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On Kronecker and Linearly Structured Covariance Matrix Estimation
KTH, School of Electrical Engineering (EES), Signal Processing. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.ORCID iD: 0000-0001-6615-6583
KTH, School of Electrical Engineering (EES), Signal Processing. KTH, School of Electrical Engineering (EES), Centres, ACCESS Linnaeus Centre.ORCID iD: 0000-0002-6855-5868
2014 (English)In: IEEE Transactions on Signal Processing, ISSN 1053-587X, E-ISSN 1941-0476, Vol. 62, no 6, p. 1536-1547Article in journal (Refereed) Published
Abstract [en]

The estimation of covariance matrices is an integral part of numerous signal processing applications. In many scenarios, there exists prior knowledge on the structure of the true covariance matrix; e. g., it might be known that the matrix is Toeplitz in addition to Hermitian. Given the available data and such prior structural knowledge, estimates using the known structure can be expected to be more accurate since more data per unknown parameter is available. In this work, we study the case when a covariance matrix is known to be the Kronecker product of two factor matrices, and in addition the factor matrices are Toeplitz. We devise a two-step estimator to accurately solve this problem: the first step is a maximum likelihood (ML) based closed form estimator, which has previously been shown to give asymptotically (in the number of samples) efficient estimates when the relevant factor matrices are Hermitian or persymmetric. The second step is a re-weighting of the estimates found in the first steps, such that the final estimate satisfies the desired Toeplitz structure. We derive the asymptotic distribution of the proposed two-step estimator and conclude that the estimator is asymptotically statistically efficient, and hence asymptotically ML. Through Monte Carlo simulations, we further show that the estimator converges to the relevant Cramer-Rao lower bound for fewer samples than existing methods.

Place, publisher, year, edition, pages
2014. Vol. 62, no 6, p. 1536-1547
Keywords [en]
Kronecker model, parameter estimation, signal processing algorithms, structured covariance estimation
National Category
Signal Processing
Identifiers
URN: urn:nbn:se:kth:diva-134096DOI: 10.1109/TSP.2014.2298834ISI: 000333025000017Scopus ID: 2-s2.0-84896455737OAI: oai:DiVA.org:kth-134096DiVA, id: diva2:664603
Funder
EU, FP7, Seventh Framework Programme, 228044Swedish Research Council, 621-2011-5847
Note

QC 20140422. Updated from manuscript to article in journal.

Available from: 2013-11-15 Created: 2013-11-15 Last updated: 2017-12-06Bibliographically approved
In thesis
1. Exploiting Prior Information in Parametric Estimation Problems for Multi-Channel Signal Processing Applications
Open this publication in new window or tab >>Exploiting Prior Information in Parametric Estimation Problems for Multi-Channel Signal Processing Applications
2013 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis addresses a number of problems all related to parameter estimation in sensor array processing. The unifying theme is that some of these parameters are known before the measurements are acquired. We thus study how to improve the estimation of the unknown parameters by incorporating the knowledge of the known parameters; exploiting this knowledge successfully has the potential to dramatically improve the accuracy of the estimates.

For covariance matrix estimation, we exploit that the true covariance matrix is Kronecker and Toeplitz structured. We then devise a method to ascertain that the estimates possess this structure. Additionally, we can show that our proposed estimator has better performance than the state-of-art when the number of samples is low, and that it is also efficient in the sense that the estimates have Cram\'er-Rao lower Bound (CRB) equivalent variance.

In the direction of arrival (DOA) scenario, there are different types of prior information; first, we study the case when the location of some of the emitters in the scene is known. We then turn to cases with additional prior information, i.e.~when it is known that some (or all) of the source signals are uncorrelated. As it turns out, knowledge of some DOA combined with this latter form of prior knowledge is especially beneficial, giving estimators that are dramatically more accurate than the state-of-art. We also derive the corresponding CRBs, and show that under quite mild assumptions, the estimators are efficient.

