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518 Chapter 9 SAMPLING RATE CONVERSION
Solution
The FIR filter that meets the specifications of this problem is exactly the same
as the filter designed in Example 9.8. Its bandwidth is π/5.
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□ EXAMPLE 9.16 A signal x(n) has a total bandwidth of 0.9π. Itisresampled by a factor of
4/3 toobtain y(m). We want to preserve the frequency band up to 0.8π and
require that the band up to 0.7π be free of aliasing. Using the Parks-McClellan
algorithm, determine the coefficients of the FIR filter that has 0.1 dB ripple in
the passband and 40 dB attenuation in the stopband.
Solution
The overall signal bandwidth is ω x,s1 =0.9π, the bandwidth to be preserved is
ω x,p =0.8π, and the bandwidth above which aliasing is tolerated is ω x,s2 =0.7π.
From (9.60) and using I =4and D =3,the FIR filter design parameters are
ω p =0.2π and ω s =0.275π. With these parameters, along with the passband
ripple of 0.1 dB and stopband attenuation of 40 dB, the optimum FIR filter
length is 58. The details and computation of design plots follow.
% Given Parameters:
I = 4; D = 3; Rp = 0.1; As = 40;
wxp = 0.8*pi; wxs1 = 0.9*pi; wxs2 = 0.7*pi;
% Computed Filter Parameters
wp = wxp/I; ws = min((2*pi/I-wxs1/I),(2*pi/D-wxs2/I));
% Filter Design
[delta1,delta2] = db2delta(Rp,As);
[N,F,A,weights] = firpmord([wp,ws]/pi,[1,0],[delta1,delta2],2);
N = ceil(N/2)*2+1; h = firpm(N,F,I*A,weights);
Hf1 = figure(’units’,’inches’,’position’,[1,1,6,3],...
’paperunits’,’inches’,’paperposition’,[0,0,6,3]);
% Filter Design Plots
[Hr,w,a,L] = Ampl_res(h); Hr_min = min(Hr); w_min = find(Hr == Hr_min);
H = abs(freqz(h,1,w)); Hdb = 20*log10(H/max(H)); min_attn = Hdb(w_min);
subplot(2,1,1); plot(w/pi,Hr,’m’,’linewidth’,1.0); axis([0,1,-0.1,I+0.1]);
set(gca,’xtick’,[0,wp/pi,ws/pi,1],’ytick’,[0,I]); grid;
ylabel(’Amplitude’,’vertical’,’cap’);
title(’Amplitude Response’,’fontsize’,TF,’vertical’,’baseline’);
xlabel(’Frequency in \pi units’,’vertical’,’middle’);
subplot(2,1,2); plot(w/pi,Hdb,’m’,’linewidth’,1.0); axis([0,1,-60,10]);
set(gca,’xtick’,[0,wp/pi,ws/pi,1],’ytick’,[-60,round(min_attn),0]); grid
ylabel(’Decibels’,’vertical’,’cap’);
xlabel(’Frequency in \pi units’,’vertical’,’middle’);
title(’Log-magnitude Response’,’fontsize’,TF,’vertical’,’baseline’);
The filter responses are shown in Figure 9.29, which shows that the designed
filter achieves the attenuation of 40 db.
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