## 4 Evaluation
### Ranging Accuracy  
#### 4.3.1 Theoretical Simulation Model  
A simplified theoretical model for the ranging is derived based on the literature for UWB propagation channels [12; 30; 31]. This model can be used to understand the different effects that occur during ranging, and to explain and predict the ranging results. The theoretical model is a combination of the path loss model and the ground reflection model. The path loss model accounts for the losses during propagation in the air for the signal, and the ground model includes the interference between the direct path and the most important reflection with the ground. Based on this analysis a simulation curve is determined to predict and interpret the experimental results.  
**Path loss model:** The magnitude of the degeneration of the signal strength during propagation can be dedicated to different influences: the transmit power of the sender (\(P_{t}\)), the gain of the antennas at sender (\(G_{t}\)) and receiver (\(G_{r}\)), the losses during propagation at the PCB (\(L_{t}\) and \(L_{r}\)), and the losses during propagation of the signal through the air (\(PL\)) depending on the length of the transmission path between sender and receiver. The link budget of the signal during propagation for the LOS path can be evaluated by using Friis path loss formula:  
\[P_{r}=P_{t}+G_{r}+G_{t}-L_{r}-L_{t}-PL. \tag{1}\]  
The losses in the PCB and the gain of the antennas will be the same for different distances/environments and are hardware specific. During the evaluation of the experiments an approximation of these factors can be made. The link margin is expressed as the difference between the receiver's sensitivity and the expected minimum receive power. The link margin should  
Figure 6: Test setup with attenuator on the path between sender and receiver.  
Figure 7: Packet/Range received ratio for different path attenuations and receive powers.  
be high enough for correct ranging. Equation (1) points out a dependence between receive power and transmit power. The link margin can be increased by increasing the transmit power. The transceiver has a maximal value of 33.5 dB in steps of 0.5 dB for the transmission power.  
The path loss component from Equation (1) is dependent on the distance between sender and receiver, the frequency, and the propagation speed. This path loss component is based on two components, a fixed value measured at the reference distance \(d_{0}=1\) m and a logarithmic factor on the relative increase in distance between sender and receiver.  
\[PL(d)=PL_{0}(d_{O})+10\ \gamma\ log_{10}(\frac{d}{d_{0}})+X_{\sigma};d>d_{0}, \tag{2}\]  
where the \(PL_{0}\) factor is the path loss at a distance of 1 m, and \(\gamma\) is the path loss exponent which has value 2 for the free space situation and is chosen 1.58 in the outdoor measurements as suggested value for farm (outdoor) environments in Reference [32]. The \(X_{\sigma}\) factor is zero-mean Gaussian random variable with standard deviation \(\sigma\) and this factor can be referred to as shadowing and captures the path loss deviation from its median value.  
**Ground reflection model:** Interference will occur at the receiving antenna between the direct, LOS, path, and other (reflected) paths with different lengths from sender to receiver. In the ground reflection model, the interference between the most important reflection path and the direct path is taken into account (see Figure 8). The difference in path length will result in distances where both signals will interfere constructively, increasing the receive power, or destructively, where they cancel each other out. The separations between sender and receiver where the signals interfere destructively are dependent on the center frequency of the UWB ranging and the height of the antenna nodes.