[论文解读] Dispersion Analysis of Infinite Constellations in Ergodic Fading Channels

Shlomi Vituri|arXiv (Cornell University)|Sep 18, 2013

Advanced Wireless Communication Techniques参考文献 18被引用 10

一句话总结

本论文分析了在遍历衰落信道中无限星座的扩展特性，推导出标量和MIMO衰落信道中Poltyrev容量与信道扩展的精确表达式。结果表明，可实现的归一化对数密度与容量之间的差距，与信道扩展除以块长的平方根成正比，并乘以误码概率的逆Q函数，从而证实了渐近最优性。

ABSTRACT

This thesis considers infinite constellations in fading channels, without power constraint and with perfect channel state information available at the receiver. Infinite constellations are the framework, proposed by Poltyrev, for analyzing coded modulation codes. The Poltyrev's capacity, is the highest achievable normalized log density (NLD) of codewords per unit volume, at possibly large block length, that guarantees a vanishing error probability. For a given finite block length and a fixed error probability, there is a gap between the highest achievable NLD and Poltyrev's capacity. The dispersion analysis quantifies asymptotically this gap. The thesis begins by the dispersion analysis of infinite constellations in scalar fading channels. Later on, we extend the analysis to the case of multiple input multiple output fading channels. As in other channels, we show that the gap between the highest achievable NLD and the Poltyrev's capacity, vanishes asymptotically as the square root of the channel dispersion over the block length, multiplied by the inverse Q-function of the allowed error probability. Moreover, exact terms for Poltyrev's capacity and channel dispersion, are derived in the thesis. The relations to the amplitude and to the power constrained fading channels are also discussed, especially in terms of capacity, channel dispersion and error exponents. These relations hint that in typical cases the unconstrained model can be interpreted as the limit of the constrained model, when the signal to noise ratio tends to infinity.

研究动机与目标

• 分析在接收端具备完美信道状态信息的标量和MIMO遍历衰落信道中无限星座的扩展特性。
• 量化有限块长下最高可实现的归一化对数密度（NLD）与Poltyrev容量之间的差距。
• 推导出在衰落信道中Poltyrev容量与信道扩展的精确表达式。
• 探讨无约束（无限星座）与功率/幅度受限衰落信道之间的关系。
• 证明无约束模型在高信噪比极限下可近似为受限模型。

提出的方法

• 采用Poltyrev的无限星座框架，将码字建模为n维空间中的格点，且无功率约束。
• 利用随机编码论证和大偏差理论，推导出在衰落信道中误码概率的渐近行为。
• 通过衰落分布的Fisher信息矩阵计算信道扩展，以捕捉二阶渐近特性。
• 对于MIMO信道，通过将衰落矩阵建模为具有i.i.d.元素的随机矩阵，将分析扩展至多天线系统。
• 使用逆Q函数将可实现NLD与Poltyrev容量之间的差距表示为块长和误码概率的函数。
• 通过取高信噪比极限，建立与功率受限系统的关联，表明无约束模型收敛至受限情形。

实验结果

研究问题

• RQ1 标量和MIMO遍历衰落信道中Poltyrev容量的精确表达式是什么？
• RQ2 在衰落信道中，可实现NLD与Poltyrev容量之间的差距如何随块长和误码概率变化？
• RQ3 无限星座在衰落信道中的信道扩展是什么？其推导过程如何？
• RQ4 无约束星座的结果与功率或幅度受限衰落信道的结果之间有何关系？
• RQ5 无约束模型能否被解释为受限衰落信道模型在高信噪比极限下的极限形式？

主要发现

• 标量衰落信道中Poltyrev容量以闭式表达，其依赖于衰落分布的矩。
• 标量衰落信道的信道扩展以对数信道增益方差的形式表达。
• 对于MIMO衰落信道，利用衰落矩阵的特征值分布推导出Poltyrev容量与扩展。
• 可实现NLD与Poltyrev容量之间的差距渐近地表现为√(D/N) ⋅ Q⁻¹(ε)，其中D为信道扩展，N为块长，ε为误码概率。
• 在高信噪比区域，无约束无限星座模型收敛至功率受限衰落信道模型。
• 推导出的扩展表达式证实，无限星座在遍历衰落信道中可实现渐近最优的NLD性能。