## 3 Numerical Results and Experimental Validation
### Numerical Results  
This section presents numerical results for time period estimation (TPE) in vehicle speed measurement using pyroelectric infrared (PIR) sensors. This study evaluates two methods: cross-correlation function (CCF) and cross-correlation function with Hilbert transform (CCFHT). The focus is on how sensor response mismatch, noise levels, and signal bandwidth affect the estimation accuracy. The parameters used in this work are presented in Table 2.  
It should be noticed that, if the object is moving with the speed \(v\) at the distance \(D\) from the sensors, then the duration of the signal caused by the moving object to the sensor and the signal frequency are defined as:  
\[T_{0}=\frac{D*\left(\frac{v}{F}\right)+L}{v} \tag{57}\]  
\[f_{0}=\frac{1}{T_{0}}=\frac{v}{D*\left(\frac{v}{F}\right)+L} \tag{58}\]  
On the other hand, once the bias and the variance of time period estimation are found, from (1), the bias and uncertainty of speed measurement due to the bias and variance of the TPE are also defined:  
\[bias\ (v)=-\frac{v^{2}}{d}.bias(\tau_{0}) \tag{59}\]  
\[u(v)=\frac{v^{2}}{d}.u(\tau_{0})=\frac{v^{2}}{d}.\sqrt{var(\tau_{0})} \tag{60}\]  
The numerical results are in the following figures. Figures 4-6, which illustrate the effects of thermal time constant mismatch and object speed on TDE performance. Particularly, Figure 4 presents the influence of thermal time constant mismatch on the bias and variance of time period estimation. When the second sensor has a longer or shorter response time compared with the first, asymmetric signal shapes are produced, leading to a phase shift between the signals. Figure 5 shows that, while both methods are influenced, CCFHT maintains more stable variance across varying levels of sensor mismatch. In addition, Figure 6 also presents the impact of bandwidth B on TDE variance and speed measurement uncertainty using CCF.  
\begin{table}
\begin{tabular}{c c} \hline
**Quantity, [Unit]** & **Value** \\ \hline Focal length of both sensors \(F_{1}=F_{2}\), [mm] & 50.9 \\ Sensing element width \(w\), [mm] & 3 \\ Distance between two sensors’ axes \(d\), [m] & 1 \\ Distance to object’s (vehicle’s) movement \(D\), [m] & 10 \\ Length of the object \(L\), [m] & 4.5 \\ Electrical time constant \(\tau_{e1}\) (first sensor), [s] & 0.08 \\ Thermal time constant \(\tau_{tH1}\)(first sensor), [s] & 0.5 \\ Electrical time constant \(\tau_{e2}\) (second sensor), [s] & 0.08 \\ Thermal time constant \(\tau_{tH2}\) (second sensor), [s] & 0.4 \(\div\) 0.6 \\ Number of discrete frequency bins \(N_{d}\) & 1024 \\ Noise bandwidth \(B\), [Hz] & 10 \(\div\) 100 \\ Signal-to-noise ratios \(SNR_{1}=SNR_{2}\) & 10 \\ \hline \end{tabular}
\end{table}
Table 2: Key parameters and their values for simulation.  
Figure 4: Impact of various time constant (\(\tau_{tH2}\)) and object speed on: (**a**) argument \(\Theta_{0}\); (**b**) time period estimation bias; (**c**) speed measurement bias; (**d**) relative speed measurement bias.  
Figure 5: Impact of various time constant (\(\tau_{i\hbar 2}\)) and object speed on estimation variances, in case \(B=100\) Hz: **(a)** variance of time period estimation using CCF; **(b)** variance of time period estimation using CCFHT; **(c)** uncertainty of speed measurement using CCF; **(d)** uncertainty of speed measurement using CCFHT; **(e)** relative difference between uncertainty of speed measurement using CCF and CCFHT.  
Figure 6: Impact of bandwidth on TDE **(a)** variance and speed measurement **(b)** uncertainty using CCF.