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
Author to whom correspondence should be addressed.
Received: 8 June 2017 / Revised: 8 July 2017 / Accepted: 13 July 2017 / Published: 15 July 2017
This paper investigates polar codes for the additive white Gaussian noise (AWGN) channel. The scaling exponent
of polar codes for a memoryless channel
characterizes the closest gap between the capacity and non-asymptotic achievable rates as follows: For a fixed
, the gap between the capacity
and the maximum non-asymptotic rate
achieved by a length-n
polar code with average error probability
. It is well known that the scaling exponent
for any binary-input memoryless channel (BMC) with
is bounded above by
. Our main result shows that
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
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
by using a superposition of
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.
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MDPI and ACS Style
Fong, S.L.; Tan, V.Y.F. Scaling Exponent and Moderate Deviations Asymptotics of Polar Codes for the AWGN Channel. Entropy 2017, 19, 364.
Fong SL, Tan VYF. Scaling Exponent and Moderate Deviations Asymptotics of Polar Codes for the AWGN Channel. Entropy. 2017; 19(7):364.
Fong, Silas L.; Tan, Vincent Y.F. 2017. "Scaling Exponent and Moderate Deviations Asymptotics of Polar Codes for the AWGN Channel." Entropy 19, no. 7: 364.
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