On the Computational Cutoff Rate, R0for the Peak-Power-Limited Gaussian Channel

Abstract

The "computational cutoff rate," R <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> , represents a practical measure of the maximum reliable data rate that can be achieved by coding over a given communication channel using a given modulation format, in contrast with the "channel capacity," <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</tex> , which represents an idealized theoretical limit on the achievable data rate. Moreover, designing signal sets with good error probabilities using the R <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> criterion results in a mathematical problem that is much more tractable than that obtained by using the probability of error itself as a criterion. Both of the above reasons establish the importance of R <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> in communications theory. This paper starts with a brief tutorial background, which reveals the origin and the significance of R <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> . Next, the problem of achieving R <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</inf> over the additive white Gaussian noise (AWGN) dispersive or nondispersive channel, using quadrature-amplitude modulation (QAM) with a peakpower constraint, is addressed. The major result is that, for both cases, the optimum transmission signal set is chosen from a discrete distribution. The solution is derived in detail for the peak-power-limited nondispersive channel, where it is shown that the optimum QAM symbols are selected independently from a probability distribution that is uniform in the phase and discrete in the radius. The solution for the corresponding peak-power-limited dispersive channel is obtained only asymptotically, for large signal-to-noise ratio (SNR), where it is shown that the QAM symbols are selected independently from a uniform distribution within a disk in the complex signal space.