# Passive RFID-Based Inventory of Traffic Signs on Roads and Urban Environments
## 3 RFID Link Budget Calculation
### Downlink (Reader to Tag) Communication
#### 3.1.2 Propagation under Real Conditions  
It is reasonable to suppose that the field radiated from the vehicle mainly arrives at the RFID tag via two distinct paths: one is the direct path from the vehicle to the tag, with length r\({}_{1}\), and the other, of length \(r_{2}\), is the path reflected from the surface of the ground (see Figure 4). Invoking (5), we model the direct field at the RFID tag as  
\[e_{1}(t)=\frac{\sqrt{60p_{T}g_{T}}}{r_{1}(t_{1})}\cos(\omega_{c}t-k_{c}r_{1}( t_{1})), \tag{6}\]  
where \(t_{1}\) is the time instant at which the wave received by the tag was emitted by the reader. Considering the vehicle as a point traveling at constant velocity \(v\) along a straight line, and after some tedious algebra, (4) yields  
\[t_{1}=t-\sqrt{\left(vt\right)^{2}+\left(1-\left(v/c\right)^{2}\right)\left(d^ {2}+\left(h_{1}-h_{2}\right)^{2}\right)}/c, \tag{7}\]  
Figure 4: Propagation paths. where \(d\) is the distance from the traffic sign to the path followed by the car. In the case of the ground-reflected path, a change in the amplitude and phase of the radiowave occurs on the reflection, which is given by the Fresnel coefficient \(\Gamma(\theta)=\rho(\theta)e^{\frac{1}{\Psi}}\) (\(\theta\)), where \(\rho\) stands for the amplitude attenuation and \(\Psi\) is the phase shift. Furthermore, observe that the Fresnel coefficient not only depends on the grazing angle \(\theta\) but also on the field polarization. The expression of the field produced by the reflected wave is then  
\[e_{2}(t)=\frac{\sqrt{60p_{T}g_{T}}}{r_{2}(t_{2})}\rho\cos(\omega_{c}t-k_{c}r_{2 }(t_{2})+\Psi), \tag{8}\]  
where \(t_{2}\) is the time instant at which the ground-reflected wave was emitted by the vehicle and can be calculated similarly to that in (\(\mathcal{T}\)). In complex environments, such as urban areas, other reflected waves \(e_{3}(t)\),..., \(e_{N}(t)\) may also reach the RFID tag. Let the \(n\)th wave be of amplitude \(\alpha_{n}(t)\) and phase \(\beta_{n}(t)\), i.e.,  
\[e_{n}(t)=\alpha_{n}(t)\cos(\omega_{c}t+\beta_{n}(t)),\]  
for \(n=3\),..., \(N\). All the waves add at the receiver, leading to either constructive or destructive interference, and the resultant field is given by  
\[e(t)=\sum_{n=1}^{N}e_{n}(t). \tag{9}\]  
Some algebra shows that  
\[e(t)=r(t)\cos(\omega_{c}t+\theta(t)), \tag{10}\]  
where  
\[r(t)=\sqrt{i(t)^{2}+q(t)^{2}}\] \[\theta(t)=\arctan(q(t)/i(t))\]  
with  
\[i=a\cdot\cos(\arctan(a_{2}/a_{1}))+\sum_{n=3}^{N}\alpha_{n}\cos(\beta_{n}) \tag{11a}\] \[t_{1}=t-\sqrt{\left(vt\right)^{2}+\left(1-\left(v/c\right)^{2}\right)\left(d^ {2}+\left(h_{1}-h_{2}\right)^{2}\right)}/c, \tag{7}\]0  
and  
\[t_{1}=t-\sqrt{\left(vt\right)^{2}+\left(1-\left(v/c\right)^{2}\right)\left(d^ {2}+\left(h_{1}-h_{2}\right)^{2}\right)}/c, \tag{7}\]1 \[t_{1}=t-\sqrt{\left(vt\right)^{2}+\left(1-\left(v/c\right)^{2}\right)\left(d^ {2}+\left(h_{1}-h_{2}\right)^{2}\right)}/c, \tag{7}\]2 (12b) \[t_{1}=t-\sqrt{\left(vt\right)^{2}+\left(1-\left(v/c\right)^{2}\right)\left(d^ {2}+\left(h_{1}-h_{2}\right)^{2}\right)}/c, \tag{7}\]3  
where, for simplicity, we have dropped the dependence with time. The rms value of the field strength is given by \(e_{rms}=r_{rms}/\surd 2\), where \(r_{rms}\) is the rms value of \(r(t)\). Then the mean received power can be calculated by (1), i.e.,  
\[t_{1}=t-\sqrt{\left(vt\right)^{2}+\left(1-\left(v/c\right)^{2}\right)\left(d^ {2}+\left(h_{1}-h_{2}\right)^{2}\right)}/c, \tag{7}\]4  
To compute \(p_{R}\), the coefficients \(\alpha\) and \(\beta\) in (11) may be also determined using ray-tracing models; however, sufficient data is not expected to be available in practice so we must characterize the data statistically. If N\(\rightarrow\)so, the central limit theorem enables us to assume that the sums \(\sum\limits_{n=1}^{N}\alpha_{n}\cos(\beta_{n})\) and \(\sum\limits_{n=1}^{N}\alpha_{n}\sin(\beta_{n})\) are zero-mean Gaussian variables with common variance \(\sigma^{2}\).