Characterizing probability distribution function for the input of a communication channel that achieves the maximum possible data rate is one of the most fundamental problems in the field of information theory. In his ground-breaking paper, Shannon showed that the capacity of a point-to-point additive white Gaussian noise channel under an average power constraint at the input, is achieved by Gaussian distribution. Although imposing a limitation on the peak of the channel input is also very important in modelling the communication system more accurately, it has gained much less attention in the past few decades. A rather unexpected result of Smith indicated that the capacity achieving distribution for an AWGN channel under peak constraint at the input is unique and discrete, possessing a finite number of mass points.
In this thesis, we study multiple access channel under peak constraints at the inputs of the channel. By extending Smith's argument to our multi-terminal problem we show that any point on the boundary of the capacity region of the channel is only achieved by discrete distributions with a finite number of mass points. Although we do not claim uniqueness of the capacity-achieving distributions, however, we show that only discrete distributions with a finite number of mass points can achieve points on the boundary of the capacity region.
First we deal with the problem of maximizing the sum-rate of a two user Gaussian MAC with peak constraints. It is shown that generating the code-books of both users according to discrete distributions with a finite number of mass points achieves the largest sum-rate in the network. After that we generalize our proof to maximize the weighted sum-rate of the channel and show that the same properties hold for the optimum input distributions. This completes the proof that the capacity region of a two-user Gaussian MAC is achieved by discrete input distributions with a finite number of mass points.
| Identifer | oai:union.ndltd.org:LACETR/oai:collectionscanada.gc.ca:OWTU.10012/7223 |
| Date | January 2013 |
| Creators | Mamandipoor, Babak |
| Source Sets | Library and Archives Canada ETDs Repository / Centre d'archives des thèses électroniques de Bibliothèque et Archives Canada |
| Language | English |
| Detected Language | English |
| Type | Thesis or Dissertation |
Page generated in 0.0016 seconds