Skip to main content
SpringerLink
Log in

Expectations of useful complex Wishart forms

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

Abstract

Complex Wishart matrices are a key part of many multidimensional signal processing algorithms, and expectations of expressions containing them are needed to quantify their performance. In this paper we derive concise formulas forE{WAW},E{W −1 AW −1}, and other useful expressions, whereW follows the complex Wishart distribution andA is a deterministic matrix.A need not be symmetric. We show how the results can be used to quantify the performance of adaptive array processing algorithms.

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

Access this article

Log in via an institution

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. T.W. Anderson,An Introduction to Multivariate Statistical Analysis, 2nd ed., New York: Wiley, 1984.

    Google Scholar 

  2. R.J. Muirhead,Aspects of Multivariate Statistical Theory, New York: Wiley, 1982.

    Google Scholar 

  3. C.G. Khatri, “Classical Statistical Analysis Based on a Certain Multivariate Complex Gaussian Distribution,”Annals of Mathematical Statistics, vol. 36, no. 1, 1965, pp. 98–114.

    Google Scholar 

  4. R.A. Wooding, “The Multivariate Distribution of Complex Normal Variables,”Biometrika, vol. 43, 1956, pp. 212–215.

    Google Scholar 

  5. N.R. Goodman, “Statistical Analysis Based on a Certain Multivariate Complex Gaussian Distribution (An Introduction),”Annals of Mathematical Statistics, vol. 34, no. 1, 1963, pp. 152–177.

    Google Scholar 

  6. M.S. Srivastava, “On the complex Wishart distribution,”Annals of Mathematical Statistics, vol. 36, no. 1, 1965, pp. 313–315.

    Google Scholar 

  7. C.I. Caldwell, “Statistics of Complex Variables,” unpublished notes, 1989.

  8. I.P. Kirsteins,Reduced-Rank Interfence Cancellation, Ph.D. dissertation. The University of Rhode Island, 1991.

  9. I.S. Reed, J.D. Mallet, and L.E. Brennan, “Rapid Convergence Rate in Adaptive Arrays,”IEEE Transactions on Aerospace and Electronic Systems, vol. AES-10, no. 6, 1974, pp. 885–863.

    Google Scholar 

  10. G.P.H. Styan, “Three Useful Expressions for Expectations Involving a Wishart Matrix and Its Inverse,” Y. Dodge (Ed.),Statistical Data Analysis and Inference, North-Holland, Elsevier, 1989.

  11. I. Olkin and H. Rubin, “A Characterization of the Wishart Distribution,”Annals of Mathematical Statistics, vol. 33, 1962, pp. 1272–1280.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Electrical and Computer Engineering, Ohio University, 45701, Athens, OH

    John A. Tague

  2. Graduate Program in Acoustics, The Pennsylvania State University, 16802, University Park, PA

    Curtis I. Caldwell

Authors
  1. John A. Tague
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Curtis I. Caldwell
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tague, J.A., Caldwell, C.I. Expectations of useful complex Wishart forms. Multidim Syst Sign Process 5, 263–279 (1994). https://doi.org/10.1007/BF00980709

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00980709

Key Words

Search

Navigation