Autocorrelations of Binary Sequences and Run Structure

TL;DR

This paper establishes a mathematical relationship between the autocorrelation properties of binary sequences and their run structure, providing formulas and characterizations that facilitate analysis of sequence correlations.

Contribution

It introduces formulas linking autocorrelation to run structure and characterizes skew-symmetric sequences through run length encoding.

Findings

Autocorrelation can be expressed via run structure.

Second order difference of autocorrelation relates to run lengths.

Skew-symmetric sequences characterized by run length encoding.

Abstract

We analyze the connection between the autocorrelation of a binary sequence and its run structure given by the run length encoding. We show that both the periodic and the aperiodic autocorrelation of a binary sequence can be formulated in terms of the run structure. The run structure is given by the consecutive runs of the sequence. Let C=(C(0), C(1),...,C(n)) denote the autocorrelation vector of a binary sequence. We prove that the kth component of the second order difference operator of C can be directly calculated by using the consecutive runs of total length k. In particular this shows that the kth autocorrelation is already determined by all consecutive runs of total length L<k. In the aperiodic case we show how the run vector R can be efficiently calculated and give a characterization of skew-symmetric sequences in terms of their run length encoding.