Minimax lower bounds for nonparametric estimation of the instantaneous frequency- and time-varying amplitude of a harmonic signal

Abstract

Estimation of the instantaneous frequency- and time-varying amplitude along with their derivatives is considered for a harmonic complex-valued signal given with an additive noise. Asymptotic minimax lower bounds are derived for the mean-squared errors of estimation provided that the phase and amplitude are arbitrary piece-wise differentiable functions of time. It is shown that these lower bounds are different only in constant factors from the optimal upper bounds of mean-squared errors of estimates given by the generalized local polynomial periodogram. The time-varying phase and amplitude are derived which are “ worst”, respectively, for estimation of the instantaneous frequency, amplitude and their derivative. These “ worst” functions can be applied in order to test the accuracy of algorithms used for estimation of the instantaneous frequency and amplitude.