Say $\alpha$ is a complex random variable, then which one of the following expressions is correct?

  1. $\mathbb{E}[\alpha^2]$ or

  2. $\mathbb{E}[\alpha \alpha^*]$?

  • 1
    $\begingroup$ The answer depends on what you want to calculate, of course. Actually, I don't know what No. 1 could be useful for. No 2. is the mean power of the random variable $\alpha$ and is probably found much more frequently than expression no 1. I might be able to give more detailed information if you elaborate on the purpose of the expression. $\endgroup$ – Deve Apr 30 '14 at 6:53
  • $\begingroup$ @Deve I have a complex random Variable $\alpha$. I want to write the expression for power of $\alpha$. I am not sure which is the best way to write? I think according to you, No 2 is right! $\endgroup$ – Mithun Apr 30 '14 at 9:31
  • $\begingroup$ Yes, that's right. I have edited your question according to your comment. I hope you agree with it. $\endgroup$ – Deve Apr 30 '14 at 11:50

The correct expression for the mean power of a complex random variable $\alpha=x+jy$ is $$ \begin{align} \bar P &= \operatorname E\left[\alpha \alpha^*\right]\\ &= \operatorname E\left[x^2 + y^2\right]\\ &= \operatorname E\left[x^2\right] + \operatorname E\left[y^2\right]\\ &= \bar P_\mathrm{x} + \bar P_\mathrm{y} \end{align} $$

In other words, the mean power of a complex random variable is the sum of the mean powers of its real and imaginary part, respectively. In contrast, expression no. 1 from your question evaluates to $$ \begin{align} \operatorname E\left[\alpha^2\right] &= \operatorname E\left[x^2 + j2xy - y^2\right]\\ &= \operatorname E\left[x^2\right] - \operatorname E\left[y^2\right] + \operatorname E\left[j2xy\right]\\ &= \bar P_\mathrm{x} - \bar P_\mathrm{y} + j2\operatorname E\left[xy\right] \end{align} $$ Note that for the special case of $x$ and $y$ being uncorrelated $\operatorname E\left[xy\right]=E\left[x\right]E\left[y\right]$. If, in addition, $x$ or $y$ has zero mean $E\left[xy\right]=0$.

  • $\begingroup$ Note that your last comment is only true if either $E[x]=0$ or $E[y]=0$ (or both), i.e. $E[xy]=0$ does not in general mean that $x$ and $y$ are uncorrelated, but that they are orthogonal. However, in the important case of zero mean RVs, these two conditions are equivalent. That's why they are often confused. $\endgroup$ – Matt L. Apr 30 '14 at 11:58
  • $\begingroup$ Your right, of course. Thank you for the hint. $\endgroup$ – Deve Apr 30 '14 at 12:03

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