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

摘要

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.