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A constructive analysis of the aperiodic binary correlation function

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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.

Published in:

IEEE Transactions on Information Theory  (Volume:15 ,  Issue: 3 )