مرکز منطقه ای اطلاع رساني علوم و فناوري - A constructive analysis of the aperiodic binary correlation function

Abstract :

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.