Abstract
Chirps arise in many signal processing applications. While chirps have been extensively studied as functions over both the real line and the integers, less attention has been paid to the study of chirps over finite groups. We study the existence and properties of chirps over finite cyclic groups of integers. In particular, we introduce a new definition of a finite chirp which is slightly more general than those that have been previously used. We explicitly compute the discrete Fourier transforms of these chirps, yielding results that are number-theoretic in nature. As a consequence of these results, we determine the degree to which the elements of certain finite tight frames are well distributed.
Highlights
A linear chirp is a function whose frequency changes linearly with time
The study of chirps has mostly been confined to the real line and the integers, in the context of integral transforms and the chirp Z-transform, respectively
Less attention has been paid to the study of chirps over finite cyclic groups, that is, to chirps over Za ≡ Z/aZ = {0, . . . , a − 1}, where a is a positive integer
Summary
A linear chirp is a function whose frequency changes linearly with time. For example, while a wave function of the form exp(2πixt) has constant frequency x, the chirp exp(2πi(xt + yt2/2)) has frequency x + yt at time t ∈ R. A − 1}, where a is a positive integer This is in contrast to wave functions which, in the context of Fourier transforms, have been studied for many decades on arbitrary locally compact abelian groups. The computation of Gauss sums [3] is equivalent to finding the discrete Fourier transform (DFT) of a finite chirp, and was a subject of great interest in the mid-nineteenth century. This connection between modern signal processing and classical number theory is emphasized in [4], in which the trace of the DFT matrix, Tr. is noted to be the canonical example of a Gauss sum.
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