Tracking analysis of a gradient-based adaptive IIR notch filter with constrained poles and zeros

In this paper, tracking performance is analyzed in detail of a plain gradient algorithm for a second-order adaptive IIR notch filter with constrained poles and zeros, which takes a chirped signal as its input. First, two sets of difference equations for frequency tracking error and mean square error (MSE) are established in terms of convergence in the mean and convergence in the mean square, respectively. Closed form expressions for the asymptotic tracking error and MSE are then derived from these difference equations. Optimum step size parameter of the algorithm is also worked out based on the minimization of the asymptotic tracking error and MSE. It is discovered that the asymptotic tracking error can be driven to zero for a positive chirp rate by selecting a proper step size value, which is a rare property for an adaptive filtering algorithm. Extensive simulations are provided to support the analytical findings.