Error Probability Bounds for Balanced Binary Relay Trees

TL;DR

This paper analyzes how the overall detection error probability decreases in a balanced binary relay tree with multiple sensors, providing bounds on the error decay rate as the number of sensors increases.

Contribution

It derives tight bounds for the total error probability at the fusion center, characterizing the convergence rate as the number of sensors grows.

Findings

Error probability bounds depend on the number of sensors N.

Total error probability converges to zero at a quantifiable rate.

Even sensors with error probabilities approaching 1/2 can achieve reliable detection.

Abstract

We study the detection error probability associated with a balanced binary relay tree, where the leaves of the tree correspond to $N$ identical and independent detectors. The root of the tree represents a fusion center that makes the overall detection decision. Each of the other nodes in the tree are relay nodes that combine two binary messages to form a single output binary message. In this way, the information from the detectors is aggregated into the fusion center via the intermediate relay nodes. In this context, we describe the evolution of Type I and Type II error probabilities of the binary data as it propagates from the leaves towards the root. Tight upper and lower bounds for the total error probability at the fusion center as functions of $N$ are derived. These characterize how fast the total error probability converges to 0 with respect to $N$, even if the individual sensors have error probabilities that converge to 1/2.