Power Spectral Density:
The Power Spectral Density (PSD) of an ergodic (i.e. time average statistics are identical to ensemble average statistics), discrete, random signal is the Fourier transform of its autocorrelation function, as given by
where xx( ) is the autocorrelation function for the discrete random signal x and is normalised frequency. This is also known as the Wiener-Khintchine theorem. To calculate this expression exactly requires knowledge of all the autocorrelation function values for the discrete random signal x. Given only a ?nite length data record, as in many practical problems such as speech and array processing, namely [ ] =0 1 -1 it is only possible to estimate this quantity.
One method to estimate the PSD is based on the Fast Fourier Transform (FFT). This method is called the periodogram, as defined by
Notice that the PSD is being estimated at N discrete-frequencies, which in terms of normalised frequencies corresponds to =0 1 ( -1) . The sampling interval T is implicity assumed, for convenience, to be unity.