The next step of our study was to estimate the CW parameters, namely, instant amplitude and phase. Let us consider a general CW model with variable amplitude *A(t)* and phase which can be written as

where *X*p and *Y*p are the Pole coordinates. Mathematically (not geophysically, indeed!), we can suppose three equivalent models for the CW phase:

where *P* is the CW period, and zero subscripts mean constant (time-independent) values. In other words, we can consider the following three models: with variable period and constant phase, variable phase and constant period, or variable both period and phase. Of course, this is a subject of a special geophysical consideration.

The CW amplitude time series can be easily computed as

The CW amplitude variations thus obtained are shown in Fig. 2 for two CW series. We can see that both CW series show very similar behavior of the CW amplitude, with some differences near the ends of the interval. In both CW series, three deep minima of the amplitude below 0.05 mas around 1850, 1925 and 2005 are unambiguously detected.

Evaluation of the CW phase is a more complicated task. Two methods of investigation of the CW phase variations were examined. The first method was developed by Malkin (2007) for Free Core Nutation modelling. The computations are made in two steps. In the first one the wavelet analysis is applied to both CW series to get the period variations. To perform the wavelet transform (WT) we used the program WWZ developed by the American Association of Variable Star Observers^{1}. Theoretical background of this method can be found in Foster (1996). Since, as discussed above, we cannot separate by a mathematical tool the phase variations from the period variations, we consider the WT output as apparent period variations *P (t).* Then we can compute the phase variations as

here is the parameter to be adjusted.

The second method we used to evaluate the phase variations is the Hilbert transform (HT). For this work we used the function hilbert from the MATLAB Signal Processing Toolbox.

Thus we computed the CW phase variations for two methods of the PM series filtering and two methods of the CW phase evaluation. The results of the computation of the CW phase variations after removing the linear trend corresponding to *P*_{0} are shown in Fig. 3. One can see similar behavior of the CW phase obtained in all four variants, with some differences near the ends of the interval. However, substantial phase jumps in the 1850s and 2000s are clearly visible in all the cases, and their epochs are contemporary with the minima of the CW amplitude as shown in Fig. 2. So, we can conclude that the well-known event in the 1920s of the simultaneous deep minimum of the CW amplitude and the large phase jump may be not unique.