arXiv 1803.040752018-03-12stat.MEeess.ASeess.SPmath.STstat.APstat.TH

瞬时频率的核估计

Kernel estimation of the instantaneous frequency

Kurt S. Riedel

发表信息：: I.E.E.E. Trans. Signal Processing 42, pp. 2644-2649 (1994)
DOI：: 10.1109/78.324730

摘要：

我们考虑核估计的瞬时频率的缓慢发展的正弦曲线在白噪声中。预期的估计误差由两项组成。系统偏置误差随着核半宽度的增加而增加，而随机误差随着核半宽度的增加而减小。对于非调制信号$g(t)$，使预期误差最小化的内核半宽按比例缩放为$h \sim \left[{ σ^2 \over N| \partial_t^2 g^{}|^2 } \right]^{1/ 5}$，其中% $A^{(\ell)}$是频率上的相干信号，$f_{\ell}$，$σ^2$是噪声方差，$N$是单位时间内的测量次数。我们表明，估计的瞬时频率对应于估计的调制信号，$A(t)\exp(iϕ(t))$的一阶导数。对于瞬时频率估计，使期望误差最小化的半宽度较大：$h_{1,3} \sim \left[{ σ^2 \over A^2N| \partial_t^3 (e^{i \tildeϕ(t)} )|^2 } \right]^{1/ 7}$。由于最佳半宽度取决于未知函数的导数，我们在估计实际信号之前首先估计这些导数。

英文摘要：

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

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