PDF based on Circular Power Spectrum

In Section 10.1.2 we discussed circular models. The theoretical PDF is based on the circular power spectrum $\rho_k$ :

$${\cal E}\left\{ \vert X_k\vert^2 \right\} = N \rho_k,\;\;\; 1\leq k \leq N,$$ (11.1)

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

$$\log p({\bf x}; \mbox{\boldmath$\rho$} ) = - \frac{1}{2} \sum_{k=1}^N \; \left\{ \log 2\pi \rho_k + \frac{\vert X_k\vert^2}{N \rho_i} \right\}.$$ (11.2)

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 (11.1), we may generate random data at the DFT output as

$$X_k = \sqrt{N \rho_k} \; u_k,$$ (11.3)

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.