# Power spectral density of a under sampled signal

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DIPAN KAR on 9 May 2020
Edited: DIPAN KAR on 19 May 2020
Hi ,
I have a under sampled signal x. which is a 2k Hz signal. This same signal is under sampled at 50Hz. I tried to extract to freq response of this signal but as the siganl is sampled below the Nyquest rate I am not able to get it. I also tried with pwelch() function but it is also not giving me correct reasult.
How should I calculate the power spectral density of this signal x?
fs = 50
t = 0:1/fs:2^22- 1/fs;
x = sin(2*pi*2000*t) ;
plot(t,x);
y = fft(x);
f = (0:length(y)-1)*50/length(y)*80;
plot(f,abs(y))
title('Magnitude')
figure(2)
num_segments = 32;
window_length = floor(length(x)/num_segments);
[PSDphase ,f] =pwelch(x ,window_length ,[] ,[] ,50 ,'twosided');
h=semilogx( f, (10* log10(PSDphase)), 'b'); %hold on;
grid on;

Navya Seelam on 12 May 2020
Increase the sampling frequency of the signal to get expected results. When a signal is undersampled , aliasing happens and hence you are not able to get correct result.

DIPAN KAR on 13 May 2020
Dear,
Thank you for the answer. But in the actual problem I can't increase the sampling frequency. So is there a solution of this kind of problem when the sampling frequency is less than that signal frequency?
Regards
Navya Seelam on 14 May 2020
Hi,
One thing to notice here is the signal x is of the form sin(2*pi*n) with n being an integer. So x is actually a zero signal or nearly zero signal (due to precision limitations) and we cannot perform any operation on a zero signal.
DIPAN KAR on 19 May 2020
Thank you for pointing out the error. But I wanted to do fft of the phase and extract the PSD from it. Is this is the correct way?
fs = 50
t = 0:1/fs:2^22- 1/fs;
x = 2*pi*2000*t;
plot(t,x);
y = fft(x);
f = (0:length(y)-1)*50/length(y)*80;
plot(f,abs(y))
title('Magnitude')
figure(2)
num_segments = 32;
window_length = floor(length(x)/num_segments);
[PSDphase ,f] =pwelch(x ,window_length ,[] ,[] ,50 ,'twosided');
h=semilogx( f, (10* log10(PSDphase)), 'b'); %hold on;
grid on;

R2020a

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