Quantifying the accuracy of adaptive tracking algorithms

The use of adaptive algorithms such as Kalman Filtering, LMS and RLS together with FIR model structures is very common and extensively analysed. In the interests of improved performance an extension of the FIR structure has been proposed in which the fixed poles are not all at the origin, but instead are chosen by prior knowledge to be close to where the true poles are. Existing FIR analysis would indicate that the noise and tracking properties of such a scheme are invariant to the choice of fixed pole location. This paper establishes both numerically and theoretically that in fact this is not the case. Instead, the dependence of fixed pole location is made explicit by deriving frequency domain expressions that are obtained by using new results on generalised Fourier series and generalised Toeplitz matrices.