TABLE OF CONTENTS Given a linear shift-invariant system with a unit sample response h( n), the input and output are related by a convolution sum
As discussed in Chap. 2, this relationship implies that Y( e j ?) = X( e j ?) H( e j ?) where H( e j ?), the frequency response of the system, is the discrete-time Fourier transform of h( n). This relationship between x( n) and y( n) may also be expressed in the z-transform domain as
where H( z), the z-transform of h( n), is the system function of the LSI system. The system function is very useful in the description and analysis of LSI systems. In this chapter, we look at the characterization of a linear shift-invariant system in terms of its system function and discuss special types of LSI systems such as linear phase systems, allpass systems, minimum phase systems, and feedback networks.
The frequency response of a linear shift-invariant system is the discrete-time Fourier transform of the unit sample response, and the system function is the z-transform of the unit sample response:
The frequency response may be derived from the system function by evaluating H( z) around the unit circle:
If the z-transform of the input to a linear shift-invariant system with a system function H( z) is
