IEEE TRANSACTIONS ON COMPUTERS, MAY 1969

Short Notes

Computation of the Fast Walsh-Fourier TransformJOHN L. SHANKS, MEMBER, IEEE

Abstract-The discrete, orthogonal Walsh functions can begenerated by a multiplicative iteration equation. Using this iterationequation, an efficient Walsh transform computation algorithm isderived which is analogous to the Cooley-Tukey algorithm for thecomplex-exponential Fourier transform.

Index Terms-Algorithm, Cooley-Tukey, Hadamard-Fourier,orthogonal, transform, Walsh-Fourier.

OPTIMIZATION OF THE DISCRETE WALSH TRANSFORMSeveral authors have treated the set of Walsh func-

tions and the Walsh-Fourier transform [1]- [4], [7].We shall discuss here the set of discrete, orthogonalWalsh functions and the derivation of a Walsh trans-form algorithm which is analogous to the Cooley-Tukeyalgorithm [5] for the complex-exponential Fouriertransform. 'The discrete Walsh functions are sampled versions of

the continuous set. As discussed here, the discrete func-tions are assumed to be infinite in extent, and are peri-odic with period M, where M is an integral power oftwo. Thus a complete orthogonal set will have M dis-tinct functions. We shall designate these functionswal (m, n). The complete set is represented over therange m =O, 1, 2, * *, M-1 and n =O, 1, 2, *M-1.The first two discrete Walsh functions are defined as

wal (0, n) = 1wal (1, n)

for n = 0, 1, 2, * ,M-1

= 1 for n=0, 1,2, . .., (M/2)-1 i= -1 for n = M/2, (M/2) + 1, * M

(1)

(2)

The remainder of the set can be generated by an itera-tive equation. Various iteration equations have beenused to generate the Walsh functions, but for our de-velopment it is convenient to introduce the multiplica-tive iterative equation (3):

wal (m, n) = wal ([m/2], 2n) wal (m -2 [m/2], n), (3)where [m/2] indicates the integer part of m/2. It is

Manuscript received November 1, 1968; revised February 6, 1969.The author is with the Pan American Petroleum Corporation,

Tulsa, Okla. 74102.' At EASTCON, 1968, and at the IEEE Workshop of FFT Pro-

cessing, October 6-8, 1968, J. E. Whelchel [9] presented the matrixderivation of the fast Fourier-Hadamard transform. The Hadamard[8], [10] and the discrete Walsh transforms are identical with thepossible exception of a permutation in the order of functions.

012 34 5 6 7o0 T T T T f T +1

-I

2

3

? 9 o

1 1lv ~ ~l I I

I 1 1

5 4 f T

6

7

Fig. 1. The eight discrete Walsh functions of length 8.

shown in the Appendix that (1), (2), and (3) generatean orthogonal set.

Fig. 1 shows the eight discrete Walsh functions oflength M=8 as generated by (1), (2), and (3). Notethat the discrete Walsh functions as defined here aresymmetric with respect to the argument (m, n). That is,

wal (m, n) = wal (n, m).Given an M-length real array f(n), we can define theWalsh transform as

M-1

F(m) = Ef(n) wal (m, n),nO0 (4)

Similarly, the inverse transform is1 M-1

f(n) =-E F(m) wal (n, m),M m=O

n =O,)1, 2, , - 1.Since the Walsh functions can have values of +1 and-1 only, computation of (4) or (5) requires no multi-plications. Furthermore, using (3) we can derive a com-putation algorithm analogous to the Cooley-Tukeyalgorithm. This algorithm will require M log2 M sum-mations to compute a complete Walsh transform ratherthan M2 as indicated by (4).The derivation of the algorithm parallels the one

given by Cooley and Tukey. Rather than present thegeneral proof, we shall outline the derivation for thespecial case M=8. Anyone familiar with the deriva-tion of the Cooley-Tukey algorithm should be able toextend the work presented here to the general case.

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IEEE TRANSACTIONS ON COMPUTERS, MAY 1969

Let us replace the indices in (4) with a set which canhave only values of 0 and 1. That is, let

m = 4j2 + 2j1 + jO,n = 4k2 + 2k1 + ko,

j2, ji, jO = 0 or 1,k2, ki, ko = 0 or 1.

(6)(7)

Using these new indices, the notation wal (m, n) be-comes

wal (j2, jl, jo; k2, ki, ko).Thus we rewrite (4) as

1 1 1

F(j2,jl, jO) = : EE f(k2, kj, ko)ko=O kl=O k2=0 (8)

n Ao0fn Al A2 A3=Fm m

2 03

5

6

0426

537

+1 ---r-

Fig. 2. Signal flow graph of 8-length discrete Walsh transforms.Multiplying factors are +1 and -1, as indicated by the solid anddotted branches, respectively.

*wal (j2,jljo; k2, k1, ko).Also, we can restate the iteration equation (3) usingthe new indices. Note that the numbers j2, j1,Jo are es-sentially the digits of the binary representation of m.Therefore, dividing m by two and keeping the integerpart is equivalent to shifting the binary representationof m one bit to the right and dropping the fractional bit.That is, if

m-->2j11o

then[m/2] 0j2j1.

Similarly, multiplying n by two is equivalent to shift-ing the binary representation of n one bit to the left. If

n k2kikothen

2n k2k1koO.An 8-length Walsh function is periodic with period 8.Thus any index (such as 2n) can be evaluated modulo8. This is equivalent to deleting any bits above thethird bit and we have

2n(modulo 8) *-> k1koO.Using these indices, (3) becomes

wal (j2, jl, jo; k2, k1, ko) = wal (0,j2,jl; k1, ko, 0)*wal (0, 0, jo; k2, kj, ko).

