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I have a hard time describing my problem, but I'll try my best. It's all about the well-known zero-mean, circularly-symmetric, multivariate complex Gaussian distribution $f(z)=\frac{1}{\pi^K\det(\mathbf{\Sigma})}\exp(-\mathbf{z}^H\mathbf{\Sigma}^{-1}\mathbf{z})$, with $\mathbf{\Sigma}$ being the complex covariance matrix.

If we now normalize $\mathbf{\Sigma}$ such that its main diagonal becomes a unity vector (i.e. we receive the correlation matrix) and also normalize the magnitude of $\mathbf{z}$ to 1 - is the resulting, adapted function still a valid probability density function then? If yes, what is the resulting distribution? Or can the result at least be used as a likelihood function, e.g. in the context of maximum-likelihood estimation?

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