passed without attenuation is the passband of the filter, and the range of frequencies
that is attenuated is the stopband. Thus the magnitude response of an ideal filter is given
by jH!j 1 in the passband and jH!j 0 in the stopband. Note that the frequency
response H! of a digital filter is a periodic function of !, and the magnitude response
jH!j of a digital filter with real coefficients is an even function of !. Therefore the
digital filter specifications are given only for the range 0 ! p.
The magnitude response of an ideal lowpass filter is illustrated in Figure 5.1(a). The
regions 0 ! !
c
and ! > !
c
are referred to as the passband and stopband, respec-
tively. The frequency that separates the passband and stopband is called the cut-off
frequency !
c
. An ideal lowpass filter has magnitude response jH!j 1 in the fre-
quency range 0 ! !
c
and has jH!j 0 for ! > !
c
. Thus a lowpass filter passes all
low-frequency components below the cut-off frequency and attenuates all high-fre-
quency components above !
c
. Lowpass filters are generally used when the signal
components of interest are in the range of DC to the cut-off frequency, but other higher
frequency components (or noise) are present.
The magnitude response of an ideal highpass filter is illustrated in Figure 5.1(b). The
regions ! ! !
c
and 0 ! < !
c
are referred to as the passband and stopband, respec-
tively. A highpass filter passes all high-frequency components above the cut-off fre-
quency !
c
and attenuates all low-frequency components below !
c
. As discussed in
Chapter 1, highpass filters can be used to eliminate DC offset, 60 Hz hum, and other
low frequency noises.
The magnitude response of an ideal bandpass filter is illustrated in Figure 5.1(c). The
regions ! < !
a
and ! > !
b
are referred to as the stopband. The frequencies !
a
and !
b
are called the lower cut-off frequency and the upper cut-off frequency, respectively. The
H(w)
H(w)
H(w)
H(w)
w
1
0
w
c
p
0
w
a
w
b
p
0
w
a
w
b
p
0
w
c
p
w
1
(a)
(b)
w
w
1
1
(c)
(d)
Figure 5.1 Magnitude response of ideal filters: (a) lowpass, (b) highpass, (c) bandpass, and
(d) bandstop
184
DESIGN AND IMPLEMENTATION OF FIR FILTERS