Capacity Optimality of AMP in Coded Systems
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This paper studies a random linear system with arbitrary input distributions, whose capacity is well known for Gaussian input distribution but still remains an open problem for non-Gaussian distributions. Based on the potential minimum mean-square error (MMSE) optimality of approximate message passing (AMP) and the mutual information and MMSE (I-MMSE) lemma, a closed form of capacity is established in the large-system limit. Furthermore, with the correctness assumption of state evolution, the achievable rate of AMP for the coded random linear system is analyzed following the code-rate-MMSE lemma. We prove that the low-complexity AMP achieves the capacity based on matched forward error control (FEC) coding. As examples, Gaussian, quadrature phase shift keying (QPSK), 8PSk, and 16 quadrature amplitude modulation (16-QAM) input distributions are studied as special instances. As comparison, we show that the proposed AMP receiver has a significant improvement in achievable rate comparing with the conventional Turbo method and the state-of-art AWGN-optimized coding 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). Numerical results are provided to verify the validity and accuracy of the theoretical results.
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