This paper investigates a large unitarily invariant system (LUIS) involving a unitarily invariant sensing matrix, an arbitrarily fixed signal distribution, and forward error control (FEC) coding. A universal Gram-Schmidt orthogonalization is considered for constructing orthogonal approximate message passing (OAMP), enabling its applicability to a wide range of prototypes without the constraint of differentiability. We develop two single-input-single-output variational transfer functions for OAMP with Lipschitz continuous local estimators, facilitating an analysis of achievable rates. Furthermore, when the state evolution of OAMP has a unique fixed point, we reveal that OAMP can achieve the constrained capacity predicted by the replica method of LUIS based on matched FEC coding, regardless of the signal distribution. The replica method is rigorously validated for LUIS with Gaussian signaling and certain sub-classes of LUIS with arbitrary signal distributions. Several area properties are established based on the variational transfer functions of OAMP. Meanwhile, we present a replica constrained capacity-achieving coding principle for LUIS. This principle serves as the basis for optimizing irregular low-density parity-check (LDPC) codes specifically tailored for binary signaling in our simulation results. The performance of OAMP with these optimized codes exhibits a remarkable improvement over the unoptimized codes and even surpasses the well-known Turbo-LMMSE algorithm. For quadrature phase-shift keying (QPSK) modulation, we observe bit error rates (BER) performance near the replica constrained capacity across diverse channel conditions.

翻译：本文研究大规模酉不变系统（LUIS），涉及酉不变感知矩阵、任意固定信号分布及前向纠错（FEC）编码。考虑采用通用格拉姆-施密特正交化方法构建正交近似消息传递（OAMP），使其能够适用于无需可微性约束的广泛原型。我们为具有Lipschitz连续局部估计器的OAMP开发了两个单输入单输出变分传递函数，便于分析可达速率。此外，当OAMP的状态演化具有唯一不动点时，我们揭示OAMP能够达到基于匹配FEC编码的LUIS复制方法所预测的约束容量，而与信号分布无关。复制方法在具有高斯信令的LUIS及具有任意信号分布的某些LUIS子类中得到严格验证。基于OAMP的变分传递函数建立了几条面积性质。同时，我们提出了LUIS的复制约束容量可达编码原理。该原理为模拟结果中针对二进制信令优化的不规则低密度奇偶校验（LDPC）码设计提供了基础。采用这些优化码的OAMP性能相比未优化码有显著提升，甚至超越著名的Turbo-LMMSE算法。对于正交相移键控（QPSK）调制，我们在多种信道条件下观察到接近复制约束容量的误比特率（BER）性能。