Due to the advantage of capacity-achieving, polar codes have been extended to the block fading channel whereas most constructions involve complex iterative-calculation. In this paper, we establish a systematic framework to analyze the error performance of polar codes in the case of block mapping and random mapping. For both the mappings, by introducing the new concept, named split polar spectrum, we derive the upper bound on the error probability of polarized channel which explicitly reveals the relationship between the diversity order L and the block-wise weight distribution of the codeword. For the special case L=2 in the block mapping, we design the enumeration algorithm to calculate the exact split polar spectrum based on the general MacWilliams identities. For arbitrary diversity order in the random mapping, with the help of uniform interleaving, we derive the approximate split polar spectrum by combining the polar spectrum and the probability of fading pattern for a specific weight. Furthermore, we propose the design criteria to construct polar codes over the block fading channel. The full diversity criterion is the primary target to achieve the diversity gain and the product distance criterion requires to maximize the product of the block-wise Hamming distance whereby obtain the coding gain. Guided by these design criteria, the construction metrics, respectively named polarized diversity weight (PDW) and simplified PDW (SPDW) are proposed to designpolar codes in both mappings. Such simple metrics can construct polar codes with similar or better performance over those based on traditional methods in block fading channel.

中文翻译：

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通过极性频谱对块衰落信道进行极性编码分集


由于容量实现的优势，极性码已扩展到块衰落信道，而大多数构造涉及复杂的迭代计算。在本文中，我们建立了一个系统框架来分析极性码在块映射和随机映射情况下的错误性能。对于这两种映射，通过引入称为分裂极谱的新概念，我们推导了极化信道错误概率的上限，明确揭示了分集阶数 L 与码字的块式权重分布之间的关系。针对块映射中L=2的特殊情况，我们设计了基于一般MacWilliams恒等式的枚举算法来计算精确的分裂极谱。对于随机映射中的任意分集阶，在均匀交织的帮助下，我们通过结合特定权重的极谱和衰落模式的概率来导出近似分裂极谱。此外，我们提出了在块衰落信道上构建极性码的设计标准。全分集准则是实现分集增益的首要目标，而乘积距离准则则要求最大化分块汉明距离的乘积，从而获得编码增益。在这些设计标准的指导下，提出了分别称为极化分集权重（PDW）和简化PDW（SPDW）的构造度量来设计两种映射中的极化码。这种简单的度量可以构造极性码，其性能与块衰落信道中基于传统方法的极性码相似或更好。