Since jo can only be 0 or 1, wal (0, 0, jo; k2, k1, ko) repre-sents either wal (0, n) or wal (1, n). The function wal(0, n) is 1.0 for all n. The function wal (1, n) is 1.0 if0

SHORT NOTES

where =1, 2, - ,p, andAo(kp_1 kp-2, . , ko) = f(kp.1, kp_2, . , ko). (16)

The general equation for the M= 2P-length discreteWalsh function iswal (jp-1, jp_2, * ,jo; kp-,, kp-2, k*ko)

P-1~~~(17)

= II (-1)'P1-ki(.i=OFrom (15) we can build a program for the general M-length discrete Walsh transform. With (17) we canevaluate a Walsh function for any m and n. Also, (17)can be used to prove the orthogonality (see Appendix)and the symmetry of the set of discrete Walsh func-tions.

APPENDIXThe orthogonality of the discrete Walsh functions as

generated by (1), (2), and (3) can be proved by usingthe general Walsh function (17). Equation (17) isderived from the iterative equation (3) and the defini-tions (1) and (2). The dot product of any two M-length Walsh functions is as follows.0(rp-l rp-2, . . . 2 rO; jp-l, jp_2, . . .*, jo)

1

= cE k** kgEkp-l kp-2 ko=O (18)*wal (r_, ,ro; k,-,, * , ko)*wal (jp-1,, jo; kp-,) . , ko) (19)

1 p-i

E E2 E rpJ{(-)r-1-iki( )iP-1-iki} (20)kp-1 kp-2 kO=O i=O

EE E J~~~1(- 1) ki(rp-1-j^+jp-1-") (21)kp-1 kp-2 k0=O i=Op-i 1

= E (_ l)ki(rp-1-2+jp-1-i) (22)i=O ki=O

p- I

II { 1 + (-1) (rp-1-i+jp-j-i) (23)i=O

Suppose each rk =jk= or 1. Then (23) becomesp-1

=112 = 2P = M.i=O

Suppose at least one rk Fjk. Then at least one term inthe product in (23) is 0 and =0. In terms of thedecimal indices, this means that4(m, 1)

M-1

ZE wal (m, n) wal (1, n) = M form = (24)n=O

=0 form l

Thus (1), (2), and (3) generate an orthogonal set.

REFERENCES[1] J. L. Walsh, "A closed set of orthogonal functions," Am J.

Math., vol. 55, pp. 5-24, January 1923.[2] J. L. Hammond and R. S. Johnson, "Orthogonal square wave

functions," J. Franklin Inst., vol. 273, pp. 211-225, March 1962.[31 K. W. Henderson, "Some notes on the Walsh functions," IEEE

Trans. Electronic Computers (Correspondence) vol. EC-13, pp.50-52, February 1964.

[4] H. F. Harmuth, "A generalized concept of frequency and someapplications," IEEE Trans. Information Theory, vol. IT-14, pp.375-382, May 1968.

[5] J. W. Cooley, and J. W. Tukey, "An algorithm for the machinecomputation of complex Fourier series," Math. of Computation,vol. 19, pp. 297-301, April 1965.

[6] G-AEC Subcommittee on Measurement Concepts, "What is theFast Fourier transform?" Proc. IEEE, vol. 55, pp. 1664-1674,October 1967.

[71 N. J. Fine, "The generalized Walsh functions," Trans. Amer.Math. Soc., vol. 69, pp. 66-77, 1950.

[8] S. W. Golomb and L. D. Baumert, "The search for Hadamardmatrices," Am. Math. Monthly, vol. 70, p. 12, January 1963.

[9] J. E. Whelchel and D. F. Guinn, "The fast Fourier-Hadamardtransform and its use in signal representation and classification,"Melpar, Inc., Fall's Church, Va., Tech. Rept. PRC 68-11, 1968.[10] W. K. Pratt, J. Kane, and H. C. Andrews, "Hadamard trans-form image coding," Proc. IEEE, vol. 57, pp. 58-68, January1969.

Threshold Logic Design of Pulse-TypeSequential Networks

RAYMOND M. KLINE, MEMBER, IEEE, ANDDONALD F. WANN, MEMBER, IEEE

Abstract-A modification of the conventional design method forpulse-type sequential logic is developed in which threshold gates areused in all of the combinational portions of the circuit. The canonicalconfiguration, memory element employed, tabular design aids, etc.,found in the conventional method were changed as necessary inorder to use more effectively the unique properties possessed bythreshold elements. A theorem providing insight for threshold gaterealization is given and a classification of sequential functions is dis-cussed in order to facilitate the understanding and use of the newprocess. Finally, examples are presented showing a significant ad-vantage of the present method in both gate and transistor economyover conventional design.

Index Terms-Logical design, realizability, sequence classifica-tion, sequential networks, switching circuit synthesis, thresholdlogic.

I. INTRODUCTIONResults of extensive research have been reported

concerning the application of threshold logic principlesin the design of combinatorial logic circuits [1]- [5]. Inthese papers, attention has been focused on generalproperties of threshold functions, on techniques fordetermining whether a given Boolean function isrealizable by means of a single gate, and on variousapplications such as the development of learning

Manuscript received October 25, 1968; revised November 18,1968.

The authors are with the Department of Electrical Engineering,School of Engineering and Applied Science, Washington University,St. Louis, Mo. 63130.

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