ISSN 0032-9460, Problems of Information Transmission, 2009, Vol. 45, No. 4, pp. 333–342.
Pleiades Publishing, Inc., 2009.
Original Russian Text
M.S. Bespalov, 2009, published in Problemy Peredachi Informatsii, 2009, Vol. 45, No. 4, pp. 43–53.
New Enumeration of Walsh Matrices
M. S. Bespalov
Vladimir State University
Received December 27, 2008; in ﬁnal form, September 10, 2009
Abstract—The discrete Walsh transform is a linear transform deﬁned by a Walsh matrix.
Three ways to construct Walsh matrices are known, which diﬀer by the sequence order of
rows and correspond to the Paley, Walsh, and Hadamard enumerations. We propose a new
enumeration of Walsh matrices and study its properties. The new enumeration is constructed
as a linear rearrangement; we obtain an eigenvector basis for it and propose a convenient-
to-generate fast implementation algorithm; the new enumeration possesses certain symmetry
properties, which make it similar to the discrete Fourier transform.
The fundamental orthogonal transforms used in digital data processing are the discrete Fourier
transform and discrete Walsh transform. The discrete Walsh transform is considered in one of
three enumerations, namely, the Paley, Walsh, and Hadamard enumeration, and is deﬁned by the
corresponding Walsh matrix, which in the technical literature [1,2] is usually denoted by Pal, Wal,
and Had, respectively. Let us agree to ﬁx the notation W , U,andH for Walsh matrices in the
Paley, Walsh, and Hadamard enumeration, respectively (thus distinguishing the Paley enumeration
and depreciating the Walsh enumeration).
Walsh matrices in any enumeration are always of order 2
. Therefore, as an index, we use not
the order of a matrix (as in [1,2]) but the corresponding exponent, which we call the level of the
Each row (with number k) of any of the Walsh matrices W , U,orH will be referred to as a
discrete Walsh function and denoted by w
(skipping an argument). Similarly (without
an argument), we denote discrete Rademacher functions.
The enumeration of Walsh matrices proposed by Paley  is considered to be the basic enumer-
ation in mathematical literature [4, 5]. Its continual analog is the Walsh transform introduced by
Fine , and other its well-known [1, 4, 5] generalizations are multiplicative Vilenkin–Chrestenson
systems. Walsh functions are deﬁned as all possible products of Rademacher functions.
In [1, p. 50], the classiﬁcation problem for Walsh matrices was posed, as well as the problem of
selecting from them new classes of matrices of interest for digital data processing. As an illustration,
all the six possible row permutations of the fourth-order Walsh matrix (preserving the ﬁrst row)
are written out, and four symmetric of them are indicated. The fourth matrix (other than the
Paley, Walsh, and Hadamard enumeration) has a structure which is symmetric and antisymmetric
horizontally, vertically, and about the center of the matrix. It is interesting that a similar symmetry
property is obeyed by a matrix obtained by deleting the ﬁrst row and ﬁrst column (which consists
Supported in part by the Analytical Departmental Target Program “Development of Scientiﬁc Potential
of the Higher School,” reg. no. 2.1.1/5568.