Simple periodogram:
Figure 3 outlines the production principle.
We compute the discrete Fourier transform of a realization of the signal's duration N. A normalisation factor affects the obtained power spectrum, representing the estimate of the PSD, which is then equal to:
\({\hat{\gamma}}_x(\omega)=\frac{1}{R_f(0)}\left|Y(\omega)\right|^2=\frac{1}{N.R_f(0)}\left|\sum_{k=1}^{N}{x(k)f(k)e^{-jk\omega}}\right|^2\) (33)
In the case of a rectangular window \(f(k)=rec\left(\frac{k}{N}\right)\) , we find the PSD's expression (32):
\({\hat{\gamma}}_x(\omega)=\frac{1}{N}\left|\sum_{k=1}^{N}{x(k)e^{-jk\omega}}\right|^2\)
Which, expanded, gives:
\({\hat{\gamma}}_x(\omega)=\sum_{k=1}^{N}{\left[\frac{1}{N}x(i)x(i-k)\right]e^{-jk\omega}}\) (34)
The term in square brackets denotes the estimate of the autocorrelation function \({\hat{R}}_x(k)\) .
Note: in definition (30), the TFD \(Y(\omega)\) of y(k) is calculated with a normalisation factor equal to \(\frac{1}{\sqrt N}\) and identical to that which will be considered for the reverse DFT.