ISSN 0032-9460, Problems of Information Transmission, 2010, Vol. 46, No. 3, pp. 253–271.
Pleiades Publishing, Inc., 2010.
Original Russian Text
M.S. Bespalov, 2010, published in Problemy Peredachi Informatsii, 2010, Vol. 46, No. 3, pp. 60–79.
Eigenspaces of the Discrete Walsh Transform
M. S. Bespalov
Vladimir State University
Received February 4, 2010; in ﬁnal form, May 4, 2010
Abstract—We reﬁne the notion of a discrete Walsh function and generalize the notion of
a discrete Walsh transform, for which we propose a method for generating a corresponding
W -matrix. We propose spectral decompositions of the discrete Walsh transform operators
in arbitrary enumerations, as well as methods for ﬁnding bases of eigenspaces, one of them
using a new direct product of matrices. We propose a notation for the fast discrete Walsh
transform algorithm in the Paley enumeration. We construct Parseval frames for eigenspaces of
the discrete Walsh transform in the Paley enumeration and demonstrate methods for applying
them in error detection and correction.
The discrete Walsh transform is one of the fundamental linear operators used in digital data
processing and data transmission through communication channels. Traditionally, three types of
the discrete Walsh transform are distinguished , namely, in the Paley, Hadamard, and Walsh enu-
meration. The Paley enumeration, proposed in , is considered [3,4] to be the main. An overview
of technological applications  also indicates the priority of the Paley enumeration. On the other
hand, due to its construction, the discrete Walsh transform in the Hadamard enumeration has
advantages in implementation, which are pointed out below.
Matrices of the discrete Walsh transform are always of order 2
; therefore, as a subscript in the
notation for them we use the exponent n, which will be referred to as the level of a matrix. We use
the notation W
for a matrix of level n of the discrete Walsh transform in the Paley and
Hadamard enumeration, respectively.
Matrices of the discrete Walsh transform in the Hadamard enumeration were introduced in 
as the Kronecker product H
of the matrix
are major examples of Hadamard matrices , which are matrices with entries 1
and −1 with the largest possible absolute value of the determinant |det H
| = N
The Hadamard conjecture (still unproved ) states that for any positive integer k there exists a
Hadamard matrix of order 4k. A characteristic property of a Hadamard matrix was proposed by
Paley , who introduced the matrices W
Usually [3,4], matrices W
are deﬁned through Walsh functions w
Nonnegative integers j and real numbers x ∈ [0, 1) can be represented in a simple binary code,
Supported in part by the Analytical Departmental Target Program “Development of the Scientiﬁc Poten-
tial of the Higher School,” project no. 2.1.1/5568.