  Recent development in compressed sensing (CS) has revealed that the use of a
special design of measurement matrix, namely the spatially-coupled matrix, can
achieve the information-theoretic limit of CS. In this paper, we consider the
measurement matrix which consists of the spatially-coupled \emph{orthogonal}
matrices. One example of such matrices are the randomly selected discrete
Fourier transform (DFT) matrices. Such selection enjoys a less memory
complexity and a faster multiplication procedure. Our contributions are the
replica calculations to find the mean-square-error (MSE) of the Bayes-optimal
reconstruction for such setup. We illustrate that the reconstruction thresholds
under the spatially-coupled orthogonal and Gaussian ensembles are quite
different especially in the noisy cases. In particular, the spatially coupled
orthogonal matrices achieve the faster convergence rate, the lower measurement
rate, and the reduced MSE.
