Scaling Exponent and Moderate Deviations Asymptotics of Polar Codes for the AWGN Channel
Scaling Exponent and Moderate Deviations Asymptotics of Polar Codes for the AWGN Channel
复制标题
AWGN 信道的极性码的缩放指数和适度偏差渐近
DOI：
10.3390/e19070364
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发表时间：
2017
期刊：
影响因子：
--
通讯作者：
V. Tan
中科院分区：
文献类型：
--
作者：
S. Fong;V. Tan
本文研究了加性白色高斯噪声（AWGN）信道下的极化码。针对无记忆信道q Y的极化码的标度指数μ| X，容量I（q Y| X）将容量和非渐近可达速率之间的最近间隙表征如下：对于固定的e ∈（0，1），容量I（q Y）之间的差距|X）和由长度为n的极化码实现的最大非渐近速率Rn *，其中平均错误概率e按n - 1 / μ缩放，即，I（q Y| X）-Rn * = Θ（n - 1 / μ）。众所周知，对于任何二进制输入无记忆通道（BMC），|X）∈（0，1）由4上界。七一四我们的主要结果表明，4。714仍然是AWGN信道的标度指数的有效上限。我们的证明技术涉及以下两个想法：（i）AWGN信道的容量可以在O（n - 1 / μ log n），通过使用由n个星座组成的输入字母表并将输入分布限制为均匀的;（ii）具有由n个星座组成的输入字母表的多址信道（MAC）的容量可以在O（n - 1 / μ log n）。此外，我们研究了极化码在中等偏差范围内的性能，其中容量的差距和错误概率都随着n的增长而消失。提出了一种极化码的显式构造，以服从AWGN信道的差距与容量之间的某种折衷和错误概率的衰减率。
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 e ∈ ( 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 e 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.