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Entropy 2017, 19(7), 364; https://doi.org/10.3390/e19070364

Scaling Exponent and Moderate Deviations Asymptotics of Polar Codes for the AWGN Channel

Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117583, Singapore
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Received: 8 June 2017 / Revised: 8 July 2017 / Accepted: 13 July 2017 / Published: 15 July 2017
(This article belongs to the Special Issue Multiuser Information Theory)
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Abstract

This paper investigates polar codes for the additive white Gaussian noise (AWGN) channel. The scaling exponent μ of polar codes for a memoryless channel q Y | X with capacity I ( q Y | X ) characterizes the closest gap between the capacity and non-asymptotic achievable rates as follows: For a fixed ε ( 0 , 1 ) , the gap between the capacity I ( q Y | X ) and the maximum non-asymptotic rate R n * achieved by a length-n polar code with average error probability ε scales as n - 1 / μ , i.e., I ( q Y | X ) - R n * = Θ ( n - 1 / μ ) . It is well known that the scaling exponent μ for any binary-input memoryless channel (BMC) with I ( q Y | X ) ( 0 , 1 ) is bounded above by 4 . 714 . Our main result shows that 4 . 714 remains a valid upper bound on the scaling exponent for the AWGN channel. Our proof technique involves the following two ideas: (i) The capacity of the AWGN channel can be achieved within a gap of O ( n - 1 / μ log n ) by using an input alphabet consisting of n constellations and restricting the input distribution to be uniform; (ii) The capacity of a multiple access channel (MAC) with an input alphabet consisting of n constellations can be achieved within a gap of O ( n - 1 / μ log n ) by using a superposition of log n binary-input polar codes. In addition, we investigate the performance of polar codes in the moderate deviations regime where both the gap to capacity and the error probability vanish as n grows. An explicit construction of polar codes is proposed to obey a certain tradeoff between the gap to capacity and the decay rate of the error probability for the AWGN channel. View Full-Text
Keywords: AWGN channel; moderate deviations; multiple access channel; polar codes; scaling exponent AWGN channel; moderate deviations; multiple access channel; polar codes; scaling exponent
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).
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Fong, S.L.; Tan, V.Y.F. Scaling Exponent and Moderate Deviations Asymptotics of Polar Codes for the AWGN Channel. Entropy 2017, 19, 364.

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