Diversity-Rate Trade-Off in an channel with fading

Abstract

We study the diversity order versus rate of an additive white Gaussian noise (AWGN) channel in the whole capacity region. We show that for both discrete and continuous inputs, Gallager抯 upper bounds on error probability have exponential diversity in the low? and high?rate regions, but only sub?exponential diversity in the mid?rate region. We then show that the upper bounds with Gaussian input provide good approximation throughout the capacity region even for finite constellations. Next, we study an AWGN channel with fading. Using a tight upper and new tight lower bounds, we show that for a slow Rayleigh fading channel there is polynomial diversity. For a fast fading channel, we obtain a new upper bound which shows exponential diversity (even for Rayleigh fading) in the low? and high?rate regions but polynomial diversity in the mid?rate region. We also provide new tight lower bounds which exhibit exponential diversity throughout. We then obtain a power allocation scheme that maximizes the error exponents. For this scheme, the diversity order is exponential throughout the capacity region. For the selective channel inversion (SCI) power?allocation scheme, the diversity order is exponential in the low? and high?rate regions but polynomial in the mid?rate region. For the fast fading case, we also provide a new upper bound on the block error probability and a power?allocation scheme that minimizes it. The diversity order of this scheme is the same as that for SCI but provides lower BER.