Sequence Pairs with Lowest Combined Autocorrelation and Crosscorrelation

TL;DR

This paper characterizes when binary sequence pairs meet a fundamental autocorrelation and crosscorrelation bound, linking it to Golay pairs, and extends the results to complex sequences, analyzing their correlation properties.

Contribution

It proves that the Pursley-Sarwate bound is exactly met by Golay complementary pairs and generalizes this characterization to complex-valued sequences.

Findings

Golay pairs meet the Pursley-Sarwate bound exactly.

The bound is generalized to complex sequences.

Asymptotic correlation factors are computed for Golay families.

Abstract

Pursley and Sarwate established a lower bound on a combined measure of autocorrelation and crosscorrelation for a pair $(f,g)$ of binary sequences (i.e., sequences with terms in $\{-1,1\}$). If $f$ is a nonzero sequence, then its autocorrelation demerit factor, $\text{ADF}(f)$, is the sum of the squared magnitudes of the aperiodic autocorrelation values over all nonzero shifts for the sequence obtained by normalizing $f$ to have unit Euclidean norm. If $(f,g)$ is a pair of nonzero sequences, then their crosscorrelation demerit factor, $\text{CDF}(f,g)$, is the sum of the squared magnitudes of the aperiodic crosscorrelation values over all shifts for the sequences obtained by normalizing both $f$ and $g$ to have unit Euclidean norm. Pursley and Sarwate showed that for binary sequences, the sum of $\text{CDF}(f,g)$ and the geometric mean of $\text{ADF}(f)$ and $\text{ADF}{(g)}$ must be at least $1$. For randomly selected pairs of long binary sequences, this quantity is typically around $2$. In this paper, we show that Pursley and Sarwate's bound is met for binary sequences precisely when $(f,g)$ is a Golay complementary pair. We also prove a generalization of this result for sequences whose terms are arbitrary complex numbers. We investigate constructions that produce infinite families of Golay complementary pairs, and compute the asymptotic values of autocorrelation and crosscorrelation demerit factors for such families.