Kernel estimation of the instantaneous frequency

We consider 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 which minimizes the expected error scales as h ∼ [ σ2 N |∂2 t g |2 ]1/5 , where σ2 is the noise variance and N is the number of measurements per unit time. We show 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: h1,3 ∼ [ σ2 A2N |∂3 t (eiφ̃ (t))|2 ]1/7 . Since the optimal halfwidths depend on derivatives of the unknown function, we initially estimate these derivatives prior to estimating the actual signal.