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 sensors. The root of the tree represents a fusion center that makes the overall detection decision. Each of the other nodes in the tree is a relay node that combines two binary messages to form a single output binary message. Only the leaves are sensors. In this way, the information from the sensors is aggregated into the fusion center via the relay nodes. In this context, we describe the evolution of the Type I and Type II error probabilities of the binary data as it propagates from the leaves toward 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 $1/2$.
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机译：我们研究了与平衡二进制中继树相关的检测错误概率，其中树的叶子对应于$ N $个相同且独立的传感器。树的根代表一个融合中心，该融合中心可以做出整体检测决策。树中的每个其他节点都是一个中继节点，该节点将两个二进制消息合并以形成单个输出二进制消息。只有叶子是传感器。这样，来自传感器的信息将通过中继节点聚合到融合中心。在这种情况下，我们描述了二进制数据从叶向根传播时I型和II型错误概率的演变。得出了融合中心总错误概率作为$ N $函数的严格上限和下限。这些特征表示相对于$ N $，总错误概率收敛到0的速度有多快，即使各个传感器的误差概率收敛到$ 1/2 $也是如此。
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