Noisy Compressive Sampling Based on Block-Sparse Tensors: Performance Limits and Beamforming Techniques

Abstract : —Compressive Sampling (CS) is an emerging research area for the acquisition of sparse signals at a rate lower than the Shannon sampling rate. Recently, CS has been extended to the challenging problem of multidimensional data acquisition. In this context, block-sparse core tensors have been introduced as the natural multidimensional extension of block-sparse vectors. The (M1,. .. , MQ)-block sparsity for a tensor assumes that Q support sets, characterized by Mq indices corresponding to the non-zero entries, fully describe the sparsity pattern of the considered tensor. In the context of CS with Gaussian measurement matrices, the Cramér-Rao Bound (CRB) on the estimation accuracy of a Bernoulli-distributed block-sparse core tensor is derived. This prior assumes that each entry of the core tensor has a given probability to be non-zero, leading to random supports of truncated Binomial-distributed cardinalities. Based on the limit form of the Poisson distribution, an approximated CRB expression is given for large dictionaries and a highly block-sparse core tensor. Using the property that the mode unfolding matrices of a block-sparse tensor follow the Multiple-Measurement Vectors (MMV) model with a joint sparsity pattern, a fast and accurate estimation scheme, called Beamformed mOde based Sparse Estimator (BOSE), is proposed in the second part of this work. The main contribution of the BOSE is to " map " the MMV model onto the Single MV model thanks to beamforming techniques. Finally, the proposed performance bounds and the BOSE are applied in the context of CS to (i) non-bandlimited multidimensional signals with separable sampling kernels and (ii) for multipath channels in a multiple-input multiple-output (MIMO) wireless communication scheme.
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Rémy Boyer, Martin Haardt. Noisy Compressive Sampling Based on Block-Sparse Tensors: Performance Limits and Beamforming Techniques. IEEE Transactions on Signal Processing, Institute of Electrical and Electronics Engineers, 2016, ⟨10.1109/TSP.2016.2600510⟩. ⟨hal-01353875⟩

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