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Sparse Random BlockBanded Toeplitz Matrix for Compressive Sensing
Xiao XUE Song XIAO Hongping GAN
Publication
IEICE TRANSACTIONS on Communications
Vol.E102B
No.8
pp.15651578 Publication Date: 2019/08/01
Online ISSN: 17451345
DOI: 10.1587/transcom.2018EBP3247
Type of Manuscript: PAPER Category: Fundamental Theories for Communications Keyword: compressive sensing, sparse random blockbanded Toeplitz matrix, restricted isometry property, measurement matrix,
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Summary:
In compressive sensing theory (CS), the restricted isometry property (RIP) is commonly used for the measurement matrix to guarantee the reliable recovery of sparse signals from linear measurements. Although many works have indicated that random matrices with excellent recovery performance satisfy the RIP with high probability, Toeplitzstructured matrices arise naturally in real scenarios, such as applications of linear timeinvariant systems. Thus, the corresponding measurement matrix can be modeled as a Toeplitz (partial) structured matrix instead of a completely random matrix. The structure characteristics introduce coherence and cause the performance degradation of the measurement matrix. To enhance the recovery performance of the Toeplitz structured measurement matrix in multichannel convolution source separation, an efficient construction of measurement matrix is presented, referred to as sparse random blockbanded Toeplitz matrix (SRBT). The sparse signal is prerandomized by locally scrambling its sample locations. Then, the signal is subsampled using the sparse random banded matrix. Finally, the mixing measurements are obtained. Based on the analysis of eigenvalues, the theoretical results indicate that the SRBT matrix satisfies the RIP with high probability. Simulation results show that the SRBT matrix almost matches the recovery performance of random matrices. Compared with the existing banded block Toeplitz matrix, SRBT significantly improves the probability of successful recovery. Additionally, SRBT has the advantages of low storage requirements and fast computation in reconstruction.

