Localizing Lung Sounds: Eigen Basis Decomposition for Localizing Sources Within a Circular Array of Sensors


Lung disorders or injury can result in changes in the production of lung sounds both spectrally and regionally. Localizing these lung sounds can provide information to the extent and location of the disorder. Difference in arrival times at a set of sensors and triangulation were previously proposed for acoustic imaging of the chest. We propose two algorithms for acoustic imaging using a set of eigen basis functions of the Helmholtz wave equation. These algorithms remove the sensor location contribution from the multi sensor recordings using either an orthogonality property or a least squares based estimation after which a spatial minimum variance (MV) spectrum is applied to estimate the source locations. The use of these eigen basis functions allows possible extension to a lung sound model consisting of layered cylindrical media. Theoretical analysis of the relationship of resolution to frequency and noise power was derived and simulations verified the results obtained. Further, a Nyquist’s criteria for localizing sources within a circular array shows that the radius of region where sources can be localized is inversely proportional to the frequency of sound.The resolution analysis and modified Nyquist criteria can be used for determining the number of sensors required at a given noise level, for a required resolution, frequency range, and radius of region for which sources need to be localized.

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  1. 1.

    Here acoustic imaging refers to obtaining the location of all sound sources, and is different from x-ray imaging and CT scans which show the lungs in a visual format.

  2. 2.

    \({\boldsymbol{a}}(\theta_v,k) \triangleq \left[e^{ik{\boldsymbol{x}}_1 sin(\phi)},\ \ldots, \ e^{ik{\boldsymbol{x}}_Q sin(\phi)}\right]^T\) where ϕ is the DOA.

  3. 3.

    The discrete form of the orthogonality relationship for exponential functions applied in Eq. 19 is valid only if there is no aliasing. A discrete number of sensors sample the imping wavefront and is analogous to sampling the function e inθ at the angular positions of the sensors. For large n, a greater number of sensors spanning the circumference of a circle is required in order to avoid aliasing.

  4. 4.

    In this paper, the subscript n can refer to either noise or the mode, whenever ambiguity arises in the equations clarifications are provided in the description of these equations.


  1. 1.

    Ward, D. B., & Williamson, R. C. (1999). Beamforming for a source located in the interior of a sensor array. In Proceedings of the fifth international symposium on signal processing and its applications, 1999. ISSPA ’99 (Vol. 2, pp. 873–876). doi:10.1109/ISSPA.1999.815810.

  2. 2.

    Moussavi, Z. (2007). Acoustic mapping and imaging of thoracic sounds. In Fundamentals of respiratory sounds and analysis (Ch. 8, pp. 51–52). Morgan and Claypool.

  3. 3.

    Mansy, H. A., Hoxie, S. J., Warren, W. H., Balk, R. A., Sandler, R. H., & Hassaballa, H. A. (2004). Detection of pneumothorax by computerized breath sound analysis. Chest, 126(4), 881S.

    Google Scholar 

  4. 4.

    Kompis, M., Pasterkamp, H., & Wodicka, G. R. (2001). Acoustic imaging of the human chest. Chest, 120(4), 1309–1321. doi:10.1378/chest.120.4.1309.

    Article  Google Scholar 

  5. 5.

    Charleston-Villalobos, S., Cortés-Rubiano, S., González-Camerena, R., Chi-Lem, G., & Aljama-Corrales, T. (2004). Respiratory acoustic thoracic imaging (rathi): Assessing deterministic interpolation techniques. Medical & Biological Engineering & Computing, 42(5), 618–626.

    Article  Google Scholar 

  6. 6.

    Charleston-Villalobos, S., Gonzalez-Camarena, R., Chi-Lem, G., & Aljama-Corrales, T. (2007). Acoustic thoracic images for transmitted glottal sounds. In Engineering in medicine and biology society, 2007. EMBS 2007. 29th annual international conference of the IEEE (pp. 3481–3484). doi:10.1109/IEMBS.2007.4353080.

  7. 7.

    Harris, F. J. (1978). On the use of windows for harmonic analysis with the discrete fourier transform. Proceedings of the IEEE, 66(1), 51–83.

