Broadband Fading Channels: Signal Burstiness and Capacity by Vijay Gautam Subramanian

The goal of this thesis is to understand how to achieve capacity for wireless fading channels with emphasis on the role of the burstiness in both time and frequency of the transmitted signals. Typical signal design constraints for a communication system are concerned primarily with regulating the transmitted power. This allows for signals of di ering burstiness, viz., ones that uniformly spread the energy over time and frequency such as direct-sequence spread spectrum signals and others that transmit large portions of the energy either in some time-slice or some bandwidth-slice. The focus of this thesis is to understand the di erence in performance between these two classes of signals. The notions of capacity per unit cost and reliability function per unit cost are used in the comparison. A natural means to constrain burstiness is to introduce peak-value constraints. An analysis of the capacity of discrete-time memoryless channels where the peak constraints assume small values is performed, and it is shown the capacity varies asymptotically as the product of one-half the peak squared and the Fisher information. In addition the shape of the reliability function for small-valued peak constraints is determined. For Rayleigh fading channels the Fisher information is zero, indicating the di culty of transmission over such channels with small peak constrained signals. A more exible means to control burstiness of signals is to constrain fourth-order moments, which are related to the variance of the energy or power of the signal. The scalar discrete-timememoryless Rayleigh fading channel is considered rst and it is shown that the capacity per unit energy equals the capacity per unit energy of the additive white gaussian noise (AWGN) channel with the same signal-to-noise ratio. Constraining the fourth moment of the input signal leads to lower capacity for the Rayleigh fading channel than for the AWGN channel. Speci cally, the Rayleigh fading channel has nite capacity per fourth-moment cost whereas the AWGN channel has in nite capacity per fourth-moment cost. iii The general wide-sense stationary and uncorrelated scattering (WSSUS) fading channel has a similar behavior in that the capacity per unit energy equals the capacity per unit energy of the AWGN channel with the same signal-to-noise ratio. The fourth-moment cost is generalized to a fourth-order functional called fourthegy which depends on the square of the signal ambiguity function. The capacity per unit fourthegy for WSSUS channels is identi ed and is nite and nonzero. It follows that signals with small values of fourthegy per unit energy are able to communicate only a small number of bits per unit energy. It shown in this thesis that for wideband fading channels the nonbursty direct-sequence spread spectrum signals have small values of fourthegy per unit energy; therefore, such signals can only achieve small rates of reliable information transfer. This result holds in general for di use WSSUS fading channels as well as specular WSSUS fading channels with many multipath components. The accompanying reliability function per unit energy and reliability function per unit fourthegy are also derived. iv DEDICATION Dedicated to my parents Jwalamukhy and Lt. Col. R. Subramanian v ACKNOWLEDGMENTS I would like to sincerely thank my thesis advisor Prof. B. Hajek for an extremely enjoyable and productive time during the course of my Ph.D. work. I would also like to acknowledge his many suggestions which have helped me broaden my academic horizons and have aided me in developing a healthy research attitude. I would like to thank Profs. R. Blahut, U. Madhow, M. M edard, D. Sarwate, and R. Sowers for agreeing to serve on my committee and for their constructive criticism. Thanks are also due to all my instructors for taking great interest in my progress. In particular, I would like to express my gratitude to Profs. U. Madhow and R. Srikant for their support. Special appreciation is due to my many friends from all walks of life who have made my stay colorful and worthwhile. My o cemates deserve special mention for having endured my ability to initiate and to indulge in loud and protracted discussions, and my talents in the art of territory usurpation. Special thanks are also due to Hemant Chaskar and Dilip Warrier for having put up with my innumerable idiosyncrasies. I would also like to acknowledge my brother, my aunt and all my other relatives who supported me despite a more than healthy dose of me. Finally, I would like to pay obeisance to God without whose blessings none of this would have been possible, and sing like one of the great Indian composers Sri Muthuswami Dikshitar, Vatapi Ganapatim Bhajeham Varanasyam Varapradam. (I worship Vatapi Ganapati, the elephant-faced giver of boons.) vi TABLE OF CONTENTS CHAPTER PAGE