This paper addresses short-length sparse superposition codes (SSCs) over the additive white Gaussian noise channel. Damped approximate message-passing (AMP) is used to decode short SSCs with zero-mean independent and identically distributed Gaussian dictionaries. To design damping factors in AMP via deep learning, this paper constructs deep-unfolded damped AMP decoding networks. An annealing method for deep learning is proposed for designing nearly optimal damping factors with high probability. In annealing, damping factors are first optimized via deep learning in the low signal-to-noise ratio (SNR) regime. Then, the obtained damping factors are set to the initial values in stochastic gradient descent, which optimizes damping factors for slightly larger SNR. Repeating this annealing process designs damping factors in the high SNR regime. Numerical simulations show that annealing mitigates fluctuation in learned damping factors and outperforms exhaustive search based on an iteration-independent damping factor.

Convolutional approximate message-passing (CAMP) is an efficient algorithm to solve linear inverse problems. CAMP aims to realize advantages of both approximate message-passing (AMP) and orthogonal/vector AMP. CAMP uses the same low-complexity matched-filter as AMP. To realize the asymptotic Gaussianity of estimation errors for all right-orthogonally invariant matrices, as guaranteed in orthogonal/vector AMP, the Onsager correction in AMP is replaced with a convolution of all preceding messages. CAMP was proved to be asymptotically Bayes-optimal if a state-evolution (SE) recursion converges to a fixed-point (FP) and if the FP is unique. However, no proofs for the convergence were provided. This paper presents a theoretical analysis for the convergence of the SE recursion. Gaussian signaling is assumed to linearize the SE recursion. A condition for the convergence is derived via a necessary and sufficient condition for which the linearized SE recursion has a unique stationary solution. The SE recursion is numerically verified to converge toward the Bayes-optimal solution if and only if the condition is satisfied. CAMP is compared to conjugate gradient (CG) for Gaussian signaling in terms of the convergence properties. CAMP is inferior to CG for matrices with a large condition number while they are comparable to each other for a small condition number. These results imply that CAMP has room for improvement in terms of the convergence properties.

Expectation propagation (EP) decoding is proposed for sparse superposition coding in orthogonal frequency division multiplexing (OFDM) systems. When a randomized discrete Fourier transform (DFT) dictionary matrix is used, the EP decoding has the same complexity as approximate message-passing (AMP) decoding, which is a low-complexity and powerful decoding algorithm for the additive white Gaussian noise (AWGN) channel. Numerical simulations show that the EP decoding achieves comparable performance to AMP decoding for the AWGN channel. For OFDM systems, on the other hand, the EP decoding is much superior to the AMP decoding while the AMP decoding has an error-floor in high signal-to-noise ratio regime.

Pilot contamination is addressed in massive multiple-input multiple-output (MIMO) uplink. The main ideas of pilot decontamination are twofold: One is to design transmission timing of pilot sequences such that the pilot transmission periods in different cells do not fully overlap with each other, as considered in previous works. The other is joint channel and data estimation via approximate message-passing (AMP) for bilinear inference. The convergence property of conventional AMP is bad in bilinear inference problems, so that adaptive damping was required to help conventional AMP converge. The main contribution of this paper is a modification of the update rules in conventional AMP to improve the convergence property of AMP. Numerical simulations show that the proposed AMP outperforms conventional AMP in terms of estimation performance when adaptive damping is not used. Furthermore, it achieves better performance than state-of-the-art methods based on subspace estimation when the power difference between cells is small.