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Chi Square Random Variable

While trying to derive the theoretical bit error rate (BER) for BPSK modulation in a Rayleigh fading channel, I realized that I need to discuss chi square random variable prior.

What is chi-square random variable?

Let there be $m$ independent and identically distributed Gaussian random variables $X_i$ with mean $0$ and variance $\sigma^2$ and we form a new random variable,

$Z = \sum_{i=1}^mX_i^2$ .

Then $Z$ is a chi square random variable with $m$ degrees of freedom.

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Deriving PDF of Rayleigh random variable

In the post on Rayleigh channel model, we stated that a circularly symmetric random variable is of the form $Z = X + jY$ , where real and imaginary parts are zero mean independent and identically distributed (iid) Gaussian random variables. The magnitude $|Z|$ which has the probability density,

$p(z) = \frac{z}{\sigma^2}e^{\frac{-z^2}{2 \sigma^2}},\ \ \ z\ge 0$

is called a Rayleigh random variable. Further, the phase $\theta$ is uniformly distributed from $[0,\ 2\pi]$ . In this post we will try to derive the expression for probability density function (PDF) for $|Z|$ and $\theta$ .

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