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 :
Data PDF of circularly stationary process
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.