## 2 FMCW Radar Range Resolution System  
The primary waveform types of FMCW radar are triangular and sawtooth waveforms. This section discusses the transmission and reception process, as well as range estimation, using the sawtooth waveform in FMCW radar.  
The system configuration of an FMCW radar is as follows. The waveform generated by the FMCW radar is modulated by a Voltage-Controlled Oscillator (VCO) after passing through a Digital-to-Analog Converter (DAC). The modulated signal is then amplified by a Power Amplifier (PA) before being transmitted. The transmitted signal is reflected by a target and received back by the radar, with a time delay corresponding to the round-trip time. The received signal is amplified by a Low Noise Amplifier (LNA) and then mixed with the transmitted signal. This mixed signal is amplified by an Intermediate Frequency (IF) amplifier, and the signal containing the beat frequency is extracted using a Low Pass Filter (LPF). The extracted signal is converted to a digital signal via an Analog-to-Digital Converter (ADC), and range information is extracted through a Fast Fourier Transform (FFT) [27; 28; 29].  
If a target is located at a distance \(R\) from the radar, and the transmitted signal with frequency \(f_{0}\) is reflected from the target and received after a delay time \(\tau\), the transmitted and received signals of the radar can be expressed as Equations (1) and (2):  
\[T(t)=A(t)Re[exp(j2\pi f_{0}t)] \tag{1}\]  
\[R(t)=A(t-\tau)Re[exp(j2\pi f_{0}(t-\tau))] \tag{2}\]  
The delay time \(\tau\) is given by Equation (3):  
\[\tau=\frac{2R}{C} \tag{3}\]  
\(R\) is the distance between the radar and the target, and \(C\) is the speed of light.  
The distance to the target can be determined from the beat frequency of the radar signal. The beat frequency can be derived from the slope of the transmitted and received sawtooth waveforms and is expressed as Equation (4):  
\[fb=\frac{BW\cdot\tau}{T_{s}} \tag{4}\]  
\(f_{b}\) is the beat frequency, \(T_{s}\) is the sweep time, and \(BW\) is the modulation bandwidth. Substituting Equation (3) into Equation (4), the distance \(R\) from the radar to the target can be calculated as Equation (5):  
\[R=\frac{f_{b}\cdot C\cdot T_{s}}{2 BW} \tag{5}\]  
Range resolution refers to the radar's ability to distinguish the distance between multiple objects. The frequency difference between the transmitted and received signals can be calculated using Equations (1) and (2), leading to Equation (6):  
\[\Delta f=\frac{2\pi\cdot BW}{T_{s}}\cdot\tau \tag{6}\]  
The frequency difference \(\Delta f\) is expressed as in Equation (6). Substituting Equation (3) into Equation (6), and considering \(2\pi\) and \(T_{s}\) as constants, the range resolution can be derived, as shown in Equation (7):  
\[\Delta R=\frac{C}{2 BW} \tag{7}\]  
Equation (7) demonstrates that range resolution is determined by the modulation bandwidth. One method to increase the modulation bandwidth is by increasing the number of FFT points, resulting in a finer division of the frequency domain and reducing the minimum resolvable frequency interval, which is referred to as frequency resolution. Frequency resolution is given by Equation (8):  
\[fr=\frac{F_{s}}{N_{fft}} \tag{8}\]  
\(F_{s}\) is the sampling rate, and \(N_{fft}\) is the number of FFT points.