    Article  Google Scholar 

  8. 8.

    Murphy, Jr., R. L. H. (1996). Method and apparatus for locating the origin of intrathoracic sounds. U.S. patent, 729,272.

  9. 9.

    McKee, A. M., & Goubran, R. A. (2005). Sound localization in the human thorax. In Instrumentation and measurement technology conference, 2005. IMTC 2005. Proceedings of the IEEE (Vol. 1, pp. 117–122). doi:10.1109/IMTC.2005.1604082.

  10. 10.

    Ozer, M. B., Acikgoz, S., Royston, T. J., Mansy, H. A., & Sandler, R. (2007). Boundary element model for simulating sound propagation and source localization within the lungs. Journal of the Acoustical Society of America, 122(1), 657–661.

    Article  Google Scholar 

  11. 11.

    Barshinger, J. N., & Rose, J. L. (2004). Guided wave propagation in an elastic hollow cylinder coated with a viscoelastic material. IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, 51(11), 1547–1556. doi:10.1109/TUFFC.2004.1367496.

    Article  Google Scholar 

  12. 12.

    Valle, C., Qu, J., & Jacobs, L. J. (1999). Guided circumferential waves in layered cylinders. International Journal of Engineering Science, 37(11), 1369–1387.

    Article  Google Scholar 

  13. 13.

    Yao, G.-J., Wang, K.-X., Ma, J., & White, J. E. (2005). Sh wavefields in cylindrical double-layered elastic media excited by a shear stress source applied to a borehole wall. Journal of Geophysics and Engineering, 2(2), 169–175. http://stacks.iop.org/1742-2140/2/169.

    Article  Google Scholar 

  14. 14.

    Abhayapala, T. D. (2006). Broadband source localization by modal space processing. In S. Chandran (Ed.), Advances in direction-of-arrival estimation (Ch. 4, pp. 71–86). Norwood: Artech House.

    Google Scholar 

  15. 15.

    Wodicka, G., Stevens, K., Golub, H., Cravalho, E., & Shannon, D. (1989). A model of acoustic transmission in the respiratory system. IEEE Transactions on Biomedical Engineering, 36(9), 925–934.

    Article  Google Scholar 

  16. 16.

    Garbacz, R., & Pozar, D. (1982). Antenna shape synthesis using characteristic modes. IEEE Transactions on Antennas and Propagation [legacy, pre-1988], 30(3), 340–350.

    Article  Google Scholar 

  17. 17.

    Harackiewicz, F., & Pozar, D. (1986). Optimum shape synthesis of maximum gain omnidirectional antennas. IEEE Transactions on Antennas and Propagation [legacy, pre-1988], 34(2), 254–258.

    Article  Google Scholar 

  18. 18.

    Abhayapala, T. D., Kennedy, R. A., & Williamson, R. C. (2000). Nearfield broadband array design using a radially invariant modal expansion. Journal of the Acoustical Society of America, 107, 392–403.

    Article  Google Scholar 

  19. 19.

    Ward, D. B., & Abhayapala, T. D. (2004). Range and bearing estimation of wideband sources using an orthogonal beamspace processing structure. In Proc. IEEE int. conf. acoust., speech, signal processing, ICASSP 2004 (Vol. 2(2), pp. 109–112).

  20. 20.

    Abhayapala, T. D., & Ward, D. B. (2002). Theory and design of high order sound field microphones using spherical microphone array. In IEEE international conference on acoustics, speech, and signal processing, 2002. Proceedings. (ICASSP ’02) (Vol. 2, pp. 1949–1952).

  21. 21.

    Ward, D. B., & Abhayapala, T. D. (2001). Reproduction of a plane-wave sound field using an array of loudspeakers. IEEE Transactions on Speech and Audio Processing, 9(6), 697–707. doi:10.1109/89.943347.

    Article  Google Scholar 

  22. 22.

    Colton, D., & Kress, R. (1998). Inverse acoustic and electromagnetic scattering theory (2nd ed.). New York: Springer.

    Google Scholar 

  23. 23.

