The well-known relationships for zero-mean circularly-symmetric complex Gaussian $z = a + jb = |z| \exp(j\varphi)$ signals are
- the amplitudes $|z| = \sqrt{a^2 + b^2}$ are Rayleigh-distributed
- the phases $\varphi = \arg(z)$ are uniformly distributed
- the pdf of such signals is fully described by their complex covariance matrix $\mathbf{\Sigma}$, which is Wishart-distributed in case it is estimated from a number of independent samples ($\rightarrow \mathbf{\hat{\Sigma}}$)
My question is two-fold:
1) What's the distribution of $z$ in case the amplitudes are constant, e.g. all normalized to 1?
2) What's the distribution of $\mathbf{\hat{\Sigma}}$ in case the same normalization is applied, i.e. the amplitudes of its elements are removed?
I figure that it's then $z = \exp(j\varphi)$, with $\varphi \sim \mathcal{U}$, i.e. I am possibly looking for the distribution the exponential of a uniformly distributed variable follows. For $\mathbf{\Sigma}$, however, I have no idea.