Power spectral density of gaussian white noise

Question

TheBeginner 2013-11-8

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https://ww2.mathworks.cn/matlabcentral/answers/105368-power-spectral-density-of-gaussian-white-noise

回答： Jeremy 2015-6-18

Hi, I just wanted to check that the matlab function "pwelch" gives a correct estimates of the PSD of a gaussian white noise. I plot the estimate of the PSD and also the variance, which is supposed to be equal to the mean of PSD. I always have a bias that I don't understand.

Here's my code :

sigma = 1000;
L = 200000;
%PSD
noise= randn(1,L) * sigma;
[PSD_noise,vect_freq_noise] = pwelch(noise,75000,[],4000);
%Display
plot(vect_freq_noise,2*PSD_noise,'-g'), %PSD
%Theoritical value of the PSD
hold on, plot([vect_freq_noise(1) vect_freq_noise(end)],[sigma^2 sigma^2],'r');
%Value of the mean
plot([vect_freq_noise(1) vect_freq_noise(end)],[mean(2*PSD_noise) mean(2*PSD_noise)],'b');

Any idea where's the mistake?

Thank you

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TheBeginner 2013-11-8

在 MATLAB Online 中打开

I have also tried to calculate myself the PSD :

sigma = 1000;
L = 200000;
noise= randn(1,L) * sigma;
PSD = abs(fft(noise)).^2/length(noise);
vect_freq = linspace(0,pi,length(PSD));
plot(vect_freq,PSD,'-.g');
hold on, plot([vect_freq(1) vect_freq(end)],[sigma^2 sigma^2],'r');
hold on, plot([vect_freq(1) vect_freq(end)],[mean(PSD) mean(PSD)],'-.b');

And it works fine. I think there's a normalization factor in pwelch that I can't quite figure out.

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Answer 1

Jeremy 2015-6-18

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在 MATLAB Online 中打开

The "density" in PSD means that the power is normalized to something, usually 1 Hz, but in this case it is the Nyquist frequewncy since there was sampling rate input into pwelch. The energy of white noise will be spread over all frequencies so you need to look at the integral of the signal:

sum(PSD_noise*.0016) % this should equal omega^2