Covariance Matrix
A Unifying Theory for
Robust Adaptive



Joseph R. Guerci
Science Applications International Corp.
Air Defense Technology Division
4001 N. Fairfax Drive, Suite 400
Arlington, VA 22201
tel: (703) 243-9830

Abstract In this paper, we unify several seemingly disparate approaches to robust adaptive beamforming through the introduction of the concept of a "covariance matrix taper (CMT)." This is accomplished by recognizing that an important class of adapted pattern modification techniques [1]-[3], are realized by the application of a conformal matrix "taper" to the original sample covariance matrix. From the Schur Product Theorem for positive (semi) definite matrices, and Kolmogorov's existence theorem, we further establish that CMTs are in fact the solution to a minimum variance optimum beamformer associated with an auxiliary stochastic process which is related to the original by a Hadamard (Schur) product. This allows us to gain deeper insight into the design of both existing pattern modification techniques and new CMTs that can, for example, simultaneously address several different design constraints such as pattern distortion due to insufficient sample support, and weights mismatch due to nonstationary interference. Also introduced is a new 2D CMT for space-time adaptive radar applications designed to provide more robust clutter cancellation. Since the CMT approach only involves a single matrix Hadamard product, it is also inherently low complexity. The practical utility of the CMT approach is illustrated through its application to both spatial and spatio-temporal adaptive beamforming examples.

[1] B.D. Carlson, "Covariance Matrix Estimation Errors and Diagonal Loading in Adaptive Arrays," IEEE Transactions on Aerospace and Electronic Systems, vol. 24, no. 4, pp. 397­401, July 1988.

[2] R.J. Mailloux, "Receive Array Pattern Modification Using Covariance Matrix Augmentation," Electronics Letters, vol. 31, no. 10, pp. 771­772, 1995.

[3] M. Zatman, "Production of Adaptive Array Troughs by Dispersion Synthesis," Electronics Letters, vol. 31, no. 25, pp. 2141­2142, 1995.



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