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Finite-Dimensional Infinite Constellations

机译：有限维度的无限星座

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摘要

In the setting of a Gaussian channel without power constraints, proposed by Poltyrev in 1994, the codewords are points in an $n$-dimensional Euclidean space (an infinite constellation) and the tradeoff between their density and the error probability is considered. The normalized log density (NLD) plays the role of the communication rate, and capacity as well as error exponent bounds for this setting are known. This paper considers the infinite constellation setting in the finite block-length (dimension) regime. A simplified expression for Poltyrev's achievability bound is found and it is shown to be closely related to the sphere converse bound and to a recently proposed achievability bound based on point processes. The bounds are then analyzed asymptotically for growing $n$: for fixed NLD, the bounds turn out to be extremely tight compared to previous error exponent analysis. For fixed error probability $ varepsilon $, it is shown that the gap of the highest achievable NLD to the optimal NLD (Poltyrev's capacity) is approximately $sqrt{{1}over{2n}}Q^{-1}(varepsilon)$ , where $Q$ is the standard complementary Gaussian cumulative distribution function, thus extending the channel dispersion analysis to infinite constellations. Connections to the error exponent of the power-constrained Gaussian channel and to the volume-to-noise ratio as a figure of merit are discussed. Finally, the new tight bounds are compared to state-of-the-art coding schemes.
机译：在1994年Poltyrev提出的无功率约束的高斯信道的设置中，码字是$ n $维欧几里德空间（无限星座）中的点，并且考虑了它们的密度与错误概率之间的折衷。归一化的日志密度（NLD）充当通信速率的角色，并且已知该设置的容量以及错误指数范围。本文考虑了有限块长（维度）方案中的无限星座设置。找到了Poltyrev可达到性界线的简化表达式，它与球形逆界和最近基于点过程提出的可达到性界线密切相关。然后渐近分析边界的增长情况，以得出$ n $：对于固定NLD，与以前的误差指数分析相比，边界变得非常紧密。对于固定错误概率$ varepsilon $，表明最高可实现NLD与最优NLD（Poltyrev的容量）之间的差距约为$ sqrt {{1} over {2n}} Q ^ {-1}（varepsilon）$ ，其中$ Q $是标准的互补高斯累积分布函数，因此将信道色散分析扩展到了无限星座。讨论了与功率受限的高斯通道的误差指数以及作为品质因数的体积噪声比的关系。最后，将新的严格界限与最新的编码方案进行比较。

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