Generalized adaptive notch and comb filters for identification of quasi-periodically varying systems

The problem of identification/tracking of quasi-periodically varying real-valued systems is considered. This problem is a generalization, to the system case, of a classical signal processing task of either elimination or extraction of nonstationary sinusoidal signals buried in noise. The solution is based on the exponentially weighted basis function (EWBF) approach. The proposed algorithms are capable of tracking slow changes in system frequencies, which means that not only the expansion coefficients in the basis function description of the analyzed system but also the basis functions themselves are adjusted in an adaptive manner. First, assuming that the system frequencies are known and constant, the running basis and fixed basis variants of the EWBF algorithm are derived, and their relationship to the classical notch filter with constrained poles and zeros is established. Next, the frequency-adaptive versions of both algorithms are obtained using the gradient search and recursive prediction error principles, respectively. Finally, the interrelated frequencies case is analyzed and two additional parameter tracking algorithms (generalized adaptive comb filters) are derived.