APPLICATION OF DISCRETE WALSH FUNCTIONS TO DIGITAL FILTER TECHNIQUES
Norman Clare Meek
E
Monterey, California
Application of Discrete Walsh Functions to Digital Filter Techniques
by
Norman Clare Meek
Thesis Advisor S. R. Parker June 1972
Approved Ion. pub-tic >io£zat>c; dlitiAhution untimlteA.
Application of Discrate Walsh Functions to
Digital Filter Techniques
by
Norman Clare Meek Lieutenant, United States Navy B.S., University of Colorado, 1965
Submitted in partial fulfillment of the requirements for the degree of
ELECTRICAL ENGINEER
from the
NAVAL POSTGRADUATE SCHOOL June 1972
ABSTRACT
Methods of generating and properties of Walsh functions are discussed with emphasis on discrete Walsh functions.
Hardware and software versions of two methods of Walsh func- tion generation are shown. The Walsh transform is explored and hardware and software realizations of a fast Walsh trans- form algorithm, particularly applicable to hardware, is in- troduced. Computer programs containing the various algo- rithms are included. A discussion of the feasibility of the use of Walsh functions to produce a digital filter with a desired frequency response is presented.
TABLE OF CONTENTS
I. HISTORICAL BACKGROUND AND SCOPE OF THESIS 6
II. PROPERTIES OF WALSH FUNCTIONS 10
III. GENERATION OF WALSH FUNCTIONS 15
IV. THE WALSH TRANSFORM 18
V. DIGITAL FILTER CONSIDERATIONS 24
VI. RESULTS AND CONCLUSIONS 31
APPENDIX A - WALSH FUNCTIONS AS PRODUCTS OF RADEMACHER FUNCTIONS 33
APPENDIX B - FREQUENCY SPECTRUM (FOURIER TRANSFORM) OF THE CONTINUOUS WALSH FUNCTIONS 36
COMPUTER PROGRAM 1 - HADAMARD MATRIX GENERATION 55
COMPUTER PROGRAM 2 - WALSH TRANSFORM AS HADAMARD MATRIX PRODUCT 60
COMPUTER PROGRAM 3 - FAST WALSH TRANSFORM 68
BIBLIOGRAPHY 7 6
INITIAL DISTRIBUTION LIST 78
FORM DD1473 79
LIST OF ILLUSTRATIONS
1. The First 16 Harr Functions, x t (t) 38
2. The First 5 Rademacher Functions, R(k,t) 39
3. The First 16 Walsh Functions, W(k,t) 40
4. Discrete Walsh Functions of Order 16 (Sequency Ordered) 41
5. Walsh Function Generator (as Rademacher Function Products) 42
6. Generation of W-.,(ll,t) 43 lb
7. Serial In/Parallel Out Fast Walsh Transform Realization 44
8. Fast Walsh Transform Realization Parallel In/Parallel Out 45
9. Discrete Walsh Functions of Order 16 (Binary Ordered) 46
10. Frequency Spectrum of the First 8 Discrete Walsh Functions, W..,(k,j) 47
11. Frequency Spectrum of the Second 8 Discrete Walsh Functions, W (k,j) 48
12. Walsh Function Resonator 49
13. Walsh Function Resonator for W„„(l4,j) 50
14. Cascade Version of Walsh Resonator 51
15. Cascade Version of W^^(l4,j) Walsh Resonator 52
16. Narrow Bandpass Filter Using Walsh Resonators 53
17. Frequency Response of Narrow Bandpass Filter Using Walsh Resonators 54
ACKNOWLEDGMENT
The assistance and advice of William J. Dejka, Modular
Concepts Division Head, Naval Electronics Laboratory Command, is gratefully acknowledged.
I. HISTORICAL BACKGROUND AND SCOPE OF THESIS
A pair of functions are said to be orthogonal if the in- tegral of the product of those functions over all space vanishes. Systems of orthogonal functions have been avail- able for many years. At the beginning of the twentieth cen- tury, there were many well known and useful orthogonal functions: the trigonometric functions which occur in Fourier series; orthogonal polynomials such as those of Legendre,
Hermite, and LaGuerre; Bessel's functions; the S turm-Lioville series; and other special functions. The general theory in- cluding all such systems of functions was essentially devel- oped prior to 1910 and it was shown that a set of such functions is by no means exceptional but can be produced at will.
Alfred Harr [Ref. 10] formulated an orthogonal system of functions in 1909 (now called the Harr functions)' which essentially take on only two values. An exemplary set of
Harr functions is shown in Figure 1. A formal expansion of an arbitrary continuous function in terms of Harr functions converges uniformly to the given function. This combination of convergence and bi-valued properties had not been pos- sessed by any other orthogonal set known to that time.
Independently, and at about the same time in early 1922,
H. Rademacher [Ref. 20] and J.L. Walsh [Ref. 24J, each exhi- bited a set of orthogonal functions taking on the values + 1 on the interval [0,1]. The Rademacher set, [Figure 2], was not complete as other non-trivial functions exist which are orthogonal to all of the members of the set. The Walsh
functions, {Figure 3] , are a complete set and exhibit similar" ities to the trigonometric functions in many of their proper- ties. The Walsh functions are discussed in detail in the remainder of this thesis.
Very little work was done with these various sets of bi- valued functions until the digital computer became a useful tool of the scientific community. Because of the dual value characteristics of the Harr, Raderaacher, and Walsh functions, the applicability to the binary configured machines seemed natural. A substantial amount of effort has been directed into application areas during the past several years and pro-
gress has been made in many fields which in ' the past have only been associated with trigonometric functions. Some of the Walsh function application areas of interest include: multiplexing, pattern recognition, image processing, electro- magnetic radiation, and digital filtering. This paper will primarily discuss the area of digital filtering although many of the results are applicable to other areas of interest.
Previously, analysis of digital filters and implementa- tion has been restricted to Laplace and Fourier methods and hence to trigonometric relationships. The implementation of a digital filter based on Walsh functions offers definite advantages unique to a two valued function but also has dis- tinct disadvantages inherent in the use of a set of discon- tinuous functions to describe a continuous function. Efforts expended thus far in the area of Walsh function digital filtering appear to be primarily concerned with obtaining the expansion of a given function in terms of the Walsh functions and the recombination of the weighted Walsh func- tions to obtain the filtered and reconstituted function. In this thesis the manipulation of the Walsh function represen- tation of the given function will be treated and compared to standard Fourier methods. It will be shown that in general there is no advantage to using Walsh functions for digital filtering when a specific frequency response is desired, but that for certain frequency responses, Walsh functions do give a decided advantage over the Fourier methods.
Section II presents some of the properties of Walsh func- tions including a modified notation method which, it is believed, is clearer and easier to use in mathematical re-
lationship s .
Sections III and IV show methods of Walsh function gen- eration and Walsh transform techniques respectively. Algo- rithms are presented which are adaptable to either hardware or software and will either generate a set of Walsh functions or will take the Walsh transform of a given function. Serial or parallel input of a sampled function is provided for.
Section V considers the use of Walsh functions to produce a digital filter with a desired frequency response. The fre- quency spectrum of discrete Walsh functions is derived, and it is shown that in general it is as complex to combine these to achieve a desired frequency spectrum output as is the case in the Fourier domain.
