Fourier transforms of finite chirps

Abstract

Chirps arise in many signal processing applications.While chirps have been extensively studied as functions over boththe real line and the integers, less attention has been paid tothe study of chirps over finite groups. We study the existence andproperties of chirps over finite cyclic groups of integers. Inparticular, we introduce a new definition of a finite chirp whichis slightly more general than those that have been previouslyused. We explicitly compute the discrete Fourier transforms ofthese chirps, yielding results that are number-theoretic innature. As a consequence of these results, we determine the degreeto which the elements of certain finite tight frames arewell distributed.