Kernel estimation of the instantaneous frequency

Abstract

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 as$h\sim {\left[\frac{{\sigma }^2}{N|{\partial }_t^2{g}^{}{|}^2}\right]}^{1/5}$, where %$A^{(\ell )}$ is the coherent signal at frequency, $f_{\ell }$, ${\sigma }^2$ 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, $A(t)\exp (i\varphi (t))$. For instantaneous frequency estimation, the halfwidth which minimizes the expected error is larger: $h_{1,3}\sim {\left[\frac{{\sigma }^2}{A^{2}N|{\partial }_t^3(e^{i\tilde{\varphi }(t)}){|}^2}\right]}^{1/7}$. Since the optimal halfwidths depend on derivatives of the unknown function, we initially estimate these derivatives prior to estimating the actual signal.

References in corpus (4)

• Minimum bias multiple taper spectral estimation
• Optimal Data-based Kernel Estimation of Evolutionary Spectra
• Function Estimation Using Data Adaptive Kernel Estimation - How Much Smoothing?
• Optimal Boundary Kernels and Weightings for Local Polynomial Regression

Cited by in corpus (1)

• Adaptive Kernel Estimation of the Spectral Density with Boundary Kernel Analysis