II. PROPERTIES OF WALSH FUNCTIONS
Walsh functions can be defined oyer the interval 10,1], in the following manner;
1 °-t - 1 w(ow u,c;t) - Cl ' ^ 0, otherwise k+ W(2k+p,t) = W(k,2t)+(-l) Pl^W(k,2t-1) (II-D
p = 0,1 -
k = 0,1,2. . .
When Walsh functions are used for digital filtering, the input function is normally sampled at some rate as are the
Walsh functions. In order to discriminate between the dis- crete (sampled) and the continuous functions and for ease in mathematical manipulation, the following notation is used:
Continuous function notation:
W(k,t)
k = 0,1,2,... index of Walsh function
O^t^l independent variable
Discrete function notation:
W (k,j) N k = 0,1,2,..., N-l index of Wash function
N = a power of 2 order of set of Walsh functions
j = 0,1,2,..., N-l independent variable
The discrete Walsh functions are formed from the contin- uous = functions by sampling WO,t) at t (j /N) , Ci • e . , from the right). Figure 4 shows the set of order 16 discrete
Walsh functions arranged as a 16 x 16 matrix.
10
:
Several other forms of notation have been used for Walsh functions. Some of the most common of these are tabulated in
Table 1 for comparison and cross reference purposes.
Discrete Continuous Walsh Corrington Harmuth j=0,l..,N-l 0-6t£l Q^t^l O^t^l jk i. ^" Vft ^ [Ref .24] [Ref. 7] 2 ~ 2 [Ref. 9]
W (0,j) W(0,t) (t) (t) Cal(0,6 8 *000
w (i,j) W(l,t) (t) (£) Sal(l,9 8 x *001 W (2,j) W(2,t) (t) Cal(l,6 g *011
W (3,j) W(3,t) (t) Sal(2 , B g *010 W (4,j) W(4,t) (t) Cal(2,6 8 *110 Wg(5,j) W(5,t) lu
Table 1. Notations for Walsh Functions
These functions form a complete, orthonormal set over the interval [0,1]. The implication of this property is that any absolutely integrable function defined over the interval can be expanded in a series of Walsh functions analogous to the Fourier expansion of such a function.
That is: N-l f(t) = I a W(k,t) CII-2) K k=0 where
a, = / f(t) W(k,t) dt (II-3) k
Assuming f(jx) is a sampled representation of f(_t), where the sample period, T * 1/N; then a. = for K.
11
k ^ N and equations (II-2) and (II-3) reduce to:
= a (k,j) f(jT) I WN CH-4) j=0 K W and N-l a, = 1/N 2 f(jT) W (k,j) (H-5) n k j=o
If the function f(t) or f(ji) is defined over the inter- val [0,T], a simple change of scale, t, = t/T will allow the use of the above equations to find the Walsh representation of the function over [0,1].
Sequency is defined as one-half the number of zero cross ings per unit time. The number of zero crossings and hence the sequency of a given Walsh function can be obtained by k+1 k+ (-p z - . (II-6)
where z is the sequency
and k is the index of the Walsh function
The set, {a,}, therefore represents the sequency spectrum of f(t) in the same sense that a set of Fourier coefficients represents a frequency spectrum. It should be noted that
the Walsh functions defined in this thesis are ordered ac-
cording to increasing sequency. This is the definition as used by Walsh in his original paper [Ref .24] and is equiva-
lent to that used by Earmuth iRef. 9J . Another method of
defining Walsh functions is as products of Rademacher func-
tions. When defined in this way, they are not ordered
according to increasing sequency but are what is termed binary ordered. Appendix A shows this method of defining
12
Walsh functions and derives algorithms for converting from one method of ordering to the other. The original sequency ordering of Walsh seems to be best suited to the purposes of communications engineering.
Another property of Walsh functions can be seen from their internal symmetries. A product of two Walsh functions, if they are both even or both odd about y, will be even. If one is odd and one is even, their product will be odd. This is also true with the symmetries about t"> "o> etc. Since this product function is completely described by its symmetry
properties about the points 2 , i=l,2,3,..., it must be concluded that this product is also a Walsh function. More- over, the index of the product is the modulo 2 sum of the indices of the two factors. Thus:
W(k, t) 'W(m, t) - W(kem,t)
where <& indicates addition modulo 2.
= i.e. W(2,t) -W(3,t) W(10 , t) -W(ll , t) 2 2 = w(io en ,t) 2 2 - w(oi ,t) 2 - W(l,t)
Since the reciprocal of a Walsh function is the same function,
WCk,t) = 1/W(k,t) the Walsh functions form a closed set under multiplication or division.
A useful property of discrete Walsh functions is that
W (j,k) = W (k,j), N N that is the matrix representation, Figure 4, is symmetric.
13
.
A Hademard matrix is an N x N matrix, each of whose ele- ments are +1 or -1 and such that the rows of the matrix are orthogonal to each other using the ordinary vector inner pro' duct. Since the discrete Walsh function representation con- forms to this definition, it is a symmetric Hadamard matrix.
If H denotes the matrix, then:
T T H'H = NI where H is H transpose
N is the order and
I is the identity matrix T but H is symmetric, so H = H , and therefore, HH = NI . This important property will be used in Section IV, The Walsh
Trans form
14
.
III. GENERATION OF WALSH FUNCTIONS
Walsh functions can be generated in several different ways, all of which depend upon the mathematical properties of the functions as discussed in Section II. Figure 5 shows a hardware realization of a Walsh function generator designed by Harmuth [Ref. 9J. This generates Walsh functions as pro- ducts of Rademacher functions as discussed in Appendix A.
Swick [Ref. 22 and 5] showed that by using the properties of symmetry, a particular Walsh function can be written down directly from left to right without making use of any preced- ing function. This method is to expand the index of the desired function into binary.
i.e. k=b b n ...b n ,N=2 , n n-1 1 and let U(k,t) = 1 in the interval [0,1/N]. Then for j = n, 2~^ n-1, ..., 1 if bj = 1(0) then W(k,t) is odd(even) about when it is considered as a function over the interval
^ J [0,2 ']. Figure 6 shows the generation of W(ll,t) by this method. Eleven in binary is 1011 and n = 4. Letting
W(ll,t) = 1 in the interval [0,1/16], then b, = 1 indicates
= an odd function about 1/16 on the interval [0,1/8] ; b „ indicates an even function about 1/8 over the interval
10,1/4]; b„ = 1 indicates an odd function about 1/4 oyer
[0,1/2] ; and b, = 1 indicates an odd function over the entire interval 10,1]
Peterson, lRef.18], devised a method similar to Swicks in which the index is expanded into reflected binary (Grey code)
15 i.e. k = g g .....g-., WCk,t) = 1 in the interval I0,1/N] n n-1 X and then for j - n, n~l,...,l; WCk,t) in the interval
r o J 2*~ ~ n "'" " 1 *"* ^ i "* £>8 i J is equal to (-1) n . w (k, t ) in the interval
-JJ [0,2 J . Referring again to Figure 6 for the generation of
W(ll,t), and using the algorithm from Appendix A for conver- sion from binary to reflected binary, 11 in reflected binary is 1110. Letting W(ll,t) = 1 in the interval [0,1/16], then
(-1) 84 = -1 and W(ll,t) in the interval [1/16,1/8] equals
-W(ll,t) in the interval [0,1/16]. Similarly, W(ll,t) in
[1/8,1/4] equals -W(ll,t) in [0,1/8]; W(ll,t), 1/4 t 1/2 equals -W(ll,t) for t 1/2; and the function from t = 1/2 to t = 1 is equal to W(ll,t) over [0,1/2].
