4.8 FREQUENCY RESPONSE OF AN LTIC SYSTEM

Filtering is an important area of signal processing. Filtering characteristics of a system are indicated by its response to sinusoids of various frequencies varying from 0 to ?. Such characteristics are called the frequency response of the system. In this section, we shall find the frequency response of LTIC systems.

In Section 2.4-4 we showed that an LTIC system response to an everlasting exponential input x( t) = e ^st is also an everlasting exponential H( s) e ^st. As before, we use an arrow directed from the input to the output to represent an input -output pair:

Setting s = j ? in this relationship yields

Noting that cos ? t is the real part of e ^{j ? t}, use of Eq. (2.40) yields

We can express H( j ?) in the polar form as

With this result, the relationship (4.73) becomes

In other words, the system response y( t) to a sinusoidal input cos ? t is given by

Using a similar argument, we can show that the system response to a sinusoid cos ( ? t + ?) is

This result is valid only for BIBO-stable systems. The frequency response is meaningless for BIBO-unstable systems. This follows from the fact that the frequency response in Eq. (4.72) is obtained by setting s = j ? in Eq. (4.71). But,...