Scaling exponent and moderate deviations asymptotics of polar codes for the AWGN channel

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 qY|X with capacity I(qY|X ) characterizes the closest gap between the capacity and non-asymptotic achievable rates as follows: For a fixed ? ? (0, 1), the gap between the capacity I(qY|X ) and the maximum non-asymptotic rate Rn achieved by a length-n polar code with average error probability ? scales as n, i.e., I(qY|X ) – Rn = ?(n ). It is well known that the scaling exponent µ for any binary-input memoryless channel (BMC) with I(qY|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(nlog 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 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. © 2017 by the authors. Licensee MDPI, Basel, Switzerland. T