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Noncircularity of Complex Random Variable

Context

Given two complex random variables, \(X\) and \(Y\):

\[ \begin{align} X &= X_R + j \cdot X_I \\ Y &= Y_R + j \cdot Y_I \\ \end{align} \]

where

\[ \begin{align} X_R \sim \mathcal{N} ( \mu, \sigma^2) &, X_I \sim \mathcal{N} ( \mu, \sigma^2) , \\ Y_R \sim \mathcal{N} ( \mu, \sigma^2) &, Y_I = Y_R, \\ \end{align} \]

The distribution functions of \(X\) and \(Y\) are 2D Gaussian and 1D Gaussian, respectively. Despite the distribution functions are so different between \(X\) and \(Y\), their expected magnitude, \(E|X|^2\) and \(E|Y|^2\), are the same:

\[ \begin{align} E|X|^2 &= E(X_R^2 + X_I^2) = 2( \mu^2 + \sigma^2 ) \\ E|Y|^2 &= E(Y_R^2 + Y_I^2) = 2( \mu^2 + \sigma^2 ) \end{align} \]

This observation highlights the fact that, when designing algorithms against complex values, relying on solely the magnitude of the signal may not be efficient.

Noncircularity

The difference between \(X\) and \(Y\) can be easily emphasized by evaluating the expected values of their square terms, \(E(X)^2\) and \(E(Y)^2\):

\[ \begin{align} E(X)^2 &= E(X_R + j X_I )^2 = 2j \cdot E(X_R X_I) = 2j \cdot E(X_R)E(X_I) = 2j \cdot \mu^2 \\ E(Y)^2 &= E(Y_R + j Y_I )^2 = 2j \cdot E(Y_R Y_I) = 2j E(Y_R)^2 = 2j \cdot (\mu^2 + \sigma^2) \end{align} \]

And a proposed metric to quantify this observation is a normalized circularity coefficient, \(k_x\):

\[ k_x = \frac{ E(X^2) }{ E|X|^2 } \]

Note: The paper in reference defines \(k_x\) under the assumption that \(X\) is zero mean. So the formula of \(k_x\) may require modification when \(X\) is not zero mean.

Reference