It should be noted that by using Swick's method but ex- panding the index in reflected binary or by- using Peterson's method with the index expanded in binary, the Walsh functions generated are in binary order instead of sequency order.
Figures 7 and 8 show hardware versions of a fast Walsh transform algorithm which will be discussed in further detail in Section IV. A single pulse input into the device of
Figure 7 or a pulse shifted through the input register of
Figure 8, will produce as an output the Walsh functions in- stead of the sequency spectral components.
Computer program 1 utilizes a set of algorithms [Ref .11,
12, and 13J to form a Hadamard matrix of Walsh functions in either binary or sequency order. These algorithms compute the value of the Walsh function at each point without refer- ence to previous points or previous functions. This is a
16 relatively slow method of function generation since it does not take advantage of the symmetry properties of the Walsh
f unc t ions .
Computer program 2 also generates a Hadamard matrix in sequency order using an algorithm based on Peterson's method of Walsh function generation.
17
IV. THE WALSH TRANSFORM
It was shown in Section II that a function can be ex- pressed in a series based on Walsh functions analogous to a
Fourier series. In this series, a set of coefficients,
{a, }, represented the sequency spectrum of the given function
The discrete or sampled form of the Walsh transpose became:
N-l = a, 1/N I f (jT) -W N (k,j) (iv-i) k j=0
If the samples of f (jx) are arranged in a column matrix,
f (0) f( T) f (2T) F =
f [(N-l)T] and premul tiplied by the Hadamard representation of the Walsh functions, equation (IV-1) is realized in matrix form as:
W = 1/N HF (IV-2) where W is a column matrix of the spectral coefficients.
Because of this property, the finite Walsh transform is often called the Hadamard transform.
From the properties of H as described in Section II, it can be readily seen that:
HW = 1/N HHF = NI/N F = F (IV-3) or that with the exception of a constant multiplier, 1/N, the same procedure is used to obtain both the transform and the inverse transform.
18
:
Some of the properties of the finite Walsh transform
[Ref. 15 J , are as follows;
if fCjT)^ >W(f)
and g(jT)« ^W(g) are transform pairs, then:
(1) The transform is linear
a-f (JT)+b-g(JT)< a-W(f)+b-W(g) for real constants a and b.
(2) The coefficient of index is the sample mean of f (j)
N-l a(0) = 1/N I f(jx)
Logical or dyadic convolution of two sequences f(j) and g(j) is defined as
N-l h(jx)= 1/N £ f (kx)g[(j where (3) h(jx)= f(j)e g(j)< "W(f)-w(g) where ® indicates logical convolution 2 and (A) f (jx)8f (jt)« >W(f) This is the analog of the Wiener-Khinchin theorem of the frequency domain. The Walsh power spectrum is the finite Walsh transform of the logical autocorrelation of f(jx). Computer program 2 utilizes the Hadamard matrix multi- plication method to give both the transform or inverse transform of an input. This method is straight forward but 19 time requirements can be decreased considerably by using fast transform algorithms much like the fast Fourier transform. Several approaches exist for fast Walsh transform imple- mentation. One method described by Shanks [Ref. 21J parallels the derivation by Cooley and Tukey for the fast Fourier trans- form, but even faster methods are available [Ref. 2, 23, and 25] which make use of the internal symmetry properties of the Walsh functions. A useful method is closely associated with Peterson's method of Walsh function generation. Instead of the reflected binary expansion of the index indicating whether the inter- val is repeated with or without a sign change, the index indicates whether the interval is added or subtracted with the adjacent interval. For example, for the second spectral component of an eighth order system: f(jT) = f f f f f f f ofl 2 3 4 5 6 7 aj = 3 nl1 (reflected binary) = = g 8 1 8 = X 3 °» 2 » 1 = - - - and a 1/8 (f +f .,) (f + f ) [ (f +f ) (f +f ) ] 2 Q 2 3 4 5 fi ? 1 t t g 8 8 g ; 8 8 3 2 3 3 2 3 Figure 7 depicts a hardware realization of this type system for an eighth order spectrum. Using this or other fast Walsh transform algorithms reduces the number of operations required to Nlog„N real additions (and/or subtractions) as compared to Nlog N complex 20 multiplications and Nlog N complex additions as required by 2 the fast Fourier transform algorithms. It is therefore much faster to go from the time to the sequency domain than from the time to the frequency domain. Also the inverse Walsh transform can be performed with the same hardware or software as the transform and is therefore much more economical if the cost is calculated in time, computer memory or hardware. Computer program 3 and Figure 8 depict software and hard- ware realization of another fast Walsh transform method. In this method, sums and differences of the sample values are formed according to a pattern. The sequency spectrum coeffi- cients are the output and in the hardware version, the se- quency corresponding to the coefficient can be found by trac- ing the most direct path from anyone of the sample input points to the desired output. When an addition is passed through, a zero is written down; when a subtraction is passed through, a one is written down starting at the higher and pro- ceeding to the lower digits. When the output terminal is reached, the digits written down are the reflected binary representation of the index of the coefficient. The software version, represented by subroutine FASWAL of computer program 3 requires the subroutine function CONVRT to convert the coefficients to the sequency order. These coefficients are in a reversed reflected binary order much as the discrete fast Fourier transform's outputs are in a reversed binary order, [Ref. 8] . 21 To find the corresponding indicies as CONVRT does where WO) = TOO W(m) is the Walsh coefficient in sequency order T(k) is the coefficient in reversed reflected binary order (1) expand m into binary m » b b , ...t n n-1 1: (2) convert to reflected binary m = 8 n g n-r" 8 l (3) reverse order of digits k = g g .