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 the construction of orthogonal approximate message passing (OAMP), which renders the results applicable to general prototypes without the differentiability restriction. For OAMP with Lipschitz continuous local estimators, we develop two variational single-input-single-output transfer functions, based on which we analyze the achievable rate of OAMP. Furthermore, when the state evolution of OAMP has a unique fixed point, we reveal that OAMP reaches the constrained capacity predicted by the replica method of the LUIS with an arbitrary signal distribution based on matched FEC coding. The replica method is rigorous for LUIS with Gaussian signaling and for certain sub-classes of LUIS with arbitrary signal distributions. Several area properties are established based on the variational transfer functions of OAMP. Meanwhile, we elaborate a replica constrained capacity-achieving coding principle for LUIS, based on which irregular low-density parity-check (LDPC) codes are optimized for binary signaling in the simulation results. We show that OAMP with the optimized codes has significant performance improvement over the un-optimized ones and the well-known Turbo linear MMSE algorithm. For quadrature phase-shift keying (QPSK) modulation, replica constrained capacity-approaching bit error rate (BER) performances are observed under various channel conditions.

翻译：摘要：本文研究一类大规模酉不变系统（LUIS），涉及酉不变感知矩阵、任意固定信号分布以及前向纠错（FEC）编码。考虑采用通用格拉姆-施密特正交化方法构造正交近似消息传递（OAMP）算法，使结果可推广至无可微性限制的一般原型。对于具有利普希茨连续局部估计器的OAMP，我们推导出两种变分单输入单输出传递函数，并基于此分析OAMP的可达速率。进一步，当OAMP的状态演化存在唯一不动点时，我们揭示OAMP可达到基于匹配FEC编码的LUIS（具有任意信号分布）的副本方法预测的约束容量。副本方法对高斯信令的LUIS及某些具有任意信号分布的LUIS子类是严格的。基于OAMP的变分传递函数，我们建立了若干面积性质。同时，针对LUIS提出了一种可达副本约束容量的编码原理，并据此优化了用于二进制信令的非规则低密度奇偶校验（LDPC）码。仿真结果表明，采用优化码的OAMP相比未优化码及著名的Turbo线性最小均方误差算法具有显著性能提升。在正交相移键控（QPSK）调制下，各种信道条件下均观察到接近副本约束容量的误比特率（BER）性能。