Synchronizing to Periodicity: The Transient Information and Synchronization Time of Periodic Sequences

We analyze how difficult it is to synchronize to a periodic sequence whose structure is known, when an observer is initially unaware of the sequence's phase. We examine the transient information T, a recently introduced information-theoretic quantity that measures the uncertainty an observer experiences while synchronizing to a sequence. We also consider the synchronization time tau, which is the average number of measurements required to infer the phase of a periodic signal. We calculate T and tau for all periodic sequences up to and including period 23. We show which sequences of a given period have the maximum and minimum possible T and tau values, develop analytic expressions for the extreme values, and show that in these cases the transient information is the product of the total phase information and the synchronization time. Despite the latter result, our analyses demonstrate that the transient information and synchronization time capture different and complementary structural properties of individual periodic sequences -- properties, moreover, that are distinct from source entropy rate and mutual information measures, such as the excess entropy.