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

,

where
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,

,

where,
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,

where is

.

For our current analysis, we know that

.

Differentiating both sides,

.

Applying this to the above equation, pdf of chi square random variable with two degrees of freedom is,
.

PDF of chi-square random variable with m degrees of freedom

The probability density function is,

, where

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.

Simulation Model

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)

Reference

[DIGITAL-COMMUNICATION: PROAKIS] Digital Communications, by John Proakis

Related posts:

1. Deriving PDF of Rayleigh random variable
2. Maximal Ratio Combining (MRC)
3. Equal Gain Combining (EGC)

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

is called a Rayleigh random variable. Further, the phase is uniformly distributed from . In this post we will try to derive the expression for probability density function (PDF) for and .

The text provided in Section 5.4.5 of [ELECTRONIC-COMMUNICATION:PRADIP] is used as reference.

Joint probability

The probability density function of is

.

Similarly probability density function of is

.

As and are independent random variables, the joint probability is the product of the individual probability, i.e,

.

The joint probability that the random variable lies between and and the random variable lies between and is,

.

Conversion to polar co-ordinate

Given that is in the Cartesian co-ordinate form, we can convert that into the polar co-ordinate where,
and
.

Figure: Cartesian co-ordinate to polar co-ordinate

The area is Cartesian co-ordinate form is equal to the area in the polar co-ordinate form.

.

Simplifying,
.

Summarizing the joint probability density function,

.

Since and are independent, the individual probability density functions are,
,

.

Simulation Model

Simple Matlab/Octave simulation model is provided for plotting the probability density of and . The script performs the following:

(a) Generate two independent zero mean, unit variance Gaussian random variables

(b) Using the hist() function compute the simulated probability density for both and

(c) Using the knowledge of the equation (which we just derived), compute the theoretical probability
density function (PDF)

(d) Plot the simulated and theoretical probability density functions (PDF) and show that they are in good agreement.

Click here to download Matlab/Octave script for simulating the probability density function (PDF) of Rayleigh random variable

Figure: Simulated/theoretical PDF of Rayleigh random variable

Figure: Simulated/theoretical PDF of uniformly distributed theta random variable

Reference

[ELECTRONIC-COMMUNICATION:PRADIP] Principles of Electronic Communications Analog – Digital, by Pradip Kumar Ghosh

Related posts:

1. Chi Square Random Variable
2. Rayleigh multipath channel model
3. BER for BPSK in Rayleigh channel