Accurate and efficient solution of Hankel matrix systems by FFT and the conjugate gradient methods

Abstract

Hankel matrix systems often arise in many problems of signal analysis. Toeplitz matrix systems is a special case of a Hankel matrix equations. Both the auto-correlation and the covariance matrix equations are different forms of the Hankel matrix equations. Trench has developed a direct method that can solve Hankel matrix equations in θ(N2) operations. In this paper, we propose an alternate algorithm. This new method is a combination of the FFT and the conjugate gradient method. The advantage of this new approach is that it is computationally robust to highly ill-conditioned and even singular matrix equations. Preliminary results indicated that for very large complex Toeplitz matrix equations, the CPU time is proportional to N as the number of unknowns as increased, as opposed to N2for conventional methods.