A constructive analysis of the aperiodic binary correlation function

The aperiodic binary correlation function or correlogram occurs as the output of the detector in the correlation detection of a binary code word in an unsynchronized code stream . 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 as a function of the code words and are developed in this paper. In particular, sufficient conditions for the general quadratic equation to possess solutions are demonstrated, as well as for the restricted cases in which and , i.e., commutative or anticommutative code word pairs, or and . 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.