Fourier Transform of Digital Signals

Abstract

This chapter presents the main properties of the Fourier transformation of digital signals. We explain the Poisson’s summation formula which is the essential formula for Fourier analysis of periodic signals. We demonstrate the Shannon theorem, which sets the conditions for which sampling takes place without loss of information. We show the Whittaker–Shannon theorem proving that an analog signal can be reconstructed from its samples if the sampling was done respecting the Shannon condition. We look into the situation where initially infinite-length signals have their support truncated by the multiplication by a finite duration window keeping only the samples lying within a time interval, a rectangular window. We study the effects induced in the frequency domain by this limitation and present the use of apodization windows introduced to minimize the drawbacks of time limitation. We present the FFT algorithm and the use of zero padding for signal interpolation.