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 independent and identically distributed Gaussian random variables with mean and variance and we form a new random variable,
Then is a chi square random variable with degrees of freedom.
There are two types of chi square distribution. The first is obtained when has a zero mean and is called central chi square distribution. The second is obtained when has a non-zero mean and is called non-central chi square distribution. Four our discussion, we will focus only on central chi square distribution.
PDF of chi-square random variable with one degree of freedom
Using the text in Chapter 2 of [DIGITAL-COMMUNICATION: PROAKIS] as reference.
The most simple example of a chi square random variable is
is a Gaussian random variable with zero mean and variance .
The PDF of is
By definition, the cumulative distribution function (CDF) of is
This simplifies to
Differentiating the above equation with respect to to find the probability density function,
Summarizing, the pdf of chi square random variable with one degree of freedom is,
PDF of chi-square random variable with two degrees of freedom
Chi square random variable with 2 degrees of freedom is,
and are independent Gaussian random variables with zero mean and variance .
In the post on Rayleigh random variable, we have shown that PDF of the random variable,
For our current analysis, we know that
Differentiating both sides,
PDF of chi-square random variable with m degrees of freedom
The probability density function is,
the Gamma function is defined as,
p an integer > 0
I do not know the proof for deriving the above equation. If any one of you know of good references, kindly let me know. Thanks. 🙂
Just for your reference, Matlab/Octave simulation model performing the following is provided
(a) Generate chi square random variables having m=1, 2, 3, 4, 5 degrees of freedom
(b) Probability density function is computed and plotted
Click here to download: Matlab/Octave script for simulating PDF of chi square random variable
Figure: PDF of chi square random variable (=1)