IEEE TRANSACTIONS ON COMPUTERS, SEPTEMBER 1970
850
CONCLUSIONS
1) DTL gates or other compatible gates that may be wired together
like those in Fig. 1 can in fact be used in two ways. Without wiring,
the gates perform normal NAND functions. Two or more outputs
wired together result in a two-level net performing AND-NOR functions.
2) Any Boolean function can be realized with wired gates.
3) If two types of transistor gates A and B are to be compared in speed,
the possibility of wiring should be taken into account. Let tA and tB
be the average propagation time for one single gate of each type. If
gates of type A cannot be wired and gates of type B can be wired, we
have to reduce tB with a factor k before comparison with tA. In some
applications it may be justified to set k as large as 2.
4) The concept of wired gates ought to be treated by modern textbooks
on switching theory and logical design.
PER E. DANIELSSON
Dept. of Elec. Engrg.
Institute of Technology
Lund, Sweden
very useful recursive structure, perhaps first developed by Lechner [5] and
shown in (17) of [4], is generated by either of the products
W'
=
bi O M bp-i 0 btP-i
Abstract-The matrix form of the Walsh functions as defined in
the above-mentioned short note [1 ] can be generated by the modulo-2
product of two generating matrices: the natural binary code, and
the transpose of the bit-reversed form of the first. As a result, the
coefficients of the Walsh transform occur in bit-reversed order.
By simply reordering the Walsh functions themselves to correspond
to generation by the product of two such code matrices, neither or
both in bit-reversed form, the Walsh coefficients occur in their
natural order.
Index Terms-Code matrix, Walsh-Fourier transform, Walsh
functions, Walsh matrix.
A useful relationship between the Walsh functions and the CooleyTukey algorithm for the fast Fourier transform has been established by
Shanks [1], upon which Lechner has also commented [2]. As with the
Cooley-Tukey algorithm, the coefficients of the Walsh-Fourier transform
occur in bit-reversed order, unless the discrete data representing the function to be transformed are first rearranged in bit-reversed order [3]. This
inconvenience can be circumvented by a very simple reordering of the
Walsh functions themselves.
The author has shown in a previous correspondence that the binary
Walsh functions in matrix form can be generated by modulo-2 multiplication of two simpler matrices representing some form of the reflected and/or
natural binary code(s) [4].
By way of review, the basic binary Walsh matrix, possessing the property that each row has one more sign change than the preceding one (from
0 to 1 or from 1 to 0 in this case), is generated by any of the products
W=ai (i) bp -i
=
ap_
P-3 bti= i' a_9
3a
(i
=
O, 1, **,p
-
1), (1)
where ai denotes the reflected binary code and bi the natural binary code
of p variables in the form of matrices of 2P rows and p columns, and where
ap-i and bp-i represent the respective matrices with the order of their
columns reversed-or, in the contemporary terminology, in bit-reversed
form. Examples are (21) and (22) of [4]. The superscript t denotes transposition of the matrix. The symbol 0 denotes matrix multiplication, modulo-2.
Similarly, a modified binary Walsh matrix possessing a very simple and
Manuscript received August 21, 1969; revised March 17,
1970.
(2)
=
in which neither of the code matrices is, or both are, in bit-reversed form.
Other forms of the Walsh matrix (some symmetrical and some unsymmetrical) can be generated by using other variations and combinations of
the foregoing code matrices and matrices representing other codes. Still
other forms can be obtained by independent reordering of the rows and/or
columns of the Walsh matrix itself.
Now, the binary matrix representation of Fig. 1 of [1] (after replacing
each 1 by 0 and each -1 by 1) is generated by either of the products
W"
=
bi (i) bp-i
=
bp-i (
(3)
bt
in which either one of the code matrices is in bit-reversed form, but not
both. (This form of W is one presented, although not obtained from a
product of code matrices, by Kaczmarz and Steinhaus [6].)
That this is so is also evident from (13) of [1], in which the k's are reversed in bit order with respect to thej's, and which could be written in the
more compact matrix form
F
Comment on "Computation of the Fast
Walsh-Fourier Transform"
I
=
[U - 2W"] -f
(4)
where f is a column matrix of the discrete values fn of the M-length real
array whose transform is desired, and where F is a column matrix of the
transform coefficients Fm. Subtracting twice the binary Walsh matrix from
the unit matrix U, all of whose elements are 1, effects the necessary transformation 0-+1, 1 - 1, corresponding to (11) of [4]. The symbol denotes
ordinary matrix multiplication.
Since only one of the generating code matrices in either product in (3)
is in bit-reversed form, it is not surprising that the coefficients Fm in (13)
of [1 ] occur in bit-reversed form.
