论文标题
通过空间耦合稀疏叠加代码与AMP解码的稀疏叠加代码
Capacity-achieving Spatially Coupled Sparse Superposition Codes with AMP Decoding
论文作者
论文摘要
稀疏的叠加代码，也称为稀疏回归代码（SPARC），是一类，以接近通道容量的速率，在AWGN通道上有效通信的有效通信。在标准的SPARC中，代码字是I.I.D的列的稀疏线性组合。高斯设计矩阵虽然在空间耦合的SPARC中，设计矩阵具有块状结构，在该结构中，高斯条目的方差可以在块上变化。精心设计的空间耦合结构可以显着增强迭代解码算法的误差性能，例如近似消息传递（AMP）。 在本文中，我们获得了一个非轴突结合，即与AMP解码的空间耦合SPARC的误差的可能性。将此绑定应用于简单的带对角设计矩阵，我们证明，与AMP解码的空间耦合SPARC达到了AWGN通道的容量。结合还强调了误差概率的衰减如何取决于空间耦合SPARC的每个设计参数。 AMP解码的一个有吸引力的特征是它的渐近平方误差（MSE）可以通过称为状态进化的确定性递归来预测。我们的结果提供了第一个证据，即MSE集中在空间耦合设计的状态进化预测上。结合状态进化预测，该结果意味着与拟议的带 - 基因对角线设计的空间耦合SPARC是能力方面的。使用用于建立主要结果的证明技术，我们还获得了用于使用空间耦合设计矩阵压缩传感的AMP MSE的浓度不等式。最后，我们提供了数值模拟结果，以证明空间耦合SPARC的有限长度误差性能。将性能与使用DVB-S2标准的LDPC代码的编码调制方案进行了比较。
Sparse superposition codes, also called sparse regression codes (SPARCs), are a class of codes for efficient communication over the AWGN channel at rates approaching the channel capacity. In a standard SPARC, codewords are sparse linear combinations of columns of an i.i.d. Gaussian design matrix, while in a spatially coupled SPARC the design matrix has a block-wise structure, where the variance of the Gaussian entries can be varied across blocks. A well-designed spatial coupling structure can significantly enhance the error performance of iterative decoding algorithms such as Approximate Message Passing (AMP). In this paper, we obtain a non-asymptotic bound on the probability of error of spatially coupled SPARCs with AMP decoding. Applying this bound to a simple band-diagonal design matrix, we prove that spatially coupled SPARCs with AMP decoding achieve the capacity of the AWGN channel. The bound also highlights how the decay of error probability depends on each design parameter of the spatially coupled SPARC. An attractive feature of AMP decoding is that its asymptotic mean squared error (MSE) can be predicted via a deterministic recursion called state evolution. Our result provides the first proof that the MSE concentrates on the state evolution prediction for spatially coupled designs. Combined with the state evolution prediction, this result implies that spatially coupled SPARCs with the proposed band-diagonal design are capacity-achieving. Using the proof technique used to establish the main result, we also obtain a concentration inequality for the MSE of AMP applied to compressed sensing with spatially coupled design matrices. Finally we provide numerical simulation results that demonstrate the finite length error performance of spatially coupled SPARCs. The performance is compared with coded modulation schemes that use LDPC codes from the DVB-S2 standard.
