 u(t) must be stationary. Furthermore, we require either that Puu(r) -> 0 on the time scale 1/e or that most energy in the spectrum of contained at baseband. When these conditions are not satisfied, the stochastics of the right-hand side of (5.127) do start to depend on 0. Solving (5.127) then requires the construction of a Fokker-Planck equation [Dem02, Risk89]. Solving (5.127) is straightforward. Since () is Gaussian, the same holds for the averaged averaged phase differences 9(t+r)-i - O(t) = efttrn(s)ds. Hence,the variance (5.128) entirely characterizes the averaged phase behavior. With (5.129) (5.130) (5.131) (5.132) we find 02 (r) by solving (5.133) This is in agreement with existing results [Dem98, Dem02]. Equation (5.133) is the starting point for computing, for instance, an oscillator's phase noise spectrum. Example (White input noise): For a Gaussian stationary white input noise source u(t), it holds that (5.134) Using (5.126) and (5.132), it is readily shown that the variance of the averaged phase behavior O(t) satisfies (5.135) In the equations above, Nu is the power spectral density level of the white noise process u(t). Equation (5.135) is solved by (5.136) Hence, the variance of the averaged stochastic phase shift increases linearly with time. 5.7 OSCILLATOR PHASE (NOISE) ANALYSIS 0.012 0.008 0.006 0.004 0.002 Figure 5.26: Variance ozo (T) of an harmonic oscillator's phase O(t) caused by the injection of a white noise current. The variance was computed by means of Monte-Carlo simulations. The dashed line marks the variance of the averaged phase shift as solved from (5.135). In order to verify (5.136), the variance (5.137) of the (non-averaged) phase shift 0(t) was computed numerically by means of Monte- Carlo simulations: (5.122) was solved for 500 randomly generated noise samples. The ISF r(T) was set equal to that of the harmonic oscillator in Fig. 5.22. The numerically computed variance is shown in Fig. 5.26. It is in close agreement with the variance of the averaged phase shift predicted by (5.