  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.
