DESIGN AND ANALYSIS OF ANALOG FILTERS A Signal ...

A Signal Processing Perspective 33
passband edges and the time-domain response, while still retaining something of the
“ideal” filter construct. Consider the family of raised-cosine lowpass filters
represented as follows:
where RC denotes raised-cosine and is a real parameter to be specified, bounded
by 0 and 1:
This family of raised-cosine lowpass filters is illustrated in Figure 2.13. The
corresponding unit impulse response is as follows:
A plot of (2.10) is shown in Figure 2.14 for three values of and
Compare Figure 2.14 with the plot in Figure 2.3.
Note that if (2.10) becomes identical with (2.2), and a plot of (2.9)
would become identical to that shown in Figure 2.2. In other words, the “ideal”
lowpass filter of Figure 2.2 and (2.2) is a special case of (2.9) and (2.10). If
then (2.9) becomes
Section 2.1 Time, Frequency, and s Domains