Asymptotic Behavior of Error Exponents in the Wideband Regime

摘要

In this paper, we investigate the fundamental tradeoff between rate and bandwidth when a constraint is imposed on the error exponent. Specifically, we consider both additive white Gaussian noise (AWGN) and Rayleigh-fading channels where the input symbols are assumed to have a peak constraint. For the AWGN channel model, the optimal values of $R_z(0)$ and $mathdot{R_z}(0)$ are calculated, where $R_z(1/B)$ is the maximum rate at which information can be transmitted over a channel with bandwidth $B$ when the error-exponent is constrained to be greater than or equal to $z$. The computation of $R_z(0)$ follows Gallager's infinite-bandwidth reliability function computation, while the computation of $mathdot{R}_z(0)$ is new and parallels Verdu's second-order calculation for channel capacity. Based on these calculations, we say that a sequence of input distributions is near optimal if both $R_z(0)$ and $mathdot{R_z}(0)$ are achieved. We show that quaternary phase-shift keying (QPSK), a widely used signaling scheme, is near optimal within a large class of input distributions for the AWGN channel. Similar results are also established for a fading channel where full channel side information (CSI) is available at the receiver.