What is the probability density function (pdf) of the dot product of M complex normal random variables?

Question

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 $\mathcal{CN}(0,c)$.

Answer 1

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 (2011). That publication is behind a pay wall. You can find the expression in equation 14 of this paper, 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 $\Psi(\omega_1,\omega_2)=[1+\tfrac{1}{4}c^2(\omega_1^2+\omega_2^2)]^{-M}$.