Decomposition of Toeplitz matrices via convex optimization

Tryphon T. Georgiou

doi:10.1109/LSP.2006.874442

Decomposition of Toeplitz matrices via convex optimization

Tryphon T. Georgiou

Electrical and Computer Engineering

Research output: Contribution to journal › Article › peer-review

6 Scopus citations

Abstract

We point out that autocovariance functions of moving average processes of any given order m can be characterized via a linear matrix inequality (LMI). This LMI-condition can be used to decompose any Toeplitz autococovariance matrix into a sum of a singular Toeplitz covariance plus the autocovariance matrix of a moving average process of order m and of maximal variance. The decomposition is unique and subsumes the Pisarenko harmonic decomposition that corresponds to m = 0. It can be used to account for mutual couplings between elements in linear antenna arrays or identify colored noise consistent with the covariance data. The same LMI-condition leads to an efficient computation of the least order of a MA-spectrum that agrees with covariance moments.

Original language	English (US)
Pages (from-to)	537-540
Number of pages	4
Journal	IEEE Signal Processing Letters
Volume	13
Issue number	9
DOIs	https://doi.org/10.1109/LSP.2006.874442
State	Published - Sep 2006

Bibliographical note

Funding Information:
Manuscript received December 28, 2005; revised February 23, 2006. This work was supported in part by the National Science Foundation and in part by the Air Force Office of Scientific Research. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. James E. Fowler.

Keywords

Convex optimization
Moving average processes
Pisarenko harmonic decomposition
Spectral analysis

Publisher link

10.1109/LSP.2006.874442

Find It

Find It at U of M Libraries

Cite this

@article{c439f741a24c4df7ae8745e8f675f189,

title = "Decomposition of Toeplitz matrices via convex optimization",

abstract = "We point out that autocovariance functions of moving average processes of any given order m can be characterized via a linear matrix inequality (LMI). This LMI-condition can be used to decompose any Toeplitz autococovariance matrix into a sum of a singular Toeplitz covariance plus the autocovariance matrix of a moving average process of order m and of maximal variance. The decomposition is unique and subsumes the Pisarenko harmonic decomposition that corresponds to m = 0. It can be used to account for mutual couplings between elements in linear antenna arrays or identify colored noise consistent with the covariance data. The same LMI-condition leads to an efficient computation of the least order of a MA-spectrum that agrees with covariance moments.",

keywords = "Convex optimization, Moving average processes, Pisarenko harmonic decomposition, Spectral analysis",

author = "Georgiou, {Tryphon T.}",

note = "Funding Information: Manuscript received December 28, 2005; revised February 23, 2006. This work was supported in part by the National Science Foundation and in part by the Air Force Office of Scientific Research. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. James E. Fowler.",

year = "2006",

month = sep,

doi = "10.1109/LSP.2006.874442",

language = "English (US)",

volume = "13",

pages = "537--540",

journal = "IEEE Signal Processing Letters",

issn = "1070-9908",

publisher = "Institute of Electrical and Electronics Engineers Inc.",

number = "9",

}

TY - JOUR

T1 - Decomposition of Toeplitz matrices via convex optimization

AU - Georgiou, Tryphon T.

N1 - Funding Information: Manuscript received December 28, 2005; revised February 23, 2006. This work was supported in part by the National Science Foundation and in part by the Air Force Office of Scientific Research. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. James E. Fowler.

PY - 2006/9

Y1 - 2006/9

N2 - We point out that autocovariance functions of moving average processes of any given order m can be characterized via a linear matrix inequality (LMI). This LMI-condition can be used to decompose any Toeplitz autococovariance matrix into a sum of a singular Toeplitz covariance plus the autocovariance matrix of a moving average process of order m and of maximal variance. The decomposition is unique and subsumes the Pisarenko harmonic decomposition that corresponds to m = 0. It can be used to account for mutual couplings between elements in linear antenna arrays or identify colored noise consistent with the covariance data. The same LMI-condition leads to an efficient computation of the least order of a MA-spectrum that agrees with covariance moments.

AB - We point out that autocovariance functions of moving average processes of any given order m can be characterized via a linear matrix inequality (LMI). This LMI-condition can be used to decompose any Toeplitz autococovariance matrix into a sum of a singular Toeplitz covariance plus the autocovariance matrix of a moving average process of order m and of maximal variance. The decomposition is unique and subsumes the Pisarenko harmonic decomposition that corresponds to m = 0. It can be used to account for mutual couplings between elements in linear antenna arrays or identify colored noise consistent with the covariance data. The same LMI-condition leads to an efficient computation of the least order of a MA-spectrum that agrees with covariance moments.

KW - Convex optimization

KW - Moving average processes

KW - Pisarenko harmonic decomposition

KW - Spectral analysis

UR - http://www.scopus.com/inward/record.url?scp=33747793363&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33747793363&partnerID=8YFLogxK

U2 - 10.1109/LSP.2006.874442

DO - 10.1109/LSP.2006.874442

M3 - Article

AN - SCOPUS:33747793363

SN - 1070-9908

VL - 13

SP - 537

EP - 540

JO - IEEE Signal Processing Letters

JF - IEEE Signal Processing Letters

IS - 9

ER -

Decomposition of Toeplitz matrices via convex optimization

Abstract

Bibliographical note

Keywords

Publisher link

Find It

Other files and links

Fingerprint

Cite this