Space System Time-Error Buildup and the Allan Deviation

Relating the standard deviation of time-error buildup, σt(T), at some time T after synchronization to a clock’s Allan deviation, σy(T), is problematic for several reasons. Notably, the stochastic integrals of various relevant noise types do not exist in closed form, and the standard deviation does not necessarily converge for the noise types of relevance for atomic clocks and crystal oscillators. Consequently, as an expedient, one often writes σt(T) = kσy(T)T, where k is a constant that depends on the noise type under consideration, as well as the statistical question of interest. Here, we consider the question of Clock Family Time-Error (CFTE) buildup and compute k for noise processes of relevance to atomic timekeeping in space. One of the interesting results of the present work is the k-value that we obtain for flicker frequency noise, which shows a dependence on the time after synchronization.