# Fingerprint Based Codebook for RIS Passive Beamforming Training
## 5 Performance Evaluation  
In this section, we evaluate the performance of the proposed fingerprint-based passive beamforming scheme. First, we compare its achievable rate, in bits/s/Hz (bps/Hz), that can be provided to the RX with the benchmarks' schemes: CE-based PBF, random configuration PBF, and random CB-PBF schemes. Using (1), the achievable rate can be expressed as:  
\[R=\log_{2}\left(1+\frac{|(\mathbf{G_{2}\theta G_{1}}+\mathbf{H_{o}})\mathbf{V}|^{2}}{\sigma _{n}^{2}}\right) \tag{3}\]  
where \(\sigma_{n}^{2}\) donates the average noise power. We consider the CE and AO schemes because they can provide optimal performance, and our FP-CB is built using them, while the random CB-PBF scheme is the conventional CB-PBF method. For localization accuracy, we consider that the positioning service can estimate RX location with an error following a normal distribution that has a 2.5-m standard deviation. This accuracy can be easily obtained using GPS or Wi-Fi services. Second, we investigate the impact of localization accuracy on our scheme, where varied accuracies are assumed. Then, we study the impact of time-variant channels on the schemes' performance. Furthermore, this work assumes RIS assists a multi-input, single-output system where the same channel parameters as [18] are considered. Furthermore, the target area is shown in Figure 3, where the fingerprint points are uniformly distributed. Moreover, when a new RX enters the network, its position can be anywhere in the area, following a uniform distribution. Around 100,000 Monte Carlo simulations are considered, and other simulation parameters are summarized in Table 2.  
Figure 4 shows the achievable rate obtained using the proposed FP-PBF scheme, which is comparable to other PBF schemes. The rates of CE-PBF and random configuration methods are the upper and lower benchmarks. Any increase in the random CB size or the number of candidate RPs for searching in our proposed FP-CB, \(C\), enhances the achievable rate because a wider coverage of space can be guaranteed. However, that will also increase the required overhead. The proposed scheme outperforms the random CB-based scheme even if the same number of RPs are used. For example, using 7 RPs, the FP-PBF method \begin{table}
\begin{tabular}{c c} \hline \hline
**Inputs**: FP codebook \(\mathbf{W}\) with \(P\) RPs, and the number of candidates RPs, \(C\). \\
1. & **for** each new RX **do** \\
2. & **Calculate** the RX estimated location, \((x^{\prime},y^{\prime},z^{\prime})\) \\
3. & **Calculate**, \(d_{P}\), for each FP point \(p\), using (2) \\
4. & **Determine the \(C\) nearest points.** \\
5. & **Configure** RIS elements using the candidate \(C\) RPs. \\
6. & **Calculate and memorize** OF for each \(c\)th RP. \\
7. & **Find \(\mathbf{C^{*}}=\mathbf{argmax_{Y}}\)** \\
8. & \(\mathbf{\theta}=\mathbf{W}[C^{*}]\) \\
9. & **end for** \\
**Output**: The passive BF vector \(\mathbf{\theta}\). \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Simulation Parameters. obtains an achievable rate of 11 bps/Hz, which is a 23% increase over the random CB-based scheme's performance. Moreover, at a certain number of RPs, the achievable rate of the proposed scheme nearly saturates using 10 RPs; it achieves 11.07 bps/Hz and obtains near performance to the CE-based method. Moreover, lower complexity and a shorter time can be guaranteed using the proposed scheme. For instance, using 10 candidate RPs for searching achieves a rate of 11.07 bps/Hz, which is lower than that obtained using the complicated CE-based scheme with only 0.4 bps/Hz; however, when considering the required system overhead, results will change as we will further discuss. First, we study the effect of the localization accuracy in Figure 5, where the proposed FP-PBF scheme achievable rate is presented versus the standard deviation of localization error, \(\sigma^{2}\), when the number of candidate RPs is varied from one to 15 RPs. Here, providing localization with low accuracy decreases the achievable rate even though the same number of RPs are used.