A new memoryless nonlinear gradient algorithm for a second-order adaptive IIR notch filter and its performance analysis

Abstract

Adaptive infinite inpulse-response (IIR) notch filters have been widely studied for many years. However, further efforts are still to be made to pursue new algorithms which work better than the conventional plain gradient algorithm but have little increase in complexity. In this paper, we employ the gradient linearization, Taylor series expansion, and calculus of variations to derive a novel memoryless nonlinear gradient algorithm for a second-order adaptive IIR notch filter, which improves the estimation performance considerably. Approximate closed-form expressions for the stability bounds on the step size parameter and the steady-state coefficient variance are also derived. Extensive simulations indicate the significant improvement that may be achieved using the new algorithm, and verify the validity of the analytical results.Index Terms-Frequency estimation, gradient algorithm, infinite impulse-response notch filtering, performance analysis.

Improving the performance analysis of a gradient‐based adaptive IIR notch filter

“…This is because the term related to δ a 2 unrelated to α cannot be neglected in the derivation of e(n). Also, v 1 (n) is the output noise of the notch filter and its variance is given by [7] Next, if Eqs. …”

“…(n) is the output noise from the partial gradient filter and its variance is derived as follows [7]:…”

“…Therefore, it is practical and many such algorithms have been proposed [3,4,611]. Also, recently, performance analysis has been attempted for the gradient algorithm with gradient linearization and nonlinearization [7]. (In regard to the analysis of the response characteristics of the lattice algorithm and the recursive prediction error algorithm, Refs.…”

“…Typical IIR notch filters known to date include the bilinear notch filter [3,4], the notch filter with constrained poles and zeros [5], and the lattice notch filter [1,2,13,14]. Algorithms for updating the filter coefficients include the sign algorithm [3], the plain gradient algorithm [3,4,6], the normalized gradient algorithm [4,6], the lattice algorithm [1,2], and the recursive prediction error algorithm [5,7]. In addition, many existing algorithms with improved performance have been developed [e.g., 711].…”

A novel algorithm for adaptive IIR notch filter based on lattice form structure

“…Adaptive IIR notch filters are very useful in many signal processing applications such as communications, active noise control, radar, sonar, and biomedical engineering [1][2][3][4][5][6][7][8][9][10]. They can be used to detect a narrowband signal in noise by estimating a few coefficients compared with the FIR notch filters and have less complexity.…”