Give

# Graduate Exam Abstract

## Zhifei Fan

Ph.D. Final
October 19, 2006, 1:50pm-4:0
Engr B4
Signal Design, Diversity, and Capacity in Multi-access Communication Systems

Abstract: In this dissertation, we exploit degrees of freedom in time and frequency to trade-off capacity and diversity in time-frequency-spreading channels. We then design signals to maximize channel performance. We consider the trade-off between multiplexing gain and diversity gain for the block-fading sub-channel model. The channel vector may be rank deficient with arbitrary covariance structure. We derive this trade-off by considering the scaling law of the ergodic capacity, which represents the multiplexing gain, and the error probability, which determines the diversity gain at high SNR. With a fixed multiplexing gain, we give an upper bound and a lower bound on the maximum diversity gain, and give an optimization procedure to get the exact maximum diversity gain. We also address the trade-off between multiplexing gain and diversity gain for the frequency-selective channel. Similarly, we derive this trade-off by considering the scaling law of the ergodic capacity, which determines the multiplexing gain, and the error probability, which determines the diversity gain at high SNR. It is proved that this trade-off only depends on the number of independent taps of the equivalent FIR channel filter. The error probability is bounded by the outage probability and the error probability without outage. The scaling law of the outage probability and the error probability without outage at high SNR are derived, as are approximations of the outage probability at both low and high SNR. Besides the theoretical research on capacity and diversity, we investigate signal design, i.e. joint analog precoder and equalizer design, for multichannel data transmission over the frequency-selective channel. The design goal is to maximize mutual information rate, minimize the mean square error, or minimize the bit error rate subject to a transmit power constraint. We assume a continuous channel model with precoder ransmissions for M subchannels that lie in an $n$-dimensional linear subspace of $L^2({\mathcal{R}})$. We first design the subspace according to the channel characteristics, and then design the precoders as functions in this subspace. After the design of the optimal precoder and equalizer, we explore the geometry of these designs. We show that all of these precoder and equalizer designs are, in fact, decompositions of a virtual two-channel problem into a system of canonical coordinates, wherein variables in the canonical message channel are correlated only pairwise with corresponding variables in the canonical measurement channel. This finding clarifies the geometry of precoder and equalizer designs and illustrates that they decompose the two-channel communication problem into what might be called the Shannon channel. We also investigate joint precoder and equalizer designs for a CDMA multi-user, multi-path system, and design the precoder to get a simplified receiver design for an MMSE equalizer on the receiver side, using a warp convergence property for special matrices.

Adviser: Louis L. Scharf