Kernel estimation of the instantaneous frequency

Considers kernel estimators of the instantaneous frequency of a slowly evolving sinusoid in white noise. The expected estimation error consists of two terms. The systematic bias error grows as the kernel halfwidth increases while the random error decreases. For a nonmodulated signal g(t), the kernel halfwidth that minimizes the expected error is proportional to h~[(σ²)/(N|∂_t² g|2)]^1/5 where σ² is the noise variance and N is the number of measurements per unit time. The author shows that estimating the instantaneous frequency corresponds to estimating the first derivative of a modulated signal, A(t)exp(iφ(t)). For instantaneous frequency estimation, the halfwidth which minimizes the expected error is larger: h_1,3~[(σ²)/(A²N|∂_t ³(e^iφ¯(t/))|sup 2/)]¹/$ u7. Since the optimal halfwidths depend on derivatives of the unknown function, the authors initially estimate these derivatives prior to estimating the actual signal