-.8 1 2 n (A) use this as the binary representation of k P =i Figure 7 shows another version of tbis same fast trans- form algorithm. In this case, the input is applied serially and after N input samples, the output (in parallel) will be the Walsh transform or the inverse transform if the input is the sequency ordered spectral coefficients. Note that as for Figure 8, the output is in reversed reflected binary order . Either of these realizations have advantages when hard- ware implemented since the ordering of the output can be predetermined and hard-wired so the reordering is not neces- sary each time the system is used. Note also that either of these units can be used to generate Walsh functions. The one 22 . in Figure 7 by putting a unit pulse into the input and the one in Figure 8 by shifting a unit pulse down through the input regis ter 23 V. DIGITAL FILTER CONSIDERATIONS One of the traditional methods of digital filtering in the frequency domain is by the use of the convolution theorem. That is, an input function is transformed into the frequency domain using either the Fourier, Laplace, or "Z" transform. This is multiplied by a weighting function, the transform of the impulse response of a desired filter, and the inverse transform is taken, yielding the desired output. The logical convolution theorem does not produce this result. Much has been written about the transition to and from the Walsh or sequency domain for the purpose of digital fil- tering but very little material can be found concerning what can be done while in the sequency domain to obtain the desired response from the filter in the time domain. It is possible to construct a filter with any desired sequency response. There is a rough correlation between se- quency and frequency responses of filters but only to the point that the higher sequencies contain more of the higher frequencies than the lower sequencies do. For example, low pass sequency filtering at a sequency cutoff of one-half the highest sequency, the sampling sequency, is equivalent to halving the sampling rate of the input function. To take advantage of the high speed and other economical properties of the Walsh functions, it would be desirable to be able to obtain a desired frequency response of a digital filter which operates in the sequency domain. 24 ' The first step in the investigation of this possibility will be to find the frequency spectrum of the discrete Walsh functions. (The frequency spectrum of the continuous Walsh functions is shown in Appendix B.) The Z-transform of a Walsh function can be written as: N-l -l Z[W (k,j)] = I W.,(k,j) -z (V-l) MN i=0 noting that for N = 1, Z[W (0,0)] = 1 (V-2) 1 and for N = 2, -1 Z[W (0,j)] = 1 + z 2 -1 z[w (i, j)] = 1-z (V-3) 2 the following recursive relationship is postulated j+P N/2 Z[W (2k+p,j)] = Z[W (k,j)][l+(-l) .Z ] (V-4) N N/2 '- p = 0,1 k 0,1,2, . . . ,N-1 j = 0,1,2, ... ,N-1 N = 2, A, 8,... This can be induced by the following argument for N a power of 2, say N = 2 . Expressing k of equation (V-4) in binary as : k - bb b . . .b (V-5) n n-l n-zn l The binary representation of k can be converted to reflected binary (see Appendix A) as; k„=bb ,b ..b. 2 n n-l n-2v 1 b b ...b n n-ln 2 CV-6) g 8 g ,g g n n-l n-2* l 25 , ,. Noting that 12 -kj = b b . . . .b.O (V-7) 1 J 2 n n-1 1 and - b (V-8) I2'kJ g g _ ..,g g n n> 1 ;l ;L then + 2.j] - b b ...b i (V-9) U 2 n n _ 1 1 and u+2-j] - 8 g 8 b (V-10) g n n-r" i i - = ,. f 1 for b- where b 1 Lo for b = 1 From Peterson's method of Walsh function generation as discussed in Section III, the higher order function is com- posed of the lower order function over the interval [0,1/2] and (-1) 1 or (-1) 1, (for 2«j and 2*j+l, respectively), times the lower order function over the interval [1/2,1], J j + l Since (-1) = (-1)1 and (-1) (-1) 1 , equation (V-4) is es tablished The Z-transform of a Walsh function will therefore be of the form, -1 2 4 N/2 ] . . Z[W (k, j) ^(1+z ) (1+z" ) (1+z" ) . (l+z" ) (V-ll) N As can be seen from equations (V-4, 7, 8, 9 and 10), the sequence of signs is determined by the reflected binary code of the index, k; that is, r n Z[W (k,j)] = [l+(-l) z" (V-12) N JJ" ] n = . . n 1,2,4, . ,N/2 r = where is the subscripted digit of 8rito [log (N/n)-lJ/>t/ \ it bg re. 2 the reflected binary representation of k. 26 , , , — evaluating m /2 .. . -imwT 1+z - 1 + e — i = /^I IWT 2 z = e T = sample period OJ = radian frequency r = 2(e-7 e -7 ) e — = 2Cos (ihwt) e (V-13) 4 UT Iff , , • i r -i r = -Z— = f = = letting x -r— ; 1/T sampling frequency £m I S s then, -m/2 1+z = 2Cos (xm)v e 7: (V-14) X03T -r— 2 z = e 2 Similarly - . TT -m/2 _ . s -lxm 1-z = 2Sm(xm)e -r + tt (V-15) ioot 2 2 z = e T~2 Therefore the frequency spectrum of a discrete Walsh function is simply a product of a group of sine and cosine functions with a linear phase response and can be written down directly upon expansion of the index, k, into reflected binary. For W (k,j), expanding k into reflected binary gives, = = k 8 8 ,g n lo 8 (N) n n-r- l ' 2 then the magnitude of the frequency spectral component is given as n n - 1 (k,j)]| - 2 (x)f x) |FlW |f (2x)...f(2 | (V-16) N n n _ 1 Cos <: x *°* f (x) = l > 8, Sin 00 for g = 1 and the phase is given by, F[W (k,j)J = -(N-l)x+m7T/2 N 27 where n m - I g P-a p For example, the frequency spectrum of (-14,j) would be, Wo 2 k = 14 = 01110_ = 01001 10in 2 g N = 32, n = 5 g g g g g = 01001 5 Zf 3 2 1 5 | = | F[W„ (14, j) 2 | Cos(x) Sin(2x)Cos(4x)Cos(8x)Sin(16x) 2 ] and the phase is, ¥ F[W (14,j)] = - 31x + 2-j = - 31x + tt z* 32 Figure 10 shows the amplitude of the frequency spectrum of the first eight Walsh functions of order 16 plotted on the interval [0^f/f ^..51. Note that 0.5 f/f is the maximum or s s Nyquist frequency for digital filter applications. In general it is only necessary to