# An Overview of Lidar Imaging Systems for Autonomous Vehicles
## 2 Basics of Lidar Imaging
### Measurement Principles
#### 2.1.3 Continuous Wave Frequency Modulated (FMCW) Approach  
In the case of the FMCW approach, the emitted instantaneous optical frequency is periodically shifted, usually by varying the power applied to the source [34]. The reflected signal is mixed with the emitted source, creating a beat frequency that is a measure of the probed distance [35]. The source is normally a diode laser to enable coherent detection. The signal is then sent to the target, and the reflected signal that arrives at the receiver, after a traveled time \(t_{OF}\), is mixed with a reference signal built from the emitter output. For a static target, the delay between the collected light and the reference causes a constant frequency difference \(f_{r}\), or beat frequency, from the mixed beams. Letting the instantaneous frequency vary under a linear law, \(f_{r}\) is directly proportional to \(t_{OF}\) and hence proportional to the target range too [26; 36; 37], following:  
\[f_{r}=slope\cdot\Delta\tau=\frac{B}{T}t_{OF}=\frac{B}{T}\frac{2R}{c}\Rightarrow R =f_{r}\frac{cT}{2B}, \tag{3}\]  
where \(B\) is the bandwidth of the frequency sweep, \(T\) denotes the period of the ramp, and \(\Delta\tau\) equals the total travelled time \(t_{OF}\). Figure 3 depicts all these parameters. In practice, the frequency difference between the outgoing and incoming components is translated into a periodic phase difference between them, which causes an alternating constructive and destructive interference pattern at the frequency \(f_{r}\), i.e., a beat signal at frequency \(f_{r}\). By using FFT to transform the beat signal in time domain to frequency domain, the peak of beat frequency is easily translated into distance. Usually, a triangular frequency modulation is used (Figure 4) rather than a ramp. The modulation frequency in this case is denoted as \(f_{m}\). Hence, the rate of frequency change can be expressed as \(2f_{m}B\)[38], and the resulting beat frequency is given by:  
\[f_{r}=\frac{4 Rf_{m}B}{c} \tag{4}\]  
Figure 3: Frequency modulation and detection in the frequency-modulated continuous-wave (FMCW) method: main parameters involved. This type of detection has the very relevant advantage of adding on the capability of measuring not only range but also, using the same signal, the velocity of the target and its sign. If the target moves, the beat frequency obtained will not only be related to \(R\), but also to the velocity \(v_{r}\) of the target relative to the sensor. The velocity contribution is taken into account by the Doppler frequency \(f_{d}\), which will affect the sweep of the beat frequency up or down (Figure 5). Thus, beat frequency components are superimposed to \(f_{r}\), following [39]:  
\[f^{+}=f_{r}+f_{d}\text{ and }f^{-}=f_{r}-f_{d}. \tag{5}\]  
In this case, range can be obtained from:  
\[R=\frac{cT}{4B}\left(f^{+}+f^{-}\right), \tag{6}\]  
while relative velocity and its direction can also be calculated using the Doppler effect:  
\[v_{r}=\frac{\lambda}{2}f_{d}=\frac{\lambda}{4}\left(f^{+}-f^{-}\right), \tag{7}\]  
showing the ability of FMCW to simultaneously measure range and relative velocity using the properties of the Fourier spectra [38, 40]. FMCW takes advantage of the large frequency bandwidth available in the optical domain and exploits it to improve the performance of the range sensor. The resolution of the technique is now related to the total bandwidth of the signal. Since the ramp period can be chosen arbitrarily,  
Figure 4: Triangular frequency modulation with time and linked amplitude signal change in the time domain. Figure 5: Triangular modulation frequency signal and beat frequency for a moving target. the FMCW method can determine \(t_{oF}\) values in the picosecond range, equivalent to millimeter or even submillimeter distances, by performing frequency measurements in the kilohertz regime, which is perfectly feasible. Resolutions of 150 \(\upmu\)m have been reported, which is an improvement of two orders of magnitude relative to the other approaches.