  This paper studies a large random matrix system (LRMS) model involving an
arbitrary signal distribution and forward error control (FEC) coding. We
establish an area property based on the so-called Turbo approximate message
passing (Turbo-AMP) algorithm. Under the assumption that the state evolution
for AMP is correct for the coded system, the achievable rate of Turbo-AMP is
analyzed. We prove that Turbo-AMP achieves the constraint capacity of the LRMS
with an arbitrary signal distribution provided that a matching condition is
satisfied. We provide related numerical results of binary signaling using
irregular low-density parity-check (LDPC) codes. We show that optimized codes
demonstrate significantly better performance over un-matched ones under
Turbo-AMP. For quadrature phase shift keying (QPSK) modulation, bit error rate
(BER) performance within 1 dB from the constrained capacity limit is observed.
