## 3 Soft Decision Fusion Based on the Subpartitioning of the Local Decision Space in Clusters  
We consider the binary detection in the clustered distributed detection system. In binary detection, there are two hypotheses: \(H_{0}\) represents the absence of the target (the signal) and \(H_{1}\) represents the presence of the target (the signal). The FC makes a global binary decision (0 or 1) by processing local decisions received from all local sensors. It is the hard decision fusion when each local sensor makes a binary hard decision (0 or 1) and delivers it to the FC. Conversely, it is the soft decision fusion when each local sensor delivers a multiple-bit decision to the FC. In this paper, we apply soft decision fusion in clusters formed by Section 2 and apply hard decision fusion between clusters. The proposed clustered detection system is shown in Figure 4.  
We assume that there are N local sensors in the distributed detection system. By using the FLS and the fuzzy c-means clustering algorithm in Section 2, M clusters are formed. In the \(m\)th cluster, there are \(n_{m}\) local sensors and \(c_{m}\) is the CH in it. N = \(n_{1}+n_{2}+\ldots+n_{M}\). We take the \(m\)th cluster for example. In the \(m\)th cluster, let \(y_{1}\), \(y_{2}\),..., \(y_{nm}\) be the statistically independent observations of \(n_{m}\) local sensors and have known probability distributions under both hypotheses (\(H_{0}\) and \(H_{1}\)). For the \(k\)th local sensor in the \(m\)th cluster, \(T_{k}\) is the sensor's threshold determined by the probability of false alarm of the \(k\)th local sensor, and \(L_{k}(y_{k})\) is the likelihood ratio test at the \(k\)th local sensor, which is given by:  
\[L_{k}(y_{k})=\frac{P(y_{k}|H_{1})}{P(y_{k}|H_{0})}\]  
Figure 4: The proposed clustered system. If a local decision \(u_{k}\) is a one-bit hard decision, then the associate decision space \(\Omega^{k}\) is partitioned into two exclusive regions, \(\Omega^{k}_{0}\) and \(\Omega^{k}_{1}\), such that:  
\[\left\{ \begin{array}{ll}u_{k}=1,&if\;\;y_{k}\in\Omega^{k}_{1}\\ u_{k}=0,&if\;\;y_{k}\in\Omega^{k}_{0}\end{array} \right. \tag{6}\]  
Furthermore, we have a probability of detection (\(P_{dk}\)) and a probability of false alarm (\(P_{jk}\)) for the \(k\)th local sensor:  
\[\left\{ \begin{array}{ll}P_{dk}=P\{u_{i}=1|H_{1}\}.\\ P_{fk}=P\{u_{i}=1|H_{0}\}.\end{array} \right. \tag{7}\]  
To improve the detection performance, a \(b\)-bit (\(b\geq 2\)) soft decision can be obtained, in which \(b\) is determined by the communication capability of the system. Its essence is the subpartitioning of the \(\Omega^{k}_{0}\) and \(\Omega^{k}_{1}\) space. For illustration purposes, the 2-bit soft decision case is considered. In the 2-bit soft decision case, \(\Omega^{k}_{0}\) and \(\Omega^{k}_{1}\) are partitioned into two exclusive regions respectively. The quantization rule and decision rule  
\[u_{k}=\left\{ \begin{array}{ll}00(\textit{decision is }H_{0}\textit{ with higher confidence, soft decision value}:0),&y_{k}\in\Omega^{k}_{00}\\ 01(\textit{decision is }H_{0}\textit{ with lower confidence, soft decision value}:1),&y_{k}\in\Omega^{k}_{01}\\ 10(\textit{decision is }H_{1}\textit{ with lower confidence, soft decision value}:2),&y_{k}\in\Omega^{k}_{10}\\ 11(\textit{decision is }H_{1}\textit{ with higher confidence, soft decision value}:3),&y_{k}\in\Omega^{k}_{11}\end{array} \right. \tag{8}\]  
where:  
\[\left\{ \begin{array}{ll}\Omega^{k}=\Omega^{k}_{0}\cup\Omega^{k}_{1},\Omega^ {k}_{0}\cap\Omega^{k}_{1}=\varnothing\\ \Omega^{k}_{0}=\Omega^{k}_{00}\cup\Omega^{k}_{01},\;\Omega^{k}_{00}\cap\Omega ^{k}_{01}=\varnothing\\ \Omega^{k}_{1}=\Omega^{k}_{10}\cup\Omega^{k}_{11},\;\Omega^{k}_{10}\cap\Omega ^{k}_{11}=\varnothing\end{array} \right. \tag{9}\]  
The subpartitioning in the 2-bit decision case is shown in Figure 5.  
Corresponding to Equation (7), we also have:  
\[\left\{ \begin{array}{ll}P_{d10k}=P\{u_{k}=10|H_{1}\}\\ P_{d11k}=P\{u_{k}=11|H_{1}\}\\ P_{f10k}=P\{u_{k}=10|H_{0}\}\\ P_{f11k}=P\{u_{k}=11|H_{0}\}\end{array} \right.