What would be the probability density function (pdf) of the complex random variable given below?


$$Z = \sum\_{i=1}^{M}{x\_{i}^{\*}y\_{i}}$$


where $x\_i, y\_i$ are independent r.v.'s with $\mathcal{CN}(0,c)$.



What is the probability density function (pdf) of the dot product of M complex normal random variables?
The joint characteristic function $\Psi(\omega\_1,\omega\_2)$ of the real and imaginary parts of $Z=z\_1+iz\_2$ is derived in: [Distribution of Inner Product of Complex Gaussian Random Vectors and its Applications](http://ieeexplore.ieee.org/document/6047542/) (2011). That publication is behind a pay wall. You can find the expression in equation 14 of this [paper](http://hosting.uop.gr/~nsagias/images/articles/J1_2014.pdf), for the most general case that each $x\_i$ and $y\_i$ can have different Gaussian distributions. For the case in the OP the characteristic function is simply $\Psi(\omega\_1,\omega\_2)=[1+\tfrac{1}{4}c^2(\omega\_1^2+\omega\_2^2)]^{-M}$.


