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.
Similar content being viewed by others
References
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.
D.R. Fuhrmann, “Application of Toeplitz Covariance Estimation to Adaptive Beamforming and Detection,”IEEE Trans. Signal Process., Vol. 39(10), 1991.
J.P. Burg, D.G. Luenberger, and D.L. Wenger, “Estimation of Structured Covariance Matrices,”Proc. IEEE, Vol. 70(9), 1982.
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.
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.
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.
A. Graham.Kronecker Products and Matrix Calculus with Applications, New York: Halsted Press, 1981.
P.A. Regalia and S.K. Mitra, “Kronecker Products, Unitary Matrices, and Signal Processing Applications,”SIAM Rev., 31(4), 1989, pp. 586–613.
U. Grenander and G. Szego,Toeplitz Forms and Their Applications, University of California Press, 1958.
R.M. Gray, “Toeplitz and Circulant Matrices II,”Technical Report, Information Systems Laboratory, Stanford Electronics Laboratories, Stanford University, Stanford, CA, 1977.
N.R. Goodman, “Statistical Analysis Based On A Certain Multivariate Complex Gaussian Distribution (An Introduction),”Ann. Math. Statist., March 1963.
F.C. Robey, “A Covariance Modeling Approach to Adaptive Beamforming and Detection,” D.Sc. Thesis, Dept. Elec. Eng., Washington University, St. Louis, MO.
H.A. D'Assumpcao, “Some New Signal Processors for Arrays of Sensors,”IEEE Trans. Inform. Theory, Vol. IT-26(4), 1980.
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.
M.A. Koerber, Private correspondence.
Author information
Authors and Affiliations
Rights 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
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00986236