35
period, 37
theorem, 37
window, 181
Scalar filtering, 119, 123
matrix, 119
Scaling, 62
factor, 42
Seismic disturbance, 45, 120, 128
events, vii
Seismology, 140
Semi-infinite interval, 6
Sequency amplitude spectrum, 98
bandwidth, 36, 123
based filtering, 119
coefficient, 43
— definition of, 12
234
SUBJECT INDEX
domain, 14, 35,82, 116
filter, 38
filtering, 115, 119, 127, 128, 130
limited spectrum, 85, 101
matrix, 182
multiplexing, vii
normalised, 47
number, 23
octave analysis, 105
order, vi, 13, 17, 19, 27, 68
ordered algorithm, 62
transform, 52, 55
range, 27
spectrum, 47, 96
time plot, 111
Sequential multiplexing system, 74, 144
Serial programmable generators, 27, 28
Series representation, 10
Shift index, 43
Short-term spectral analysis, 109, 126
“Shuffling” matrix, 97
Shift theorem, viii, 3, 17, 46, 76, 89, 93
Side-band filter, 144
Side-lobes, 47
Signal detection, 185
flow diagrams, 20, 54
processing, 18, 26, 62, 140, 166
processing — medical, 182
-to-noise ratio, 123, 172
time function, 1
Sine function, 18
Sine-cosine functions, 2, 3, 7, 13, 18, 42,
93
series, 13
Single-dimensional transformation (see
Transformation)
Sinusoidal carrier, 160
function, 43
series, 47
waveform, 3, 14, 86
Slant transform, 80, 149
Software generation, 26
Sparse matrix, 60, 97
Spectral analysis, 18, 85, 109, 126, 148,
167
components, 13
decomposition, 4, 14, 86, 166
Imaging Inc., 171
points, 98
Spectrometer — infra-red, 167
Walsh, 167
Spectroscopy, 140, 167
Spectrum — compressed, 104
Speech processing, 180
recognition, 181
signals — non-stationarity of, 181
synthesis, 105, 180
Square waves, 72
Stacked magnetic dipoles, 157
Stair-step approximation, 185
Statistical analysis, 140
redundancy, 153
Step-shaped signal, 144
Sub-area coding, 176
Suboptimal filtering, 119
transform, 180
Subroutine, 63
Switching functions, 140
Symmetry of Walsh function, 12
relationship, 12, 46, 63
Symmetrical transform, 41
Symposium — Hatfield, viii
— Washington, viii, 140
Synchronisation, 144
Synthesis, 14, 34
T
Taper window. 111
TDM (see Time division multiplexing)
Television transmission, 65, 148, 179
Threshold criteria, 33
filtering, 64, 120, 171
level, 35, 36, 120, 153
limit, 146
ratio v/c, 155
Time base, 7, 20, 21, 73, 86, 128
— dyadic, 62
delay function, 93, 162
division multiplexing, 141
domain, 5, 184
— limited function, 85
signal, 6
matrix, 181
sequency domain, 126
series, 13, 128
— economic, 128
shift — circular, 42
Transfer function, 135, 172
SUBJECT INDEX
235
Transform, 12
— BIFORE, 12, 60, 104
coefficients, 59
— comparative timing, 63
complex Walsh, 60
— direct, 124
— fast Fourier, 35, 52
— Haar, 79
—Walsh, 20, 52, 59, 64
— finite, 41
— finite discrete Haar, 78
— finite discrete Walsh, 41
filtering, 148
—Fourier, 98, 124, 146, 166, 182
— generalised, 60
—Haar, vii, 166, 171, 184
— “in place” algorithm, 55, 63
— inverse, 62, 68, 124
— one dimensional, 63, 80
operator, 41, 93, 109
— orthogonal, 54, 60
pair, 40, 41
products, 46
programming, 63
— R, 59, 184
— running, 109
— slant, 80, 149
spectroscopy, 140
— symmetrical, 41
— sub optimal, 180
—Walsh, vii, 42, 60, 166
Transformation, 63
— analog, 65, 148
— digital, 64, 68
— hardware, 64
— hybrid, 65
matrix, 120
— non-linear, 185
— parallel, 69
— partial, 69
— picture, 69
time, 72
— two-dimensional, 63, 69, 80, 81
Transformed domain, 5, 6
matrix, 52
Transient signal, 99
Translation matrix, 94
Transmission rate, 148
Transversal filter, 115, 127
Trapezium rule, 41
Trend determination, 128
Trigonometric multiplying factor, 62
Two-dimensional transformation, 63,
69, 80, 81
array, 63
filtering, 115, 120, 126, 132, 135, 150,
179
spectra, 109
y
Variable-word length, 147
Vector filtering, 119
matrix, 118
Ventricular fibrillation, 183, 185
Vocoder, 105
W
WAL function (see Walsh function)
Walsh and Haar function relationship,
76
Walsh carrier, 161
codes, 146
discrete convolution, 93
domain, 64, 93
filtering, 89, 171
function, 7, 10, 12, 14, 17, 31
— amplitude, 9
— coefficient, 127, 128
— continuous, 143
— definition, 20, 22, 25
— derivation from Boolean synth-
esis, 25
— from difference equation, 20
— from Hademard matrix, 24
— from Rademacher function, 22
— discrete, 7
— electromagnetic radiation of, vii,
140, 155
expansion, 47
— generation, 26, 69
— multiplication property of, 26
— normalised, 12
— recursive definition, vi
relationships, 46
— symmetry of, 1 2
— Kaczmarz order, vi, 17
236
SUBJECT INDEX
matched filtering, 172
matrix, 70
order, 17
radiation, 155
reconstruction, 36
related masks, 167
series, 10, 12, 14, 189
— discrete, 7
spectral analysis, 85, 101
spectrum, 89, 96, 99
spectrum — pitch synchronous, 181
synthesis, 14
terms, 34, 76
to Fourier conversion, 49
transform, vii, 42, 60, 166, 171
algorithm, 20
— BIFORE, 12, 60, 104
characteristics^ 49
definition, 40
discrete, 41
—fast, 20, 52, 59, 64
of channel statistics, 148
spectrometer, 167
transformation, 40
waveform, 155
waveform synthesis, 33
Wavelength-to-aperture ratio, 178
Weighting function, 111, 134
Weights — filter, 124
Wiener filtering, 80, 115, 117, 124, 133
Khintchine method, 89
theorem, 85, 89, 96
process, 119
Word length, 153, 175, 177
recognition, 181
World Organisation of General Systems
and Cybernetics, viii
Z
Zero crossing, 7, 13, 17, 82, 86, 104, 185
order component, 117
— order hold, 36
phase shift, 44
sequency term, 64
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