Sequence Pairs with Lowest Combined Autocorrelation and Crosscorrelation

For a pair $(f,g)$ of sequences of length $\ell$ whose terms are in $\{-1,1\}$, Pursley and Sarwate established a lower bound on a combined measure of crosscorrelation and autocorrelation for $f$ and $g$. They showed that the sum of the mean square crosscorrelation between $f$ and $g$ and the geometric mean of $f$'s mean square autocorrelation and $g$'s mean square autocorrelation must be at least $1$. For randomly selected binary sequences, this quantity is typically about $2$. In this paper, we show that Pursley and Sarwate's bound is met precisely when $(f,g)$ is a Golay complementary pair. This result generalizes to sequences whose terms are arbitrary complex numbers.