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

D.P. Feldman and J.P. Crutchfield, Synchronizing to Periodicity:
The Transient Information and Synchronization Time of Periodic
Sequences. *Advances in Complex Systems*. **7**(3-4):
329-355, 2004.

## Abstract

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 $\TI$,
a recently introduced information-theoretic quantity that measures the
uncertainty an observer experiences while synchronizing to a sequence.
We also consider the synchronization time $\ST$, which is the average
number of measurements required to infer the phase of a periodic signal.
We calculate $\TI$ and $\ST$ for all periodic sequences up to and
including period $23$. We show which sequences of a given period
have the maximum and minimum possible $\TI$ and $\ST$ 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.

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