If the Walsh functions in Fig. 1 of [1] are rearranged in the order given
by (2), corresponding to (16) of [4], then (4) can be rewritten as
F'= [U-2W'] f
(5)
Then (12) and (13) of [1] could also be rewritten as
wal (12, jl,jo; k2, kl, ko)
F'(j2, j, jo)
=
=
( -l)j2k2( -l)ljk( l)joko
1
1
1
k2=O
k1=O
ko=O
(6)
E (-l)j2k2 E (_ )'sk' E (- l)jokof(k2, kl, ko),
(12,1l,
O =
Oor 1; k2, kl, ko = 0 or 1),
(7)
and the coefficients of the transform would then occur in the natural order
instead of bit-reversed order.
Thus, in machine computation as described in [3], the computation
may be done "in place," yet requires no "shuffling" of either the initial data
sequence or the resulting spectrum.
The signal flow graph for (5) or (7) differs from that in Fig. 2 of [1 ] only
in that the coefficients Fm at the right are now in their natural order, i.e., the
same order as the dataf, at the left.
KEITH W. HENDERSON
Stanford Linear Accelerator
Stanford University
Stanford, Calif. 94305
Center1
REFERENCES
[1] J. L. Shanks, "Computation of the fast Walsh-Fourier transform," IEEE Trans.
Computers (Short Notes), vol. C-18, pp. 457-459, May 1969.
[2] R. J. Lechner, "Comment on 'Computation of the fast Walsh-Fourier transform',"
IEEE Trans. Computers (Correspondence), vol. C-19, p. 174, February 1970.
M1Operated under contract with the U. S. Atomic Energy Commission.
851
CORRESPONDENCE
[3] W. T. Cochran, J. W. Cooley, D. L. Favin, H. D. Holmes, R. A. Kaenel, W. M. Lang,
G. C. Maling, D. E. Nelson, C. E. Rader, and P. D. Welch, "What is the fast Fourier
transform?" Proc. IEEE, vol. 55, pp. 1664-1674, October 1967.
[4] K. W. Henderson, "Some notes on the Walsh functions," IEEE Trans. Electronic
Computers (Correspondence), vol. EC-1 3, pp. 50-52, February 1964.
[5] R. Lechner, "A transform theory for functions of binary variables," in Theory of
Switching, Harvard Computation Lab., Cambridge, Mass., Progress Rept. BL-30, Sec.
X, pp. 1-37, November 1961.
[6] S. Kaczmarz and H. Steinhaus, Theorie der Orthogonalreihen, 2nd ed. New York:
Chelsea, 1951, pp. 125-134.
Correction to "Pictorial Output with a Line Printer"'
In some of the February 1970 issues of this TRANSACTIONS (and in some
of the reprints) there was an unfortunate printing error. Table I of the
above-mentioned Short Note by I. D. G. Macleod was published incorrectly. Table I in correct form is given below.
THE EDITOR
Manuscript received July 21, 1970.
IEEE Trans. Computers (Short Notes), vol. C-19, pp. 160-162, February 1970.
'
TABLE I
DENSITY CODES EMPLOYED
Hybrid Solution of Partial Differential Equations
Chan ([1], ref. 18) and Vichnevetsky ([2], ref. 46) have referred to a
"Memorandum No. 8" of the M.I.T. Electronic Systems Lab., entitled
"Hybrid Solution of Initial Value Problems for Partial Differential
Equations," attributed to this author. Persons trying to access this work
face difficulties. Indeed, the undersigned is not aware of such a memorandum. Apparently it is identical to the M.S. thesis of the same title written
under Prof. L. A. Gould and completed in August, 1964. The Electronic
Systems Lab. has exhausted its supply of this work and the original vellums have not been saved. It is therefore necessary to request a copy of the
M.S. thesis either from the M.I.T. Library Microfilm Division or, as long
as his supply lasts, from this author.
H. S. WITSENHAUSEN
Bell Telephone Labs., Inc.
Murray Hill, N. J. 07974
REFERENCES
[1] S. K. Chan, "The serial solution of the diffusion equation using nonstandard hybrid
techniques," IEEE Trans. Computers, vol. C-18, pp. 786-799, September 1969.
[2] R. Vichnevetsky, "Analog hybrid solution of partial differential equations in the
nuclear industry," Simulation, vol. I1, pp. 269-281, December 1968.
Manuscript received January 5, 1970.
Overprinted Character
Combinations
blank
+
1
Z
X
A
M
00=
0+
0 +'
0+'
O +1' . =0.79
ox'.-
OX'.HC
OX'.HB
OX'.HBV
OX'.HBVA
Estimated Density
Values
0.0
0.15
0.22
0.25
0.29
0.33
0.37
0.40
0.42
0.45
0.53
0.56
0.60
0.64
0.67
0.85
0.89
0.93
0.97
1.00