Multivariate Gaussian in Frequency Domain

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 and satisfy

 (16.4)

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

 (16.5)

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. Let be the mean of . Let be the DFT of . Then, consider the PDF

 (16.6)

This is a useful and very tractable PDF.

Subsections
Baggenstoss 2017-05-19