On minimax lower bound for time-varying frequency estimation of harmonic signal

Estimation of the instantaneous frequency and its derivatives is considered for a harmonic signal with a time-varying phase and time-invariant amplitude. The asymptotic minimax lower bound is derived for the mean squared error of estimation provided that the phase is an arbitrary m-times piece-wise differentiable function of time. It is shown that this lower bound is different only in a constant factor from the upper bound for the mean squared errors of the local polynomial periodogram with optimal window size.