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 $\mathcal{CN}(0,c)$.
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asked Jan 7, 2017 at 7:26
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1 Answer
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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 $\Psi(\omega_1,\omega_2)=[1+\tfrac{1}{4}c^2(\omega_1^2+\omega_2^2)]^{-M}$.
answered Jan 7, 2017 at 9:04
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$\begingroup$ What does OP stand for? $\endgroup$ Commented Jan 8, 2017 at 21:57
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1$\begingroup$ lifewire.com/what-does-o-p-stand-for-2483372 $\endgroup$ Commented Jan 9, 2017 at 7:25