Vernotte François
Laboratoire d'Astrophysique de l'Observatoire de Besançon, (UMR CNRS 6091) France.
IEEE Trans Ultrason Ferroelectr Freq Control. 2002 Apr;49(4):508-13. doi: 10.1109/58.996570.
It is well-known that low frequency noises (flicker FM and random walk FM) are not stationary; it is not possible to define either the mean value or the (true) variance. Therefore, the use of a stationary approach yields convergence problems unless a low cut-off frequency is introduced, the physical meaning of which is not clear. As an example, in the case of random walk FM, the mean frequency of an oscillator does not converge if the analysis duration tends toward infinity. However, linear drifts appear if a phase sequence of random walk FM is observed over a duration smaller than the inverse of its low cut-off frequency. Moreover, the estimators, which are devoted to these non-stationary processes (i.e., the Hadamard variance), are insensitive to linear frequency drifts and converge for lower frequency noises (f(-4) FM). The moment condition explains the link between insensitivity to drifts and convergence for low frequency noises in a stationary approach. This condition may be summarized by the following consideration: the divergence effect of a low frequency noise for the lowest frequencies induces a false drift with random drift coefficients; the lower the low cut-off frequency, the higher the variance of the coefficients of this drift. These variances may be known by theoretical calculations. The order of the drift is directly linked to the power law of the noise. The moment condition will be demonstrated and applied for creating new estimators (new variances) and for simulating low frequency noises with a very low cut-off frequency.
众所周知，低频噪声（闪烁调频和随机游走调频）是非平稳的；既无法定义其均值，也无法定义（真实）方差。因此，使用平稳方法会产生收敛问题，除非引入一个低截止频率，但其物理意义并不明确。例如，在随机游走调频的情况下，如果分析时长趋于无穷大，振荡器的平均频率不会收敛。然而，如果在小于其低截止频率倒数的时长内观察随机游走调频的相位序列，就会出现线性漂移。此外，用于这些非平稳过程的估计器（即哈达玛方差）对线性频率漂移不敏感，并且对于较低频率噪声（f(-4)调频）会收敛。矩条件解释了在平稳方法中对漂移不敏感和对低频噪声收敛之间的联系。这个条件可以通过以下考虑来概括：低频噪声在最低频率处的发散效应会导致具有随机漂移系数的虚假漂移；低截止频率越低，这种漂移系数的方差就越高。这些方差可以通过理论计算得知。漂移的阶数与噪声的幂律直接相关。将对矩条件进行证明，并应用于创建新的估计器（新的方差）以及模拟具有非常低截止频率的低频噪声。
