Goal-Oriented Wireless Sensor Networks: Error Exponent and Outage Analyses With Distributed Hypothesis Testing

This paper presents an analytical framework for characterizing and evaluating the rate-threshold (RT) region in distributed hypothesis testing (DHT) against independence, assuming a binary symmetric source (BSS) model. We consider a single sensor–single decision center (DC) setting, where a lossy source encoder (SE) compresses its observations using a binary linear block code and transmits them over a fading wireless channel. The DC, in turn, has perfect side information via an ideal (error-free, non-fading) channel and performs hypothesis testing using the Neyman–Pearson criterion, as shown in the graphical example. In a finite sample regime, we derived lower and upper bounds on Type-II error probability and determined the sufficient number of samples (SNS) and the corresponding decision threshold, for which the bounds are tight and yield an exponential approximation to Type-II error probability. Next, we define the admissible RT region by considering a critical relationship between the source coding rate and the decision threshold, an essential factor that influences the achievable Type-II error exponent. The impact of channel fading is then analyzed through the lens of the lossy source–channel separation theorem, leading to an outage probability expression. Numerical results are provided under the Nakagami fading model, illustrating how the outage probability is affected by the fading severity, decision threshold, and the correlation level between the source and side information.