A constructive analysis of the aperiodic binary correlation function

The aperiodic binary correlation function or correlogram
X \ast Y
occurs as the output of the detector in the correlation detection of a binary code word
Y
in an unsynchronized code stream
X
. In most applications, such as self-synchronizing data links, the correlation properties are determined by the desired system characteristics, and the problem is to find codes that approximate the desired correlation. As a partial solution to this problem, the general algebraic properties of
X \ast Y
as a function of the code words
X
and
Y
are developed in this paper. In particular, sufficient conditions for the general quadratic equation
X \ast Y = W \ast Z
to possess solutions are demonstrated, as well as for the restricted cases in which
W = \pm Y
and
Z = \pm X
, i.e., commutative or anticommutative code word pairs, or
Y = X
and
Z = W
. The principal tool used in this development is a repeated Kronecker product of palindromic factor code words, which we call pseudo-Rademacher-Walsh (PRW) codes since the simplest examples of such products are the normal Rademacher-Walsh codes. The PRW codes are used as a basis for constructing arbitrarily many nontrivial solutions to each of the four possible correlogram identities.