Question on FFT PSD function

[neoflash]

power spectral density fft

There is a matlab embeded example for FFT PSD analysis:

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t = 0:.001:.25;
x = sin(2*pi*50*t)
X = fft(x,256)

% why PSD is |x(f)|^2/256 ? should not that be divided by T=256*0.001 ??
PSD = X.*conj(X)/(256);

f = 1000/256*(0:127);
plot(f,PSD(1:128))
title('Power spectral density')
xlabel('Frequency (Hz)')
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As stated above, matlab calculated x's fft and derive its PSD by using:

PSD = X.*conj(X)/(256);

I am very very curious about why it is divided by 256, not 256*0.001??

Since PSD should be defined as |X(f)|²/T, T is the length of computing length in time domain.

thanks.

 

[hamidr_karami]

matlab psd fft

the formula |X|^2/T,uses in continues domain
and 0.001 is samples in time

 

[neoflash]

psd fft matlab

??

 

[claudiocamera]

power spectral density fourier transform

In spectrum estimation PSD can be defined basically using non parametric estimation:

considering the periodogram method we have DTFTestimated = (X.X*)/N where N is the number of points from the DTFT.

In you case N= 256.

The square of the modulus of Fourier Transform is the estimated power density, when the periodogram is applied it has the dimension of normalized energy authoug some autors will call it as the power spectral density (PSD)

Regards