The generalized linear system (GLS) has been widely used in wireless communications to evaluate the effect of nonlinear preprocessing on receiver performance. Generalized approximation message passing (AMP) is a state-of-the-art algorithm for the signal recovery of GLS, but it was limited to measurement matrices with independent and identically distributed (IID) elements. To relax this restriction, generalized orthogonal/vector AMP (GOAMP/GVAMP) for unitarily-invariant measurement matrices was established, which has been proven to be replica Bayes optimal in uncoded GLS. However, the information-theoretic limit of GOAMP/GVAMP is still an open challenge for arbitrary input distributions due to its complex state evolution (SE). To address this issue, in this paper, we provide the achievable rate analysis of GOAMP/GVAMP in GLS, establishing its information-theoretic limit (i.e., maximum achievable rate). Specifically, we transform the fully-unfolded state evolution (SE) of GOAMP/GVAMP into an equivalent single-input single-output variational SE (VSE). Using the VSE and the mutual information and minimum mean-square error (I-MMSE) lemma, the achievable rate of GOAMP/GVAMP is derived. Moreover, the optimal coding principle for maximizing the achievable rate is proposed, based on which a kind of low-density parity-check (LDPC) code is designed. Numerical results verify the achievable rate advantages of GOAMP/GVAMP over the conventional maximum ratio combining (MRC) receiver based on the linearized model and the BER performance gains of the optimized LDPC codes (0.8~2.8 dB) compared to the existing methods.

翻译：广义线性系统（GLS）在无线通信中广泛应用，用于评估非线性预处理对接收机性能的影响。广义近似消息传递（AMP）是GLS信号重构领域的先进算法，但其仅适用于具有独立同分布（IID）元素的测量矩阵。为突破这一限制，针对酉不变测量矩阵建立了广义正交/向量AMP（GOAMP/GVAMP），该算法在无编码GLS中已被证明具有复制贝叶斯最优性。然而，由于GOAMP/GVAMP的复杂状态演化（SE）机制，对于任意输入分布而言，其信息论极限仍是开放性难题。针对该问题，本文分析了GLS中GOAMP/GVAMP的可达速率，确立了其信息论极限（即最大可达速率）。具体而言，我们将GOAMP/GVAMP的全展开状态演化转换为等效的单输入单输出变分状态演化（VSE），并利用VSE及互信息与最小均方误差（I-MMSE）引理推导出GOAMP/GVAMP的可达速率。进一步提出最大化可达速率的最优编码原理，并据此设计了一类低密度奇偶校验（LDPC）码。数值结果验证了GOAMP/GVAMP相较于基于线性化模型的传统最大比合并（MRC）接收机的可达速率优势，以及优化LDPC码相比现有方法获得的误码率性能增益（0.8~2.8 dB）。