Adaptive notch filtering for the retrieval of sinusoids in noise

In this paper, an adaptive notch filter is developed (employing a frequency domain and time domain analysis) for the enhancement and tracking of sinusoids in additive noise, colored or white. The notch filter is implemented as a constrained infinite impulse response filter with the constraint enforced by a single parameter termed the debiasing parameter. The resulting notch filter requires few parameters, facilitates the formation of the desired band rejection filter response, and also leads to various useful implementations (cascade, parallel). For the adaptation of the filter coefficients, the stochastic Gauss-Newton algorithm is used. The convergence of this updating procedure is established by studying the associated differential equation. Also, it is shown that the structure present in the problem enables truncation of the gradient, thereby reducing the complexity of adapting the filter coefficients. Simulation results are presented to substantiate the analysis, and to demonstrate the potential of the notch filtering technique.