Autocorrelations of Binary Sequences and Run Structure

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 <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> , C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> ,..., C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> ) denote the autocorrelation vector of a binary sequence and Δ the difference operator. We prove that the th component of Δ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> (C) can be directly calculated by using the consecutive runs of total length k. In particular, this shows that the th autocorrelation is already determined by all consecutive runs of total length l <; k.In the aperiodic case, we show how the run vector can be efficiently calculated and give a characterization of skew-symmetric sequences in terms of their run length encoding.