Page 238 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 51 FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS

Thus, from Eq. (5.67)

which is a rational function of o. The result (5.85) is the same as the Laplace transform
counterpart H(s) = Y(s)/X(s) with s = jo [Eq. (3.40)], that is,

5.6 FILTERING

One of the most basic operations in any signal processing system is filtering. Filtering is
the process by which the relative amplitudes of the frequency components in a signal are
changed or perhaps some frequency components are suppressed. As we saw in the
preceding section, for continuous-time LTI systems, the spectrum of the output is that of
the input multiplied by the frequency response of the system. Therefore, an LTI system
acts as a filter on the input signal. Here the word "filter" is used to denote a system that
exhibits some sort of frequency-selective behavior.

A. Ideal Frequency-Selective Filters:
An ideal frequency-selective filter is one that exactly passes signals at one set of
frequencies and completely rejects the rest. The band of frequencies passed by the filter is
referred to as the pass band, and the band of frequencies rejected by the filter is called the
stop band.
The most common types of ideal frequency-selective filters are the following.
1. Ideal Low-Pass Filter:

An ideal low-pass filter (LPF) is specified by

which is shown in Fig. 5-5(a). The frequency o, is called the cutoff frequency.
2. Ideal High-Pass Filter:

An ideal high-pass filter (HPF) is specified by

which is shown in Fig. 5-5(b).