异向编织的渐近分析与空间耦合
Asymptotic Analysis and Spatial Coupling of Counter Braids
摘要：
计数器辫（CB）是Lu等人在2007年提出的一种新型计数器架构，用于高速链路上的每流测量，可以使用消息传递（MP）以低复杂度解码。CB实现与流大小分布的熵下限相匹配的渐近压缩率（在最优解码下）。在本文中，我们将空间耦合的概念应用到CB中，以改善原始CB的性能，并分析了由此产生的空间耦合CB（SC-CB）的性能。我们引入了一个等价的二部图表示的CB相同的迭代迭代有限长度和渐近性能。基于这种等价表示，我们然后分析了单层CB和SC-CB在Lu等人提出的MP解码算法下的渐近性能。特别地，我们导出了非耦合系统的势阈值，并证明了它等于面积阈值。我们还推导出麦克斯韦解码器的CB和证明的潜在阈值是上界的麦克斯韦解码阈值。我们表明，在扩展的MP外部信息传递曲线下的面积等于零精确的面积阈值。这一点，结合麦克斯韦解码器和仿真结果的分析，导致我们的猜想，即潜在的阈值等于麦克斯韦解码阈值，因此最大后验解码阈值的下限。有趣的是，SC-CB没有显示出MP解码阈值的阈值饱和到空间耦合的低密度奇偶校验码和其他耦合系统的潜在阈值特性的众所周知的现象。然而，SC-CB产生比它们的未耦合对应物更好的MP解码阈值。最后，我们还考虑SC-CB作为一种压缩感知方案，并表明可以实现低欠采样因子。
英文摘要：
A counter braid (CB) is a novel counter architecture introduced by Lu et al. in 2007 for per-flow measurements on high-speed links which can be decoded with low complexity using message passing (MP). CBs achieve an asymptotic compression rate (under optimal decoding) that matches the entropy lower bound of the flow size distribution. In this paper, we apply the concept of spatial coupling to CBs to improve the performance of the original CBs and analyze the performance of the resulting spatially-coupled CBs (SC-CBs). We introduce an equivalent bipartite graph representation of CBs with identical iteration-by-iteration finite-length and asymptotic performance. Based on this equivalent representation, we then analyze the asymptotic performance of single-layer CBs and SC-CBs under the MP decoding algorithm proposed by Lu et al.. In particular, we derive the potential threshold of the uncoupled system and show that it is equal to the area threshold. We also derive the Maxwell decoder for CBs and prove that the potential threshold is an upper bound on the Maxwell decoding threshold. We show that the area under the extended MP extrinsic information transfer curve is equal to zero precisely at the area threshold. This, combined with the analysis of the Maxwell decoder and simulation results, leads us to the conjecture that the potential threshold is equal to the Maxwell decoding threshold and hence a lower bound on the maximum a posteriori decoding threshold. Interestingly, SC-CBs do not show the well-known phenomenon of threshold saturation of the MP decoding threshold to the potential threshold characteristic of spatially-coupled low-density parity-check codes and other coupled systems. However, SC-CBs yield better MP decoding thresholds than their uncoupled counterparts. Finally, we also consider SC-CBs as a compressed sensing scheme and show that low undersampling factors can be achieved.
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