1. Definition and Principles:
Frequency Modulated Continuous Wave (FMCW) is an advanced sensing scheme based on coherent detection technology. Unlike traditional ToF (Time of Flight), which measures distance by timing pulsed signals, FMCW emits a continuous laser beam whose frequency is modulated over time in a controlled manner. Utilizing the principle of wave interference, it measures the frequency difference (i.e., beat frequency) between the echo and the currently transmitted signal, thereby enabling simultaneous detection of target distance and velocity.

Figure 1: Schematic diagram of FMCW principle

As shown in the figure above, the working principle of FMCW involves a laser outputting a linearly frequency-modulated optical carrier at approximately 200 THz, which is split by a coupler into transmitted light (Tx) and local oscillator light (LO). The Tx light illuminates the target and reflects back to form the echo light (Rx). The Rx and LO are coherently mixed in a coupler (optical mixer). Due to the frequency difference, the intensity of the combined light after interference fluctuates periodically at a beat frequency of < 500 MHz; subsequently, a photodetector directly converts this optical intensity variation into a beat electrical signal of the same frequency.

2. Calculation of Distance and Velocity

Figure 2: Calculation of FMCW distance and velocity

(Image source: https://www.indie.inc/wp-content/uploads/2025/01/IND-0124-Microtech-Ventures-Whitepaper-LiDAR.pdf)

Distance Calculation ($r$):
The total round-trip time of light between the radar and the target is $\Delta t$, and the one-way time is $\Delta t/2$; combined with the speed of light $c$ (approximately $3 \times 10^8 \, \text{m/s}$), the formula for calculating the target distance is: $$r = \frac{\Delta t}{2} \times c$$

Velocity Calculation ($v$):
The relative motion between the target and the radar generates a Doppler frequency shift $f_d$; combined with the wavelength of light $\lambda$, the formula for calculating the relative velocity of the target is: $$v = \frac{f_d \times \lambda}{2}$$

3. Modulation Paradigms:
The core of FMCW lies in the active modulation of the laser frequency. Depending on the requirements of the application scenario, the transmitted frequency can be shifted up and down according to specific functional patterns, forming different waveforms:

• Sine Wave: Frequency transitions smoothly. Commonly used in short-range precision sensors, it can effectively reduce high-frequency harmonic interference.
• Square Wave (FSK): Rapidly switches between discrete frequencies. Mainly used to simplify digital processing workflows or for identification in specific scenarios.
• Triangle Wave: Frequency rises and falls symmetrically. By comparing the frequency shifts in the rising and falling segments, distance and velocity can be decoupled, making it the mainstream choice for automotive 4D LiDAR.
• Sawtooth Wave: Frequency sweeps linearly in one direction. It has the simplest structure and is suitable for basic ranging, but suffers from velocity ambiguity in high-dynamic scenarios.

(Image source: https://en.wikipedia.org/wiki/Square_wave_(waveform)#/media/File:Waveforms.svg)