Consider a random vector $X=[X_1, X_2, \ldots , X_n]$ where $X_i$ ($i \in 1, 2,\ldots, n$) are independent complex Gaussian random variables with zero mean and variance $\sigma_i^2$, i.e., $X_i \sim CN(0, \sigma_i^2)$.

How can I find expectation of norm of $X$, where the norm of $X$ is given by \begin{equation} \|X\|=\sqrt{\sum_{i=1}^n |X_i|^2} \end{equation}

Any help regarding this problem is really appreciated. Thanks.

  • 1
    $\begingroup$ The way $\text{“}CN\text{''}$ is usually defined, if $X\sim CN(0,\sigma^2)$ then the real and imaginary parts of $X$ are independent and each is distributed as $N(0,\sigma^2/2).$ Thus the square of the norms is distributed as $(\sigma^2/2) \chi^2_{2n}. \qquad$ $\endgroup$ – Michael Hardy May 30 '17 at 11:30
  • $\begingroup$ I can understand that if the random variables are i.i.d, i.e., same variance $\sigma^2$, the norm is chi-square distributed, however the problem is the variance for each random variable is different. Can you help me regarding this? Thanks $\endgroup$ – Sabyasachi G May 30 '17 at 12:39

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