Optimal ambiguity functions and Weil's exponential sum bound

TL;DR

This paper establishes an optimal bound on the ambiguity functions of Bjorck CAZAC sequences using Weil's exponential sum bound, with implications for communications and radar applications.

Contribution

It provides the first optimal magnitude bound for the ambiguity functions of Bjorck CAZAC sequences, leveraging Weil's exponential sum bound.

Findings

Bound |A_p(u)|  2/0 4/p outside (0,0)

Bound is proven to be optimal given the CAZAC constraints

Uses Weil's exponential sum bound linked to the Riemann hypothesis for finite fields

Abstract

Complex-valued periodic sequences, u, constructed by Goran Bjorck, are analyzed with regard to the behavior of their discrete periodic narrow-band ambiguity functions A_p(u). The Bjorck sequences, which are defined on Z/pZ for p>2 prime, are unimodular and have zero autocorrelation on (Z/pZ)\{0}. These two properties give rise to the acronym, CAZAC, to refer to constant amplitude zero autocorrelation sequences. The bound proven is |A_p(u)| \leq 2/\sqrt{p} + 4/p outside of (0,0), and this is of optimal magnitude given the constraint that u is a CAZAC sequence. The proof requires the full power of Weil's exponential sum bound, which, in turn, is a consequence of his proof of the Riemann hypothesis for finite fields. Such bounds are not only of mathematical interest, but they have direct applications as sequences in communications and radar, as well as when the sequences are used as coefficients of phase-coded waveforms.