Performance of Linear Field Reconstruction Techniques with Noise and Correlated Field Spectrum

Conference paper

Abstract

We consider a wireless sensor network that is deployed over an area of interest for environmental monitoring. As often in the practice, the sensor nodes are randomly distributed on the area and their measurements are noisy. Furthermore, the sensors measure a multidimensional physical field (signal), with correlated spectrum, which can be approximated as bandlimited. A central controller, the sink node, is in charge of reconstructing the field from the sensor measurements, which represent an irregular sampling of the signal. We assume that the sink uses a linear reconstructing technique, and we take as performance metric of the reconstruction quality the mean square error (MSE) of the estimate. We then carry out an asymptotic analysis as the number of sensors and the number of the field harmonics go to infinity, while their ratio is kept constant. In particular, we approximate the MSE on the reconstructed field as a function of the eigenvalues of the matrix representing the sampling system. We validate our approximation against numerical results, for some of the most common spectrum correlation models.

Keywords

Signal reconstruction Irregular sampling Sensor networks Random matrix theory 

Notes

Acknowledgments

This work was supported partially by the Italian Ministry for University and Research through the PRIN CARTOON 2006 project and partially by the European Commission in the framework of the FP7 Network of Excellence in Wireless COMmunications NEWCOM++ (contract n. 216715).

References

  1. 1.
    R. Bellman. Introduction to Matrix Analysis. McGraw-Hill, 2nd edition, 1970.Google Scholar
  2. 2.
    H.G. Feichtinger, K. Gröchenig, and T. Strohmer. Efficient numerical methods in nonuniform sampling theory. Numerische Mathematik, 69:423–440, 1995.MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    D. Ganesan, S. Ratnasamy, H. Wang, and D. Estrin. Coping with irregular spatio-temporal sampling in sensor networks. ACM SIGCOMM, pp. 125–130, Jan. 2004.Google Scholar
  4. 4.
    S.M. Kay. Fundamentals of Statistical Signal Processing: Estimation Theory. Prentice Hall, 1993.Google Scholar
  5. 5.
    V.A. Marčenko and L.A. Pastur. Distribution of eigenvalues for some sets of random matrices. USSR Sbornik, 1:457–483, 1967.CrossRefGoogle Scholar
  6. 6.
    A. Nordio, C.F. Chiasserini, and A. Muscariello. Signal compression and reconstruction in clustered sensor networks. In IEEE ICC, Beijing, China, May 2008.Google Scholar
  7. 7.
    A. Nordio, C.F. Chiasserini, and E. Viterbo. The impact of quasi-equally spaced sensor layouts on field reconstruction. In International Symposium on Information, Processing in Sensor Networks (IPSN 2007), Cambridge, MA, Apr. 2007.Google Scholar
  8. 8.
    A. Nordio, C.F. Chiasserini, and E. Viterbo. Performance of linear field reconstruction techniques with noise and uncertain sensor locations. IEEE Transactions on Signal Processing, 56(8):3535–3547, Aug. 2008.MathSciNetCrossRefGoogle Scholar
  9. 9.
    A. Nordio, C.F. Chiasserini, and E. Viterbo. Reconstruction of multidimensional signals from irregular noisy samples. IEEE Transactions on Signal Processing, 56(9):4247–4285, Sep. 2008.MathSciNetCrossRefGoogle Scholar
  10. 10.
    W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery. Numerical Recipes. Cambridge University Press, 2nd edition, 1997.Google Scholar
  11. 11.
    H. Shin and J.H. Lee. Capacity of multiple-antenna fading channels: Spatial fading correlation, double scattering, and keyhole. IEEE Transactions on Information Theory, 49,10(10):2636–2647, Oct. 2003.Google Scholar
  12. 12.
    A. Tulino and S. Verdú. Random matrices and wireless communications. Foundations and Trends in Communications and Information Theory, 1(1), Jul. 2004.Google Scholar
  13. 13.
    D. Voiculescu. Limit laws for random matrices and free products. Inventiones Mathematicae, 104(1):201–220, 1991.MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    J. Wishart. The generalized product moment distribution in sample from a normal multivariate population. Biometrika, 20:32–52, 1928.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Politecnico di TorinoTorinoItaly

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