## 6 Numerical Results
### Performance Evaluation of the Proposed LRT-BER Fusion Rule  
Figure 3 shows the receiver operating characteristic (ROC) curves of the proposed LRT-BER fusion rule in Equation (16). The false alarm probabilities and the detection probabilities are assumed to be identical for all sensors. In this experiment, we use \(P_{fa}=0.1\) and \(P_{d}=0.6\). We consider two different SNRs of the received signals, \(\overline{\gamma}_{b}=-5\) dB and \(\overline{\gamma}_{b}=5\) dB. The number of sensors is \(N=10\) and \(N=20\), respectively. BPSK modulation is used, and the Rayleigh fading envelope is assumed. In Figure 3, both the closed-form performance evaluation in Equations (29) and (30) and the CLT approximation performance evaluation in Equations (31) and (32) are included. The label \"CF\"denotes the closed-form performance evaluation, and the label \"CLT\" denotes the CLT approximation performance evaluation. We can see that as the system false alarm probability increases, the system detection probability increases. Note that there is a gap between the CLT approximation and the closed-form performance evaluation under the same condition. This is because the CLT approximation performance evaluation is based on the assumption with a large number of sensors. However, for the setup of Figure 3, there is only 10 or 20 sensors. It can be observed that when the number of sensors increases, such a gap decreases. Furthermore, the gap under the SNR \(\gamma_{b}=-5\) dB is smaller than that under the SNR \(\overline{\gamma}_{b}=5\) dB. Besides, the detection performance of a single sensor system has been marked in the intersect of the two lines \(P_{d}=0.6\) and \(P_{fa}=0.1\). The fusion system based on LRT-BER fusion rule can significantly enhance the system detection performance under the case of received SNR \(\bar{\gamma}_{b}=5\) dB. However, for \(\bar{\gamma}_{b}=-5\) dB and \(N=10\), the fusion system only slightly outperforms a single sensor system. This is because the degradation due to the communication channel leads to larger transmission error between sensors and the FC. To examine the relationship between the detection performance of the proposed LRT-BER Equation (16) and the physical layer specifications (modulation, reception mode and the channel characteristics), several experiments are given. Figure 4 gives the system detection probability as a function of SNR. The system false alarm probability is \(P_{\rm FA}=0.01\). The number of sensors is \(N=20\). The closed-form performance evaluation is used. In terms of the statistic characteristics of the flat and slow fading channel, Rayleigh fading, Ricean fading with factor \(K=2\) and Nakagami-\(m\) fading with \(m\) = 2 are considered. Figure 5 describes the ROC curves. The false alarm probability is \(P_{\rm FA}=0.01\), and the SNR is set to \(\overline{\gamma}_{b}=-5\) dB. Other parameters are the same as they are in Figure 4.  
Figure 3: ROC curves for the likelihood ratio test (LRT)-bit error (BER) fusion rule based on central limit theorem (CLT) and closed-form (CF) performance evaluation with \(N\) = 10 and \(N\) = 20. Average received SNRs are assumed to be SNR = \(-5\) dB and SNR = 5 dB. Each local detection index is assumed to be equal to each other (\(P_{fa}\) = 0.1, \(P_{d}\) = 0.6). BPSK modulation is used, and the Rayleigh fading envelope is assumed. From these two figures, it can be observed that the LRT-BER fusion rule with coherent BPSK modulation performs better than that with the other modulations for the identical statistic characteristics of the channel. This is because the BER of the BPSK modulation is smaller than that of 2 FSK and DPSK modulations, which can be derived from Equations (7), (9) and (10). For identical modulation, the  
Figure 4: Detection probability of the LRT-BER fusion rule as a function of different received SNRs for BPSK, 2 FSK and DPSK modulations with \(N\) = 20. Rayleigh fading, Ricean fading with factor \(K\) = 2 and Nakagami-\(m\) fading with \(m\) = 2 are considered.