We propose a three-module extension of score-based VAMP (SC-VAMP) for signal recovery in nonlinear channels, where the received signal is obtained by applying a nonlinearity to a linear mixture of the transmitted signal, followed by additive Gaussian noise. The key idea is to introduce a latent variable representing the output of the linear mixing stage, which decomposes the inference problem into three modules: a likelihood module that handles the nonlinear observation via Gauss--Hermite quadrature, a coupling module that enforces the linear constraint between the transmitted signal and the latent variable via LMMSE estimation, and a denoiser module that incorporates the code constraint using belief propagation (BP) decoding. Each module exchanges extrinsic scalar-Gaussian messages with Onsager corrections derived from posterior variances that are computed in closed form or to quadrature accuracy. Numerical experiments with BPSK-modulated LDPC codewords transmitted through a hyperbolic tangent channel demonstrate that the proposed method achieves a clear waterfall in bit error rate (BER), with the gap to the capacity estimate narrowing as the block length increases from 128 to 2304. The framework provides a modular receiver architecture applicable to a broad class of nonlinear channels. Since only the likelihood module depends on the channel nonlinearity, the architecture readily adapts to other channel models by replacing a single module while leaving the coupling and decoder modules unchanged.

翻译：针对非线性信道中的信号恢复问题，我们提出基于得分的VAMP（SC-VAMP）的三模块扩展方案，其中接收信号通过将非线性函数应用于发送信号的线性混合后叠加加性高斯噪声获得。核心思想是引入表征线性混合阶段输出的潜变量，从而将推理问题分解为三个模块：通过高斯-埃尔米特求积处理非线性观测的似然模块、利用线性最小均方误差（LMMSE）估计强制执行发送信号与潜变量间线性约束的耦合模块，以及采用置信传播（BP）解码嵌入码约束的降噪模块。每个模块交换带有昂萨格修正项的外标量高斯消息，其中修正项源自以闭合形式或求积精度计算的后验方差。通过双曲正切信道传输BPSK调制LDPC码字的数值实验表明，所提方法在误码率（BER）上呈现清晰的水瀑特性，且当块长度从128增至2304时，与容量估计值的差距逐渐缩小。该框架为广泛非线性信道类提供了模块化接收机架构。由于仅似然模块依赖于信道非线性特性，该架构通过替换单一模块即可便捷适配其他信道模型，而耦合模块与译码模块保持不变。