Joint Extremum Compression and Detection of a Time-Delayed Signal for Distributed Sensing

We study the problem of joint compression and detection in distributed sensing systems, motivated by applications such as device-to-device connectivity in IoT networks and distributed radar. In such systems, spatially separated sensors must collaboratively decide whether their observations stem from a common underlying signal, while communicating over highly bandwidth-limited links. We consider a fundamental, insightful model in which one sensor (the encoder) observes a continuous-time realization of a stationary bandlimited Gaussian process, while the other sensor (the decoder) observes a delayed and noisy version of that signal, with an unknown delay. The encoder is allowed to transmit only a $k$-bit message to the decoder to assist in making a binary decision: either the observations are statistically independent, or they are time-shifted noisy versions of the same signal. We propose a low-complexity extremum-based scheme that exploits the structure of the signal to enable reliable decision-making under tight communication constraints. We derive nonasymptotic upper bounds on the false alarm and mis-detection probabilities of our method, as well as a simplified asymptotic bound for the latter. Representative simulations demonstrate that the proposed scheme outperforms the prevalent 1-bit-per-sample quantization baseline and a Fisher-information-based compression benchmark, while closely approaching an information-theoretic (nonrealizable) rate-distortion benchmark.