Local polynomial periodograms for signals with the time-varying frequency and amplitude

The estimation of the time-varying IF Ω(t) and amplitude A(t) of harmonic signals is considered. The nonparametric local polynomial approximation (LPA) is used in order to develop two new estimators, which enable one to estimate Ω(t), Ω⁽¹⁾ (t), ..., Ω((m_ω-1))(t) and A(t), A⁽¹⁾ (t), ..., A((m_A))(t), where m_ω and m _A are the degrees of the LPA of the phase and the amplitude. The a priori amplitude information about Ω(t), A(t) and their derivatives can be incorporated in order to improve the estimates. The estimators of the IF and the amplitude have forms of generalized periodograms and Fourier transforms respectively, where a polynomial function appears in the degree of the complex exponent usual for the Fourier transform. The asymptotic bias and covariance are obtained for the new estimators. In particular, it is shown that one of the new estimators ensures complete compensation of the disturbing influence of the time-varying amplitude for the estimation of the IF and vice versa