Capacity Optimality of OAMP: Beyond IID Sensing Matrices and Gaussian Signaling

Abstract

This paper studies a large unitarily invariant system (LUIS) involving a unitarily invariant sensing matrix, an arbitrary signal distribution, and forward error control (FEC) coding. We develop a universal Gram-Schmidt orthogonalization for orthogonal approximate message passing (OAMP). Numerous area properties are established based on the state evolution and minimum mean squared error (MMSE) property of OAMP in an un-coded LUIS. As a byproduct, we provide an alternative derivation for the constrained capacity of a LUIS. Under the assumption that the state evolution for OAMP is correct for the coded system, the achievable rate of OAMP is analyzed. We prove that OAMP achieves the constrained capacity of the LUIS with an arbitrary signal distribution provided that a matching condition is satisfied. Meanwhile, we elaborate a capacity-achieving coding principle for LUIS, based on which irregular low-density parity-check (LDPC) codes are optimized for binary signaling in the numerical 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, capacity-approaching bit error rate (BER) performances are observed under various channel conditions.

Capacity Optimal Generalized Multi-User MIMO: A Theoretical and Practical Framework

“…When CSI is only available at the receiver, the Gaussian capacity of MU-MIMO was obtained with Gaussian signaling [14]. However, for arbitrary input distributions, only the constrained capacity of P2P-MIMO was derived with independent and identically distributed (IID) channel matrices [15], [16] or unitarilyinvariant channel matrices [17], [18] by using the adaptive interpolation method and random matrix theory [15]- [17], or using the MMSE and decoupling properties of approximate message passing (AMP)-type algorithms [18], [19]. At present, it is unknown about the constrained capacity of MU-MIMO with arbitrary input distributions and unitarily-invariant channel matrices.…”

“…In [18], it provided the achievable rate analysis for OAMP in P2P coded linear systems with unitarily-invariant matrices and arbitrary signal distributions. In addition, the capacity optimality of OAMP was rigorously proven, where the achievable rate of OAMP achieves the system capacity under the constraints of channel matrices and input signals [18]. Furthermore, it was shown that OAMP outperforms AMP in physical random access channels [30] and the conventional Turbo receivers in P2P channels [18], [21].…”

“…In addition, the capacity optimality of OAMP was rigorously proven, where the achievable rate of OAMP achieves the system capacity under the constraints of channel matrices and input signals [18]. Furthermore, it was shown that OAMP outperforms AMP in physical random access channels [30] and the conventional Turbo receivers in P2P channels [18], [21]. However, most existing works on OAMP are for un-coded systems or P2P channels.…”

“…4) Difference from P2P-MIMO in [18], [19]: In [18], [19], AMP and OAMP were proven to be capacity optimal for MIMO with arbitrary input signaling, but it is limited to P2P channels that only involve one user. This paper studies a more complicated GMU-MIMO that involves multiple users.…”

“…As a result, the design of capacity-achieving receivers becomes much more difficult. In summary, the results in P2P MIMO [18], [19] cannot be straightforwardly applied to the capacity region analysis and capacity-achieving receiver design for GMU-MIMO.…”