about psd estimation by FFT

Hi,

In topic "Power Spectral Density Estimates Using FFT (https://www.mathworks.com/help/signal/ug/power-spectral-density-estimates-using-fft.html)", why are squared dft values multipled by (1/(fs*N))?

fs = 1000;

t = 0:1/fs:1-1/fs;

x = cos(2*pi*100*t) + randn(size(t));

N = length(x);

xdft = fft(x);

xdft = xdft(1:N/2+1);

psdx = (1/(fs*N)) * abs(xdft).^2;

psdx(2:end-1) = 2*psdx(2:end-1);

freq = 0:fs/length(x):fs/2;

Answer 1

[Askic V]

It is because of Parseval's theorem, which expresses the energy of a signal in time-domain in terms of the average energy in its frequency components

In this case, n and k are integers, but actually these are samples i.e. discrete time intervals, so they can be considered as n*deltaT, in such case it is

deltaT is a sample time period i.e. deltaT = 1/Fs where Fs is a sampling frequency.

That is why 1/(fs*N)!

[Abdullah Talha Sözer]

Thanks