How does a millimeter wave FMCW radar calculate range and velocity of a target?

FMCW Radar Signal Processing

FMCW is the dominant radar waveform for automotive, industrial, and consumer mmWave radar because it provides simultaneous range and velocity measurement with simple, continuous-transmission hardware.

Range-Doppler Processing

(1) Range FFT (fast-time FFT): for each chirp: the beat signal (mixer output) is sampled by the ADC at N_samples points during the chirp duration T_chirp. An FFT of these N_samples produces a range profile: each FFT bin corresponds to a range. Range per bin: delta_R = c / (2B) (the range resolution, determined by the chirp bandwidth). Number of range bins: N_range = N_samples / 2. Maximum unambiguous range: R_max = N_range × delta_R = f_s × c × T_chirp / (4B), where f_s is the ADC sample rate. Example: B = 4 GHz, N_samples = 512, f_s = 20 Msps: delta_R = 3e8 / (2×4e9) = 3.75 cm. R_max = 256 × 3.75 cm = 9.6 m. For longer range: use longer chirps or lower bandwidth. (2) Doppler FFT (slow-time FFT): a frame consists of N_chirps consecutive chirps. For each range bin: extract the complex value from the range FFT across all chirps. An FFT of these N_chirps values produces the Doppler spectrum for that range bin. Velocity per bin: delta_v = lambda / (2 × N_chirps × T_chirp). Maximum unambiguous velocity: v_max = lambda / (4 × T_chirp). Example: lambda = 3.9 mm (77 GHz), N_chirps = 128, T_chirp = 40 us: delta_v = 3.9e-3 / (2×128×40e-6) = 0.38 m/s. v_max = 3.9e-3 / (4×40e-6) = 24.4 m/s (87.8 km/h). For higher v_max: use shorter chirps (but this reduces the maximum range). (3) The output is a 2D range-Doppler map: an N_range × N_chirps matrix where each cell contains the signal magnitude at a specific range and velocity. Targets appear as peaks in this map. Detection is performed by CFAR (Constant False Alarm Rate) threshold applied to the range-Doppler map.

Angular Measurement

Range and velocity are measured along the radar line-of-sight. Angular position requires multiple receive antennas: (1) Receive phased array: multiple RX antennas spaced at lambda/2 (1.95 mm at 77 GHz). The phase difference between RX channels for a target at angle theta: delta_phi = 2×pi × d × sin(theta) / lambda. An FFT across RX channels (angle FFT) produces the angular spectrum. Angular resolution: delta_theta ≈ lambda / (N_RX × d) radians. For N_RX = 4 with d = lambda/2: delta_theta ≈ 1/(4×0.5) = 0.5 radians = 28.6°. For N_RX = 12: delta_theta ≈ 9.5° (much better). (2) MIMO radar: use M TX antennas and N RX antennas to create a virtual array of M×N elements. Each TX transmits a uniquely coded waveform (time-division, frequency-division, or code-division multiplexing). The receiver separates the TX signals and processes each TX-RX pair as a virtual element. For 3 TX × 4 RX: 12 virtual elements. Angular resolution improves by a factor of 3 (the virtual array is 3× larger than the physical RX array). (3) The complete processing chain: for each frame of N_chirps: Range FFT → range profiles for each RX channel. Doppler FFT → range-Doppler maps for each virtual element. Angle FFT → range-Doppler-angle 3D data cube. CFAR detection → list of detected targets with range, velocity, and angle.

Hardware Implementation

(1) Radar SoC: modern automotive radar uses single-chip solutions: TI AWR1843 (3 TX, 4 RX, 77 GHz, 4 GHz BW). Infineon AURIX radar platform. NXP TEF82xx. The SoC integrates: chirp synthesizer (PLL), TX PA, RX LNA/mixer, ADC, and radar DSP. The entire radar sensor fits on a small PCB (30 × 30 mm) with an on-board antenna array. (2) ADC requirements for FMCW: the beat frequency is typically 1-50 MHz (much lower than the RF frequency). ADC sample rate: 10-50 Msps. Resolution: 10-12 bits. This is a very modest ADC requirement compared to communication receivers. (3) Antenna: the antenna is typically integrated on the radar PCB as a patch array or slot array on a high-frequency laminate (Rogers). For 77 GHz: element spacing = 1.95 mm. A 12 × 8 array is only 23 × 16 mm. Gain: 18-25 dBi.

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