  This paper addresses the reconstruction of sparse signals from spatially
coupled, linear, and noisy measurements. A unified framework of rigorous state
evolution is established for developing long-memory message-passing (LM-MP) in
spatially coupled systems. LM-MP utilizes all previous messages to compute the
current message while conventional MP only uses the latest messages. The
unified framework is utilized to propose orthogonal approximate message-passing
(OAMP) for spatially coupled systems. The framework for LM-MP is used as a
technical tool to prove the convergence of state evolution for OAMP. Numerical
results show that OAMP for spatially coupled systems is superior to that for
systems without spatial coupling in the so-called waterfall region.
