PDF based on Circular Power Spectrum

In Section 9.1.2 we discussed circular models. The theoretical PDF is based on the circular power spectrum $ \rho_k$ :

$\displaystyle {\cal E}\left\{ \vert X_k\vert^2 \right\} = N \rho_k,\;\;\; 1\leq k \leq N,$ (10.1)

where $ X_k$ are the DFT coefficients of the length-$ N$ input data $ {\bf x}$. The PDF of $ {\bf x}$ for a circular power spectral process is written in the frequency domain

$\displaystyle \log p({\bf x};$   $\displaystyle \mbox{\boldmath$\rho$}$$\displaystyle ) = - \frac{1}{2} \sum_{k=1}^N \; \left\{ \log 2\pi \rho_k + \frac{\vert X_k\vert^2}{N \rho_i} \right\}.$ (10.2)

Although written using the DFT coefficients $ X_k$, it is a PDF defined on $ {\bf x}$. Given a circular power spectrum $ \rho$, 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

$\displaystyle X_k = \sqrt{N \rho_k} \; u_k,$ (10.3)

where $ u_k$ is a Gaussian random variable with mean zero and variance 1. When $ X_k$ is the zero or Nyquist bin, $ u_k$ is real, otherwise, it is a complex Gaussian random variable.

Baggenstoss 2017-05-19