This paper studies a generalization of sparse superposition codes (SPARCs) for communication over the complex additive white Gaussian noise (AWGN) channel. In a SPARC, the codebook is defined in terms of a design matrix, and each codeword is a generated by multiplying the design matrix with a sparse message vector. In the standard SPARC construction, information is encoded in the locations of the non-zero entries of the message vector. In this paper we generalize the construction and consider modulated SPARCs, where information is encoded in both the locations and the values of the non-zero entries of the message vector. We focus on the case where the non-zero entries take values from a phase-shift keying (PSK) constellation. We propose a computationally efficient approximate message passing (AMP) decoder, and obtain analytical bounds on the state evolution parameters which predict the error performance of the decoder. Using these bounds we show that PSK-modulated SPARCs are asymptotically capacity achieving for the complex AWGN channel, with either spatial coupling or power allocation. We also provide numerical simulation results to demonstrate the error performance at finite code lengths. These results show that introducing modulation to the SPARC design can significantly reduce decoding complexity without sacrificing error performance.

中文翻译：

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复杂 AWGN 信道的调制稀疏叠加码


本文研究了稀疏叠加码 (SPARC) 的推广，用于通过复杂的加性高斯白噪声 (AWGN) 信道进行通信。在 SPARC 中，码本是根据设计矩阵定义的，并且每个码字是通过将设计矩阵与稀疏消息向量相乘而生成的。在标准 SPARC 结构中，信息被编码在消息向量的非零条目的位置中。在本文中，我们概括了这种结构并考虑了调制 SPARC，其中信息在消息向量的非零条目的位置和值中进行编码。我们重点关注非零条目从相移键控 (PSK) 星座中获取值的情况。我们提出了一种计算高效的近似消息传递（AMP）解码器，并获得了预测解码器错误性能的状态演化参数的分析界限。使用这些界限，我们表明，通过空间耦合或功率分配，PSK 调制的 SPARC 可以渐近地实现复杂 AWGN 信道的容量。我们还提供数值模拟结果来演示有限代码长度下的错误性能。这些结果表明，在 SPARC 设计中引入调制可以显着降低解码复杂性，而不会牺牲错误性能。