It was demonstrated that, as a nonlinear implementation of Slepian-Wolf Coding, Distributed Arithmetic Coding (DAC) outperforms traditional Low-Density Parity-Check (LPDC) codes for short code length and biased sources. This fact triggers research efforts into theoretical analysis of DAC. In our previous work, we proposed two analytical tools, Codebook Cardinality Spectrum (CCS) and Hamming Distance Spectrum, to analyze DAC for independent and identically-distributed (i.i.d.) binary sources with uniform distribution. This article extends our work on CCS from uniform i.i.d. binary sources to biased i.i.d. binary sources. We begin with the final CCS and then deduce each level of CCS backwards by recursion. The main finding of this article is that the final CCS of biased i.i.d. binary sources is not uniformly distributed over [0, 1). This article derives the final CCS of biased i.i.d. binary sources and proposes a numerical algorithm for calculating CCS effectively in practice. All theoretical analyses are well verified by experimental results.

中文翻译：

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独立和同分布二进制源的分布式算术编码的码本基数谱

已经证明，作为 Slepian-Wolf 编码的非线性实现，分布式算术编码 (DAC) 在短代码长度和偏置源方面优于传统的低密度奇偶校验 (LPDC) 代码。这一事实引发了对 DAC 理论分析的研究工作。在我们之前的工作中，我们提出了两种分析工具，即码本基数谱 (CCS) 和汉明距离谱，来分析 DAC 的具有均匀分布的独立和同分布 (iid) 二进制源。本文将我们在 CCS 上的工作从统一的 iid 二进制源扩展到有偏差的 iid 二进制源。我们从最终的 CCS 开始，然后通过递归向后推导出 CCS 的每个级别。本文的主要发现是有偏差的 iid 二进制源的最终 CCS 在 [0, 1) 上不是均匀分布的。本文推导出有偏差的 iid 二进制源的最终 CCS，并提出了一种在实践中有效计算 CCS 的数值算法。所有的理论分析都得到了实验结果的很好的验证。