From the multiple frequency tracker to the multiple frequency smoother

The problem of extraction/elimination of nonstationary sinusoidal signals from noisy measurements is considered. This problem is usually solved using adaptive notch filtering (ANF) algorithms. It is shown that the accuracy of frequency estimates can be significantly increased if the results obtained from ANF are backward-time filtered by an appropriately designed lowpass filter. The resulting adaptive notch smoothing (ANS) algorithm can be employed to perform many off-line signal processing tasks, such as elimination of sinusoidal interference from a prerecorded signal. In the single sinusoid case, we show that when the unknown signal frequency drifts according to the random-walk model, the optimally tuned ANS algorithm is, under Gaussian assumptions, statistically efficient, i.e., it attains the Cramer-Rao type lower smoothing bound, which limits accuracy of any frequency estimation scheme.