plot the first N/2 spectrums, since by symmetry, the N/2 remaining spectrums, (Figure 11), can be obtained by reversing the f/f axis of the lower order frequency spectrums. One method often used for digital filtering in the fre- quency domain is to use digital resonators in the form of comb filters with zeros introduced to give the desired spec- trum response. A Walsh function resonator can be defined as a digital circuit whose pulse response is a Walsh function. A Walsh resonator can be constructed as shown in Figure 12. Here, the sign on each input to the N input summer is deter- mined by the sign of the particular Walsh function at that point, A resonator is shown in Figure 13. j. W« 9 (14,j) 28 The z-transf orm of W (k,j), equation (V-ll), leads to another and simpler realization of the Walsh resonator. This method, shown in Figure 14, uses cascaded sections of lower order filters. The W o0 Cl4,j) resonator of this type is shown in Figure 15. This is the method which led to the development of the fast Walsh transform algorithm discussed in Section IV. The hardware realization of that algorithm shown in Figure 7 is simply a full set of N Walsh resonators. Referring again to Figures 10 and 11, it can be seen that various frequency responses can be obtained by cascading and paralleling various combinations of Walsh resonators. There is no method known that will give the classic filter frequency responses such as Butterworth, Chebychev, or elliptic using this method. A function minimization of the mean square error between a desired frequency response and the frequency res- ponse of cascaded/parallel Walsh resonators can give the necessary combination whose frequency response is- as close to the desired response as is possible with a given number of resonators. This method will lead to results comparable to that of using a digital comb filter and resonators in the frequency domain, [Ref. 8J. As an example, a filter is desired with a narrow band- width at approximately f/f = .25, using three Walsh resona- tors. Using the W ,(7,k) resonator as a starting approxima- tion, it can be seen that the bandwidth can be decreased and sidelobe magnitudes decreased by cascading it with notch filters constructed from W (6,k) and W ,(9,k) resonators. lb1£ lb1 29 The notch filters are simply resonators whose output is sub- tracted from the input in a feed forward manner. This is made possible by the linear phase characteristics of the Walsh resonators. Figure 16 shows a block diagram of such a filter and Figure 17 shows the frequency response of the filter. The 3db bandwidth is about .04f/f avid the nearest sidelobe is s down approximately 16db from the main lobe. Neither of the diagrams takes into account the delay errors which would probably be introduced by such a' system but these errors can be eliminated much in the same way as they are in frequency domain digital filters. Filters with higher Q's and better sidelobe suppression can be obtained by using higher order Walsh resonators and/or more of them in the system. 30 VI. RESULTS AND CONCLUSIONS Many computer runs of both, an on-line and off-line nature using the Walsh transforms for digital filtering have been made with various types of inputs and filter responses. After this work, it was concluded that the use of Walsh functions is not the absolute answer to digital filtering or to the many other fields of application being investigated. However, they do appear to have definite applications to assist in overcoming certain problems. The areas, other than multiplexing [Ref. 4, 14, 15 and 17], where greater benefits may be obtained as compared to the conventional methods appear to be where either the input or output function is of a discontinuous nature. One area where this has been put to u'se is in the field of image pro- cessing [Ref. 1, 3 and 19]. Here the input function, nor- mally two dimensional, may have noise bursts included in it which may be attenuated by low pass sequency filtering or may have low contrast areas which can be enhanced by high pass sequency filtering. In the first case, the input has discon- tinuities, the noise bursts; in the second, the output re- quires the presence of discontinuities, the image edges. It has been shown that digital image processing by multi-dimen- sional Walsh methods gives essentially equivalent results to Fourier methods but with the speed increase inherent in the sequency domain. 31 Image processing and the field of pattern recognition iRef . 6 J have much in common and again if the pattern to be processed is of a discontinuous form, it is believed that Walsh functions may again be superior to Fourier methods. Another signal which has characteristics that lend themselves to Walsh functions is one of the "infinitely" clipped type. Infinitely clipped signals also only take on two values and can be represented by Walsh functions easily. The use of Walsh functions in this area, especially for application to infinitely clipped speech processors and vocoders should be investigated further. Walsh functions will probably gain even greater status in the fields of image processing and multiplexing but with the exception of very specialized applications, their use to filter sinusoids will probably remain quite limited. APPENDIX A Walsh Functions as Products of Rademacher Products The Rademacher functions can be defined as follows: A i, o^t^i R(0,t) =. \o, other wise 1, 0**f R(l,t) 1, |-t±l , o therwise then R(n,t) = R(n-l,2t)+ R(n-l,2t-l) (A-l) n = 2,3,4,.... Each Rademacher function is a replica of the preceding func- tion with the sequency doubled. Referring to Figure 2 and Figure 3, the first 5 Rademacher functions are W(0,t), W(l,t) W(3,t), W(7,t), and W(15,t). That is, the Rademacher func- tions are the Walsh functions numbered 2 -1, n = 0,1,2,.... Defining a different method of ordering the Walsh functions, W(m,t), and expressing m in binary form, m = g n 8 n-l-" g 2 g l Then : 7 gj V. u,t) = R ^> t ) CA-2) b YT i.e. WC4,t) = W(100 ,t) = R(3,t) b 2 b W(5,t) = W(101 ,t) = R(3,t) «R(l,t) b 2 b 33 : : that W(4,t) = W(7,t) and WC5,t),= W(6,t), the func- Noting b tions W(m,t), are not in the same order as W(k,t). To find the index, m, in W(m,t) which corresponds to the index, k, in W(k,t), it is necessary to represent W(k,t) as the product of Rademacher functions, remembering that R(i,t)= W(2 -l,t). • = From the product rule, W(p , t ) W (q , t) W(p ing k in its binary digits: k = b b ,b „. . .b. the desired J ° n n-1 n-2 1 relationship can be found. From (A-2) n n g j 8 W(m,t) = R(j,t) J = "p|*W(2 -l,t) j b J"j" j=l j-l but 2 in binary is of the form 111«»»*1 and k must be the modulo 2 sum of several factors of this type. This can be express ed as b b _...b= g g g ...g i &§ ,6g ,gfa n n-1 1 -^n^n^n n g~n-l n-l* * n-ie* ' *e l n digits n-1 digits Adding the right side gives, b b ••.b ,- g„(gx& « b * -, • )...(g n n-1 'n n g«n-li)(g g„n-1 g„ -2 n or V 8 n * 8 n-l*---**j+l«V Therefore g = b to n n b b «n» « a -l- n* «„-! „-! or = b $ b gb n-li n n-1, and 34 & J J J-l n+1 of m . . , is the Thus the binary representation , g g _ . g ^ reflected binary (Grey code) representation of k * binary That is, W(bb ...b ,t) = W(gg .....g-pt) (A-3) n n-1 1 n n-J- * b The following algorithms can be used to convert from binary to reflected binary or vice versa. = = = . g. b . e b . ; b . g .e b . ; b (A-4) J 3 3-1 3 3 3 + 1 n+. This is illustrated by the following diagram. b b . b b n n-1 n-2 2 ^ x b ,_-bV V \ Vb b n+1 n n-1 3 2 8 8 6 8 8 n n-l n-2 2 1 b b b n+1xi n n-l 3 2 7 b b . n n-1 n-2 »/ h Figure 9 shows a set of discrete Walsh functions genera- ted as the product of Rademacher functions and left in binary order . 35 : , t APPENDIX B Frequency Spectrum (Fourier Transform) of the Continuous Walsh. Functions The Fourier transform, F(w), of a function is defined as, 00 . iwt i = /_1 F( W ) = / e" f(t) dt _00 w = radian frequency therefore iwt F (w) = / e~ W(k,t)dt k and by the definition of W(k,t), 1 1 ,1WC K _ 1WC F (w)= / e~ W(k,2t)dt+(-l) / e W(k , 2t-l) d Zk or k F (w)= 1/2 F (w/2) [l+(-l) e i|] 2k k Similarly k F (W) = 1/2 F (^/2)[l-(-l) e ~] 2k 1 k Therefore , iw 2e F_(w) = i(l-e" ) = Sin(w/2) u iw T w ? ~.iv F (w) = 1/212! 4 Sin(w/4)](l-e if) 1 w/2 4i -iw . 2 . ... = — e — Smc Cw/4) w 2 2 8Ci) 2 - F 9 (w) e"if Sin (w/8) Sin(w/4) 2 F^Cw) = — e"^| Sin Cw/8) CosCw/4) j w ^ By induction, a general form of F, (w) can be established as follows 36 expanding k in reflected binary, k g g g " n n-l-*' l let n m « I g j-l J then, . . n+l . N m . t -. >. v 2 (1) -iw _. 2. , n+l f )= — . e _. 2 w Sin ( w k ( / n-1 S - 8 j+1 (1 g ) j + 1 "[7 Sin j (w/2 )-Cos J (w/2 ) J-l 37 Xo -^ .(1)1 1 (a) 1 •:-» 1 ° 2 2 ( >o r -> 2 1 (3) 2 CO 1 x< ~4-> 3 (2) x J=L 3 (3)1 3 JZZL CO 1 3 JZZL 1 5 vC ) X ° JZZL 3 3 (7)1 3 nn. (8)1 3 (O 1 x ZL > t AXIS trt" 1 6 - 'ig. 1. The first 16 Harr Functions, xX i )(t) 38 R(0,t) R(l,t) o R(2,t) R(3,t) R(4,t) 1 < • 4- 1 AXIS +H 1 -H> t 1 _1_ 5 1 6 8 t 8 8 4 8 Fig. 2. The First Five Rademacher Functions, R(k,t). 39 1 1 -I W(0,t) --> K(8 t t)r 1U W(l,t) » W(9,t) -1 -1 1 W(2,t) > W(10,t) --*. -1 -1 1 1 W(3,t) •> W(ll,t) -- t ^» -1 -1 1 1 W(4,t) * W(12,t) -1 -1 1 1 W(5,t) -> W(13,t) .-£-> It 1 W(6,t) -> W(14,t) -1 1 W(7,t) * W(15,t) ^_ -1 -1 j AXIS 1- *t AXIS ly—) »— 1 1 1 H 1 1 > r- O 1 1 — 13 5 _3 2 i 8 7 77 VTi 8 «* 8 ^ Fie. 3. The First 16 Walsh Functions, W(k,t). 40 H A D A M A P. D M A T R I X (SEQUENCY ORDERED) 0RrF.n= 16 W(K,J)= J= 9 10 11 12 13 Ik 15 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 2 1 1 1 1 -1 -1 -1 -1 J. -1 -1 -1 1 1 1 1 3 1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 k 1 1 -1 -1 -1 -1 I 1 1 1 -1 -1 -1 -1 1 1 5 1 1 -1 -1 -1 -1 1 1 -1 -1 1 1 1 1 - JL -1 6 1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 1 7 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 8 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 9 1 -1 -1 1 1 -1 -1 1 -1 1 1 -1 -1 1 1 -1 10 1 -1 -1 1 -1 1 1 -1 -1 1 1 -1 1 -1 -1 1 11 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 -1 1 1 -1 12 1 -1 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 -1 1 13 1 -1 1 -1 -1 1 -1 1 -1 1 -1 X 1 -i 1 -1 Ik 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 l -1 1 -1 15 1 -1 1 -1 1 -1 1 -1 - 1 -1 1 -1 i -l 1 K = J = 1 2 3 h 5 6 7 8 g 10 11 12 13 III 15 Fig. 4. Discrete Walsh Functions of Order 16 (Sequency Ordered) 41 o WCl,t) -o W(2,t) Clock Fig. 5. Walsh Function Generator(as Rademacher Function Products) . 42 1 1/16 (J 1 1/8 -1 1 1/4 -1 1 1/2 —i5s» -1 1 1 u .A 1 -1 Fig. 6. Generation of W,g(ll,t). 43 . CM CO m vO ccj cfl CO 9 c> +il !p Pri? .P^v StS .£L I r-K I CO I a H 0) I I o Cu B o o H 4-» O CU a, ±6 0* CO <^ A V+ a) CO 0J CO T3 H M I Q< O >-. g CO CO >mH CO CJ CD N 4-i cu * P< cr1 -H c cu a •H CO 3 I I I Qt J-i I H •H Fig. 7. Serial In/Parallel out Fast Walsh Transform Real iza t ion 44 rd "d >^ 1 C/3 a) M •H 4J /^N 1 "O m t-» /-\ M O - m •r~» •I ) o a) a Fig. 8. Fast Walsh Transform Realization Parallel in/ Parallel out 45 H A D A M A R D M A T R ! (BINARY ORDERED) ORDER= 1G \ J /a V J = 1 2 3 k 5 G 7 8 9 10 11 12 13 n» 15 K= 1 1 1 1 1 1 1 1 1 1 1 1 1 1 i 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -i -1 2 1 1 1 1 -1 -1 -1 -1 1 1 1 1 _i -1 -i -1 3 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 i 1 l» 1 1 -1 -1 1 1 -1 -I 1 1 -1 -1 1 1 -i -1 5 1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 -1 -1 i 1 G 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 i 1 7 1 1 -1 -1 -1. -1 1 1 -1 -1 1 1 1 1 -l -1 8 1 -1 1 -1 i -1 1 -1 1 -1 1 -1 1 -1 i -1 9 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -i 1 10 1 -1 1 -1 -1 1 -1 1 1 -1 1 -I -1 1 -l 1 11 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 1 -1 i -1 12 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -l 1 13 1 -1 -1 1 1 -1 -1 1 -1 1 1 -1 -1 1 i -1 Ik 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 -1 1 i -1 15 1 -1 -1 1 -1 1 1 -1 -1 1 1 -1 1 -1 -i 1 — Ki' — J = 1 2 3 h 5 G 7 8 9 10 11 12 13 m 15 Fig. 9. Discrete Walsh Functions of Order 16 CBinary Ordered) 46 I w Co,j) W 16 16C1,J) A V WW\A W 16 (2,j) W (4 16 }j ) W 16 (6,j) f/f Fig. 10. Frequency Spectrum of First Discrete Walsh Functions, W,,-(k,j). 