Is there a generalised and exact formula to compute $\mathbb{E} \{ |x|^n \}$, where $x$ is a complex, circularly symmetric Gaussian random variable, i.e. $x = x_R + jx_I$, $x_R\sim \mathcal{N}(0,\sigma^2)$,$x_I\sim \mathcal{N}(0,\sigma^2)$, and $n \in \mathbb{Z}^{+}$?
Thanks, L
You are looking for: $$\mathbb E(X_R^2+X_I^2)^\frac{n}2=\sigma^n\mathbb E(U_R^2+U_I^2)^\frac{n}2$$where $U_R$ and $U_R$ have standard normal distribution.
If $X_R$ and $X_I$ are independent then so are $U_R$ and $U_I$ and in that case $U_R^2+U_I^2$ is distributed according to the chi-squared distribution with 2 degrees of freedom, having PDF $$\frac12 e^{-\frac{x}2}$$ on $(0,\infty)$.
So to be found is actually:$$\frac12\sigma^n\int_0^{\infty}x^{\frac{n}2}e^{-\frac12x}dx$$
Substitute $y=\frac12x$ and give it a try yourself.