Zhang et al. present the problem of computing the output signal-to-interference-plus-noise ratio (SINR) as a generalized eigenvalue problem involving two symmetric positive definite Toeplitz matrices, the correlation matrix of the signal and the correlation matrix of noise plus interference. The significance of this problem in signal processing is in determining the direction of arrival of the interferences. The formulation is based upon the Capon beamformer . The authors are also updating the work of Magi et al. .
The authors exploit the Toeplitz structure of the two correlation matrices to prove theorems about the eigenvectors computed above and the roots of a related filter function. The most important of these theorems is the last one, which shows that if the maximum or minimum eigenvalue is distinct, then all roots of the filter function lie on the unit circle of the complex plane. Subsequent sections of the paper explain the application of their result to beamforming and give a simple three-sensor array example.
I suggest that the authors read the work of Speiser and Van Loan , which gives an alternative formulation in terms of the generalized singular value decomposition. Their formulation of the generalized Toeplitz eigenproblem may be the best for exploiting Toeplitz structure, but it produces a nonsymmetric problem and the symmetry in this problem is easy to preserve. Overall, this paper gives an interesting update on a generalized eigenvalue problem arising in beamforming.