Multivariate Random Signals

A real time signal $x(t)$ can be considered as a random process and its samples $x_m\;\;(m=0, \cdots, N-1)$ a random vector:

$$X=[ x_0, \cdots, x_{N-1} ]^T$$

The mean vector of $X$ is

$$M_X\stackrel{\triangle}{=}E(X)=[E(x_0), \cdots, E(x_{N-1}) ]^T =[\mu_{x_0},\cdots, \mu_{x_{N-1}} ]^T$$

The covariance matrix of $X$ is

$$\Sigma_X\stackrel{\triangle}{=}E[ (X-M_X)(X-M_X)^T ]=E(XX^T)-... ... .. & \sigma_{ij}^2 & .. \\ .. & .. & .. \end{array} \right]$$

where $\sigma_{ij}^2\stackrel{\triangle}{=}E(x_ix_j)-\mu_{x_i}\mu_{x_j}$ is the covariance of two random variables $x_i$ and $x_j$. When $i=j$, $\sigma_{ij}^2$ becomes the variance of $x_i$, $\sigma_i^2\stackrel{\triangle}{=}E(x_i^2)-\mu_{x_i}^2$.

The correlation matrix of $X$ is

$$R_X\stackrel{\triangle}{=}E(XX^T) =\left[ \begin{array}{ccc}... ...& .. .. & r_{ij} & .. \\ .. & .. & .. \end{array} \right]$$

where $r_{ij}=\sigma_{ij}^2+\mu_{x_i}\mu_{x_j}$. Note that as $\sigma_{ij}=\sigma_{ji}$ and $r_{ij}=r_{ji}$, both $\Sigma_X=\Sigma^T_X$ and $R_X=R^T_X$ are symmetric matrices (Hermitian if $X$ is complex).

A signal vector $X$ can always be easily converted into a zero-mean vector $X'=X-M_X$ with all of its information (or dynamic energy) conserved. In the following, without loss of generality, we will assume $M_X=0$ and therefore $\Sigma_X=R_X$.

Before reading on, it is highly recommended to review the basics of multivariate probability theory