# Affine transformation on circularly symmetric Gaussian distribution

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 $$\mathbf{z}$$?

• 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?
– MBaz
Apr 21 at 16:05