47 W (8,j) W 16 16 C9,j) w (io,j) w (ii,j) 16 16 W (12,j) W 16 16 (13,j) s£±L W (14,j) l6 i l ^/y\A(\ Y f/f o j_ i_ £ 8 4 8 Fig. 11. Frequency Spectrum of Second 8 Discrete Walsh Functions , W, , (k, j ) . 48 Shift Register W (k,n-l) in ... -Hn-2 jiN-1 J— W (k,n-2) N WN (k,n-3) out W (k,2) N %(k,l) W N (k,0) Fig. 12. Walsh Function Resonator. 49 3 I W - *. 1 I 4- V H-4- > I 4-V » l 4-4- I l*t )!ll^ 144 I u pCD Hen M CO QJ O •H •-H J! w H o 4J J-l td a fi o w a) CO rH cd CO bO •H 50 1 CN S3 I M + r-1 U o 1 • • ca a o CN1 1 w 1 N pf* .-I I w 1 I o L— fl I- 1 o •rl 1 W M CU toO > a) I + CJ CO N M II bO o 51 u o •p rt o to a) lVvI to i-l CO 1-1 >>\ CN CO 14-t o o •H CO H > cu 'd ct} o CO V! ua) inH 60 •H «g 52 \ CO M O o en I-l 01 P> M CD H•u •r< pM CO CO v£> CO p. PI cO o u u CO !3 vO M •— o toO ^J •u •rl »v CO vC C3 v_^ o ^O CO iH CD Js ri 3 53 Fig. 17. Frequency Response of Narrow Bandpass Filter Using Walsh Resonators. 54 « <' ' f • —•* 1 i • « ——' 4 «• —;' COMPUTER PROGRAM 1 Hadamard Matrix Generation * » * T * * CO *— u. > * —4 * .1 ""> CD • af! * <. cr> tl f- * X 3 < H- U! < Z * * ZD ) •," >- ££ * » =r b.. «> C _J DC LJ fc * i— a- U. Ll! (X. ~> * a: I C) * 1 * * CJ !Y t— • 1 1 0.1 C3 U •— * * CO CD X T • LJ • . * * —1 fc— h- on O r * o ^ • r Z Ll! _]_) 3K * vC 00 t- — :r o » * * u. «-i z < • o c CO * >- * c «« o" : o ru u. ca * * 00 »— UJ + CO * * »- * i- r a «-i a > * * x LULUS' —< lu or o • <, * _J * to •* Z " ' 3k - cd it a tr f- < I- O or 13 CO Q a *• •> >- —» - ' * J" u_ a: U h- Z --< CO (J 'I "— * < 03 U. U. Z> < aJ ; * _i ro h- a z - *" 3k % . . - * lY CJ *Y c • or <^ < \ Z ' ^^ X < *"• .0 ''I "" — * O -0 LJ f- CO CO uJ _J 2 ^-T z ' i . • (Y . ..'--<-! •— -^ »- -* [ 4 i/) : * £ ' 7 5" fV no C"< + * or o r- »— Z CD 2 * LJ o C^ 3 Lj O o c / ',_, q * ,-J <— t-1 «— ,-i 7 I x * 3k < Z o c ..' c_ e^ LJ c • L «- cv UUl J u u u u U U L U U O CJ u 55 * i - i t —« < . >i —» X a < V O 5: -< — < X o ^» — IP 2L ^ - o »-< Ci: z K — UJ «. UJ /~\ « c \ 1 »-t ~> "3 *~ 1 . ( <. o c > c C C L' ) Z: •— .-.: o < C. jG O Ci y 15 _< •i in co u. U i-l_ u l_L U. IX. u U. Ul i— r^ oo -^ 56 •1 •< « '• 1« • i* %» > • i•1* 1 i -I u < oj :s q^ o rr Ui 23 LO LJ a•» UJT UJ H- rr •— CD (.0 1- a.j X UJ UJ o oc X < — • go o «~» z UJ X (J rr ^ •— Ul ^ h- c — CJ C£ _! >: Q in 1 •— •— •— p—~. •— •—• •— i v-., — U' L^ \ _J 2. S *— — i •— i- :y — ^ i V CO Q- U) o lO O LO o L. (_» U L U cv. ex. 57 tI1 « « i« i *« « , '• CO U QZJ a> or:C z Z> — (£) Zi CO Ll — ^^ h- 1— UJ 2: — X h- - rr — 5_ ^ z_- u < uJ u_ * •— O ~x» cr z: a o: ^ z> UJ u < *— -z. ~~. b. ^ < Cr * -J v • en CD UJ LC ^-< - Q UJ . UJ Q U' o < _J ii 2 o -J < o «H CJ rr r cu • _i or - -* s UJ < a: UJ «-i >_ * i * < z> •— «— h- 2 t— »— t-i 1 ; >— - ^— V V 2 _J . . n 2 — U_ II II M II N ii It II II 2 u. U —' UI z. *— c, c — _J 5 2 •— »— •— 2 V 5. •— Ni cj a. UJ o o o O U CJ u OJ i '• '> ; Q UJ CO — CO2 (T. •— 1- CJ 3z. U. ££ LJ X. »"» CJ X < ^ 5: —X OUJ z <1 3 or L. c O ijj CVJ < T **• «. oc f— — O 2: CO «H •» © UJ + o • 4 t— V ^ — ; 7 — O P 7 7 ^ CD 3 ~ •— i- a- -> or — rr Li.' U- CD 1 2 r u. z> u. f— 3 D X — — o i- £> t- Q *- < X II ii L < UJ < CJ ^ »— CT LCI X ^- -- a. TY rr- Ct LJ o o U U CJ CJ 59 i * - COMPUTER PROGRAM 2 Walsh Transform as Hadamard Matrix Product * * * * * * ***** * * * * * cd CO • * * y: < _J * * or ul ^.j * * ( U C U UUUUUULJUULi c_ c • CJ c c_ u c. U CJ L U ! U C UCUL'UUU 60 < oc OC 00— 0C>— CD CD CO < < <- _J m _1 c o o o u vC Co u CI c> r\ '/ to CD o C3 in o o .:> c" o rv r " \0 61 « 00 CD < _J o Ul 00 «s rv Hen ta c. — il ii C, <-. o ..". »-» •-» —. ^ i < r-v 00 r (— t— -j »— ~~- < _! fi 1 / " CO> «O-• o C > C ) ex. 62 1 1 CD go3 yGO CD •— I— C.J 2: ~o • U. Z *» O «-( X GO • GO D£ + _J LlI ^-~ < h- ^ 2 UJ * a fu u_ ^^ en > c CO o »— »-» _j llI O N < C/3 Lij « X m rn »^. •"• < •— ^ *-~ 2 rv- —~ 2 o •> G0 U c ^ _~i Ld GO CsJ h- •** t~ UJ T~! -I + t II ii t— j ii II J x \ zo O C i o < u u u IT! ru en IT. 63 ! ;•i . > (t — « i1 « 3« ! i \ 00 cr OLU Of. C or u < ?:. \ o X OH o: «h • < > < 5: 1 z: o «h r. o o o C O C • J ui CJ CJ r-l ^ T—". 64 < — • —« «i —! 1 1 « I. t1i ! . • " or Ll X < or.o CY< CO =) X 0- ZJ cn 00 o +— o i r bJ in cr; * « •> * + CO LJ cr, *-< C Co X X X > *-< V s O X CT *-* 2 ~ c> ~ to i:. > _J <£ -*-* •— — CD — o CJ _J — cr rvi — t- > cn C. DC Cj CO 2 r^ CL r^ -^ jo -** LO o •» _- .' > : - o •- c\i cr j CO o uo c LO c ; tf • cj u o •r-t rj 0J c 65 • ti ( " eno (— 3 D Z ^v i— ~> c •* 3 r CO V* —~S >0 7' V- CD a: Z \ O «- * — tv < Ct uJ Z '-» -» *-• o ~> > r:~ : ll II ll •: »-- ;.- r Z> •— < or < cr u! lj cd 2" CO * * * c: 1 .. c n i ^ l li. o o c: u u 66 — — )< Q nr bJ CD 3 U UIx in X X X en \ _i •s Q to O •— rr «. • • ~ »- i "3 o a. ro k-. —. - ui I t— Ui LJ i UJ L>J Z x h- id _l x -j > en v. * c z> n. '— — i— X X — n. t~ — •«-< T—i CD X c i a \ 3 o •-» 00 if c in «- ^ o O «* v o o o X X X > X » "rH «-H t~ er in 1 Z : \ O «- IT Lf > :•-• < • II < «» L_' •> 4- -3- :*• ro rx I ^ (X _J I— C3 ii » \- *— c. — a. to uj rr. — i— i— T *~ »—»-»— ti i^j a-. H ii » «-< LP II Ui UJ *^ >— + jn h- V— K- H a. 77. T 7 5 •» r ~3 ~5 .- • :• ~ — a . . :r : :r \ rr e> Or. tr :r LY r — a: c — — Ul —Off S3 I * CD CD O — C^ f") -^ Lf: na; r^ cc o^ 1 O If O O Lf> O IT c o o c j : > c c : o c u Li 67 ' « .• • < . 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Co r^ «N O o V O o X X x x i - «-< IT X Lf LP •> ii cu II M • < • ro 'o ~D _J O "3 _ — Li _J >C D sC CL — Q CO UJ m — z — z <- »— h O II L_ c — if: ii 11 • »-< ii Lu •— Ul Hr U, o t- r.. ro ^- tf) vO r^ :<" c O Lf) C ) lD c ifi a j c^ O cj o o O C.i o o o o T-l If If) Si' vO r^- r- oc o, c *— «—1 «—• x— ^- T- T-. »-. .-( 75 BIBLIOGRAPHY 1. Andrews, H. C., "Digital Image Processing," Symposium on Walsh Functions , Naval Research Laboratory, Wash., D. C, 1970. 