On Minimal Eigenvalues of a Class of Tridiagonal Matrices

It is known that the worst case near-far resistance of optimum multiuser detectors for asynchronous Gaussian multiple-access channels can be expressed in terms of a class of block-tridiagonal matrices, and the minimal eigenvalues of such a class of block-tridiagonal matrices serve as a good measure of the worst case near-far resistance. In this paper, we focus on the two-user scenario where each block-tridiagonal matrix under consideration is a tridiagonal matrix. We derive closed-form expressions for the minimal eigenvalues of such a class of tridiagonal matrices in terms of the largest real solution of a trigonometric equation in [0,pi]. We also obtain lower bounds and upper bounds on the minimal eigenvalues which improve on previously known results in the literature.