Kernel estimation of the instantaneous frequency
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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 non-modulated signal, g(t), the kernel halfwidth which minimizes the expected error scales ash ∼[σ^2 N| ∂_t^2 g^|^2 ]^1/ 5, where coherent signal at frequency, f_ℓ, σ^2 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)(iϕ(t)). For instantaneous frequency estimation, the halfwidth which minimizes the expected error is larger: h_1,3∼[σ^2 A^2N| ∂_t^3 (e^i ϕ̃(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.
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