TABLE OF CONTENTS We have seen that the discrete-time Fourier transform (DTFT) of a system's impulse response (the frequency response of the system) exists whenever the system is bounded-input bounded-output stable or, equivalently, whenever the region of convergence (ROC) of the system's transfer function includes the unit circle. Then, the frequency response of the system is simply its transfer function H( z) evaluated on the unit circle z = 1, that is, z = e j?. In this chapter, we discuss the geometric relationship between the poles and zeros of the z-transform of a signal and its corresponding Fourier transform. It is possible to estimate qualitatively the frequency response of a system just by looking at the pole-zero plot of its transfer function. For example, referring back to Figure 13.1, complex poles close to the unit circle will tend to raise the magnitude of the frequency response at nearby frequencies.
We will look in some detail at the connections between the frequency responses of first-order and second-order systems and their time-domain impulse and step responses. Finally, the analysis and design of discrete-time infinite impulse response filters and finite impulse response filters will be discussed. In particular, the window design technique of finite impulse response filters will be introduced.
If we write the rational transfer function H( z) of a causal stable system in...
