Kernel estimation of the instantaneous frequency

TL;DR

This paper develops kernel estimators for the instantaneous frequency of a slowly evolving sinusoid in white noise, analyzing the bias-variance trade-off and optimal kernel halfwidths for accurate estimation.

Contribution

It introduces a method to estimate the instantaneous frequency by optimizing kernel halfwidths based on derivative estimates, improving accuracy over previous approaches.

Findings

Optimal kernel halfwidth scales with noise variance and signal derivatives.

Estimating derivatives prior to frequency improves estimation accuracy.

Derived explicit formulas for halfwidths minimizing expected error.

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