Kalman Filters for Time Delay of Arrival-Based Source Localization
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Kalman Filters for Time Delay of Arrival-Based Source Localization Ulrich Klee, Tobias Gehrig, and John McDonough Institut f¨ur Theoretische Informatik, Universit¨at Karlsruhe, Am Fasanengarten 5, 76131 Karlsruhe, Germany Received 9 February 2005; Revised 13 October 2005; Accepted 17 October 2005 In this work, we propose an algorithm for acoustic source localization based on time delay of arrival (TDOA) estimation. In earlier work by other authors, an initial closed-form approximation was first used to estimate the true position of the speaker followed by a Kalman filtering stage to smooth the time series of estimates. In the proposed algorithm, this closed-form approximation is eliminated by employing a Kalman filter to directly update the speaker’s position estimate based on the observed TDOAs. In particular, the TDOAs comprise the observation associated with an extended Kalman filter whose state corresponds to the speaker’s position. We tested our algorithm on a data set consisting of seminars held by actual speakers. Our experiments revealed that the proposed algorithm provides source localization accuracy superior to the standard spherical and linear intersection techniques. Moreover, the proposed algorithm, although relying on an iterative optimization scheme, proved efficient enough for real-time operation. Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.
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INTRODUCTION
Most practical acoustic source localization schemes are based on time delay of arrival estimation (TDOA) for the following reasons: such systems are conceptually simple. They are reasonably effective in moderately reverberant environments. Moreover, their low computational complexity makes them well-suited to real-time implementation with several sensors. Time delay of arrival-based source localization is based on a two-step procedure. (1) The TDOA between all pairs of microphones is estimated, typically by finding the peak in a cross-correlation or generalized cross-correlation function [1]. (2) For a given source location, the squared error is calculated between the estimated TDOAs and those determined from the source location. The estimated source location then corresponds to that position which minimizes this squared error. If the TDOA estimates are assumed to have a Gaussiandistributed error term, it can be shown that the least-squares metric used in Step (2) provides the maximum likelihood (ML) estimate of the speaker location [2]. Unfortunately, this least-squares criterion results in a nonlinear optimization problem that can have several local minima. Several authors have proposed solving this optimization problem with standard gradient-based iterative techniques. While such
techniques typically yield accurate location estimates, they are typically computationally intensive and thus ill-suited for real-time implementation [3, 4]. For any pair of microphones, the surface on which the TDOA is constant is a hyperboloid of two sheets. A second class of algorithms seeks to exploit this fact by grouping all microphones
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