Joint Data Compression and Time-Delay Estimation for Distributed Systems via Extremum Encoding

Motivated by the ubiquity of mobile devices and their potential capabilities as distributed systems (e.g., for localization), we consider a time-delay estimation (TDE) problem in which there are two non-colocated sensors and communication constraints between them. When the communication bandwidth is particularly limited, there is a need for compression techniques that are specifically tailored for the TDE application. For the discrete-time version of this problem, we propose such a joint compression-estimation strategy based on what we term “extremum encoding”, whereby the (time-) index of the maximum of the observed signal in a finite observation window is sent from one sensor to another. Subsequent joint processing of the encoded message with the locally observed, time-delayed, noisy signal gives rise to our proposed time-delay “maximum-index”-based estimator. We analyze the performance of the proposed scheme in the asymptotic regime of large message size and delay spread, but with their ratio fixed. We derive the error probability exponent for this estimator, and its consistency. We validate the analysis via simulations, and further demonstrate the performance gains over traditional alternatives.