MUSIC for Single-Snapshot Spectral Estimation: Stability and Super-resolution
| Title: | MUSIC for Single-Snapshot Spectral Estimation: Stability and Super-resolution |
|---|---|
| Authors: | Liao, Wenjing, Fannjiang, Albert |
| Publication Year: | 2014 |
| Collection: | Computer Science Mathematics |
| Subject Terms: | Computer Science - Information Theory, Mathematics - Numerical Analysis |
| More Details: | This paper studies the problem of line spectral estimation in the continuum of a bounded interval with one snapshot of array measurement. The single-snapshot measurement data is turned into a Hankel data matrix which admits the Vandermonde decomposition and is suitable for the MUSIC algorithm. The MUSIC algorithm amounts to finding the null space (the noise space) of the Hankel matrix, forming the noise-space correlation function and identifying the s smallest local minima of the noise-space correlation as the frequency set. In the noise-free case exact reconstruction is guaranteed for any arbitrary set of frequencies as long as the number of measurements is at least twice the number of distinct frequencies to be recovered. In the presence of noise the stability analysis shows that the perturbation of the noise-space correlation is proportional to the spectral norm of the noise matrix as long as the latter is smaller than the smallest (nonzero) singular value of the noiseless Hankel data matrix. Under the assumption that frequencies are separated by at least twice the Rayleigh Length (RL), the stability of the noise-space correlation is proved by means of novel discrete Ingham inequalities which provide bounds on nonzero singular values of the noiseless Hankel data matrix. The numerical performance of MUSIC is tested in comparison with other algorithms such as BLO-OMP and SDP (TV-min). While BLO-OMP is the stablest algorithm for frequencies separated above 4 RL, MUSIC becomes the best performing one for frequencies separated between 2 RL and 3 RL. Also, MUSIC is more efficient than other methods. MUSIC truly shines when the frequency separation drops to 1 RL or below when all other methods fail. Indeed, the resolution length of MUSIC decreases to zero as noise decreases to zero as a power law with an exponent much smaller than an upper bound established by Donoho. Comment: Studies on the super-resolution of the MUSIC algorithm have been added in Section 4 and Section 5.4 |
| Document Type: | Working Paper |
| Access URL: | http://arxiv.org/abs/1404.1484 |
| Accession Number: | edsarx.1404.1484 |
| Database: | arXiv |
| FullText | Text: Availability: 0 CustomLinks: – Url: http://arxiv.org/abs/1404.1484 Name: EDS - Arxiv Category: fullText Text: View this record from Arxiv MouseOverText: View this record from Arxiv – Url: https://resolver.ebsco.com/c/xy5jbn/result?sid=EBSCO:edsarx&genre=article&issn=&ISBN=&volume=&issue=&date=20140405&spage=&pages=&title=MUSIC for Single-Snapshot Spectral Estimation: Stability and Super-resolution&atitle=MUSIC%20for%20Single-Snapshot%20Spectral%20Estimation%3A%20Stability%20and%20Super-resolution&aulast=Liao%2C%20Wenjing&id=DOI: Name: Full Text Finder (for New FTF UI) (s8985755) Category: fullText Text: Find It @ SCU Libraries MouseOverText: Find It @ SCU Libraries |
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| Items | – Name: Title Label: Title Group: Ti Data: MUSIC for Single-Snapshot Spectral Estimation: Stability and Super-resolution – Name: Author Label: Authors Group: Au Data: <searchLink fieldCode="AR" term="%22Liao%2C+Wenjing%22">Liao, Wenjing</searchLink><br /><searchLink fieldCode="AR" term="%22Fannjiang%2C+Albert%22">Fannjiang, Albert</searchLink> – Name: DatePubCY Label: Publication Year Group: Date Data: 2014 – Name: Subset Label: Collection Group: HoldingsInfo Data: Computer Science<br />Mathematics – Name: Subject Label: Subject Terms Group: Su Data: <searchLink fieldCode="DE" term="%22Computer+Science+-+Information+Theory%22">Computer Science - Information Theory</searchLink><br /><searchLink fieldCode="DE" term="%22Mathematics+-+Numerical+Analysis%22">Mathematics - Numerical Analysis</searchLink> – Name: Abstract Label: Description Group: Ab Data: This paper studies the problem of line spectral estimation in the continuum of a bounded interval with one snapshot of array measurement. The single-snapshot measurement data is turned into a Hankel data matrix which admits the Vandermonde decomposition and is suitable for the MUSIC algorithm. The MUSIC algorithm amounts to finding the null space (the noise space) of the Hankel matrix, forming the noise-space correlation function and identifying the s smallest local minima of the noise-space correlation as the frequency set. In the noise-free case exact reconstruction is guaranteed for any arbitrary set of frequencies as long as the number of measurements is at least twice the number of distinct frequencies to be recovered. In the presence of noise the stability analysis shows that the perturbation of the noise-space correlation is proportional to the spectral norm of the noise matrix as long as the latter is smaller than the smallest (nonzero) singular value of the noiseless Hankel data matrix. Under the assumption that frequencies are separated by at least twice the Rayleigh Length (RL), the stability of the noise-space correlation is proved by means of novel discrete Ingham inequalities which provide bounds on nonzero singular values of the noiseless Hankel data matrix. The numerical performance of MUSIC is tested in comparison with other algorithms such as BLO-OMP and SDP (TV-min). While BLO-OMP is the stablest algorithm for frequencies separated above 4 RL, MUSIC becomes the best performing one for frequencies separated between 2 RL and 3 RL. Also, MUSIC is more efficient than other methods. MUSIC truly shines when the frequency separation drops to 1 RL or below when all other methods fail. Indeed, the resolution length of MUSIC decreases to zero as noise decreases to zero as a power law with an exponent much smaller than an upper bound established by Donoho.<br />Comment: Studies on the super-resolution of the MUSIC algorithm have been added in Section 4 and Section 5.4 – Name: TypeDocument Label: Document Type Group: TypDoc Data: Working Paper – Name: URL Label: Access URL Group: URL Data: <link linkTarget="URL" linkTerm="http://arxiv.org/abs/1404.1484" linkWindow="_blank">http://arxiv.org/abs/1404.1484</link> – Name: AN Label: Accession Number Group: ID Data: edsarx.1404.1484 |
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| RecordInfo | BibRecord: BibEntity: Subjects: – SubjectFull: Computer Science - Information Theory Type: general – SubjectFull: Mathematics - Numerical Analysis Type: general Titles: – TitleFull: MUSIC for Single-Snapshot Spectral Estimation: Stability and Super-resolution Type: main BibRelationships: HasContributorRelationships: – PersonEntity: Name: NameFull: Liao, Wenjing – PersonEntity: Name: NameFull: Fannjiang, Albert IsPartOfRelationships: – BibEntity: Dates: – D: 05 M: 04 Type: published Y: 2014 |
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