Sampling and Fourier Analysis

Abstract

Up to now, we have studied the fundamental definitions and properties of discrete-time signals and systems with only parenthetical reference to the analogous theory for continuous-time signals and systems. We have thereby demonstrated that the discrete-time theory stands on its own without the necessity of any direct reference to continuous-time theory, given that we assume the form of the discrete-time input sequences. It may well be that our input is digitally generated, for example, and that the output from our filter is to be processed or displayed without conversion to a continuous-time signal. In that case, we need not consider the relationship between discrete- and continuous-time signals. However, the real physical world is continuous in time and space, and our input is usually derived from continuous-time (or equivalent) signals. The output data must often be returned to this form, as well. Therefore, in this chapter we will study the effects of conversion from continuous- to discrete-time signals, and vice versa, and will relate the z transform to the Fourier transform. In particular, the convergence, properties, and application of the discrete-time Fourier transform will be investigated in some detail.