Tracking properties of a gradient-based second-order adaptive IIR notch filter with constrained poles and zeros

Gradient-type adaptive IIR notch filters have many attractive merits for various real-life applications since they require a small number of computations and yet demonstrate practical performance. However, it is generally quite difficult to assess their performance analytically. Their tracking properties, in particular, have not yet been investigated. In this paper, the tracking performance of a plain gradient (PG) algorithm is analyzed in detail for a second-order adaptive IIR notch filter with constrained poles and zeros, which takes a linear chirp signal as its input. First, two sets of difference equations for the frequency tracking error and mean square error (MSE) are established in the sense 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. An optimum step-size parameter for the algorithm is also evaluated based on the minimization of the asymptotic tracking error or the tracking MSE. It is discovered that the asymptotic tracking error may be driven to zero for a positive chirp rate by selecting a proper step size, which is an interesting property for a real-valued adaptive filtering algorithm. Extensive simulations are performed to support the analytical findings.