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Frequency Sampling Design Techniques 349
This problem is known as an optimization problem, and it is solved using
linear programming techniques. We demonstrate the effect of transition
band sample variation on the design using the following example.
□ EXAMPLE 7.15 Using the optimum design method, design a better lowpass filter of Example
7.14.
Solution
Let us choose M = 40sothat we have one sample in the transition band
△
0.2π > Hrs = [ones(1,5),0.5,zeros(1,29),0.5,ones(1,4)];
>> k1 = 0:floor((M-1)/2); k2 = floor((M-1)/2)+1:M-1;
>> angH = [-alpha*(2*pi)/M*k1, alpha*(2*pi)/M*(M-k2)];
>> H = Hrs.*exp(j*angH);
>> h = real(ifft(H,M));
From the plots of this design in Figure 7.27, we observe that the minimum
stopband attenuation is now 30 dB, which is better than the naive design attenuation
but is still not at the acceptable level of 50 dB. The best value for T 1
was obtained by varying it manually (although more efficient linear programming
techniques are available, these were not used in this case), and the near
optimum solution was found at T 1 =0.39. The resulting filter is a obtained
using the following MATLAB script.
% T1 = 0.39
>> M = 40; alpha = (M-1)/2;
>> Hrs = [ones(1,5),0.39,zeros(1,29),0.39,ones(1,4)];
>> k1 = 0:floor((M-1)/2); k2 = floor((M-1)/2)+1:M-1;
>> angH = [-alpha*(2*pi)/M*k1, alpha*(2*pi)/M*(M-k2)];
>> H = Hrs.*exp(j*angH); h = real(ifft(H,M));
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