Random Matrix Theory: Applications in Wireless Communication

Random Matrix Theory has wide applications in Communications and Signal Processing. As a part of the talk I shall give a glimpse of application of random matrix in wireless communication. Again in the area of wireless communication there are several problems where in Random Matrix Theory comes in handy to solve problems. As an example we can consider the calculation of channel capacity of MIMO system.

Basic Idea of Digital Communication:

We assume that the noise is Additive White Gaussian Noise (AWGN).

We shall consider a MIMO system where there are $t$ transmitters and $r$ receiver antennas. Thus the vector channel can be represented as

(1)
$$y = Hx + n$$

We assume that the noise is uncorrelated. Hence,

$E\{xx^{+}\} = I_r$

In general Wireless Channel has fading. We shall consider Gaussian channel with Rayleigh Fading.

Now, we have a H which is random but independent of x and n. Each entry of H are independent of each other and are assumed to be Gaussian random variables with zero mean and real and imaginary part being independent with equal variance = 1/2. This is equivalent to assuming the magnitude to be Rayleigh distributed with $E\{Magnitude\} = 1$ and phase being uniformly distributed.

Capacity of the Channel is defined as maximum Mutual Information.

We can show that the mutual information of the channel is,

$I(x; (y,H)) = I(x; H) + I(x; y|H) = I(x; y|H)$

As, x and H are independent, $I(x;H) = 0$