A frequency tracker based on a Kalman filter update of a single parameter adaptive notch filter

Randall Ali; Toon van Waterschoot
DAFx-2023 - Copenhagen
In designing a frequency tracker, the goal is to follow the continual time variation of the frequency from a particular sinusoidal component in a noisy signal with a high accuracy and a low sample delay. Although there exists a plethora of frequency trackers in the literature, in this paper, we focus on the particular class of frequency trackers that are built upon an adaptive notch filter (ANF), i.e. a constrained bi-quadratic infinite impulse response filter, where only a single parameter needs to be estimated. As opposed to using the conventional least-mean-square (LMS) algorithm, we present an alternative approach for the estimation of this parameter, which ultimately corresponds to the frequency to be tracked. Specifically, we reformulate the ANF in terms of a state-space model, where the state contains the unknown parameter and can be subsequently updated using a Kalman filter. We also demonstrate that such an approach is equivalent to doing a normalized LMS filter update, where the regularization parameter can be expressed as the ratio of the variance of the measurement noise to the variance of the prediction error. Through an evaluation with both simulated and realistic data, it is shown that in comparison to the LMS-updated frequency tracker, the proposed Kalmanupdated alternative, results in a more accurate performance, with a faster convergence rate, while maintaining a low computational complexity and the ability to be updated on a sample-by-sample basis.