Asymptotically Optimal Blind Calibration of Uniform Linear Sensor Arrays for Narrowband Gaussian Signals

An asymptotically optimal blind calibration scheme of uniform linear arrays for narrowband Gaussian signals is proposed. Rather than taking the direct Maximum Likelihood (ML) approach for joint estimation of all the unknown model parameters, which leads to a multi-dimensional optimization problem wi...

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Veröffentlicht in:	IEEE transactions on signal processing Jg. 68; S. 5322 - 5333
Hauptverfasser:	Weiss, Amir, Yeredor, Arie
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	New York IEEE 2020 The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Schlagworte:	Asymptotic properties Calibration Computer simulation Covariance matrices Covariance matrix Cramér-Rao lower bound direction-of-arrival Estimates gain estimation Iterative methods Linear arrays Lower bounds Mathematical models maximum likelihood Maximum likelihood estimates Maximum likelihood estimation Narrowband Offsets Optimization OWL Parameters phase estimation self calibration Sensor array processing Sensor arrays Transmission line matrix methods weighted least squares
ISSN:	1053-587X, 1941-0476
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Abstract	An asymptotically optimal blind calibration scheme of uniform linear arrays for narrowband Gaussian signals is proposed. Rather than taking the direct Maximum Likelihood (ML) approach for joint estimation of all the unknown model parameters, which leads to a multi-dimensional optimization problem with no closed-form solution, we revisit Paulraj and Kailath's (P-K's) classical approach in exploiting the special (Toeplitz) structure of the observations' covariance. However, we offer a substantial improvement over P-K's ordinary Least Squares (LS) estimates by using asymptotic approximations in order to obtain simple, non-iterative, (quasi-)linear Optimally-Weighted LS (OWLS) estimates of the sensors gains and phases offsets with asymptotically optimal weighting, based only on the empirical covariance matrix of the measurements. Moreover, we prove that our resulting estimates are also asymptotically optimal w.r.t. the raw data, and can therefore be deemed equivalent to the ML Estimates (MLE), which are otherwise obtained by joint ML estimation of all the unknown model parameters. After deriving computationally convenient expressions of the respective Cramér-Rao lower bounds, we also show that our estimates offer improved performance when applied to non-Gaussian signals (and/or noise) as quasi-MLE in a similar setting. The optimal performance of our estimates is demonstrated in simulation experiments, with a considerable improvement (reaching an order of magnitude and more) in the resulting mean squared errors w.r.t. P-K's ordinary LS estimates. We also demonstrate the improved accuracy in a multiple-sources directions-of-arrivals estimation task.
AbstractList	An asymptotically optimal blind calibration scheme of uniform linear arrays for narrowband Gaussian signals is proposed. Rather than taking the direct Maximum Likelihood (ML) approach for joint estimation of all the unknown model parameters, which leads to a multi-dimensional optimization problem with no closed-form solution, we revisit Paulraj and Kailath's (P-K's) classical approach in exploiting the special (Toeplitz) structure of the observations' covariance. However, we offer a substantial improvement over P-K's ordinary Least Squares (LS) estimates by using asymptotic approximations in order to obtain simple, non-iterative, (quasi-)linear Optimally-Weighted LS (OWLS) estimates of the sensors gains and phases offsets with asymptotically optimal weighting, based only on the empirical covariance matrix of the measurements. Moreover, we prove that our resulting estimates are also asymptotically optimal w.r.t. the raw data, and can therefore be deemed equivalent to the ML Estimates (MLE), which are otherwise obtained by joint ML estimation of all the unknown model parameters. After deriving computationally convenient expressions of the respective Cramér-Rao lower bounds, we also show that our estimates offer improved performance when applied to non-Gaussian signals (and/or noise) as quasi-MLE in a similar setting. The optimal performance of our estimates is demonstrated in simulation experiments, with a considerable improvement (reaching an order of magnitude and more) in the resulting mean squared errors w.r.t. P-K's ordinary LS estimates. We also demonstrate the improved accuracy in a multiple-sources directions-of-arrivals estimation task.
Author	Yeredor, Arie Weiss, Amir
Author_xml	– sequence: 1 givenname: Amir orcidid: 0000-0002-3104-6939 surname: Weiss fullname: Weiss, Amir email: amirweiss15@gmail.com organization: Department of Computer Science and Applied Mathematics, Faculty of Mathematics and Computer Science, Weizmann Institute of Science, Rehovot, Israel – sequence: 2 givenname: Arie orcidid: 0000-0003-4296-6850 surname: Yeredor fullname: Yeredor, Arie email: arie@eng.tau.ac.il organization: School of Electrical Engineering, Faculty of Engineering, Tel-Aviv University, Tel-Aviv, Israel
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SubjectTerms	Asymptotic properties Calibration Computer simulation Covariance matrices Covariance matrix Cramér-Rao lower bound direction-of-arrival Estimates gain estimation Iterative methods Linear arrays Lower bounds Mathematical models maximum likelihood Maximum likelihood estimates Maximum likelihood estimation Narrowband Offsets Optimization OWL Parameters phase estimation self calibration Sensor array processing Sensor arrays Transmission line matrix methods weighted least squares
Title	Asymptotically Optimal Blind Calibration of Uniform Linear Sensor Arrays for Narrowband Gaussian Signals
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