On the Computational Cutoff Rate, R0for the Peak-Power-Limited Gaussian Channel
Adel A. M. Saleh, J. Salz · IEEE Transactions on Communications · 1987
The "computational cutoff rate," R0, 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,"C, which represents an idealized theoretical limit on the achievable data rate. Moreover, designing signal sets with good error probabilities using the R0criterion 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 R0in communications theory. This paper starts with a brief tutorial background, which reveals the origin and the significance of R0. Next, the problem of achieving R0over 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.