Form of the statistics

The general form of the statistics of interest is

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

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

$\displaystyle z_m = {\bf x}^\prime {\bf P}_m {\bf x} + {\bf p}_m^\prime {\bf x}+ q_m, \;\;\; 1\leq m \leq M,$ (8.1)

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

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

$ \{{\bf P}_m\}$ are $ M$ real symmetric8.1$ 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,

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

where $ N\times 1$ mean vector

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

and $ N\times N$ covariance matrix

$\displaystyle {\bf E}(({\bf x}-$$\displaystyle \mbox{\boldmath$\mu$}$$\displaystyle )({\bf x}-$$\displaystyle \mbox{\boldmath$\mu$}$$\displaystyle )^\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

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


$\displaystyle \tilde{\bf P}_m = {\bf C} \; {\bf P}_m \; {\bf C}^\prime,$ (8.2)

$\displaystyle \tilde{\bf p}_m = {\bf C} \; {\bf p}_m + 2 {\bf C} {\bf P}_m$   $\displaystyle \mbox{\boldmath$\mu$}$$\displaystyle _x,$ (8.3)

$\displaystyle \tilde{q}_m = q_m + {\bf p}_m^\prime$   $\displaystyle \mbox{\boldmath$\mu$}$$\displaystyle _x +$   $\displaystyle \mbox{\boldmath$\mu$}$$\displaystyle _x^\prime {\bf P}_m$   $\displaystyle \mbox{\boldmath$\mu$}$$\displaystyle _x,$ (8.4)

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

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

Baggenstoss 2017-05-19