Assume the complex random vector $\mathbf{y} \in \mathbb{C}^N$ is circularly symmetric distributed i.e. $\mathbf{y} \sim \mathcal{CN}(\mathbf{0},\mathbf{R})$ $$ p_{\mathbf{y}}(\mathbf{y})=\pi^{-N}\textrm{det}(\mathbf{R})^{-1}\exp\{-\mathbf{y}^H \mathbf{R}^{-1}\mathbf{y} \} $$

Now if $\mathbf{y}$ undergoes an affine transformation as $$\mathbf{z}=\mathbf{a}^H\mathbf{y}+\mathbf{b}$$ what would be the pdf of the randomvector $\mathbf{z}$?

I know $\mathbf{z}$ can not be circularly symmetric anymore , because the mean is modified by the addition of $\mathbf{b}$, whereas a circularly symmetric distributed random vector should have zero mean. What would be the pdf of the randomvector $\mathbf{z}$?

  • 2
    $\begingroup$ Wouldn't this question be a better fit for one of the statistics or math stackexchange sites? In any case, RV transformations are covered in any decent probability textbook, have you looked at any? $\endgroup$
    – MBaz
    Apr 21 at 16:05


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