## Data PDF of circularly stationary process

For the circularly stationary process, equation (9.6) may also be used but the matrix will not be Toeplitz, it will be circulant. A nice property of circulant matrices is that their eigenvectors are the DFT basis functions. As a result, (9.6) can be written in the discrete frequency-domain in terms of the circular power spectrum :

 (9.7)

Note well that although (9.7) is written in terms of , it is a PDF of . It is only necessary to substitute (9.5) for to write the PDF in terms of . Then, when integrated over the -dimensional space of , it gives 1. The circular stationary process has some very nice properties. It can be shown that the assumption of a circular stationary process means that the DFT coefficients are independent random variables. This has to do with the fact that the eigenvectors of the covariance matrix of , which is a circulant matrix, are the DFT basis vectors. An entire class of PDFs is created when an arbitrary positive function is used - the expression remains a PDF defined in (See eq. 16.5). In other words, if we assume the DFT bins are independent and obey (9.4) for any function , we obtain an exact and tractable expression for the PDF of a circularly stationary process (See Eq. 16.5). This model can be used to approximate the PDF of any stationary process whose power spectrum is known.

Baggenstoss 2017-05-19