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Window Design Techniques 339
The ideal bandstop impulse response can also be determined from the ideal
lowpass impulse response using a method similar to Figure 7.19. We can now
implement the Kaiser window design and check for the minimum stopband
attenuation. This is shown in the following MATLAB script.
>> M = 45; As = 60; n=[0:1:M-1];
>> beta = 0.1102*(As-8.7)
beta = 5.6533
>> w_kai = (kaiser(M,beta))’; wc1 = pi/3; wc2 = 2*pi/3;
>> hd = ideal_lp(wc1,M) + ideal_lp(pi,M) - ideal_lp(wc2,M);
>> h = hd .* w_kai; [db,mag,pha,grd,w] = freqz_m(h,[1]);
>> subplot(2,2,1); stem(n,hd); title(’Ideal Impulse Response’)
>> axis([-1 M -0.2 0.8]); xlabel(’n’); ylabel(’hd(n)’)
>> subplot(2,2,2); stem(n,w_kai);title(’Kaiser Window’)
>> axis([-1 M 0 1.1]); xlabel(’n’); ylabel(’w(n)’)
>> subplot(2,2,3); stem(n,h);title(’Actual Impulse Response’)
>> axis([-1 M -0.2 0.8]); xlabel(’n’); ylabel(’h(n)’)
>> subplot(2,2,4);plot(w/pi,db); axis([0 1 -80 10]);
>> title(’Magnitude Response in dB’);grid;
>> xlabel(’frequency in pi units’); ylabel(’Decibels’)
The β parameter is equal to 5.6533, and, from the magnitude plot in Figure 7.21,
we observe that the minimum stopband attenuation is smaller than 60 dB.
Clearly, we have to increase β to increase the attenuation to 60 dB. The required
value was found to be β =5.9533.
0
Magnitude Response in dB
Decibels
60
0 1/3 2/3 1
frequency in π units
FIGURE 7.21 Bandstop filter magnitude response in Example 7.11 for β =
5.6533
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