Skip to main content
Log in

Covariance structures for multidimensional data

  • Published:
Multidimensional Systems and Signal Processing Aims and scope Submit manuscript

Abstract

Certain covariance constraint sets useful in a wide class of signal processing problems are described. The constraint sets define structure in the covariance matrices of naturally multidimensional data organized into column-vector form. A notation characterizing these constraint sets is established. The structures involve a hierarchy of sub-blocks within the matrix, and include block-circulant and block-Toeplitz matrices and their respective generalizations. An example describing the structured covariance for wideband data from a uniform linear array of unstructured subarrays is included.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. D. Gray, B. Anderson, and P. Sim, “Estimation of Structured Covariances with Application to Array Beamforming,”Circuits, Systems, Signal Process., Vol. 6(4); 1987, pp. 421–447.

    Google Scholar 

  2. D.R. Fuhrmann, “Application of Toeplitz Covariance Estimation to Adaptive Beamforming and Detection,”IEEE Trans. Signal Process., Vol. 39(10), 1991.

  3. J.P. Burg, D.G. Luenberger, and D.L. Wenger, “Estimation of Structured Covariance Matrices,”Proc. IEEE, Vol. 70(9), 1982.

  4. M.I. Miller and D.L. Snyder, “The Role of Likelhood and Entropy in Incomplete-Data Problems: Applications to Estimating Point-Process Intensities and Toeplitz Constrained Covariances;Proc. IEEE, Vol. 75(7), 1987.

  5. P.J. Tourtier and L.L. Scharf, “Maximum Likelihood Identification of Correlation Matrices for Estimation of Power Spectra at Arbitrary Resolutions,”Proc. ICASSP87, pp. 2066–2069.

  6. D.R. Fuhrmann and T.A. Barton, “Estimation of Block-Toeplitz Covariance Matrices,”Research Monograph ESSRL-91-08, Dept. Elec. Eng., Washington University, St. Louis, MO, 1990.

    Google Scholar 

  7. A. Graham.Kronecker Products and Matrix Calculus with Applications, New York: Halsted Press, 1981.

    Google Scholar 

  8. P.A. Regalia and S.K. Mitra, “Kronecker Products, Unitary Matrices, and Signal Processing Applications,”SIAM Rev., 31(4), 1989, pp. 586–613.

    Google Scholar 

  9. U. Grenander and G. Szego,Toeplitz Forms and Their Applications, University of California Press, 1958.

  10. R.M. Gray, “Toeplitz and Circulant Matrices II,”Technical Report, Information Systems Laboratory, Stanford Electronics Laboratories, Stanford University, Stanford, CA, 1977.

    Google Scholar 

  11. N.R. Goodman, “Statistical Analysis Based On A Certain Multivariate Complex Gaussian Distribution (An Introduction),”Ann. Math. Statist., March 1963.

  12. F.C. Robey, “A Covariance Modeling Approach to Adaptive Beamforming and Detection,” D.Sc. Thesis, Dept. Elec. Eng., Washington University, St. Louis, MO.

  13. H.A. D'Assumpcao, “Some New Signal Processors for Arrays of Sensors,”IEEE Trans. Inform. Theory, Vol. IT-26(4), 1980.

  14. T. Anderson, “Estimation of Covariance Matrices Which Are Linear Combinations or Whose Inverses Are Linear Combinations of Given Matrices,” in (R.C. Bose et al., ed.),Essays in Probability and Statistics, Chapel Hill, NC: University of North Carolina Press, 1970, pp. 1–24.

    Google Scholar 

  15. M.A. Koerber, Private correspondence.

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Barton, T.A., Fuhrmann, D.R. Covariance structures for multidimensional data. Multidim Syst Sign Process 4, 111–123 (1993). https://doi.org/10.1007/BF00986236

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00986236

Key Words

Navigation