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Verification Decoding of High-Rate LDPC Codes With Applications in Compressed Sensing

机译：高速LDPC码的验证解码及其在压缩感知中的应用

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摘要

This paper considers the performance of $(j,k)$-regular low-density parity-check (LDPC) codes with message-passing (MP) decoding algorithms in the high-rate regime. In particular, we derive the high-rate scaling law for MP decoding of LDPC codes on the binary erasure channel (BEC) and the $q$ -ary symmetric channel ($q$-SC). For the BEC and a fixed $j$, the density evolution (DE) threshold of iterative decoding scales like $Theta (k^{-1})$ and the critical stopping ratio scales like $Theta (k^{-j/(j-2)})$. For the $q$-SC and a fixed $j$, the DE threshold of verification decoding depends on the details of the decoder and scales like $Theta (k^{-1})$ for one decoder. Using the fact that coding over large finite alphabets is very similar to coding over the real numbers, the analysis of verification decoding is also extended to the compressed sensing (CS) of strictly sparse signals. A DE-based approach is used to analyze the CS systems with randomized-reconstruction guarantees. This leads to the result that strictly sparse signals can be reconstructed efficiently with high probability using a constant oversampling ratio (i.e., when the number of measurements scales linearly with the sparsity of the signal). A stopping-set-based approach is also used to get stronger (e.g., uniform-in-probability) reconstruction guarantees.
机译：本文考虑了高速率状态下具有消息传递（MP）解码算法的$（j，k）$常规低密度奇偶校验（LDPC）码的性能。特别地，我们推导了用于在二进制擦除信道（BEC）和$ q $ -ary对称信道（$ q $ -SC）上进行LDPC码的MP解码的高速率缩放定律。对于BEC和固定的$ j $，迭代解码的密度演化（DE）阈值像$ Theta（k ^ {-1}）$之类，而临界停止比率像$ Theta（k ^ {-j /（ j-2）}）$。对于$ q $ -SC和固定的$ j $，验证解码的DE阈值取决于解码器的细节，并且一个解码器的缩放比例为$ Theta（k ^ {-1}）$。利用大有限字母编码与实数编码非常相似的事实，验证解码的分析也扩展到严格稀疏信号的压缩感知（CS）。基于DE的方法用于分析具有随机重建保证的CS系统。这导致使用恒定的过采样率（即，当测量的数量与信号的稀疏度成线性比例时），可以高概率有效地重构严格稀疏的信号。还使用基于停止集的方法来获得更强的（例如概率统一）重建保证。

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