| The interference channel, first considered by Shannon, models the communication between transmitters and receivers in which each transmitter wants to send information reliably to its intended receiver in the presence of interference from the other transmitters. The performance of this model is characterized by its capacity region, i.e., the set of rate points that can be simultaneously achieved with arbitrarily small probability of error. The capacity region of the general interference channel remains an open problem. This thesis presents results concerning particular cases of this problem. We divide the thesis into the following three parts: (1) Gaussian interference channels: We describe the model and summarize the cases that have been solved by other researchers. Then we establish the optimality of two extreme points in the achievable region of the general Gaussian interference channel. Also we prove that the class of degraded Gaussian interference channels is equivalent to the class of Z (one-sided) interference channels. Finally, we show a curious example of a Z-interference channel with degraded message sets, in which the effects of interference can be completely eliminated. (2) A strengthened Entropy Power Inequality: We prove a strengthened version of Shannon's Entropy Power Inequality for the case where one of the random vectors involved is Gaussian. In particular we show that if we add independent Gaussian noise to an arbitrary multivariate random variable, the entropy power of the resulting random variable is a concave function of the variance (power) of the added noise. This fact is precisely what allows us to establish the optimality of the new points in part 1. As a byproduct of this development we present an analog of the isoperimetric inequality in the domain of entropy powers. (3) Discrete interference channels: We establish the capacity region of two classes of discrete memoryless interference channels. |
