Complex Random Variable

Expectation

The expectation of a CRV $Z$ can be defined in terms of the expectations of its real and imaginary parts:

$$ E[Z] = E[\Re (Z)] +i E[ \Im (Z) ], \tag{1}$$

provided that the two real expectations $E[ \Re (Z)]$ and $E[ \Im (Z) ]$ are finite.

 

 

Variance

 

$$ \begin{align*} \text{Var}[Z] &= E[|Z-E[Z]|^2] \\ &= E[(Z-E[Z])(Z-E[Z])^*] \\ &= E[(Z-E[Z])(Z^*-E[Z^*])] \hspace{1cm} \left( E[Z^*] = (E[Z])^* \right) \\ &= E[(Z-E[Z])Z^*] - E[(Z-E[Z])]E[Z^*] \hspace{1cm} \left( E[Z^*] \text{ is scalar, so can be taken out} \right) \\ &= E[ZZ^*] - E[Z]E[Z^*] \hspace{1cm} \left( E[(Z-E[Z])] = E[Z] - E[Z] = 0 \right) \\ &= E[|Z|^2] - |E[Z]|^2 \end{align*} $$

 

This can be also be written using Eq. (1),

$$ \begin{align*} \text{Var}[Z] &= E[|Z|^2] - |E[Z]|^2 \\ &= E[ \left( \Re (Z) + i \Im (Z) \right) \left( \Re (Z) - i\Im(Z) \right)] - \left( E[ \Re (Z)] + i E[ \Im (Z) ] \right) \left( E[ \Re (Z)] - i E[ \Im (Z) ] \right) \\ &= E[(\Re   (Z))^2 + (\Im (Z))^2] - \left( (E[ \Re (Z) ])^2 + (E[ \Im (Z)])^2 \right) \\ &= \left( E[ (\Re (Z))^2] - (E[\Re (Z)])^2 \right) + \left( E[ (\Im (Z))^2] - (E[\Im (Z)])^2 \right) \\ &= \text{Var}[ \Re (Z) ] + \text{Var} [ \Im (Z)] \end{align*} $$ 

 

$$ \begin{align*} \therefore \text{Var}[Z] &:= E[|Z-E[Z]|^2] \\ &=E[|Z|^2] - |E[Z]|^2 \\ &= \text{Var}[\Re (Z)] + \text{Var} [\Im (Z)] \end{align*} $$

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Covariance

The covariance measures how much two random variables are correlated, and its sign shows the tendency in the linear relationship between the variables.

The covariance between the complex random variables $Z$ and $W$ is defined by

$$ \begin{align*} \text{Cov}[Z,W] = C_{Z,W}  &:= E[ (Z-E[Z])(W-E[W])^*] \\ &= E \left[ (Z-E[Z]) ( W^* - E[W^*]) \right] \\ &= E \left[ ZW^* - ZE[W^*]- W^* E[Z] + E[Z]E[W^*] \right] \\ &= E[ZW^*] - E[Z]E[W^*] - E[Z]E[W^*] + E[Z]E[W^*] \\ &= E[ZW^*] - E[Z]E[W^*] \end{align*} $$

 

$$ \text{Cov}[Z,W]  = C_{Z,W} = E[ZW^*] - E[Z]E[W^*] $$

The covariance betwen two CRVs is a complex scalar.

 

Properties

• Conjugate Symmetry

$$ \text{Cov}[Z,W] = \left( \text{Cov} [Z,W] \right)^* $$

• Sesquilinearity

$$ \begin{align*} \text{Cov}[\alpha Z, W] &= E[\alpha Z W^*]- E[\alpha Z]E[W^*] \\ &= \alpha E[ZW^*] - \alpha E[Z]E[W^*] \\ &= \alpha \text{Cov}[Z,W] \end{align*} $$

$$ \begin{align*} \text{Cov}[Z, \beta W] &= E[Z \beta^* W^*] - E[Z]E[\beta^* W^*] \\ &= \beta^* E[ZW^*] - \beta^* E[Z]E[W^*] \\ &= \beta^* \text{Cov}[Z,W] \end{align*} $$

$$ \begin{align*} \text{Cov}[Z_1 + Z_2, W] &= E[ (Z_1 + Z_2) W^*] - E[Z_1+Z_2] E[W^*] \\ &= E[Z_1 W^*] - E[Z_1]E[W^*] + E[Z_2 W^*] - E[Z_2]E[W^*] \\ &= \text{Cov}[Z_1, W] + \text{Cov}[Z_2, W] \end{align*} $$

$$ \text{Cov}[Z, W_1+W_2] = \text{Cov}[Z, W_1] + \text{Cov}[Z, W_2] $$

• Relation with Variance

$$ \begin{align*} \text{Cov}[Z, Z] &= E[Z Z^*] - E[Z]E[Z^*] = E[|Z|^2] - |E[Z]|^2 \\ &= \text{Var}[Z] \end{align*} $$

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Characteristic Function

For a real random variable $X$, the characteristic function $\Phi_X : \mathbb R \rightarrow \mathbb C $ is defined as the expected value of $e^{i \omega X}$, a function of $X$, where $ \omega \in \mathbb R $ is the unspecified argument of the characteristic function.

$$ \begin{cases} \Phi_X : w \mapsto E[ e^{i \omega X} ] \\ \Phi_X = \int_{-\infty}^\infty e^{i \omega x} f_X(x)dx \end{cases} $$ 

Hence, the characteristic function $\Phi_X$ is the Fourier transform of the pdf $f_X (x)$ with sign reversal in the complex exponential, i.e.

$$ \Phi_X(\omega) = \left( \mathcal{F} f_X (x) \right) (-\omega). $$

Then, the pdf $f_X(x)$ can be expressed in terms of $ \Phi_X(x)$:

$$ \begin{align*} f_X(x) &= \left( \mathcal{F}^{-1} \Phi_X(- \omega) \right) (x) \hspace{1.5cm} \left( \left( \mathcal{F} f_X(x) \right) = \Phi_X(-\omega) \right) \\ &= \frac{1}{2\pi} \int_{-\infty}^\infty e^{i\omega x} \Phi_X(-\omega) d\omega \\ &= \frac{1}{2\pi} \int_{-\infty}^\infty e^{-i \omega x} \Phi_X (\omega) d\omega \hspace{1cm} \left( \text{substitute } \omega \text{ for } -\omega \right) \end{align*} $$

 

Characteristic Function of a CRV

The characteristic function $\Phi_Z: \mathbb{C} \rightarrow \mathbb{C}$ of a complex random variable $Z$ is defined as

$$ \begin{align*} \Phi_Z (\omega) &:= E[e^{i \Re(w^* Z)}] \\ &= E[e^{i \left( \Re(\omega) \Re (Z) + \Im(\omega)\Im(Z) \right) }] \end{align*} $$