Multivariate Gaussian in Frequency Domain

It is possible to compute the PDF of ${\bf x}$ in the frequency domain. Let $X_k,\; k=0,1\ldots N-1$ be the DFT of ${\bf x}$. Note that

$$\sum_{i=0}^{N-1} \; x_i^2 = \frac{1}{N} \sum_{i=0}^{N-1} \; \vert X_i\vert^2.$$

Therefore, for zero-mean Gaussian data with variance 1,

$$\log p({\bf x}) = -\frac{N}{2} \log (2\pi) - \frac{1}{2N} \sum_{i=0}^{N-1} \; \vert X_i\vert^2$$

Although this is written in terms of $X_k$, it is a PDF of ${\bf x}$.

This distribution can be extended to arbitrary mean and power spectra, however we must be clear about what we are assuming. Let $H_\rho$ be defined as the case when the DFT coefficients $X_k$ are independent zero-mean complex Gaussian RVs and satisfy

$$\rho_k = \frac{1}{N} \; {\cal E}\left\{ \vert X[k]\vert^2\right\}, \; 0\geq k < N,$$ (17.4)

where $\rho_k$ are the power spectrum values. There are $N/2+1$ unique values of $\rho_k$, being a symmetric sequence. Note that the proper definition of power spectrum is through the Wiener-Kinchine theorem in which the power spectrum is defined as the Fourier transform of the autocorrelation function. In that case, the DFT of ${\bf x}$ will neither exhibit independent coefficients nor will the power spectrum be given by (17.4). These relationships will only be approximate. Hypothesis $H_\rho$, however , is more useful to us when using finite-length data samples and the DFT. The PDF of ${\bf x}$ under $H_\rho$ is precicely given by

$$\log p({\bf x}\vert H_\rho) = - \frac{1}{2} \sum_{i=0}^{N-1} \; \left\{ \log (2\pi\rho_i) + \frac{\vert X_i\vert^2}{N\rho_i} \right\}$$ (17.5)

It is important to note that (17.5) is an exact PDF of ${\bf x}$ under $H_\rho$ however it is only an approximate PDF of ${\bf x}$ for a stationary Gaussian process with power spectrum $\{\rho_0, \rho_1 \ldots \rho_{N/2}\}$.

This is easily extended to an arbitrary mean. Let ${\cal E}({\bf x}) = {\bf y}= [y_1, \ldots y_N]^\prime$ be the mean of ${\bf x}$. Let $Y_k, \; 0\leq k <N$ be the DFT of ${\bf y}$. Then, consider the PDF

$$\log p({\bf x}; {\bf y}, \rho_0,\ldots \rho_{N/2+1}) = - \frac{1}... ...\; \left\{ \log (2\pi\rho_i) + \frac{\vert X_i - Y_i\vert^2}{N\rho_i} \right\}.$$ (17.6)

This is a useful and very tractable PDF.