Minimax lower bound for time-varying frequency estimation of harmonic signal

Estimation of the instantaneous frequency and its derivatives is considered for a harmonic complex-valued signal with the 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 piecewise 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 the optimal window size. The time-varying phases “worst” for estimation of the instantaneous frequency and its derivatives are obtained as a solution of the minimax problem