## 6 Numerical Results
### Performance Comparison with the Other Fusion Rules  
In this experiment, the detection performance of MRC, EGC, two-stage CV and LRT-CS is compared to the proposed LRT-BER fusion rule. Figure 6 gives the ROC curves of these fusion rules. The number of sensors is \(N=6\). We assume that the local detection performance indices are equal to each other (\(P_{fa}=0.1\) and \(P_{d}=0.6\)). The received SNR is set to \(\bar{\gamma}_{b}=5\) dB, BPSK modulation and Rayleigh fading are considered. Since there is no closed-form performance evaluation for MRC and LRT-CS fusion rules proposed in [20], here the CLT approximation performance evaluation is employed. It can be observed that LRT-BER and LRT-CS exhibit a small deviation and perform better than MRC, EGC and two-stage CV. The performance of the MRC is worse than that of the other fusion rules. Figure 6: Comparison of ROC curves for LRT-BER, maximum ratio combiner (MRC), equal gain combiner (EGC) and LRT-channel statistics (CS) with six sensors, \(\overline{\gamma}_{b}=5\) dB, \(P_{fa}=0.1\), \(P_{d}=0.6\). BPSK modulation and Rayleigh fading assumption are considered. In addition to the CLT approximation, the deflection coefficient in Equation (43) for the mentioned fusion rules is given in Figure 7. A zoom-in display of the deflection coefficient under the low SNR scenario is also included. The local detection performance indices are assumed to be identical for all sensors. In this experiment, we use \(P_{fa}=0.1\) and \(P_{d}=0.6\). BPSK modulation is employed. The number of sensors is \(N=6\) and \(N=18\), respectively. Figure 7 shows that the EGC method that requires the minimum amount of information performs better than the MRC under a wide range of received SNR. The only exception is that MRC slightly outperforms EGC under very low SNR (e.g., SNR \(<-5\) dB). Furthermore, LRT-CS and the proposed LRT-BER present a small deviation. This is consistent with the results in Figure 6.  
In practice, sensors have different local detection performance. Furthermore, the received SNRs are not identical. Thus, we examine the detection performance of MRC, EGC, two-stage CV, LRT-CS, as well as the proposed LRT-BER fusion rule under the case of different local detection performance and different SNRs. Figure 8 gives the ROC curves for \(N=10\) sensors by using the CLT approximation performance evaluation. The solid lines are the detection performance of the mentioned fusion rules with the setting of local detection probability \(P_{d}=[0.43,0.75,0.68,0.74,0.75,0.32,0.75,0.68,0.70,0.70]\). In order to illustrate the system performance improvement caused by one sensor, the case with the fourth sensor local detection probability \(P_{d}(4)=0.99\) (the local detection probability of Sensor 4 changes from \(0.74\) to \(0.99\)) is also included in Figure 8. The dashed lines give the results with such case. The average SNRs are \(\bar{\gamma}_{b}=[5.6,8.0,6.0,8.5,6.8,10.0,5.0,5.0,5.1,5.1]\) dB. The local false alarm probabilities are assumed to be equal to each other, which are set to \(P_{fa}=0.1\). BPSK is employed, and Rayleigh fading is assumed. From the figure, we can see that when the local detection probabilities, as well as the received SNRs are not equal to each other, the proposed LRT-BER fusion rule outperforms significantly the other fusion rules. The detection performance of LRT-CS is better than MRC, EGC and two-stage CV. Figure 7: Deflection coefficient of MRC, EGC, LRT-CS, two-stage CV and LRT-BER statistics with six and eighteen sensors, respectively, \(P_{fa}=0.1\), \(P_{d}=0.6\). BPSK modulation and the Rayleigh fading assumption are considered. Furthermore, the MRC fusion rule presents the worst detection performance. Further, when one of the sensors improves the detection probability from \(P_{d}(4)=0.74\) to \(P_{d}(4)=0.99\), the improvement of the system detection probability \(P_{\text{D}}\) (the difference between the solid line and the dashed line in Figure 8) due to the increasing of local detection probability of one sensor is larger than that of LRT-CS.