## 4 Vehicle Detection Using Ultrasonic Sensors
### Analysis with a Single Vehicle
The travel time of an ultrasonic wave over this distance is given by Equation (3). This is the minimum detection interval, because the ultrasonic sensor cannot emit a new ultrasonic wave during this period:  
\[\text{Minimum detection interval}=\frac{2(nl_{l}+l_{g})}{v_{u}} \tag{3}\]  
Thus, the available detection interval \(t_{l}\) can be written as:  
\[\text{Available detection interval}\rightarrow\frac{2(nl_{l}+l_{g})}{v_{u}}<t_{i }<\frac{l_{v}+l_{d}}{v_{v}\left\lceil\frac{m}{2}\right\rceil} \tag{4}\]  
If the detection interval of an ultrasonic sensor is outside these bounds, the reliability of the collected data cannot be guaranteed. We can now determine an appropriate detection interval for an ultrasonic sensor in a road environment. If power saving is more important, a longer detection interval can be applied, whereas a shorter interval can be used when sufficient power resources are available. This shorter detection interval  
\(m=3,l_{v}=3.5\) m and \(\theta=90\). As shown in Figure 5, the maximum detection interval decreases linearly with shorter \(l_{u}\), whereas it decreases logarithmically with higher \(\mathbf{v}_{v}\). According to Equation (3), the distance traveled by an ultrasonic wave and the ultrasonic speed \(\
u_{u}\) influence the minimum detection interval. Figure 6 illustrates the variation in the minimum detection interval with temperature for different numbers of lanes. Parameter values of \(l_{l}=3.5\) m and \(l_{g}=0\) were used. As expected, the minimum detection interval decreases for shorter travel distances. Because the speed of an ultrasonic wave increases with temperature, the minimum detection interval also decreases with higher temperature. However, the variation with temperature is insignificant and is thus neglected in our later analysis. Figures 5 and 6 show that the maximum and minimum detection intervals are affected by specific parameters. Among these, \(l_{u}\) significantly influences both detection intervals. Therefore, it is necessary to determine the detection interval that is least affected by variations in this parameter. Figure 7 shows the maximum and minimum detection interval on a one-lane road with respect to the detection distance \(l_{u}\), maximum and minimum detection intervals vary significantly with the detection distance. Thus, for reliable vehicle detection, the available detection interval should be from \(20\) ms to \(111\) ms, the interval that is not affected by the variation in detection distance. Because Equation (3) uses the maximum value of \(l_{u}\) (_i.e._, \(2(nl_{l}+l_{g})\)), the minimum detection interval is not  
Figure 5: Variation in the maximum detection interval. Figure 6: Variation in the minimum detection interval. influenced by changes in \(l_{u}\). However, Equation (2) varies with \(l_{u}\), so Equation (4) should include the following additional restrictions:  
\[\text{Available detection interval}\rightarrow\frac{2\left(nl_{l}+l_{g}\right)}{v _{u}}<t_{i}<\frac{l_{v}+l_{d}}{v_{v}\left|\frac{m}{2}\right|} \tag{5}\]  
\(l_{d}=\text{width of minimum }l_{u}\);  
\(l_{v}=\text{minimum vehicle length}\);  
\(v_{v}=\text{maximum vehicle speed}\). First, for the maximum detection interval, the minimum value of \(l_{d}\) (_i.e._, the width given by the minimum \(l_{u}\)) should be used, because Figure 7 indicates that larger values lead to unreliable vehicle detection. The minimum of \(l_{d}\) varies according to the sensor position and road environment, but this information can be easily measured. Second, the vehicle length \(l_{v}\) should be minimized. Thus, we may use the length of a typical compact car. Finally, the vehicle speed \(v_{v}\) should be the maximum speed on the target road. The speed limit, or a slightly larger value, can be used for \(v_{v}\), as people often drive above the speed limit [16]. The main reason for these restrictions on \(l_{v}\) and \(v_{v}\) is that they affect the maximum detection interval. As we have mentioned, the sampling rate of an ultrasonic sensor is much lower than that of other sensors.