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Closed-form estimation of the speed of propagating waves from time measurements

  • Paolo Annibale
  • Rudolf Rabenstein
Article

Abstract

The propagating speed of waves depends on the physical properties of the transmitting material. Since these properties can vary along the propagation path, they cannot be determined from local measurements. However, mean values of the propagation speed can be obtained from time measurements, either between distributed sources and sensors (Time Of Arrival, TOA) if both are synchronized or otherwise from time differences between distributed sensors (Time Difference Of Arrivals, TDOA). This contribution investigates the required assumptions for speed estimation from time measurements and provides closed-form solutions for the synchronized and unsynchronized case. Furthermore the achievable accuracy is determined in terms of Cramer-Rao bounds. The analysis is carried out for the propagation of sound waves in air, where the propagation speed varies with the air temperature. Example results from loudspeaker-microphone recordings are provided. However the closed-form relations apply also to the propagation of other types of waves in linear regimes. This manuscript extends previous work by the authors by providing closed-form solutions and by a parallel treatment of the TOA and the TDOA measurements.

Keywords

Propagation speed Speed of sound Time of arrival  Time difference of arrivals Source localization Closed-form 

Notes

Acknowledgments

The assistance of B.Sc. Carlos Carrillo in performing the acoustic measurements is gratefully acknowledged.

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Chair of Multimedia Communications and Signal ProcessingUniversity Erlangen-NurembergErlangenGermany

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