## 6 Simulation Results  
In this section, we present computer simulation results to illustrate the performance of the proposed quantization scheme. Specifically, we use MATLAB to simulate distributed detection in a wireless sensor network with a parallel configuration, where \(N\) sensors make observations of a random Gaussian signal, and each sensor quantizes its observations into one bit of information, which is sent to the fusion center with a binary symmetric channel. The fusion center forms a global decision regarding the change of the scale parameter from \(\sigma_{0}\) to \(\sigma_{1}\). Our simulation is carried out with the following setups: \(\sigma_{0}=1\), \(\sigma_{1}=2\), \(P_{FA}\in\{0.1,0,01\}\), \(\tilde{p}\in\{0,0.1,0.2\}\). Figure 2 plots the detection probabilities of the unquantized case and the one-bit GLRT detector with the optimum threshold \(\tau^{\star}\)_versus_ the number of sensors. In the figure, a solid line and a dashed line represent the theoretical asymptotic performance, while the plus marks and circle marks represent the performance of the Monte Carlo (MC) experiments obtained by averaging over \(10^{4}\) independents runs. \begin{table}
\begin{tabular}{c c} \hline \hline
**Cross-Over Probability** & \(\tau^{\star}\) \\ \hline \(\tilde{p}=0\) & 1.4821\(\sigma_{0}\) \\ \(\tilde{p}=10^{-2}\) & 1.4109\(\sigma_{0}\) \\ \(\tilde{p}=10^{-1}\) & 1.1736\(\sigma_{0}\) \\ \(\tilde{p}=0.2\) & 1.0785\(\sigma_{0}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Optimum quantization threshold in perfect/BSC channels. It is observed that the theoretical asymptotic analysis provides a good approximation of the experimental performance with an increasing number of sensors. When the probability of false alarm (\(P_{FA}\)) is reduced, more sensors are required for the simulation to match asymptotic analysis. We provide simulation results to illustrate the performance of the GLRT used along with the proposed one-bit quantization scheme. Figure 3 depicts the detection probabilities of the GLRT (assuming error-free transmission between the sensors and the FC) _versus_ the number of sensors. As a benchmark, the performance of the unquantized detector is also reported. In our simulation, for the proposed scheme, we consider three choices of the threshold, include two heuristic choices, including \(\tau=0.5\sigma_{0}\) and \(\tau=\sigma_{0}\), as well as the optimum threshold \(\tau^{\star}\) selected from Table 1. We assume that the standard deviation under \(H_{0}\) and \(H_{1}\) is \(\sigma_{0}=1\) and \(\sigma_{1}=2\), respectively. In the figure, the performance of the Monte Carlo (MC) experiments is obtained by averaging over \(10^{4}\) independent runs. It is observed  
Figure 2: Performance the proposed one-bit generalized likelihood ratio test (GLRT) detector with the optimum threshold \(\tau^{\star}\) and the unquantized detector, when \(P_{FA}=0.1\) (**a**) and \(0.01\) (**b**). that the theoretical asymptotic analysis provides a good approximation of the experimental performance when the number of sensors is increasing. Meanwhile, the two heuristic thresholds lead to considerable performance loss. Figure 4 plots the receiver operating characteristics (ROC) curves of the unquantized detector and the one-bit GLRT detector in BSC channels with several cross-over probabilities \(\tilde{p}\). The unquantized case provides a reference on the optimal detection performance. It is seen that channel errors have a notable impact on the the one-bit detector. As shown in (13), the proposed quantization scheme requires knowledge of the crossover probability \(\tilde{p}\), which needs to be estimated in practice. Let the estimated crossover probability be denoted by \(\hat{p}\). It is of interest to examine how sensitive the proposed quantization scheme is to a mismatch between \(\tilde{p}\) and \(\hat{p}\).