New Sequences Design from Weil Representation with Low Two-Dimensional Correlation in Both Time and Phase Shifts

TL;DR

This paper introduces a new sequence construction based on Weil representation that achieves low two-dimensional correlation in both time and phase, with bounded autocorrelation and spectral properties, useful for signal processing.

Contribution

It presents a novel sequence design using Weil representation with explicit bounds on autocorrelation and spectral magnitude, enhancing sequence efficiency and correlation properties.

Findings

Sequences have autocorrelation bounded by 2√p for non-zero shifts.

Pairwise sequences have ambiguity function bounded by 4√p.

Fourier spectrum magnitude is at most 2.

Abstract

For a given prime $p$, a new construction of families of the complex valued sequences of period $p$ with efficient implementation is given by applying both multiplicative characters and additive characters of finite field $\mathbb{F}_p$. Such a signal set consists of $p^2(p-2)$ time-shift distinct sequences, the magnitude of the two-dimensional autocorrelation function (i.e., the ambiguity function) in both time and phase of each sequence is upper bounded by $2\sqrt{p}$ at any shift not equal to $(0, 0)$, and the magnitude of the ambiguity function of any pair of phase-shift distinct sequences is upper bounded by $4\sqrt{p}$. Furthermore, the magnitude of their Fourier transform spectrum is less than or equal to 2. A proof is given through finding a simple elementary construction for the sequences constructed from the Weil representation by Gurevich, Hadani and Sochen. An open problem for directly establishing these assertions without involving the Weil representation is addressed.