PROJECT - FOR EE4TK4 DIGITAL COMMUNICATION SYSTEMS

Channel Capacity for SISO Rayleigh Fading Channels for Binary PSK Modulation

Project aims to

• understand the basic concepts of the entropy, differential entropy and mutual information
• be proficient at using commonly-used binary PSK modulation and the Gaussian distribution.
• develop the ability to optimize mutual information subject to binary PSK modulation for a SISO Rayleigh fading channel.

PROBLEM -

Let us consider a wireless communication system having a single transmitting antenna and single receiving antennas with flat fading, i.e, single input and single output (SISO). For such a system, the discrete-time baseband-equivalent channel model can be represented as

Y = HX + η,                                                   (1)

where H is the real channel coefficient from the transmitter to the receiver. Now, our transmission and reception schemes are described as follows. During the first time slot, one bit training signal X = √(P_T) is sent for transmission to estimate the channel using the least square error (LSE) criterion, i.e.,

y_T = H√(P_t) + ηT

H^ˆ = arg min_H|y_t - H√(P_t)|                             (2)

During the second time slot, the signal X = x is randomly chosen from the binary PSK constellation X = {-A/2, A/2} for transmission through the channel model (1), i.e.,

y_D = Hx + η_D

where P(X = - A/2) = p and P(X = A/2) = q = 1 - p. Then, using the estimate of channel h obtained from (2), we attempt to recover the transmitted data x, i.e.,

y = H^ˆx + ξ                                                   (3)

where ξ is related to x, η_T and η_D. In this project, we make the following assumption:

• The channel coefficient H is not known at the receiver and constant during two time slots, after which it randomly changes to a new independent value that is fixed for another two time slots, and so on.
• η_T and η_D are independently Gaussian distributed with each having zero mean and variance σ².
• h is Rayleigh distributed., i.e., the probability density function of h is f_H(h) = 2he^{-h^2}, h ≥ 0.

Project is to seek for a solution to the following problem:

Problem 1: For each signal to noise ratio SNR = 1/σ², find optimal input probability p_opt(SNR) and signal transmission power P_D,opt(SNR) that maximize mutual information I(X; Y ) subject a total power constraint P_T + P_D = 1. In addition, draw two curves: P_D,opt(SNR) vs SNR (dB) and the channel capacity C(SNR) = max I(X; Y ) vs SNR (dB).