Why does the frequency spectra change when the sampling frequency is changed? - matlab

Question

Here is my code for generating a triangular waveform in the time domain and for generating its corresponding fourier series/transform (I don't know whether its series or transform because matlab only has fourier transform function but since the signal is periodic, references say that the fourier counterpart must be called fourier series).
x = 0;
s = 50; % number of sinusoidal components
fs = 330; % hertz
dt = 1/fs; % differential time
t = [0:dt:4]; % seconds
const = 2 / (pi^2);
for k = 1:2:s,
x = x + (((-1)^((k - 1) / 2)) / (k^2)) * sin(4*pi*k*t);
end
x = const * x;
% amplitude = max(x) = 0.2477
% period = 0.5 seconds
f = linspace(-fs/2,fs/2,length(x));
xk = fftshift(fft(x));
figure;
subplot(3,1,1);
plot(t,x);
grid on;
xlabel('time(seconds)');
title('Time Domain');
subplot(3,1,2);
plot(f,abs(xk));
grid on;
xlabel('frequency(hertz)');
title('Magnitude Spectrum');
subplot(3,1,3);
plot(f,angle(xk));
grid on;
xlabel('frequency(hertz)');
title('Phase Spectrum');
And here is the generated plots for the time domain signal, magnitude spectrum, and phase spectrum.
link:
fs = 330hz
My problem is when I changed the sampling frequency (fs which is currently equal to 330 hz) to another value, the plots of the magnitude and phase spectra change.
Here is the plots of the magnitude and phase spectra when the sampling frequency is equal to 400 hz:
link:
fs = 400 hz
Can you explain why does this happen? And what can I do in order to get a constant plots for the magnitude and phase spectra given any sampling frequency?

Answer 1

I can't get your pictures to load over my proxy, but the spectrum of a FFT will be have a bigger "gap" in the middle at a higher sampling rate. A fundamental property of sampling is that it introduces copies of your original spectrum; you may have learned this if you studied the discrete-time Fourier transform. At a higher sampling rate, these copies are farther apart.
Additionally, your sampling points will be in different places at different sampling rates, so you may get different lobing behavior.
Incidentally, you are getting the discrete Fourier transform in Matlab -- you are giving it a finite sequence of discrete points, not a continuous, inifintely long signal.
If you want the plots to look the same, just make their x-axes match.

Answer 2

It because that the spectra by DFT/FFT are indeed the sampled and normalized version of the original analog spectra, therefore, with the sampling step changes, the sampling step in frequency domain also changes, thus the spectra lines that you see also change since the original spectra are not constant. Another factor may be the aliasing effects, since the analog spectra of triangular waveform are infinite in theory.