Instantaneous frequency estimation at low signal-to-noise ratios using time-varying notch filters

This paper is aimed at finding parametric signal models that perform well at modelling noisy tonals at low signal-to-noise ratios (SNRs). We focus on models that are applied to a segment of data at a time, rather than work their way through the data in a sequential manner as typified by the adaptive methods. Inspired by notch filter theory, we extend the well-known time-varying AR (TVAR) models to include the effects of additive noise, and arrive at two types of time-varying notch filter (TVNF). The first one, like the TVAR model, employs a basis expansion of the filter coefficients. For the second one, we utilise the fact that tonal instantaneous frequencies (IFs) are directly proportional to the angles of the roots of the denominator polynomial, and perform a basis expansion of the IFs.Adaptive notch filters are well known and have been successfully applied in several fields. By application to simulated signals and a section of a dolphin whistle recording, it is shown that the TVNFs perform better than the TVAR model, and are useful tools for low SNR IF estimation. TVNF estimation employs a regularised Gauss–Newton type iterative search algorithm, which exhibits rapid and reliable convergence. Model order determination by Akaike's final prediction error (FPE) criterion is also discussed along with the selection of notch filter design and regularisation parameters.