A constructive analysis of the aperiodic binary correlation function
Gustavus J. Simmons · IEEE Transactions on Information Theory · 1969
The aperiodic binary correlation function or correlogramX \ast Yoccurs as the output of the detector in the correlation detection of a binary code wordYin an unsynchronized code streamX. 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 ofX \ast Yas a function of the code wordsXandYare developed in this paper. In particular, sufficient conditions for the general quadratic equationX \ast Y = W \ast Zto possess solutions are demonstrated, as well as for the restricted cases in whichW = \pm YandZ = \pm X, i.e., commutative or anticommutative code word pairs, orY = XandZ = 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.