The price of certainty: "waterslide curves" and the gap to capacity

The classical problem of reliable point-to-point digital communication is to achieve a low probability of error while keeping the rate high and the total power consumption small. Traditional information-theoretic analysis uses `waterfall' curves to convey the revolutionary idea that unboundedly low probabilities of bit-error are attainable using only finite transmit power. However, practitioners have long observed that the decoder complexity, and hence the total power consumption, goes up when attempting to use sophisticated codes that operate close to the waterfall curve. This paper gives an explicit model for power consumption at an idealized decoder that allows for extreme parallelism in implementation. The decoder architecture is in the spirit of message passing and iterative decoding for sparse-graph codes. Generalized sphere-packing arguments are used to derive lower bounds on the decoding power needed for any possible code given only the gap from the Shannon limit and the desired probability of error. As the gap goes to zero, the energy per bit spent in decoding is shown to go to infinity. This suggests that to optimize total power, the transmitter should operate at a power that is strictly above the minimum demanded by the Shannon capacity. The lower bound is plotted to show an unavoidable tradeoff between the average bit-error probability and the total power used in transmission and decoding. In the spirit of conventional waterfall curves, we call these `waterslide' curves.Comment: 37 pages, 13 figures. Submitted to IEEE Transactions on Information Theory. This version corrects a subtle bug in the proofs of the original submission and improves the bounds significantl