Peak Sidelobe Level and Peak Crosscorrelation of Golay-Rudin-Shapiro Sequences

TL;DR

This paper establishes new exponential upper bounds on the peak sidelobe level and crosscorrelation of Golay-Rudin-Shapiro sequences, improving understanding of their correlation properties for applications in communications.

Contribution

It provides tighter exponential bounds on the peak sidelobe level and crosscorrelation for Golay-Rudin-Shapiro sequences, extending previous results to all sequences generated by the recursion.

Findings

Peak sidelobe level bounded by 5(1.658967...)^{n-4}

Bound applies to all Rudin-Shapiro sequences and their Golay-Rudin-Shapiro family

Bounds are tight in the exponential growth rate for large n

Abstract

Sequences with low aperiodic autocorrelation and crosscorrelation are used in communications and remote sensing. Golay and Shapiro independently devised a recursive construction that produces families of complementary pairs of binary sequences. In the simplest case, the construction produces the Rudin-Shapiro sequences, and in general it produces what we call Golay-Rudin-Shapiro sequences. Calculations by Littlewood show that the Rudin-Shapiro sequences have low mean square autocorrelation. A sequence's peak sidelobe level is its largest magnitude of autocorrelation over all nonzero shifts. H{\o}holdt, Jensen, and Justesen showed that there is some undetermined positive constant $A$ such that the peak sidelobe level of a Rudin-Shapiro sequence of length $2^n$ is bounded above by $A(1.842626\ldots)^n$, where $1.842626\ldots$ is the positive real root of $X^4-3 X-6$. We show that the peak sidelobe level is bounded above by $5(1.658967\ldots)^{n-4}$, where $1.658967\ldots$ is the real root of $X^3+X^2-2 X-4$. Any exponential bound with lower base will fail to be true for almost all $n$, and any bound with the same base but a lower constant prefactor will fail to be true for at least one $n$. We provide a similar bound on the peak crosscorrelation (largest magnitude of crosscorrelation over all shifts) between the sequences in each Rudin-Shapiro pair. The methods that we use generalize to all families of complementary pairs produced by the Golay-Rudin-Shapiro recursion, for which we obtain bounds on the peak sidelobe level and peak crosscorrelation with the same exponential growth rate as we obtain for the original Rudin-Shapiro sequences.