g uncertainty on the positions of the zero crossings. Let us be given a stationary zero-mean Gaussian noise process u(t). This noise process is entirely characterized by its autocorrelation function (5.120) By introducing (5.121) we can write the oscillator's phase (noise) equation (5.107) as (5.122) Because of the random nature of n(t,0), (5.122) is called a stochastic differential equa- tion (SDE). For constant 0, the autocorrelation function of n(t,0), i.e. = - = , (5.123) is periodic in T. Hence, n(t,0) is cyclostationary with respect to T (see section 3.5.3 in chapter 3). This cyclostationarity together with the dependence of n(t).) on 0 compli- cates solving (5.122). Averaging reduces (5.122) to an SDE with a Gaussian stationary right-hand side that is independent of 0. Indeed, the application of the ideal lowpass averaging operator (5.90) to n(t,0) results in an averaged Gaussian noise process (). Results from cyclostationary noise theory (again, see section 3.5.3 in chapter 3) allow us to prove that the autocorrelation function of () is equal to (5.124) (5.125) (5.126) The latter transition is a consequence of the periodicity of the ISF r(t)¹³. Equation (5.126) shows that (t) is stationary and without explicit dependencies on 0. Hence, the averaged equation corresponding with (5.122) equals (5.127) an SDE with a stationary right-hand side that is independent of the averaged phase 0. 13The equivalence of (5.125) and (5.126) is of major importance in resolving the difference between two widely publicized theories [Dem00a, Haji98] on oscillator phase noise behavior. In [Vana02d], it is shown that both theories yield similar results for stationary input signals u(t). This is no longer true for non- stationary inputs -as is the case in the oscillator injection locking problem- with [Dem00a] being the one that yields correct results. 5.7.3 Averaging in the presence of random perturbations At this point, it is important to mention that the results above are not universally valid. Firstly, the input noise process