Approximation of Stable and Unstable Systems via Frequency Response Interpolation

Abstract

Discrete time transfer functions are approximated by finite impulse response (FIR) interpolations. The interpolation occurs at equally spaced points around the unit circle, and is computationally easy. Transfer functions approximated by FIR interpolations must be analytic in an annulus which contains in its interior the unit circle. As the number of interpolation points and, simultaneously, the interpolation’s degree increase, the L ∞ approximation error decreases geometrically. When interpolation with delay is used, the results extend to approximating unstable systems. If delay is not addded to an unstable system these methods will not allow good approximation. The discussion includes possible applications, such as approximate inverses for line equalization, control systems and on-line modelling.