## 5 Performance Analysis and Discussion
### Closed-Form Performance Analysis for Identical Local Detection Performance Indices  
If the local detection performance indices are equal to each other, _i.e._, \(P_{d_{i}}=P_{d}\) and \(P_{fa_{i}}=P_{fa}\), for \(i=1,\ldots,N\), the fusion statistic Equation (17) can be written as:  
\[\Lambda^{\prime}_{\text{BER}}=N^{\prime}_{1}{\log}\frac{P_{d}+P_{e }/(1-P_{e})(1-P_{d})}{P_{fa}+P_{e}/(1-P_{e})(1-P_{fa})}\] \[+\underbrace{(N-N^{\prime}_{1})}_{N^{\prime}_{0}}\log\frac{P_{d}+ (1-P_{e})/P_{e}(1-P_{d})}{P_{fa}+(1-P_{e})/P_{e}(1-P_{fa})}\] \[=N^{\prime}_{1}C_{1}+C_{2} \tag{18}\]  
where \(N^{\prime}_{0}=|S_{0}|\) and \(N^{\prime}_{1}=|S_{1}|\) with \(|\cdot|\) denoting the cardinality, that is \(N^{\prime}_{0}\) is the number of \(y_{i}=0\) in all \(N\) demodulated data, whereas \(N^{\prime}_{1}\) is the number of \(y_{i}=1\) in all \(N\) demodulated data. \(C_{1}\) and \(C_{2}\) are constants given as:  
\[C_{1}=\log\left[\frac{P_{d}+P_{e}/(1-P_{e})(1-P_{d})}{P_{fa}+P_{ e}/(1-P_{e})(1-P_{fa})}\right.\] \[\left.\cdot\frac{P_{fa}+(1-P_{e})/P_{e}(1-P_{fa})}{P_{d}+(1-P_{e })/P_{e}(1-P_{d})}\right] \tag{19}\]  
and:  
\[C_{2}=N\log\left[\frac{P_{d}+(1-P_{e})/P_{e}(1-P_{d})}{P_{fa}+(1-P_{e})/P_{e}( 1-P_{fa})}\right] \tag{20}\]  
The fusion statistic \(\Lambda^{\prime}_{\text{BER}}\) is an affine function of \(N^{\prime}_{1}\). Under both hypotheses \(H_{1}/H_{0}\), \(N^{\prime}_{1}=\sum_{i=1}^{N}y_{i}\) has a binomial distribution, _i.e._, \(N^{\prime}_{1}\sim\mathcal{B}(N,P_{j})\), for \(j\in\{0,1\}\). Here, the success probability under \(H_{1}\) hypothesis is:  
\[P_{1}=P(y_{i}=1\mid H_{1})=(1-P_{e})P_{d}+P_{e}(1-P_{d}) \tag{21}\]  
Similarly, the probability of \(y_{i}=1\) under \(H_{0}\) hypothesis is:  
\[P_{0}=P(y_{i}=1\mid H_{0})=(1-P_{e})P_{fa}+P_{e}(1-P_{fa}) \tag{22}\]  
As such, under the N-P criterion, given a predefined system false alarm probability \(\alpha\), we have:  
\[P_{\text{FA}}=\sum_{k\in\kappa}\binom{N}{k}P_{0}^{k}(1-P_{0})^{N-k}\leq\alpha \tag{23}\]  
where \(\kappa=\{k:\,k=\sum_{i=1}^{N}y_{i}\geq T\}\) and \(T\) represents the fusion threshold of the statistic \(N^{\prime}_{1}\). The threshold \(T\) can be obtained by maximizing all of the solutions that satisfy Inequality in Equation (23). Correspondingly, the detection probability of the fusion statistic \(N^{\prime}_{1}\) can be written as:  
\[P_{\text{D}} =\Pr(N^{\prime}_{1}=k{\geq}T|H_{1})\] \[+\underbrace{(N-N^{\prime}_{1})}_{N^{\prime}_{0}}\log\frac{P_{d}+ (1-P_{e})/P_{e}(1-P_{d})}{P_{fa}+(1-P_{e})/P_{e}(1-P_{fa})}\]0For the fusion statistic \(\Lambda^{\prime}_{\text{BER}}=N^{\prime}_{1}C_{1}+C_{2}\), when \(C_{1}>0\), _i.e._, \((P_{d}-P_{fa})(1-2P_{e})>0\), the detection probability of \(\Lambda^{\prime}_{\text{BER}}\) is:  
\[+\underbrace{(N-N^{\prime}_{1})}_{N^{\prime}_{0}}\log\frac{P_{d}+ (1-P_{e})/P_{e}(1-P_{d})}{P_{fa}+(1-P_{e})/P_{e}(1-P_{fa})}\]1 \[+\underbrace{(N-N^{\prime}_{1})}_{N^{\prime}_{0}}\log\frac{P_{d}+ (1-P_{e})/P_{e}(1-P_{d})}{P_{fa}+(1-P_{e})/P_{e}(1-P_{fa})}\]2 \[+\underbrace{(N-N^{\prime}_{1})}_{N^{\prime}_{0}}\log\frac{P_{d}+ (1-P_{e})/P_{e}(1-P_{d})}{P_{fa}+(1-P_{e})/P_{e}(1-P_{fa})}\]3 \[+\underbrace{(N-N^{\prime}_{1})}_{N^{\prime}_{0}}\log\frac{P_{d}+ (1-P_{e})/P_{e}(1-P_{d})}{P_{fa}+(1-P_{e})/P_{e}(1-P_{fa})}\]4  
where \(T^{\prime}\) is the threshold of \(\Lambda^{\prime}_{\text{BER}}\). The condition \((P_{d}-P_{fa})(1-2P_{e})>0\) is satisfied when the detection probability \(P_{d}\) is larger than the false alarm probability \(P_{fa}\) and the BER \(P_{e}\) is less than \(1/2\). This condition is easily satisfied for a practical decode-then-fuse target detection system. It can be observed from Equations (24) and (25) that the detection probability is equal for the fusion statistic \(N^{\prime}_{1}\) and \(\Lambda^{\prime}_{\text{BER}}\). The constants \(C_{1}\) and \(C_{2}\) will lead to different thresholds for \(N^{\prime}_{1}\) and \(\Lambda^{\prime}_{\text{BER}}\).