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326 Chapter 7 FIR FILTER DESIGN
4. Using the minimum stopband attenuation, the transition bandwidth
can be accurately computed. It is shown in the accumulated amplitude
response plot in Figure 7.9. This computed exact transition bandwidth is
ω s − ω p = 1.8π
M
which is less than half the approximate bandwidth of 4π/M.
Clearly, this is a simple window operation in the time domain and
an easy function to analyze in the frequency domain. However, there are
two main problems. First, the minimum stopband attenuation of 21 dB is
insufficient in practical applications. Second, the rectangular windowing
being a direct truncation of the infinite length h d (n), it suffers from the
Gibbs phenomenon. Ifweincrease M, the width of each side lobe will
decrease, but the area under each lobe will remain constant. Therefore, the
relative amplitudes of side lobes will remain constant, and the minimum
stopband attenuation will remain at 21 dB. This implies that all ripples
will bunch up near the band edges. It is shown in Figure 7.10.
Since the rectangular window is impractical in many applications,
we consider other fixed window functions that provide a fixed amount
Integrated Amplitude Response
Integrated Amplitude Response
M = 7
1
0
−1 0 1
frequency in π units
M = 51
1
0
−1 0 1
frequency in π units
Integrated Amplitude Response
Integrated Amplitude Response
M = 21
1
0
−1 0 1
frequency in π units
M = 101
1
0
−1 0 1
frequency in π units
FIGURE 7.10
Gibbs phenomenon
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