# Signal Detection for Ambient Backscatter Communication with OFDM Carriers
## 4 Numerical Results
First, the value \(p^{*}\) is probably the best solution we has achieved through numerical results. The \(p^{*}\)-curve grows sub-linearly versus small value of SNR (e.g, \(\gamma\leq 1\)) simultaneously, while it decays slowly and remains constant as a function of \(\gamma\geq 6\). In a certain sense, the vector \(\mathbf{z}\) represents the relative difference between two input signals. Due to weak backscattered signals, if it has any small component, \(p=2\) makes them negligible. On the other hand, the optimal power order is around 1, which is more irritated by small values. For instance, when \(\gamma\leq 1\), the proposed detector returns \(p^{*}\approx 1\) rather than \(p=2\). In this case, it prefers returning the number of non-zero values of the vector \(\mathbf{z}=\{z(n)\}\) rather than tolerates them. Figure 2: _Cont._  
Figure 4 demonstrates the theoretical BER performance of the conventional energy detector with \(p=2\) and the proposed detector with optimal power order \(p^{*}\). We set \(N_{cp}=64\), \(N=512\), and \(\alpha=0.01\). We can see that the energy detector performs worse than the proposed one because of the inaccurate Gaussian approximation. We also see that the improvement here becomes more prominent when the target probability of false alarm is smaller, as well as the achieved SNR is relatively higher. The BER becomes flat even for high SNR value. For the proposed detector, it can be found that the increasing SNR yields reduced BER, especially when SNR is small (e.g., SNR \(\leq 2\) dB). For larger SNR, the BER performance remains unchanged. This phenomenon may be caused by the strong direct-link interference [13]. Moreover, since the reader tries to distinguish between two bits by taking a sufficiently large number of samples, i.e., \(D\), the value of \(D\) we consider here is large enough to apply the CLT, thus, the probability of miss detection and the probability of false alarm are moderate, i.e., they are not changed with \(D\). Consequently, the overall BER in (21) does not become much different. The detector  
Figure 3: The optimum value of power order \(p\) versus \(\gamma\). It shows the effect of \(\gamma\) on \(p^{*}\) at several different SNR levels, which comes from solving procedure of (26). We generated an SNR vector of 21 equally-spaced points between 0 and 10. The \(p^{*}\)-curve is aimed to choose the appropriate \(p\) value for the test statistic. Figure 2: Illustration of CDFs under \(H_{0}\) and \(H_{1}\). Note that the theoretical analysis shows that the test statistic \(t\sim\Gamma(k_{i},\theta_{i})\) under \(H_{i}\), while the simulation approximation gives us \(t\sim\mathcal{N}(\mathbb{E}(t),\mathrm{Var}(t))\), where \(\mathbb{E}(t)\) and \(\mathrm{Var}(t)\) are given in (14) and (15), respectively. (**a**) Simulated CDFs for \(t\) under \(H_{0}\) when \(\gamma=0\). (**b**) Simulated CDFs for \(t\) under \(H_{1}\) when \(\gamma=0\). must reach the error probabilities uniformly over a whole uncertainty set with various \(D\). As SNR increases, it hits the SNR wall [17] while the required sample complexity meets our performance target. Figure 5 depicts the curves of BER versus SNR with several value of \(N_{cp}\) for the proposed detector. We set \(\gamma=5\) dB and \(\alpha=0.01\). The BER approaches 0.5 at small \(N_{cp}\) and there exist little gaps between BER curves. We observe that the BER decreases as \(N_{cp}\) increases. However, the smaller \(N_{cp}\) offers higher data rate from the relationship  
\[R_{\text{tag}}=\frac{f_{s}}{(N+N_{cp})}, \tag{28}\]  
where \(R_{\text{tag}}\) is the tag rate [12]. Obviously, if we fix the number of OFDM carriers \(N\), \(R_{\text{tag}}\) decreases as the CP length increases, while the BER decreases, as illustrated in Figure 5. Thus, there exists a trade-off between the BER and the data rate \(R_{\text{tag}}\). Figure 4: BER versus SNR \(\gamma\). We observe that BER achieves the maximum at \(\gamma=0\). The designed detector can perform well even when the SNR is high.