Frequency-Domain Analysis - Introduction to Digital Signal Processing and Filter Design - page 125

Magnitude response of Chebyshev I bandpass filter

1

Passband

0.8

0.6

0.4

0.2

Stopband Stopband

0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalized frequency w/pi

( a )

Magnitude response of Chebyshev II bandstop filter

1.4

1.2

1

Passband Passband

0.8

0.6

0.4

0.2

Stopband

0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalized frequency w/pi

( b )

Figure 3.10 Approximation of ideal bandpass and bandstop digital filters.

well as in the stopband, which is equiripple in nature, whereas in Figure 3.9b,
the magnitude of a highpass filter is approximated by a Butterworth type of
approximation, which shows that the magnitude in the passband is “nearly flat”
and decreases monotonically as the frequency decreases from the passband.
Figure 3.10a illustrates a Chebyshev type I approximation of an ideal band-
pass filter, which has an equiripple error in the passband and a monotonically
decreasing response in the stopband, whereas in Figure 3.10b, we have shown a
Chebyshev type II approximation of an ideal bandstop filter; thus, the error in
the stopband is equiripple in nature and is monotonic in the passband. The exact
definition of these criteria and the design of filters meeting these criteria will be
discussed in the next two chapters.

3.3.1 Time-Domain Analysis of Noncausal Inputs

Let the DTFT of the input signal x(n) and the unit impulse response h(n) of
a discrete-time system be X(e jω ) and H(e jω ) , respectively. The output y(n)
is obtained by the convolution sum x(n) = k =−∞ h(k)x(n k) ,

which shows that the convolution sum is applicable even when the input signal

∗ h(n) = y(n) −