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504 Chapter 9 SAMPLING RATE CONVERSION
Input Signal x(n)
Output y(n): I = 5, Filter Length = 31
1
1
Amplitude
0
Amplitude
0
−1
−1
0 4 8 12 16
n
0 20 40 60 80
m
Output y(n): I = 5, Filter Length = 51 Output y(n): I = 5, Filter Length = 81
1
1
Amplitude
0
Amplitude
0
−1
0 20 40 60 80
m
FIGURE 9.21 Signal plots in Example 9.9
−1
0 20 40 60 80
m
subplot(2,2,2); Hs2 = stem(m,y,’filled’); set(Hs2,’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=31’,’fontsize’,TF);
xlabel(’n’,’vertical’,’middle’); ylabel(’Amplitude’);
The signal stem plots are shown in Figure 9.21. The upper left-hand plot shows
a segment of the input signal x(n), and the upper right-hand plot shows the
interpolated signal y(n) using the filter of length 31. The plot is corrected for
filter delay and the effect of its transient response. It is somewhat surprising that
the interpolated signal is not what it should be. The signal peak is more than
one, and the shape is distorted. A careful observation of the filter response plot
in Figure 9.20 reveals that the broad transition width and a smaller attenuation
has allowed some of the spectral energy to leak, creating a distortion.
To investigate this further, we designed filters with larger orders of 51 and
81, as detailed in the following MATLAB script.
% Interpolation with Filter Design: Length M = 51
M = 51; F = [0,wp,ws,pi]/pi; A = [I,I,0,0];
h = firpm(M-1,F,A,weights); y = upfirdn(x,h,I);
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