Frequency Domain Representation of Discrete-Time Signals and Systems

Abstract

Signals, continuous-time, or discrete-time occur in the time domain. We, therefore, described rather elaborately discrete-time signals and systems in the time domain. For easier and more efficient ways to analyze such signals and systems, we next introduced the Z-transform, which is an alternative representation of discrete-time signals and systems. The Z-transform maps a discrete-time signal or an LTI discrete-time system from the discrete-time domain into a complex plane. In this plane, the discrete-time signals and systems are represented by their poles and zeros. There is another domain in which a discrete-time signal or equivalently an LTI discrete-time system can be represented. This domain is the frequency domain. We can visualize a signal more easily in the frequency domain than in the time domain. For instance, a sum of sinusoidal signals with differing frequencies is hard to identify in the time domain individually. On the other hand, such a signal can be easily identified individually in the frequency domain. This is illustrated in Figs. 4.1a and b. The discrete-time signal is shown in Fig. 4.1a. It consists of three sinusoids at frequencies 13, 57, and 93 Hz with amplitudes 1, 1.5, and 2, respectively, at a sampling frequency of 500 Hz. It is hard to discern the individual sinusoids from the figure. Figure 4.1b shows the discrete-time Fourier transform (DTFT) representation of the signal in Fig. 4.1a. One can clearly distinguish the three components in frequency and relative amplitude. The DTFT also greatly aids in the design of LTI discrete-time systems. This chapter deals with the representation of discrete-time signals and systems in the frequency domain. More specifically, we will define a mapping known as the discrete-time Fourier transform that characterizes discrete-time signals and systems in the frequency domain. As a consequence, we will show the relationship between the Z-transform and DTFT. We will also observe that the DTFT characterizes an LTI discrete-time system in the frequency domain. Because of this property, we can define the filtering operations. Filters such as lowpass, highpass, bandpass, bandstop, etc., can be characterized more efficiently in the frequency domain. It further leads us to the design of such filters (Tables 4.1, 4.2 and 4.3).