### Abstract

Subsampled (or partial) Fourier matrices were originally introduced in the compressive sensing literature by Candès et al. Later, in papers by Candès and Tao and Rudelson and Vershynin, it was shown that (random) subsampling of the rows of many other classes of unitary matrices also yield effective sensing matrices. The key requirement is that the rows of U, the unitary matrix, must be highly incoherent with the basis in which the signal is sparse. In this paper, we consider acquisition systems that - despite sensing sparse signals in an incoherent domain - cannot randomly subsample rows from U. We consider a general class of systems in which the sensing matrix corresponds to subsampling of the rows of matrices of the form Φ = RU (instead of U), where R is typically a low-rank matrix whose structure reflects the physical/technological constraints of the acquisition system. We use the term "structurally-subsampled unitary matrices" to describe such sensing matrices. We investigate the restricted isometry property of a particular class of structurally-subsampled unitary matrices that arise naturally in application areas such as multiple-antenna channel estimation and sub-nyquist sampling. In addition, we discuss an immediate application of this work in the area of wireless channel estimation, where the main results of this paper can be applied to the estimation of multiple-antenna orthogonal frequency division multiplexing channels that have sparse impulse responses.

Original language | English (US) |
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Title of host publication | 2009 47th Annual Allerton Conference on Communication, Control, and Computing, Allerton 2009 |

Pages | 1005-1012 |

Number of pages | 8 |

DOIs | |

State | Published - Dec 1 2009 |

Externally published | Yes |

Event | 2009 47th Annual Allerton Conference on Communication, Control, and Computing, Allerton 2009 - Monticello, IL, United States Duration: Sep 30 2009 → Oct 2 2009 |

### Publication series

Name | 2009 47th Annual Allerton Conference on Communication, Control, and Computing, Allerton 2009 |
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### Other

Other | 2009 47th Annual Allerton Conference on Communication, Control, and Computing, Allerton 2009 |
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Country | United States |

City | Monticello, IL |

Period | 9/30/09 → 10/2/09 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Computer Science(all)
- Control and Systems Engineering
- Communication

### Cite this

*2009 47th Annual Allerton Conference on Communication, Control, and Computing, Allerton 2009*(pp. 1005-1012). [5394883] (2009 47th Annual Allerton Conference on Communication, Control, and Computing, Allerton 2009). https://doi.org/10.1109/ALLERTON.2009.5394883

}

*2009 47th Annual Allerton Conference on Communication, Control, and Computing, Allerton 2009.*, 5394883, 2009 47th Annual Allerton Conference on Communication, Control, and Computing, Allerton 2009, pp. 1005-1012, 2009 47th Annual Allerton Conference on Communication, Control, and Computing, Allerton 2009, Monticello, IL, United States, 9/30/09. https://doi.org/10.1109/ALLERTON.2009.5394883

**A restricted isometry property for structurally-subsampled unitary matrices.** / Bajwa, Waheed U.; Sayeed, Akbar M.; Nowak, Robert.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

TY - GEN

T1 - A restricted isometry property for structurally-subsampled unitary matrices

AU - Bajwa, Waheed U.

AU - Sayeed, Akbar M.

AU - Nowak, Robert

PY - 2009/12/1

Y1 - 2009/12/1

N2 - Subsampled (or partial) Fourier matrices were originally introduced in the compressive sensing literature by Candès et al. Later, in papers by Candès and Tao and Rudelson and Vershynin, it was shown that (random) subsampling of the rows of many other classes of unitary matrices also yield effective sensing matrices. The key requirement is that the rows of U, the unitary matrix, must be highly incoherent with the basis in which the signal is sparse. In this paper, we consider acquisition systems that - despite sensing sparse signals in an incoherent domain - cannot randomly subsample rows from U. We consider a general class of systems in which the sensing matrix corresponds to subsampling of the rows of matrices of the form Φ = RU (instead of U), where R is typically a low-rank matrix whose structure reflects the physical/technological constraints of the acquisition system. We use the term "structurally-subsampled unitary matrices" to describe such sensing matrices. We investigate the restricted isometry property of a particular class of structurally-subsampled unitary matrices that arise naturally in application areas such as multiple-antenna channel estimation and sub-nyquist sampling. In addition, we discuss an immediate application of this work in the area of wireless channel estimation, where the main results of this paper can be applied to the estimation of multiple-antenna orthogonal frequency division multiplexing channels that have sparse impulse responses.

AB - Subsampled (or partial) Fourier matrices were originally introduced in the compressive sensing literature by Candès et al. Later, in papers by Candès and Tao and Rudelson and Vershynin, it was shown that (random) subsampling of the rows of many other classes of unitary matrices also yield effective sensing matrices. The key requirement is that the rows of U, the unitary matrix, must be highly incoherent with the basis in which the signal is sparse. In this paper, we consider acquisition systems that - despite sensing sparse signals in an incoherent domain - cannot randomly subsample rows from U. We consider a general class of systems in which the sensing matrix corresponds to subsampling of the rows of matrices of the form Φ = RU (instead of U), where R is typically a low-rank matrix whose structure reflects the physical/technological constraints of the acquisition system. We use the term "structurally-subsampled unitary matrices" to describe such sensing matrices. We investigate the restricted isometry property of a particular class of structurally-subsampled unitary matrices that arise naturally in application areas such as multiple-antenna channel estimation and sub-nyquist sampling. In addition, we discuss an immediate application of this work in the area of wireless channel estimation, where the main results of this paper can be applied to the estimation of multiple-antenna orthogonal frequency division multiplexing channels that have sparse impulse responses.

UR - http://www.scopus.com/inward/record.url?scp=77949640018&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77949640018&partnerID=8YFLogxK

U2 - 10.1109/ALLERTON.2009.5394883

DO - 10.1109/ALLERTON.2009.5394883

M3 - Conference contribution

AN - SCOPUS:77949640018

SN - 9781424458714

T3 - 2009 47th Annual Allerton Conference on Communication, Control, and Computing, Allerton 2009

SP - 1005

EP - 1012

BT - 2009 47th Annual Allerton Conference on Communication, Control, and Computing, Allerton 2009

ER -