Problem
Cellular telephony has forever changed the way people communicate with one another, being one of those exciting technologies that has directly and significantly influenced our everyday life. The characteristics of wireless signal changes as it travels from the transmitter antenna
to the receiver antenna. These characteristics depend upon the distance between the two antennas, the path(s) taken by the signal, and the environment (buildings and other objects) around the path. The profile of received signal can be obtained from that of the transmitted signal if we have a model of the medium between the two. This model of the medium is called channel modeling.

Rayleigh fading is a statistical model for the effect of a propagation environment on a radio signal, such as that used by wireless devices. Rayleigh fading is a reasonable model when there are many objects in the environment that scatter the radio signal before it arrives at the
receiver.

Rayleigh fading is exhibited by the assumption that the real and imaginary parts of the response are modelled by independent and identically distributed zero-mean Gaussian processes so that the amplitude of the response is the sum of two such processes.

Question 1

Random variable R is Rayleigh distributed if R = √(X² + Y²), where X~N (0,σ²) and Y~N(0,σ²) are independent normal random variables.
Derive theoretically the probability density function (PDF) and cumulative density function (PDF) of the Rayleigh distribution as well as its amplitude, and plot the figures in MATLAB.

Question 2

If a channel is not changing with time, it does not fade and instead remains at some particular level. Separate instances of the channel in this case will be uncorrelated with one another, owing to the assumption that each of the scattered components fades independently. Once relative motion is introduced between any of the transmitter, receiver, and scatterers, the fading becomes correlated and varying in time.

The normalised autocorrelation function of a Rayleigh faded channel with motion at a constant velocity is a zeroth-order Bessel function of the first kind

R(τ) = j₀(2f_dτ)

at delay τ when the maximum doppler shift is f_d .

The Jakes model is a well-known and popularly used channel model in simulating a Rayleigh fading channel. The Jakes model has the following characteristics:
? approximate the Rayleigh fading process by summing a set of complex sinusoids;

? the sinusoids are weighted so as to produce an accurate approximation of desired channel Doppler spectrum;

? Jakes shows that the theoretical Doppler spectrum for the isotropic scattering mobile radio channel can be well approximated by a summation of relatively small number of sinusoids with frequencies and relative phases of the sinusoids set according to a specific formulation.

Write a MATLAB program to simulate a wireless channel based upon the Jakes model with Doppler shifts of 10 Hz, 100 Hz, and 1000 Hz. You should plot the amplitude of the wireless fading and compare with the theoretical results in Question 1. Comment on the comparison.