Multirate Digital Signal Processing, Oversampling of Analog-to-Digital Conversion, and Undersampling of Bandpass Signals - Digital Signal Processing: Fundamentals and Applications - page 560

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FIGURE 12.3B

Spectrum before downsampling and spectrum after downsampling using the anti-aliasing filter.

% Generate 2048 samples
fs ¼ 8000; % Sampling rate

N ¼ 2048; % Number of samples

M ¼ 2; % Downsample factor

n ¼ 0:1:N-1;

x cos(5*n*pi/8);

% Compute single-sided amplitude spectrum
% AC component will be doubled, and DC component will be kept at the same value
X ¼ 2*abs(fft(x,N))/N;X(1) ¼ X(1)/2;
% Map the frequency index up to the folding frequency in Hz
f ¼ [0:1:N/2-1]*fs/N;
% Downsampling
y ¼ x(1:M:N);
NM ¼ length(y); % Length of the downsampled data
% Compute the single-sided amplitude spectrum for the downsampled signal
Y ¼ 2*abs(fft(y,NM))/length(y);Y(1) ¼ Y(1)/2;
% Map the frequency index to the frequency in Hz
fsM ¼ [0:1:NM/2-1]*(fs/M)/NM;
subplot(2,1,1);plot(f,X(1:1:N/2));grid; xlabel( ' Frequency (Hz) ' );
subplot(2,1,2);plot(fsM,Y(1:1:NM/2));grid; xlabel( ' Frequency (Hz) ' );
figure

¼ 5*sin(n*pi/4) þ