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FIR Filter Designs for Sampling Rate Conversion 505
delay = (M-1)/2; % Delay imparted by the filter
m = delay+1:1:50*I+delay+1; y = y(m); m = 1:81; y = y(81:161);
subplot(2,2,3); Hs3 = stem(m,y,’filled’); set(Hs3,’markersize’,2,’color’,’m’);
axis([-5,85,-1.2,1.2]); set(gca,’xtick’,[0:4:16]*I,’ytick’,[-1,0,1]);
title(’Output y(n): Filter length=51’,’fontsize’,TF);
xlabel(’n’,’vertical’,’middle’); ylabel(’Amplitude’);
% Interpolation with Filter Design: Length M = 81
M = 81; F = [0,wp,ws,pi]/pi; A = [I,I,0,0];
h = firpm(M-1,F,A,weights); y = upfirdn(x,h,I);
delay = (M-1)/2; % Delay imparted by the filter
m = delay+1:1:50*I+delay+1; y = y(m); m = 1:81; y = y(81:161);
subplot(2,2,4); Hs3 = stem(m,y,’filled’); set(Hs3,’markersize’,2,’color’,’m’);
axis([-5,85,-1.2,1.2]); set(gca,’xtick’,[0:4:16]*I,’ytick’,[-1,0,1]);
title(’Output y(n): Filter length=81’,’fontsize’,TF);
xlabel(’n’,’vertical’,’middle’); ylabel(’Amplitude’);
The resulting signals are also shown in lower plots in Figure 9.21. Clearly, for
large orders, the filter has better lowpass characteristics. The signal peak value
approaches 1, and its shape approaches the cosine waveform. Thus, a good filter
design is critical even in a simple signal case.
□
9.5.2 DESIGN SPECIFICATIONS
When we replace H I (ω) byafinite-order FIR filter H(ω), we must allow
for a transition band; thus, the filter cannot have a passband edge up to
π/I. Towards this, we define
• ω x,p as the highest frequency of the signal x(n) that we want to preserve
• ω x,s as the full signal bandwidth of x(n),—i.e., there is no energy in
x(n) above the frequency ω x,s .
Thus, we have 0 < ω x,p < ω x,s < π. Note that the parameters ω x,p
and ω x,s ,asdefined are signal parameters, not filter parameters; they are
shown in Figure 9.22a. The filter parameters will be defined based on ω x,p
and ω x,s .
From equation (9.48), these signal parameter frequencies for v(m)
become ω x,p /I and ω x,s /I, respectively, because the frequency scale is
compressed by the factor I. This is shown in Figure 9.22b. A linear-phase
FIR filter can now be designed to pass frequencies up to ω x,p /I and to
suppress frequencies starting at (2π − ω x,s )/I. Let
ω p =
( ωx,p
I
)
( ) 2π − ωx,s
and ω s =
I
(9.50)
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