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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, Toeplitz-structured matrices arise naturally in real scenarios, such as applications of linear time-invariant 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 block-banded Toeplitz matrix (SRBT). The sparse signal is pre-randomized 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.

- Publication
- IEICE TRANSACTIONS on Communications Vol.E102-B No.8 pp.1565-1578

- Publication Date
- 2019/08/01

- Publicized
- 2019/02/18

- Online ISSN
- 1745-1345

- DOI
- 10.1587/transcom.2018EBP3247

- Type of Manuscript
- PAPER

- Category
- Fundamental Theories for Communications

Xiao XUE

Xidian University

Song XIAO

Xidian University

Hongping GAN

Xidian University

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Xiao XUE, Song XIAO, Hongping GAN, "Sparse Random Block-Banded Toeplitz Matrix for Compressive Sensing" in IEICE TRANSACTIONS on Communications,
vol. E102-B, no. 8, pp. 1565-1578, August 2019, doi: 10.1587/transcom.2018EBP3247.

Abstract: 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, Toeplitz-structured matrices arise naturally in real scenarios, such as applications of linear time-invariant 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 block-banded Toeplitz matrix (SRBT). The sparse signal is pre-randomized 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.

URL: https://global.ieice.org/en_transactions/communications/10.1587/transcom.2018EBP3247/_p

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@ARTICLE{e102-b_8_1565,

author={Xiao XUE, Song XIAO, Hongping GAN, },

journal={IEICE TRANSACTIONS on Communications},

title={Sparse Random Block-Banded Toeplitz Matrix for Compressive Sensing},

year={2019},

volume={E102-B},

number={8},

pages={1565-1578},

abstract={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, Toeplitz-structured matrices arise naturally in real scenarios, such as applications of linear time-invariant 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 block-banded Toeplitz matrix (SRBT). The sparse signal is pre-randomized 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.},

keywords={},

doi={10.1587/transcom.2018EBP3247},

ISSN={1745-1345},

month={August},}

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TY - JOUR

TI - Sparse Random Block-Banded Toeplitz Matrix for Compressive Sensing

T2 - IEICE TRANSACTIONS on Communications

SP - 1565

EP - 1578

AU - Xiao XUE

AU - Song XIAO

AU - Hongping GAN

PY - 2019

DO - 10.1587/transcom.2018EBP3247

JO - IEICE TRANSACTIONS on Communications

SN - 1745-1345

VL - E102-B

IS - 8

JA - IEICE TRANSACTIONS on Communications

Y1 - August 2019

AB - 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, Toeplitz-structured matrices arise naturally in real scenarios, such as applications of linear time-invariant 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 block-banded Toeplitz matrix (SRBT). The sparse signal is pre-randomized 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.

ER -