It is possible to compute the PDF of in the frequency domain. Let
be the DFT of . Note that
Therefore, for zero-mean Gaussian data with variance 1,
Although this is written in terms of , it is a PDF of .
This distribution can be extended to arbitrary mean and power spectra, however we must be clear about
what we are assuming. Let be defined as the
case when the DFT coefficients are independent zero-mean complex Gaussian RVs
where are the power spectrum values. There are unique values of ,
being a symmetric sequence.
Note that the proper definition of power spectrum is through the Wiener-Kinchine theorem
in which the power spectrum is defined as the Fourier transform of the autocorrelation function.
In that case, the DFT of will neither exhibit independent
coefficients nor will the power spectrum be given by (16.4). These
relationships will only be approximate.
Hypothesis , however , is more useful to us when using finite-length
data samples and the DFT.
The PDF of under is precicely given by
It is important to note that (16.5) is an exact PDF of under however
it is only an approximate PDF of for a stationary Gaussian process with power spectrum
This is easily extended to an arbitrary mean.
be the mean of . Let
be the DFT of .
Then, consider the PDF
This is a useful and very tractable PDF.