Closed-form estimation of the speed of propagating waves from time measurements
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.
KeywordsPropagation speed Speed of sound Time of arrival Time difference of arrivals Source localization Closed-form
The assistance of B.Sc. Carlos Carrillo in performing the acoustic measurements is gratefully acknowledged.
- Annibale, P., Filos, J., Naylor, P. A., & Rabenstein, R. (2012a). Geometric inference of the room geometry under temperature variations. In Proceedings of international symposium on communications, control and signal processing, (ISCCSP), Rome, Italy: IEEE (2012).Google Scholar
- Annibale, P., Filos, J., Naylor, P. A., & Rabenstein, R. (2012b). TDOA-based speed of sound estimation for air temperature and room geometry inference. IEEE Transactions on Audio, Speech, Language and Signal Processing, 21(2), 234–246. doi: 10.1109/TASL.2012.2217130.
- Annibale, P., & Rabenstein, R. (2010). Acoustic source localization and speed estimation based on time-differences-of-arrival under temperature variations. In Proceedings of European signal processing conference (EUSIPCO). Aalborg, Denmark.Google Scholar
- Annibale, P., & Rabenstein, R. (2012a). Sound speed estimation from time of arrivals: Derivation and comparison with TDOA-based estimation. In Proceedings of European signal processing conference (EUSIPCO). Bucharest, Romania.Google Scholar
- Annibale, P., & Rabenstein, R. (2012b). Speed of sound and air temperature estimation using the TDOA-based localization framework. In Proceedings of international conference on acoustics, speech and signal processing (ICASSP), Kyoto, Japan: IEEE.Google Scholar
- Antonacci, F., Canclini, A., Filos, J., Sarti, A., & Naylor, P. (2012). Exact localization of planar acoustic reflectors in three-dimensional geometries. In International workshop on acoustic signal enhancement (IWAENC 2012). Aachen, Germany.Google Scholar
- Blackstock, D. (2000). Fundamentals of physical acoustics. New York: Wiley.Google Scholar
- Chen, J., Yao, K., Tung, T., Reed, C., & Chen, D. (2002a). Source localization and tracking of a wideband source using a randomly distributed beamforming sensor array. International Journal of High Performance Computing Applications, 16(3), 259–272. doi: 10.1177/10943420020160030601.CrossRefGoogle Scholar
- Chen, J. C., Hudson, R. E., & Yao, K. (2002b). Maximum-likelihood source localization and unknown sensor location estimation for wideband signals in the near-field. IEEE Transactions on Signal Processing, 50(8), 1843–1854.Google Scholar
- Cheung, K. W., So, H. C., Ma, W. K., Chan, Y. T. (20006). A constrained least squares approach to mobile positioning: algorithms and optimality. EURASIP Journal on Applied Signal Processing, 2006, 150–150. doi: 10.1155/ASP/2006/20858.
- Compagnoni, M., Bestagini, P., Antonacci, F., Sarti, A., & Tubaro, S. (2012). Localization of acoustic sources through the fitting of propagation cones using multiple independent arrays. IEEE Transactions on Audio, Speech, and Language Processing, 20(7), 1964–1975. doi: 10.1109/TASL.2012.2191958.CrossRefGoogle Scholar
- Compagnoni, M., Bestagini, P., Antonacci, F., Sarti, A., Tubaro, S. (2013). TDOA-based acoustic source localization in the space-range reference frame. Multidimensional Systems and Signal Processing (accepted).Google Scholar
- Contini, A., Canclini, A., Antonacci, F., Compagnoni, M., Sarti, A., & Tubaro, S. (2012). Self-calibration of microphone arrays from measurement of times of arrival of acoustic signals. In Proceedings of international symposium on communications, control and signal processing, (ISCCSP), Rome, Italy: IEEE.Google Scholar
- Crocco, M., Del Bue, A., Bustreo, M., & Murino, V. (2012). A closed form solution to the microphone position self-calibration problem. In 37th international conference on acoustics, speech, and signal processing (ICASSP 2012), Kyoto, Japan.Google Scholar
- Dokmanic, I., Lu, Y., & Vetterli, M. (2011). Can one hear the shape of a room: The 2D polygonal case. In Proceedings of internatinal conference on acoustics, speech and signal processing (ICASSP), Prague, Czech Republic: IEEE.Google Scholar
- Liu, Z., Ruan, X., & He, J. (2013). Efficient 2-D DOA estimation for coherent sources with a sparse acoustic vector-sensor array. Multidimensional Systems and Signal Processing, 24, 105–120. doi: 10.1007/s11045-011-0158-z.
- Mahajan, A., & Walworth, M. (2001). 3-D position sensing using the differences in the time-of-flight from a wave source to various receivers. IEEE Transactions on Speech and Audio Processing, 17(1), 91–94.Google Scholar
- Markovic, D., Hofmann, C., Antonacci, F., Kowalczyk, K., Sarti, A., & Kellermann, W. (2012). Reflection coefficient estimation by pseudospectrum matching. In International workshop on acoustic signal enhancement (IWAENC 2012) (pp. 181–184). Aachen, Germany.Google Scholar
- Oyzerman, A., & Amar, A. (2012). An extended spherical-intersection method for acoustic sensor network localization with unknown propagation speed. In Electrical electronics engineers in Israel (IEEEI), 2012 IEEE 27th convention of (pp. 1–4). doi: 10.1109/EEEI.2012.6377094.
- Perez-Lorenzo, J., Viciana-Abad, R., Reche-Lopez, P., Rivas, F., & Escolano, J. (2012). Evaluation of generalized cross-correlation methods for direction of arrival estimation using two microphones in real environments. Applied Acoustics, 73(8), 698–712. doi: 10.1016/j.apacoust.2012.02.002. http://www.sciencedirect.com/science/article/pii/S0003682X12000278.
- Reed, C.W., Hudson, R., & Yao, K. (1999). Direct joint source localization and propagation speed estimation. In Proceedings of international conference on acoustics, speech and signal processing, (ICASSP), (Vol. 3, pp. 1169–1172). IEEE (1999).Google Scholar
- Tervo, S., & Tossavainen, T. (2012). 3-D room geometry estimation from room impulse responses. In Proceedings of the 37th international conference on acoustics, speech, and signal processing, (ICASSP 2012) (pp. 513–516). Kyoto, Japan, March 25–30 2012.Google Scholar
- Yang, B., & Kreissig, M. (2011). An introduction to consistent graphs and their signal processing applications. In Proceedings of international conference on acoustics, speech and signal processing (ICASSP) (pp. 2740–2743). Prague, Czech Republic: IEEE.Google Scholar
- Zheng, Z., & Li, G. (2012). Fast DOA estimation of incoherently distributed sources by novel propagator. Multidimensional Systems and Signal Processing, 1–9. doi: 10.1007/s11045-012-0185-4.