In Section 9.1.2 we discussed
circular models. The theoretical PDF
is based on the circular power spectrum :
where are the DFT coefficients of the length-
input data .
The PDF of for a circular power spectral process is
written in the frequency domain
Although written using the DFT coefficients ,
it is a PDF defined on .
Given a circular power spectrum
we can generate data by generating complex
FFT output bins with the specified power spectrum,
then inverting the FFT to obtain a time-series.
Using the definition (10.1), we may generate
random data at the DFT output as
where is a Gaussian random variable
with mean zero and variance 1.
When is the zero or Nyquist bin,
is real, otherwise, it is a complex Gaussian random variable.