Wednesday, 9 May 2012

Complex Gaussian Random Vector

Consider an $N$ Dimensional Complex Gaussian Random Vector $Z \in \mathbb{C}^N$, Such That

$$Z=X+jY$$, where $X \in \mathbb{R}^N$ and $Y \in \mathbb{R}^N$ are $N$ Dimensional Real Gaussian Random Vectors .

Covariance matrix $Q$ of $Z$ in terms of Auto Covariance and Cross Covariance Matrices of $X$ and $Y$ is Defined as:

$$Q=E((Z-E(Z))(Z-E(Z))^H)$$

So:
$$Q=(\Sigma_{XX}+\Sigma_{YY})-j(\Sigma_{XY}-\Sigma_{XY}^T)$$

Define $2N$ Dimensional Gaussian Random Vector $\widetilde{Z}$, as:

$$\widetilde{Z}= \begin{bmatrix} X\\ Y \end{bmatrix}$$

The Covariance Matrix of $\widetilde{Z}$ is Denoted as $\widetilde{K}$, Defined as:

$$\widetilde{K}=E((\widetilde{Z}-E(\widetilde{Z}))(\widetilde{Z}-E(\widetilde{Z}))^T)$$

So $\widetilde{K}$, In terms of Auto Covariance and Cross Covariance Matrices of $X$ and $Y$ is:

$$\widetilde{K}=\begin{bmatrix} \Sigma_{XX} & \Sigma_{XY} \\ \\ \Sigma_{XY}^T & \Sigma_{YY} \end{bmatrix}$$

Now if $Z$ is Circularly Symmetric, Prove That:

$$\widetilde{K}=\frac{1}{2}\begin{bmatrix} Re(Q) & -Im(Q)\\ \\ Im(Q) & Re(Q) \end{bmatrix}$$

1 comment:

1. Plzz Notice that $Re(Q)$ Is Symmetric and $Im(Q)$ is Skew Symmetric.