2. Andrews, H. C. and Caspari, K. L., "A Generalized Technique for Spectral Analysis," IEEE Trans . on Computers , Vol. C-19, No. 1, pp. 16-25, January 1970. 3. Andrews, H. C. and Pratt, W. K., "Application of Fourier-Hadamar d Transformation to Bandwidth Compression," Symposium on Picture Bandwidth Compression , MIT, April, 1969. 4. Ballard, A. H., "Orthogonal Multiplexing," Space Aeronautics , Part 2, pp. 50-60, November 1962. 5. Byrnes, J. S. and Swick, D. A., "Instant Walsh Functions," SIAM Review , January 1970. 6. Carl, J. W., "An Application of Walsh Functions to Image Classification," Symposium on Walsh Functions , Naval Research Laboratory, Wash., D. C.,1970. 7. Corrington, "M. S., and Adams, R. N., " Advanced Analytical and Signal Processing Techniques; Applic a tion of Walsh Functions to Nonlinear Analysis ," Technical Report, ASTIA //AD-277942, 1962. 8. Gold, B. and Rader, C. M., Digital Processing of Signals , McGraw-Hill, New York, 1969. 9. Harmuth, H. F., Transmission of Information by Ortho- gonal Functions , Spr ingler-Ver lag , Berlin/New York, 1969. 10. Harr, A., "Zur Theorie der orthogonalen Funktionensy s- teme," Erste Mitteilung. Math. Ann ., Vol. 69, pp 331-371, 1909. 11. Hubner, H., et al, "Algorithm 388 - Rademacher Function," Comm. ACM 13 , p. 510, August 1970. 12. Hubner, H., et al , "Algorithm 389 - Binary Ordered Walsh Functions," Comm. ACM 13 , p. 511, August 1970. 13. Hubner, H., et al, "Algorithm 390 - Sequency Ordered Walsh Functions," Comm. ACM 13, p. 512, August 1970. 76 14. Huebner, H., "Analog and Digital Multiplexing by Means of Walsh Functions," Symposium on Walsh Functions , Naval Research Laboratory, Wash., D.C., 1970. 15. Huebner, H., "On the Transmission of Walsh Multiplexed Signals," Symposium on Walsh Functions , Naval Research Laboratory, Wash., D.C., 1970. 16. Kennett, B. L. N., "A Note on the Finite Walsh Trans- form," IEEE Trans. Info. Theory , IT-6 , pp 489-491, July 1970. 17. Lueke, H. D., "Binary Orthogonal Functions in Communi- cations," Internationale Ele ck tr onis che Rundschau , Vol. 29, No. 1, pp 1-3, January 1970. 18. Peterson, H. L., "Generation of Walsh Functions," Symposium on Walsh Functions , Naval Research Laboratory, Wash., D. C, 1970. 19. Pratt, W. K., and Andrews, H. C., "Hadamard Transform Image Coding," Proceedings of the IEEE , Vol. 57, No. 1, pp 58-68, January 1969. 20. Rademacher, H., "Einige Satze uber Reihen von , allegemeinen Or thogonalf unkt ionen " Math . Ann . , Vol. I >7, pp 112-138, 1922. 21. Shanks, J. L., "Computation of the Fast Walsh-Fourier Transform," IEEE Trans, on Computers , pp 457-459, May 1969. 22. Swick, D. A., "Walsh Function Generation," IEEE Trans on Info. Theory , Vol. IT-15 , No. 1, p. 167, January 1969 . 23. Vandivere, E. F., and Carrick, R. L., " Fast Walsh Trans - " form , Telcom Inc. Report TAM 2.1.2, 13 November 1967. 24. Walsh, J. L., "A Closed Set of Normal Orthogonal Func- tions," Amer . Jour. Math . , Vol. 45, pp 5-24, 1923. 25. Whelchel, J. E. and Guinn, D. F., "Fast Fourier-Hadamard Transform and its Use in Signal Representation and Classification," EASCON Record , pp 561-573, 1968. 77 n INITIAL DISTRIBUTION LIST No . Copies 1. Defense Documentation Center 2 Cameron Station Alexandria, Virginia 22314 2. Library, Code 0212 Naval Postgraduate School Monterey, California 93940 3. Dr. S. R. Parker, Code 52 Department of Electrical Engineering Naval Postgraduate School Monterey, California 93940 4. William J. Dejka, Code 4300 Advanced Modular Concepts Division Head Naval Electronics Laboratory Center San Diego, California 92152 5. Lt. Norman Clare Meek, USN P. 0. Box 67 Palmer Lake, Colorado 80133 78 Security Classification DOCUMENT CONTROL DATA -R&D Security clas silicotion ol titlo, body ol abstract and indexing annotation must be entered when the overall report Is classified) originating ACTIVITY (Corporate author) is. REPORT SECURITY C L A S SI F I C A T I Or Unclassified Naval Postgraduate School 2b. GROUP Monterey, California 93940 3 REPOR T TITLE Application of Discrete Walsh Functions to Digital Filter Techniques 4 DESCRIPTIVE NOTES (Type ol report and.inclusive dates) Electrical Engineer's Thesis; June 1972 S- AU THOR(S) (First name, middle initial, last name) Norman C. Meek «. REPORT DATE 7«. TOTAL NO. OF PAGES 76. NO. OF REFS June 1972 80 25 »a. CONTRACT OR GRANT NO. 9a. ORIGINATOR'S REPORT NUMBER(S) b. PROJEC T NO. 9b. OTHER REPORT NOISI (Any other numbers that may be assigned this report) 10. DISTRIBUTION STATEMENT Approved for public release, distribution unlimited. 11. SUPPLEMENTARY NOTES 12. SPONSORING MILI TAR Y ACTIVITY Naval Postgraduate School Monterey, California 93940 13. ABSTRACT Methods of generating and properties of Walsh functions are discussed with emphasis on discrete Walsh functions. Hardware and software versions of two methods of Walsh func- tion generation are shown. The Walsh transform is explored and hardware and software realizations of a fast Walsh trans- form algorithm, particularly applicable to hardware, is introduced. Computer programs containing the various algorithms are included. A discussion of the feasibility of the use of Walsh functions to produce a digital filter with a desired frequency response is presented. F R M (PAGE 1 ) DD , N°o v e 8 1473 79 S/N 01 01 -807-681 1 Security Classification 1-3140' Security Classification LINK B KEY WO R OS Walsh Function Walsh Transform Harr Function Rademacher Function Hadamard Transform Walsh Fourier Transform Digital Filter F 'BACK, . N oT 65 1473 80 • 101 0O7-68;> 1 Security Classification - A -j 1 4 & Thesis 135464 M42 Meek c.l Application of discrete Walsh functions to digi- tal filter techniques. 10 AUG 73 ? 11 1 n 30 JUN76 ^ 2 3 4 7 2 ' iS, JUL £7 l'4ii55 / Thesis 135464 M42 Meek c.' Application of discrete Walsh functions to digi- tal filter techniques. thesM42 Application of discrete Walsh functions 3 2768 000 98315 9 DUDLEY KNOX LIBRARY