OAMP的容量最优性：超越IID感知矩阵和高斯信号
On Capacity Optimality of OAMP: Beyond IID Sensing Matrices and Gaussian Signaling
AI总结：
本文证明OAMP在大型酉不变系统中可达到replica方法预测的约束容量，提出变分传递函数与编码原理，优化LDPC码后性能超越Turbo-LMMSE，QPSK下BER接近容量。
摘要：
本文研究了一个大型酉不变系统（LUIS），该系统涉及酉不变感知矩阵、任意固定的信号分布以及前向纠错（FEC）编码。我们考虑了一种通用的Gram-Schmidt正交化方法来构造正交近似消息传递（OAMP），使其能够适用于广泛的原型，而不受可微性的约束。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，从而便于分析可达速率。此外，当OAMP的状态演化具有唯一不动点时，我们揭示出OAMP能够达到由基于匹配FEC编码的LUIS的replica方法所预测的约束容量，无论信号分布如何。对于高斯信号以及具有任意信号分布的某些LUIS子类，replica方法得到了严格验证。基于OAMP的变分传递函数，我们建立了若干面积性质。同时，我们提出了一个用于LUIS的replica约束容量可达编码原理。该原理作为优化非规则低密度奇偶校验（LDPC）码的基础，这些码在我们的仿真结果中专门针对二进制信号进行定制。使用这些优化码的OAMP性能相比未优化的码有显著提升，甚至超过了著名的Turbo-LMMSE算法。对于正交相移键控（QPSK）调制，我们观察到在不同信道条件下，误码率（BER）性能接近replica约束容量。
英文摘要：
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.
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