Finally, we also investigate the frequency estimation scenario, where the data is a one-dimensional temporal sequence which we model as a spatial multi-sensor response. The line-frequency estimation problem is studied when some of the frequencies are known; through experimental data we show that our approach can be beneficial. The second frequency estimation paper explores the analysis of pulse spin-locking data sequences, which are encountered in nuclear resonance experiments. By introducing a novel modeling technique for such data, we develop a method for estimating the interesting parameters of the model. The technique is significantly faster than previously available methods, and provides accurate estimation results.

Abstract [sv]

Denna doktorsavhandling behandlar parameterestimeringsproblem inom flerkanals-signalbehandling. Den gemensamma förutsättningen för dessa problem är att det finns information om de sökta parametrarna redan innan data analyseras; tanken är att på ett så finurligt sätt som möjligt använda denna kunskap för att förbättra skattningarna av de okända parametrarna.

I en uppsats studeras kovariansmatrisskattning när det är känt att den sanna kovariansmatrisen har Kronecker- och Toeplitz-struktur. Baserat på denna kunskap utvecklar vi en metod som säkerställer att även skattningarna har denna struktur, och vi kan visa att den föreslagna skattaren har bättre prestanda än existerande metoder. Vi kan också visa att skattarens varians når Cram\'er-Rao-gränsen (CRB).

Vi studerar vidare olika sorters förhandskunskap i riktningsbestämningsscenariot: först i det fall då riktningarna till ett antal av sändarna är kända. Sedan undersöker vi fallet då vi även vet något om kovariansen mellan de mottagna signalerna, nämligen att vissa (eller alla) signaler är okorrelerade. Det visar sig att just kombinationen av förkunskap om både korrelation och riktning är speciellt betydelsefull, och genom att utnyttja denna kunskap på rätt sätt kan vi skapa skattare som är mycket noggrannare än tidigare möjligt. Vi härleder även CRB för fall med denna förhandskunskap, och vi kan visa att de föreslagna skattarna är effektiva.

Slutligen behandlar vi även frekvensskattning. I detta problem är data en en-dimensionell temporal sekvens som vi modellerar som en spatiell fler-kanalssignal. Fördelen med denna modelleringsstrategi är att vi kan använda liknande metoder i estimatorerna som vid sensor-signalbehandlingsproblemen. Vi utnyttjar återigen förhandskunskap om källsignalerna: i ett av bidragen är antagandet att vissa frekvenser är kända, och vi modifierar en existerande metod för att ta hänsyn till denna kunskap. Genom att tillämpa den föreslagna metoden på experimentell data visar vi metodens användbarhet. Det andra bidraget inom detta område studerar data som erhålls från exempelvis experiment inom kärnmagnetisk resonans. Vi introducerar en ny modelleringsmetod för sådan data och utvecklar en algoritm för att skatta de önskade parametrarna i denna modell. Vår algoritm är betydligt snabbare än existerande metoder, och skattningarna är tillräckligt noggranna för typiska tillämpningar.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2013. p. xiv, 37
Series
Trita-EE, ISSN 1653-5146 ; 2013:040
Keywords
Array signal processing, covariance matrix, damped sinusoids, direction of arrival estimation, frequency estimation, Kronecker, NQR, NMR, parameter estimation, persymmetric, signal processing algorithms, structured covariance estimation, Toeplitz, Array, signalbehandling, kovariansmatris, dämpad sinus, riktningbestämning, frekvensskattning, Kronecker, NQR, NMR, parameterestimering, persymmetrisk, algoritm, strukturerad kovariansmatris, Toeplitz
National Category
Signal Processing
Identifiers
urn:nbn:se:kth:diva-134034 (URN)978-91-7501-916-1 (ISBN)
Public defence
2013-12-06, Q2, Osquldas väg 10, KTH, Stockholm, 13:15 (English)
Opponent
Supervisors
Funder
EU, FP7, Seventh Framework Programme, 228044Swedish Research Council, 621-2011-5847
Note

QC 20131115

Available from: 2013-11-15 Created: 2013-11-15 Last updated: 2013-11-15Bibliographically approved

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