Decomposition of Toeplitz matrices via convex optimization

Tryphon T. Georgiou

Research output: Contribution to journalArticlepeer-review

6 Scopus citations


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 languageEnglish (US)
Pages (from-to)537-540
Number of pages4
JournalIEEE Signal Processing Letters
Issue number9
StatePublished - 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.


  • Convex optimization
  • Moving average processes
  • Pisarenko harmonic decomposition
  • Spectral analysis


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