Application of the moment condition to noise simulation and to stability analysis

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.