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Window Design Techniques 343
Ideal Impulse Response
Hann Window
1
0.5
1
0.8
hd(n)
0
w(n)
0.6
−0.5
−1
0 5 10 15 20
n
0.4
0.2
0
0 5 10 15 20
n
Actual Impulse Response
Amplitude Response
1
1
0.5
h(n)
0
Hr
0
−0.5
−1
0 5 10 15 20
n
−1
−1 0 1
frequency in π units
FIGURE 7.24 FIR Hilbert transformer design in Example 7.13
>> subplot(2,2,2); stem(n,w_han);title(’Hann Window’)
>> axis([-1 M 0 1.2]); xlabel(’n’); ylabel(’w(n)’)
>> subplot(2,2,3); stem(n,h);title(’Actual Impulse Response’)
>> axis([-1 M -1.2 1.2]); xlabel(’n’); ylabel(’h(n)’)
>> w = w’; Hr = Hr’;
>> w = [-fliplr(w), w(2:501)]; Hr = [-fliplr(Hr), Hr(2:501)];
>> subplot(2,2,4);plot(w/pi,Hr); title(’Amplitude Response’);grid;
>> xlabel(’frequency in pi units’); ylabel(’Hr’); axis([-1 1 -1.1 1.1]);
The plots are shown in Figure 7.24. Observe that the amplitude response is
plotted over −π ≤ ω ≤ π.
□
The SP toolbox provides a function called fir1 which designs conventional
lowpass, highpass, and other multiband FIR filters using window
technique. This function’s syntax has several forms, including:
• h = fir1(N,wc) designs an Nth-order (N = M − 1) lowpass FIR
filter and returns the impulse response in vector h. Bydefault this is
a Hamming-window based, linear-phase design with normalized cutoff
frequency in wc which is a number between 0 and 1, where 1 corresponds
to π rad/sample. If wc is a two-element vector, i.e., wc = [wc1 wc2],
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