  Motivated by the proliferation of mobile devices, we consider a basic form of
the ubiquitous problem of time-delay estimation (TDE), but with communication
constraints between two non co-located sensors. In this setting, when joint
processing of the received signals is not possible, a compression technique
that is tailored to TDE is desirable. For our basic TDE formulation, we develop
such a joint compression-estimation strategy based on the notion of what we
term "extremum encoding", whereby we send the index of the maximum of a
finite-length time-series from one sensor to another. Subsequent joint
processing of the encoded message with locally observed data gives rise to our
proposed time-delay "maximum-index"-based estimator. We derive an exponentially
tight upper bound on its error probability, establishing its consistency with
respect to the number of transmitted bits. We further validate our analysis via
simulations, and comment on potential extensions and generalizations of the
basic methodology.
