Example of Aliasing:
Example: Consider the signal x(n) = anu(n) , a < 1.
a) Determine the spectrum X(ω)
b) If x(n) is applied to a decimator that reduces the sampling rate by a factor of 2 determine the output spectrum
c) Show that the spectrum in part (b) is simply the Fourier transform of x(2n)
d) Plot the spectra of x(n) and x(2n) for a = 0.905
Solution:
a) The spectrum of x(n) is given by its DTFT
This spectrum is not band-limited but we may pretend it is. This may also be obtained as X(ω)
b = [1]; %Numerator coefficient
a1 = [1, -0.905]; a2 = [1, -0.819]; %Denominator coefficients
w = -pi: pi/256: pi; %A total of 512 points
[X1w] = freqz(b, a1, w); [X2w] = freqz(b, a2, w)
subplot(2, 1, 1), plot(w, abs(X1w)); legend ('Spectrum of x(n)');
xlabel('Frequency \omega, rad/sample'), ylabel('Magnitude of X1(\omega)'); grid
subplot(2, 1, 2), plot(w, abs(X2w)); legend ('Spectrum of x(2n)');
xlabel('Frequency \omega, rad/sample'), ylabel('Magnitude of X2(\omega)'); grid
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