Power spectral density of gaussian white noise

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

[TheBeginner]

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.

Answer 1

[Jeremy]

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