It is sometimes important to know the relationship between various distributions. This can be useful if there is a function available for one distribution and it can be used to derive other distributions. In the context of Wireless Communications it is important to know the relationship between the Uniform, Gaussian and Rayleigh distribution.

According to Central Limit Theorem the sum of a large number of independent and identically distributed random variables has a Gaussian distribution. This is used to model the amplitude of the in-phase and quadrature components of a wireless signal. Shown below is the model for the received signal which has been modulated by the Gaussian channel coefficients g1 and g2.

r=g1*a1*cos(2*pi*fc*t)+g2*a2*sin(2*pi*fc*t)

The envelope of this signal (sqrt(g1^2+g2^2)) as a Rayleigh distribution. Now if you only had a function for Uniform Distribution you can generate Rayleigh Distribution using the following routine.

```
clear all
close all
M=10000;
N=100;
for n=1:M;
x1=rand(1,N)-0.5;
x2=rand(1,N)-0.5;
y1=mean(x1);
y2=mean(x2);
z(n)=sqrt(y1^2+y2^2);
end
hist(z,20)
```

Note: Here a1 and a2 can be considered constants (at least during the symbol duration) and its really g1 and g2 that are varying.

#### Author: John (YA)

John has over 15 years of Research and Development experience in the field of Wireless Communications. He has worked for a number of companies around the world including Qualcomm Inc. USA. He has an MS in Electrical Engineering from Virginia Tech USA and has published his work in international journals and conferences.

x1 and x2 should be normal not uniform so use randn instead of rand.