# Stationary Target Identification in a Traffic Monitoring Radar System
## 2 Traffic Monitoring Radar System
### Radar Signal Processing for Target Information Extraction  
In this section, we describe the signal processing technique used in our traffic monitoring radar system. The schematic block diagram of our radar system is depicted in Figure 4.  
In this block diagram, the radar system consists of the waveform generator, voltage-controlled oscillator (VCO), transmit antenna element (Tx), four receiving antenna elements (Rx), frequency mixer, low-pass filter (LPF), analog-to-digital converter (ADC), and digital signal processor (DSP). First, in the FMCW radar system, a waveform whose frequency changes linearly with time is made from the waveform generator. The detailed frequency change pattern of the waveform over time is shown in Figure 5.  
Figure 4: Schematic block diagram for radar signal processing. Figure 3: Antenna system of our traffic monitoring radar: one transmit antenna element and four receiving antenna elements. As shown in the figure, the frequency-modulated waveform can be divided into four areas (i.e., \(A_{1}\), \(A_{2}\), \(A_{3}\), and \(A_{4}\)) according to the change of time. In \(A_{1}\), the transmitted waveform can be expressed as:  
\[s_{t}^{(1)}(t)=\alpha_{t}\cos\left(2\pi\left(f_{c}-\frac{\Delta B}{2}\right)t+ \pi\frac{\Delta B}{\Delta T}t^{2}\right)\ \left(0\leq t\leq\Delta T\right), \tag{1}\]  
where \(\alpha_{t}\), \(\Delta B\), and \(\Delta T\) are the amplitude, bandwidth, and sweep time of the transmitted waveform. When \(s_{t}^{(1)}(t)\) is transmitted and reflected from \(L\) targets, the received signal \(s_{r}^{(1)}(t)\) can be expressed as:  
\[s_{r}^{(1)}(t)=\sum_{l=1}^{L}\left\{\alpha_{t_{l}}\cos\left(2\pi\left(f_{c}+f_{ d_{l}}-\frac{\Delta B}{2}\right)\left(t-t_{d_{l}}\right)+\pi\frac{\Delta B}{ \Delta T}\left(t-t_{d_{l}}\right)^{2}\right)\right\}, \tag{2}\]  
where \(\alpha_{t_{l}}\left(l=1,\,2,\,\ldots,\,L\right)\) is the amplitude of the received signal reflected from the \(l_{th}\) target. In addition, \(f_{d_{l}}\) is the Doppler shift caused by the relative velocity between the radar and the \(l_{th}\) target and \(t_{d_{l}}\) is the time delay caused by the relative distance between the radar and the \(l_{th}\) target. Then, \(s_{t}^{(1)}(t)\) and \(s_{r}^{(1)}(t)\) are multiplied by the frequency mixer and passed through the LPF, as shown in Figure 4. The final output of the filter with high frequency components disappearing can be expressed as:  
\[s_{L}^{(1)}(t)=\frac{1}{2}\alpha_{t}\sum_{l=1}^{L}\alpha_{t_{l}}\cos\left(2\pi \left(\left(\frac{\Delta B}{\Delta T}t_{d_{l}}-f_{d_{l}}\right)t+\left(f_{c}+ f_{d_{l}}-\frac{\Delta B}{2}\right)t_{d_{l}}-\frac{\Delta B}{2\Delta T}t_{d_{l}} {}^{2}\right)\right). \tag{3}\]  
As given by Equation (3), the filter output has the form of a sum of sinusoids with different frequencies. These frequencies contain range and velocity information for each target. Therefore, if the Fourier transform is applied to the time-domain filter output in Equation (3), it is possible to estimate the frequencies containing the distance and velocity information of the target. The frequency estimated from the Fourier transform is given as:  
\[\hat{f}_{l}^{(1)} = \frac{\Delta B}{\Delta T}t_{d_{l}}-f_{d_{l}} \tag{4}\] \[= \frac{\Delta B}{\Delta T}\frac{2R_{l}}{c}-\frac{2v_{l}}{c}f_{c},\]  
where \(R_{l}\) and \(v_{l}\) are the relative distance and relative velocity between the radar and the \(l_{th}\) target. This frequency is also called the beat frequency because it indicates the frequency difference between the transmitted and received waveforms. Figure 5.0 Frequency change of the waveform over time. However, from Equation (4), \(R_{l}\) and \(v_{l}\) of the \(l_{th}\) target are coupled to each other, so they cannot be separated. Therefore, the additional waveform in \(A_{2}\) of Figure 5 is needed.