## 2 Materials and Methods
### PIR Signal Modeling for Time Period Estimation  
To apply the above framework to vehicle speed estimation using PIR sensors, we model the signal behavior of the sensors as a vehicle passes through their fields of view. In this study, a vehicle speed measurement is designed using two pyroelectric infrared (PIR) sensor modules placed in a known geometric configuration (Figure 2a). The system is designed to minimize centering errors by ensuring the parallel alignment between the optical axes of the sensor modules and maintaining an accurate distance \(d\) between the FOV center axes of the sensors [27; 28]. As a vehicle moves past these sensors, it causes changes in the sensors' output signals, which are then analyzed to estimate time delay between the two signals and consequently to determine the speed of the moving vehicle (Figure 2b). The fundamental principle behind this system is that the time period between the sensor outputs can be used to compute speed using the equation:  
\[v=d/\tau_{0} \tag{6}\]  
where \(\tau_{0}\) is the time period between their output signals. To analyze the performance of different TPE methods, we first define the mathematical representation of the signals. The two output signals of the PIR sensors over a monitoring window [0, \(T\)] can be modeled as \(x_{1}(t)\) and \(x_{2}(t)\) for \(0\leq t\leq T\), where \(x_{2}(t)\) is essentially a time-shifted version of \(x_{1}(t)\): \(x_{2}(t)~{}\sim~{}x_{1}(t-\tau_{0})\). Table 1 presents notations of several quantities related to PIR sensor signals that are used in this analysis. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Quantity** & **Notation in Time Domain** & **Notation in Frequency Domain** \\ \hline Difference between & & \\ infrared fluxes from the & \(\phi(t)\) & \(\Phi(f)\) \\ interested object incoming & \(\phi(t)\) & \(H_{1}(f)\) \\ on 2 elements of PIR sensor & & \\ Impulse response of PIR 1 & \(h_{1}(t)\) & \(H_{1}(f)\) \\ Impulse response of PIR 2 & \(h_{2}~{}(t)\) & \(H_{2}(f)\) \\ Output signal of PIR 1 & \(x_{1}(t)\) & \(X_{1}(f)\) \\ Output signal of PIR 2 & \(x_{2}(t)\) & \(X_{2}(f)\) \\ Noise signal at PIR 1 & \(n_{1}(t)\) & \(N_{1}(f)\) \\ Noise signal at PIR 2 & \(n_{2}(t)\) & \(N_{2}(f)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Time and frequency domain notations of quantities related to PIR sensor signals. Figure 1: Graphical illustration of a practical objective function. In PIR sensor-based speed measurement, the differential infrared flux detected by the two sensing elements of each PIR sensor is denoted as a function over time \(\phi(t)\)[29; 30]. This flux variation is caused by the movement of a heat emitting object (vehicle) across the sensor's FOV. If we define the differential flux from the object for the first sensor as \(\phi(t)\), the second sensor \(\phi(t-\tau_{0})\). As a moving heat source (e.g., a vehicle) passes through the sensor's field of view (FOV), the incident infrared radiation varies over time, producing a structured alternating signal due to the differential sensing mechanism. Given that vehicles typically follow a consistent motion trajectory within the monitored region, the resulting \(\phi(t)\) can be approximated as a sinusoidal signal within the observation window. This sinusoidal assumption is justified by the periodic nature of the sensor's response to smoothly moving sources and enables a more tractable mathematical representation for time period estimation. Treating \(\phi(t)\) as a sinusoid simplifies its frequency domain analysis, allows the use of Fourier-based techniques, and enhances computational efficiency in estimating time period via cross-correlation methods. \[\phi(t)=\phi_{0}\cdot\sin(2\pi f_{0}t)\Phi(f)=\frac{\phi_{0}}{2j}\delta(f-f_{0 })-\frac{\phi_{0}}{2j}\delta(f+f_{0}) \tag{7}\]  
Figure 2: Views of the speed measurement systems using two PIR sensors (**a**); and illustration of its working principle (**b**).