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

### Anant Sahai and Pulkit Grover

###
EECS Department

University of California, Berkeley

Technical Report No. UCB/EECS-2008-1

January 1, 2008

### http://www.eecs.berkeley.edu/Pubs/TechRpts/2008/EECS-2008-1.pdf

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 explicit models for the communication channel to study the power spent in transmission. The resulting bounds are expressed using `waterfall' curves that 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, but is further idealized in that it allows for more computational power than is currently known to be implementable. 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. The bound is shown to be order optimal by showing the existence of codes that can achieve similarly shaped waterslide curves under the proposed idealized model of decoding.

**Author Comments:** Also submitted to the IEEE Transactions on Information Theory

BibTeX citation:

@techreport{Sahai:EECS-2008-1, Author = {Sahai, Anant and Grover, Pulkit}, Title = {The price of certainty: "waterslide curves" and the gap to capacity}, Institution = {EECS Department, University of California, Berkeley}, Year = {2008}, Month = {Jan}, URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/2008/EECS-2008-1.html}, Number = {UCB/EECS-2008-1}, Note = {Also submitted to the IEEE Transactions on Information Theory}, Abstract = { 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 explicit models for the communication channel to study the power spent in transmission. The resulting bounds are expressed using `waterfall' curves that 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, but is further idealized in that it allows for more computational power than is currently known to be implementable. 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. The bound is shown to be order optimal by showing the existence of codes that can achieve similarly shaped waterslide curves under the proposed idealized model of decoding.} }

EndNote citation:

%0 Report %A Sahai, Anant %A Grover, Pulkit %T The price of certainty: "waterslide curves" and the gap to capacity %I EECS Department, University of California, Berkeley %D 2008 %8 January 1 %@ UCB/EECS-2008-1 %U http://www.eecs.berkeley.edu/Pubs/TechRpts/2008/EECS-2008-1.html %F Sahai:EECS-2008-1