  This paper proposes Bayes-optimal convolutional approximate message-passing
(CAMP) for signal recovery in compressed sensing. CAMP uses the same
low-complexity matched filter (MF) for interference suppression as approximate
message-passing (AMP). To improve the convergence property of AMP for
ill-conditioned sensing matrices, the so-called Onsager correction term in AMP
is replaced by a convolution of all preceding messages. The tap coefficients in
the convolution are determined so as to realize asymptotic Gaussianity of
estimation errors via state evolution (SE) under the assumption of orthogonally
invariant sensing matrices. An SE equation is derived to optimize the sequence
of denoisers in CAMP. The optimized CAMP is proved to be Bayes-optimal for all
orthogonally invariant sensing matrices if the SE equation converges to a
fixed-point and if the fixed-point is unique. For sensing matrices with
low-to-moderate condition numbers, CAMP can achieve the same performance as
high-complexity orthogonal/vector AMP that requires the linear minimum
mean-square error (LMMSE) filter instead of the MF.
