## 3 Soft Decision Fusion Based on the Subpartitioning of the Local Decision Space in Clusters
\tag{10}\]  
To simplify the calculation, we let the subpartition satisfy Equation (11) according to the probability of detection and the probability of false alarm:  
\[\left\{ \begin{array}{ll}P_{d10k}=P_{d11k}=\frac{P_{jk}}{2}\\ P_{f10k}=P_{f11k}=\frac{P_{f1k}}{2}\end{array} \right. \tag{11}\]  
Figure 5: The subpartitioning figure in the 2-bit decision case. According to the knowledge of the \(k\)th local sensor's probability of false alarm and the signal-to-noise ratio (SNR), which is defined as the ratio of signal power to the noise power, we can get \(P_{dk}\) and the range of sub-spaces. Let t\({}_{0}\) be the threshold between \(\Omega_{00}^{k}\) and \(\Omega_{01}^{k}\), t\({}_{1}\) be the threshold between \(\Omega_{01}^{k}\) and \(\Omega_{10}^{k}\), and \(t_{2}\) be the threshold between \(\Omega_{10}^{k}\) and \(\Omega_{11}^{k}\), then the calculation of the \(k\)th local sensor's thresholds (\(t_{0}\), \(t_{1}\), \(t_{2}\)) in the 2-bit soft decision case is summarized using two steps. Step 1: Given the false alarm probability of the \(k\)th local sensor \(P_{\delta k}\), then t\({}_{1}\), which is not only the threshold between \(\Omega_{01}^{k}\) and \(\Omega_{10}^{k}\) but also the threshold between \(\Omega_{0}^{k}\) and \(\Omega_{1}^{k}\), can be obtained under the Neyman-Pearson criterion. \(P_{dk}\) can be obtained using t\({}_{1}\). Step 2: Then, we can calculate t\({}_{0}\) by making \(P_{f10k}=P_{f11k}=P_{\delta k}/2\), and calculate \(t_{2}\) by making \(P_{d10k}=P_{d11k}=P_{dk}/2\). Considering that the communication channel is non-ideal without memory, we assume that the bit error rate of the \(k\)th local sensor is \(P_{\delta k}\), which is the same for every bit. Then, for the \(k\)th local sensor, the probability of the bit being transmitted correctly is \(1-P_{\delta k}\). In the 2-bit decision case, the station transition diagram is shown in Figure 6. We can obtain that the probability of a 2-bit soft decision being transmitted correctly by the \(k\)th local sensor is \((1-P_{\delta k})^{2}\), the probability of one bit being transmitted wrongly is \(P_{\delta k}(1-P_{\delta k})\), and the probability of two bits all being transmitted wrongly is \((P_{\delta k})^{2}\). Then, we can get the possible transition matrix of Figure 6:  
\[PT_{k}=\left[\begin{array}{cccc}(1-P_{\delta k})^{2}&(1-P_{\delta k})P_{ \delta k}&(1-P_{\delta k})P_{\delta k}&(P_{\delta k})^{2}\\ (1-P_{\delta k})P_{\delta k}&(1-P_{\delta k})^{2}&(P_  
where:  
\[\left\{ \begin{array}{ll}P_{d10k}=P\{u_{k}=10|H_{1}\}=P_{d11k}=P\{u_{k}=11|H_{1} \}=\frac{P_{dh}}{2}\\ P_{m10k}=P\{u_{k}=00|H_{1}\}=P_{m01k}=P\{u_{k}=01|H_{1}\}=\frac{1-P_{dh}}{2}\\ P_{f10k}=P\{u_{k}=10|H_{0}\}=P_{f11k}=P\{u_{k}=11|H_{0}\}=\frac{P_{fk}}{2}\\ P_{10k}=P\{u_{k}=00|H_{0}\}=P_{11k}=P\{u_{k}=01|H_{0}\}=\frac{1-P_{fk}}{2}\\ \end{array} \right.\]  
Therefore, the probability of a false alarm and the probability of detection of the CH can be obtained using:  
\[P_{fc_{m}}=\sum_{\mathrm{L}(\mathbf{U_{m}})>T_{m}}\quad P(\mathbf{U_{m}}|H_{0})PT+\ \lambda\sum_{\mathrm{L}(\mathbf{U_{m}})=T_{m}}\quad P(\mathbf{U_{m}}|H_{0}) \tag{15}\]  
and:  
\[P_{dc_{m}}=\sum_{\mathrm{L}(\mathbf{U_{m}})>T_{m}}\quad P(\mathbf{U_{m}}|H_{1})+\ \lambda\sum_{\mathrm{L}(\mathbf{U_{m}})=T_{m}}\quad P(\mathbf{U_{m}}|H_{1}) \tag{16}\]  
According to the probability of a false alarm of local sensor \(c_{m}\) (CH), we can obtain \(T_{m}\) and \(\lambda\). Then, the probability of detection of the CH can be obtained using Equation (16). To get a binary decision in the \(m\)th cluster when knowing \(T_{m}\) and \(\lambda\) after calculation, we can follow:  
\[\left\{ \begin{array}{ll}u_{c_{m}}=1,&if\ \ \mathrm{L}(\mathbf{U_{m}})>T_{m}\ \ or\ \ \lambda\geq 0.5\ when\ \ \mathrm{L}(\mathbf{U_{m}})=T_{m}\\ u_{c_{m}}=0,&if\ \ \mathrm{L}(\mathbf{U_{m}})<T_{m}\ \ or\ \ \lambda<0.5\ when\ \ \mathrm{L}(\mathbf{U_{m}})=T_{m}\\ \end{array} \right. \tag{17}\]  
In the same way, for a \(b\)-bit soft decision, \(b_{n}=2^{(b-1)}-1\), the quantization rule and decision rule:  
\[u_{k}=\left\{