Sparsity-Based Recovery of Finite Alphabet Solutions to Underdetermined Linear Systems

Abstract

We consider the problem of estimating a deterministic finite alphabet vector f from underdetermined measurements y = Af , where A is a given (random) n × N matrix. Two new convex optimization methods are introduced for the recovery of finite alphabet signals via 1-norm minimization. The first method is based on regularization. In the second approach, the problem is formulated as the recovery of sparse signals after a suitable sparse transform. The regularization-based method is less complex than the transform-based one. When the alphabet size p equals 2 and (n, N) grows proportionally, the conditions under which the signal will be recovered with high probability are the same for the two methods. When p > 2, the behavior of the transform-based method is established. Experimental results support this theoretical result and show that the transform method outperforms the regularization-based one.

Low-complexity detector for very large and massive MIMO transmission

“…O(L M ). Remark 2: It is worth noting that the equivalence between the ℓ 0 -norm and ℓ 1 -norm in (P 1 ) hold only for the noiseless case as shown in [11], and not in our case. That is why performance loss are obtained as will be shown hereafter.…”

“…This gain becomes more important for 16-QAM constellation, it is about 3dB at BER 10 −2 , and 3.7dB at BER 10 −2 . The reason for the BER gain growth with the constellation size is that, the accuracy of the ℓ 0 -norm relaxation by the ℓ 1 -norm highly depends on the system dimensions and the sparsity threshold of the signal vector [11].…”

Low-complexity half-sparse decomposition-based detection for massive MIMO transmission

“…Source separation Since the output of the first decomposition is a sparse vector that contains a majority of zero elements, the detection of the original information can be seen as a sparse source decoding. In this context, the authors in [7] demonstrated that the vector x can be recovered by resolving the following optimization problem:…”

“…Previous works proved that source separation is possible in the underdetermined case thanks to basis pursuit (BP) technique [6]. Following this approach, a sparse representation was proposed in [7] to define a successful separation method for a conditioned dimension system. Underdetermined noisy MIMO system with finite alphabet was dealt with as an application case of [7] in [1] and [8], where the problem was formulated as a basis pursuit denoising (BPDN) problem with relaxed constraints.…”

“…Following this approach, a sparse representation was proposed in [7] to define a successful separation method for a conditioned dimension system. Underdetermined noisy MIMO system with finite alphabet was dealt with as an application case of [7] in [1] and [8], where the problem was formulated as a basis pursuit denoising (BPDN) problem with relaxed constraints. In [8], the 0 -norm was relaxed into the 1 -norm to obtain a minimization problem which is solved thanks to an iterative algorithm subject to a spherical search space.…”