Chapter 10 - Fourier Analysis of Discrete-Time Signals and Systems

Abstract

This chapter discusses the Fourier representation of discrete-time signals and systems. Although theoretically useful, the discrete-time Fourier transform (DTFT) is computationally not feasible. It is the Fourier series of discrete-time signals that makes the Fourier representation computationally feasible. If the region of convergence of the Z-transform of a signal or of the transfer function of a discrete system includes the unit circle, then the DTFT of the signal or the frequency response of the system is easily found. Duality in time and frequency can be used to obtain the Fourier representation of most discrete-time signals and systems. Two computational disadvantages of the DTFT are: the direct DTFT is a function of a continuously varying frequency and the inverse DTFT requires integration. The Fourier series coefficients constitute a periodic sequence of the same period as the signal; thus both are periodic. Moreover, the Fourier series and its coefficients are obtained as sums and the frequency used is discretized. Thus, they can be obtained by computer. To take advantage of this, the spectrum of an aperiodic signal resulting from the DTFT is sampled so that in the time domain there is a periodic repetition of the original signal. For finite-support signals, a periodic extension that gives the discrete Fourier transform or DFT can be obtained. The significance of this result is that frequency representations of discrete-time signals are computed algorithmically.