## 4 Decision Fusion Rule with Channel Errors Characterized by BER
For a flat and slow Nakagami-\(m\) fading channel, the BER with coherent 2 FSK modulation, coherent BPSK modulation and noncoherent DPSK modulation is given as:  
\[P_{e}= \begin{cases}\frac{1}{\pi}\int_{0}^{\frac{\pi}{2}}\left(1+\frac{0.5 \overline{\gamma}_{b}}{m\sin^{2}\phi}\right)^{-m}\mathrm{d}\phi&\text{coherent 2 FSK modulation}\\ \frac{1}{\pi}\int_{0}^{\frac{\pi}{2}}\left(1+\frac{\overline{\gamma}_{b}}{m \sin^{2}\phi}\right)^{-m}\mathrm{d}\phi&\text{coherent BPSK modulation}\\ \frac{1}{2}\int_{0}^{\frac{\pi}{2}}\left(1+\frac{\overline{\gamma}_{b}}{m} \right)^{-m}\mathrm{d}\phi&\text{noncoherent DPSK modulation}\\ \end{cases}  \tag{10}\]where \(m\) is the Nakagami parameter. The detailed derivation of these BERs can be found in [27]. The main task of this paper is to construct a fusion rule according to the decoded results \(y_{i}\), for 1\(\leq\)\(i\)\(\leq\)\(N\). Here, we only consider the N-P criterion, since the prior probability of the hypotheses is usually unknown _a priori_. The optimal LRT-based fusion rule for N-P criterion can be written as:  
\[\frac{P(\boldsymbol{y}\mid H_{1})}{P(\boldsymbol{y}\mid H_{0})}\stackrel{{ H_{1}}}{{\underset{<}{\sim}}}T_{\rm N} \tag{11}\]  
where \(T_{\rm N}\) is the fusion threshold of the LRT fusion statistic. The LRT statistic (11) based on the knowledge of channel BER can be further written as:  
\[\Lambda= \frac{P\left(\boldsymbol{y}|H_{1}\right)}{P\left(\boldsymbol{y}| H_{0}\right)}=\prod_{i=1}^{N}\frac{P(y_{i}|H_{1})}{P(y_{i}|H_{0})}\] \[= \prod_{i=1}^{N}\frac{\sum_{u_{i}}P(y_{i}\mid u_{i})P(u_{i}\mid H_ {1})}{\sum_{u_{i}}P(y_{i}\mid u_{i})P(u_{i}\mid H_{0})}\] \[= \prod_{i=1}^{N}\frac{P(y_{i}\mid u_{i}=1)P_{d_{i}}+P(y_{i}\mid u_ {i}=0)(1-P_{d_{i}})}{P(y_{i}\mid u_{i}=1)P_{fa_{i}}+P(y_{i}\mid u_{i}=0)(1-P_{ fa_{i}})} \tag{12}\]  
where \(P_{d_{i}}\) is the detection probability and \(P_{fa_{i}}\) is the false alarm probability of the \(i\)-th sensor. The above equation is obtained by following the fact that \(H_{0}/H_{1}\rightarrow\boldsymbol{x}\rightarrow\boldsymbol{u}\rightarrow \boldsymbol{y}\rightarrow{u_{0}}\) forms a Markov chain in Figure 2. The local binary decision results are transmitted by using the BPSK, 2 FSK or DPSK modulation scheme. The demodulated data at the FC are also binary, _i.e._, \(y_{i}\in\{0,1\}\). Expanding the fusion statistic \(\Lambda\) in Equation (12) with respect to the one-bit error probability, we have:  
\[\Lambda=\prod_{i=1}^{N}\frac{\left(1-P_{e}\right)^{y_{i}}P_{e}^{(1-y_{i})}P_{ d_{i}}+P_{e}^{y_{i}}\left(1-P_{e}\right)^{(1-y_{i})}\left(1-P_{d_{i}}\right)}{ \left(1-P_{e}\right)^{y_{i}}P_{e}^{(1-y_{i})}P_{fa_{i}}+P_{e}^{y_{i}}\left(1-P _{e}\right)^{(1-y_{i})}\left(1-P_{fa_{i}}\right)} \tag{13}\]  
Note that in Equation (13), the equation holds \(y_{i}\) is either zero or one, and the facts \(P(y_{i}=0|u_{i}=1)=P(y_{i}=1|u_{i}=0)=P_{e}\) and \(P(y_{i}=1|u_{i}=1)=P(y_{i}=0|u_{i}=0)=1-P_{e}\) according to Equation (6) are employed. For Equation (13), two cases (\(P_{e}=0\) and \(0\leq P_{e}\leq 1/2\)) are considered as follows. 1. If \(P_{e}=0\), _i.e._, the communication channel is error-free, \(\Lambda_{i}\) in the BER-based fusion statistic \(\Lambda=\prod_{i=1}^{N}\Lambda_{i}\) can be reduced to: \[\Lambda_{i}= \begin{cases}\frac{P_{d_{i}}}{P_{fa_{i}}},&y_{i}=1(\text{or }u_{i}=1),P_{e}=0\\ \frac{1-P_{d_{i}}}{1-P_{fa_{i}}},&y_{i}=0(\text{or }u_{i}=0),P_{e}=0\end{cases} \] (14) This fusion statistic is the CV statistic in [9]. Note that the CV statistic instead of the two-stage CV statistic is the optimal fusion statistic under error-free communication channels. 2. If \(0<P_{e}\leq 1/2\), _i.e._, the local decisions are subject to transmission errors, the fusion statistic (13) can be written as: \[\Lambda=\prod_{i=1}^{N}\frac{P_{d_{i}}+P_{e}^{2y_{i}-1}(1-P_{e})^{1-2y_{i}}(1- P_{d_{i}})}{P_{fa_{i}}+P_{e}^{2y_{i}-1}(1-P_{e})^{1-2y_{i}}(1-P_{fa_{i}})}\] (15)It can be observed that the fusion statistic for the setup \(P_{e}=0\) can be regarded as the CV statistic under the ideal transmission channels.