# Chipless Radio Frequency Identification (RFID) Sensor for Angular Rotation Monitoring
## 3 Performance Assessment and Design Improvement  
Let us consider the two-dipole structure reported in the inset Figure 2b. The cross polar response as a function of frequency is reported in Figure 3a for a 90-degree span of \(\phi\). As is apparent, the peak values of the cross polar are localized at two frequencies \(f_{1}\) and \(f_{2}\) (i.e., 5.25 GHz and 7.35 GHz) with very slight shifts around them. It is important to mention that even when the maximum of the cross polar is not exactly at \(f_{1}\) or \(f_{2}\), the value at those frequencies is selected. This is because there is not much difference between the real maximum and the value assumed at \(f_{1}\) and \(f_{2}\), and also due to the simplification of both the algorithm and the hardware used in the decoding. Therefore, the cross polar value at \(f_{1}\) and \(f_{2}\) is the codification of the angular position of the tag (Figure 3b). Figure 2: Cross polar response of the tested structures (shown in the insets): (**a**) Single-dipole; (**b**) two dipoles; (**c**) maximum level of the cross polar for the two-dipole case. The employed dielectric has a permittivity \(\varepsilon_{\tau}=\)\(2.8-\mathrm{j}0.003\) and thickness equal to 1.5 mm, whereas the periodicity values are \(T_{x}=1.7\) cm and \(T_{y}=1.5\) cm. In order to understand the reliability of a decoding process based on the values assumed by the cross polar in correspondence of a certain number of frequencies \(f_{\textit{res}}\), it is necessary to define a distance between the coding of two different angular positions, namely \(\phi_{i}\) and \(\phi_{j}\). This distance function _Dist_ is defined as:  
\[Dist\big{(}\phi_{i},\phi_{j}\big{)}=\sum_{m=1}^{f_{\textit{res}}}\sqrt{\big{(} Xpol_{AB}(f_{n},\phi_{i})-Xpol_{AB}\big{(}f_{n},\phi_{j}\big{)}\big{)}^{2}}. \tag{3}\]  
where \(X_{pol}(f_{n},\phi_{i})\) is the codification in terms of cross polar magnitude of the reflected field. Then it is possible to evaluate the distance of a codification with all the others in order to check if any ambiguity is present. Let us fix two thresholds, (_thres\({}_{1}\)_ and _thres\({}_{2}\)_), and consider two singular angular positions, namely \(\phi_{A}\) and \(\phi_{B}\). If the value of _Dist\((\phi_{A},\phi_{B})\)_ is greater than _thres\({}_{1}\)_, they are unambiguously encoded. If _Dist\((\phi_{A},\phi_{B})\)_ is between _thres\({}_{1}\)_ and _thres\({}_{2}\)_, they are critically encoded. If _Dist\((\phi_{A},\phi_{B})\)_ is lower than _thres\({}_{2}\)_, they are not reliable. This information is clearly represented in the color map of Figure 4a for the case of the two-dipole structure. Each pixel corresponds to the distance between the two referred angular positions. It is well visible that some angle codifications are not univocally defined (such as 60 degrees and 20 degrees), because the distance between their encoding is below the imposed threshold. In order to improve the performance of the encoding scheme, it is necessary to increase the number of involved frequencies. The introduction of a third dipole is obviously beneficial, as illustrated in Figure 4b. The uncertainty has been considerably reduced and only angular values within the interval (\(55^{\circ}\), \(80^{\circ}\)) suffer from a certain degree of ambiguity. To remove the remaining inefficiencies a fourth  
Figure 4: (a) Chart mapping the distance for the two-dipole structure, and (b) three-dipole structure. The three quantized levels of \(x=Dist(\phi_{A},\phi_{B})\) refer to the value assumed by the cross polar at\(f_{1}\) and \(f_{2}\) and by considering _thres\({}_{1}=6\)_ dB and _thres\({}_{2}=3\)_ dB. The unit cell is shown in the respective insets. Figure 3: (a) Cross polar of the two-dipole structure at 5-degree steps within a 90-degree span of \(\phi\), and (b) the correspondent values for each angular position at \(f_{1}\) and \(f_{2}\). dipole is added.