# Signal Detection for Ambient Backscatter Communication with OFDM Carriers
## 4 Numerical Results  
In [12], the authors compared the energy detector with a benchmark design in which the reader detected the tag bit by distinguishing between two different orders of the average power of the received signal \(y(n)\), where  
\[\mathbf{z}\triangleq\frac{1}{N+C_{cp}}\sum_{n=1}^{N+N_{cp}-1}|y(n)|^{2}. \tag{27}\]  
They also showed that the their design was comparable to the benchmark design in terms of complexity, but the performance was better in terms of transmission rate and BER. Thus, we compare our proposed approach with [12] (i.e., \(p=2\)), which is referred as the \"conventional\" energy detector. A summary description of simulation parameters is given in Table 1. For comparison purpose, we keep all the parameter as the same for both detection schemes. In fact, if the tag backscatters the information at a lower rate than the ambient signals, we can design a receiver that can separate two signals [5; 12]. In the following, we briefly describe some metrics that we used to evaluate the proposed statistic test. The test must be a sufficient statistic for our energy detection problem and contains all the information required to distinguish two hypotheses \(H_{0}\) and \(H_{1}\). \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline OFDM bandwidth & \(f_{s}=20\) MHz \\ Number of paths & \(N_{sr}=N_{st}=8\), \(N_{tr}=1\) \\ Attenuation value & \(\eta=1\) \\ CP length & \(N_{cp}=16,32,64,128\) \\ Number of carriers & \(N=8N_{cp}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation ParametersFirst, we check the validity of the Gaussian approximation for the proposed test. In fact, since the test statistic (11) follows the Gamma distribution under both hypotheses, the length \(D\) must be large enough to apply the CLT while not very large to keep the approximation meaningful. In the first run, Figure 2 gives us a case study on this approximation. 2. Second, in order to find the optimal \(p\) for the test (11) instead of using \(p=2\), we solve (25) to obtain an adaptive power order. The result is shown in Figure 3. The purpose of this result is to observe how the \(p^{*}\) changes for maximizing the probability of correct detection according to the changes in the SNR. Thus, we may have a certain strategy to select \(p\) for a given SNR and a false alarm rate. 3. Third, with different settings (e.g., SNR and \(N_{cp}\)), we observe how much the BER changes when using our test statistics and the conventional ones in terms of our ability to solve \(p^{*}\) with high correct detection probability \(P_{d}\). The results are given in Figures 4 and 5. 4. Finally, we provide the median receiver operating characteristics (ROC) curve for our detector design as predicted by our aforementioned analysis. Following the above construction, in order to verify the accuracy of approximating the simulated PDFs (i.e., Gamma distribution) by the theoretical approximation (i.e., Q-function), Figure 2 illustrates those PDFs when \(\gamma=0\) dB, \(N_{cp}=64\), and \(N=512\), thus, \(L=8\) and \(D=57\). We observe that the Gamma approximation fits well in most cases considered. The accuracy of the approximation increases when \(p\) decreases. This approximation may be adequate for practical energy detectors since we can improve it by increasing the length \(D\) in the detection. Thus, we have to select appropriate values of length \(D\) and \(P_{f}\) to guarantee high probability of detection, while keeping the Gaussian approximations to be valid. Figure 3 shows the optimum value of \(p\) in (25) with different fixed values of \(P_{f}=\alpha\). The value \(p^{*}\) maximizing \(P_{d}\) decreases when \(\gamma\) increases. We also plot a small subgraph at the right hand side of Figure 3 to illustrate the \(p^{*}\) value (in vertical axis) versus small SNR (in horizontal axis) because we observe that \(p^{*}\)-curve has a big jump in its value for \(\gamma\) in the range of \((0,1.5)\). We offer some brief comments.