    Jones, H. M., Kennedy, R. A., & Abhayapala, T. D. (2002). On dimensionality of multipath fields: Spatial extent and richness. In IEEE international conference on acoustics, speech, and signal processing, 2002. Proceedings. (ICASSP ’02) (Vol. 3, pp. 2837–2840). doi:10.1109/ICASSP.2002.1005277.

  24. 24.

    Schmidt, R. (1986). Multiple emitter location and signal parameter estimation. IEEE Transactions on Antennas and Propagation [legacy, pre-1988], 34(3), 276–280.

    Article  Google Scholar 

  25. 25.

    Owsley, N. L. (1985). Sonar array processing. In S. Haykin (Ed.), Array signal processing. Englewood Cliffs: Prentice Hall.

    Google Scholar 

  26. 26.

    Stewart, G. W. (1977). On the perturbation of pseudo-inverses, projections and linear least squares problems. SIAM Review, 19(4), 634–662. http://www.jstor.org/stable/2030248.

    MathSciNet  MATH  Article  Google Scholar 

  27. 27.

    Li, F., & Vaccaro, R. J. (1992). Performance degradation of doa estimators due to unknown noise fields. IEEE Transactions on Signal Processing, 40(3), 686–690. doi:10.1109/78.120813.

    Article  Google Scholar 

  28. 28.

    Oppenheim, A. V. (1993). Array signal processing: Concepts and techniques. Englewood Cliffs: PTR Prentice Hall.

    Google Scholar 

  29. 29.

    Dudgeon, D. E. (1977). Fundamentals of digital array processing. Proceedings of the IEEE, 65(6), 898–904.

    Article  Google Scholar 

  30. 30.

    Kummer, W. H. (1992). Basic array theory. Proceedings of the IEEE, 80(1), 127–140. doi:10.1109/5.119572.

    Article  Google Scholar 

  31. 31.

    Heinz, G., Peterson, L. J., Johnson, R. W., & Kerk, C. J. (2003). Exploring relationships in body dimensions. Journal of Statistics Education, 11(2).

  32. 32.

    Bresler, Y., & Macovski, A. (1986). On the number of signals resolvable by a uniform linear array. IEEE Transactions on Acoustics, Speech, and Signal Processing, 34(6), 1361–1375 (see also IEEE Transactions on Signal Processing).

    Google Scholar 

  33. 33.

    Li, F., Liu, H., & Vaccaro, R. J. (1993). Performance analysis for doa estimation algorithms: Unification, simplification, and observations. IEEE Transactions on Aerospace and Electronic Systems, 29(4), 1170–1184. doi:10.1109/7.259520.

    Article  Google Scholar 

  34. 34.

    Rice, D. A. (1983). Sound speed in pulmonary parenchyma. Journal of Applied Physiology, 54(1), 304–308.

    Google Scholar 

  35. 35.

    Gavriely, N., Palti, Y., & Alroy, G. (1981). Spectral characteristics of normal breath sounds. Journal of Applied Physiology, 50(2), 307–314.

    Google Scholar 

  36. 36.

    Gavriely, N., Nissan, M., Rubin, A. H., & Cugell, D. W. (1995). Spectral characteristics of chest wall breath sounds in normal subjects. Thorax, 50(12), 1292–1300.

    Article  Google Scholar 

  37. 37.

    Royston, T. J., Zhang, X., Mansy, H. A., & Sandler, R. H. (2002). Modeling sound transmission through the pulmonary system and chest with application to diagnosis of a collapsed lung. Journal of the Acoustical Society of America, 111(4), 1931–1946.

    Article  Google Scholar 

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Correspondence to S. M. A. Salehin.

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Salehin, S.M.A., Abhayapala, T.D. Localizing Lung Sounds: Eigen Basis Decomposition for Localizing Sources Within a Circular Array of Sensors. J Sign Process Syst 64, 205–221 (2011). https://doi.org/10.1007/s11265-009-0435-3

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  • Localization
  • Lung sounds
  • Helmholtz equation
  • Basis decomposition
  • Cylindrical harmonics
  • Nyquist’s criteria
  • Resolution