Form of the statistics

The general form of the statistics of interest is

$${\bf z}= T({\bf x}) = [z_1\; z_2 \ldots z_M]^\prime ,$$

where $\{z_m\}$ are $M$ quadratic forms,

$$z_m = {\bf x}^\prime {\bf P}_m {\bf x} + {\bf p}_m^\prime {\bf x}+ q_m, \;\;\; 1\leq m \leq M,$$ (9.1)

and ${\bf x}$ is the $N$-by-1 real input data vector

$${\bf x}=[x_1\; x_2 \ldots x_N]^\prime,$$

$\{{\bf P}_m\}$ are $M$ real symmetric^9.1$N$-by-$N$ matrices, $\{{\bf p}_m\}$ are $N$-by-$1$ vectors, and $\{q_m\}$ are scalars.

The challenge is to determine the joint PDF of ${\bf z}$ under a specified Gaussian hypothesis, that is,

$$p({\bf z}; {\bf R}_x,$$   $$\mbox{\boldmath$\mu$}$$$$_x, \{ {\bf P}_m\}, \{ {\bf p}_m\}, \{q_m\}),$$

where $N\times 1$ mean vector

$${\bf E}({\bf x}) =$$   $$\mbox{\boldmath$\mu$}$$$$_x$$

and $N\times N$ covariance matrix

$${\bf E}(({\bf x}-$$$$\mbox{\boldmath$\mu$}$$$$)({\bf x}-$$$$\mbox{\boldmath$\mu$}$$$$)^\prime) = {\bf R}_x.$$

Note that there is no loss of generality in assuming that ${\bf R}_x={\bf I}_N$ and $\mu$$_x={\underline{\bf0}}$, where ${\bf I}_N$ is the $N$-by-$N$ identity matrix and ${\underline{\bf0}}$ is the $N$-by-1 vector of zeros. Call this the white Gaussian noise (WGN) assumption $H_0$. This is because we can write

$$\begin{array}{r} p({\bf z}; {\bf R}_x, \mbox{\boldmath$\mu$}_x, ... ... \{ \tilde{\bf P}_m\}, \{ \tilde{\bf p}_m\}, \{ \tilde{\bf q}_m\}), \end{array}$$

where

$$\tilde{\bf P}_m = {\bf C} \; {\bf P}_m \; {\bf C}^\prime,$$ (9.2)

$$\tilde{\bf p}_m = {\bf C} \; {\bf p}_m + 2 {\bf C} {\bf P}_m \mbox{\boldmath$\mu$} _x,$$ (9.3)

$$\tilde{q}_m = q_m + {\bf p}_m^\prime \mbox{\boldmath$\mu$} _x + \mbox{\boldmath$\mu$} _x^\prime {\bf P}_m \mbox{\boldmath$\mu$} _x,$$ (9.4)

and ${\bf C}$ is the Cholesky decomposition of ${\bf R}_x$ ,

$${\bf R}_x = {\bf C}^\prime \; {\bf C}.$$