Capacity Optimality of AMP in Coded Systems

Abstract

This paper studies a random linear system with arbitrary input distributions, whose constrained capacity is recently derived in literature. However, how to find a practical encoder and receiver to achieve this capacity still remains an open problem. In this paper, we establish an area property for AMP in coded systems. With the correctness assumption of state evolution, the achievable rate of AMP for the coded random linear system is analyzed following the code-rate-minimum mean-square error (MMSE) lemma. We prove that the low-complexity AMP achieves the constrained capacity based on matched forward error control (FEC) coding. As a byproduct, we provide an alternative concise derivation for the constrained capacity by taking advantage of the properties of AMP. As examples, Gaussian, quadrature phase shift keying (QPSK), 8PSK, and 16 quadrature amplitude modulation (16-QAM) inputs are studied as special instances. We show that the designed AMP receiver has a significant improvement in achievable rate comparing with the conventional Turbo method and the state-of-art separate detection and decoding scheme. Irregular low-density parity-check (LDPC) codes are designed for AMP to obtain capacity-approaching performances (within 1 dB away from the capacity limit).