How does FMCW measure range?

FMCW radar transmits a chirp: a signal whose frequency increases linearly from f_start to f_start + BW over time T (the chirp duration). The return signal from a target at range R arrives with a time delay tau = 2R/c. At the receiver, the return signal is mixed with the current transmitted signal (dechirping). Because the TX frequency has changed during the round-trip delay, the mixer output is a constant-frequency tone called the beat frequency: f_b = (BW/T) times tau = 2R times BW/(cT). Each target at a different range produces a different beat frequency. An FFT of the beat signal produces a range profile, with each peak corresponding to a target. Range resolution is determined by the sweep bandwidth: delta_R = c/(2 times BW). For 77 GHz automotive radar with 4 GHz bandwidth: delta_R = 3 times 10 to the 8 divided by (2 times 4 times 10 to the 9) = 3.75 cm. This excellent resolution is achieved with very low peak power (milliwatts to watts), unlike pulsed radar which requires high peak power for the same resolution.

How does FMCW measure velocity?

Velocity measurement uses the Doppler shift between consecutive chirps. A single chirp provides range information (beat frequency). A moving target causes a small phase change between consecutive chirps due to the changing range. By transmitting N chirps and performing a second FFT across the N chirp phases at each range bin (the slow-time FFT), the Doppler frequency is extracted: v = lambda times f_doppler / 2. The result is a 2D range-Doppler map: each pixel contains range and velocity information for every target. Maximum unambiguous velocity: v_max = lambda / (4 times T_chirp), where T_chirp is the chirp repetition interval. For 77 GHz with 40 microsecond chirp interval: v_max = 3.9 mm times 1/(4 times 40 microseconds) = 24.4 m/s (88 km/h). Increasing the chirp rate (shorter T_chirp) increases v_max but reduces maximum range. Velocity resolution: delta_v = lambda / (2 times N times T_chirp), inversely proportional to the total observation time.

Why is FMCW preferred for automotive radar?

FMCW dominates automotive radar for several reasons. Low peak power: FMCW transmits continuously at milliwatts to watts, compared to pulsed radar which needs kilowatts of peak power for the same range performance. This enables single-chip CMOS/SiGe implementations at low cost. High resolution: 4 GHz bandwidth at 77 GHz gives 3.75 cm range resolution, sufficient to separate closely spaced targets (vehicles, pedestrians, cyclists). Simultaneous range and velocity: the range-Doppler processing inherently provides both measurements for every target in the scene. Simple hardware: the receiver is just a mixer and low-frequency ADC (beat frequencies are typically 1 to 20 MHz for automotive ranges). No high-speed ADC needed. Integration: modern automotive radar sensors (TI AWR series, Infineon, NXP) integrate the entire radar on a single chip: VCO, TX, RX, ADC, and DSP. A complete radar module is smaller than a credit card and costs under 10 dollars in volume. MIMO: by using multiple TX and RX antennas with orthogonal waveforms (time-division or code-division), a 3-TX 4-RX chip creates a virtual 12-element array for angular resolution.

Radar

FMCW

/eff-em-see-dub/ — Frequency-Modulated CW

Chirp radar: linear freq sweep, beat frequency = range. R = c×f_b×T/(2×BW). ΔR = c/(2×BW). 77 GHz, 4 GHz BW: ΔR = 3.75 cm. Velocity: Doppler across chirps, 2D range-Doppler FFT. Low peak power (mW-W). Single-chip CMOS/SiGe. MIMO: 3TX×4RX = 12 virtual elements. Auto radar, level sensing, vital signs, gesture.

ΔR: c/(2BW)

77GHz/4GHz: 3.75 cm

Cost: <$10 chip

Understanding FMCW Radar

FMCW radar has revolutionized sensing by making high-resolution radar affordable and compact. The key insight is brilliant in its simplicity: by transmitting a frequency-swept signal and mixing the return with the current transmission, range information is encoded as a low-frequency beat tone that can be digitized with a cheap, slow ADC.

A pulsed radar achieving 3.75 cm resolution would need sub-nanosecond pulses at kilowatt peak power. An FMCW radar achieves the same resolution with milliwatts of continuous power and a standard audio-rate ADC. This is why every car now has radar sensors, and why FMCW is expanding into gesture recognition, vital signs monitoring, and industrial sensing.

FMCW Equations

Beat frequency (range):
f_b = 2R×BW/(c×T)
R = c×f_b×T/(2×BW)

Resolution:
ΔR = c/(2×BW)
4 GHz: 3.75 cm, 1 GHz: 15 cm

Velocity (Doppler):
v = λ×f_d/2
v_max = λ/(4×T_chirp)
77 GHz, T=40μs: v_max=24.4 m/s

Max range:
R_max = f_s×c×T/(4×BW)
f_s=10MHz, T=40μs, BW=4GHz: R=75m

FMCW Application Comparison

Application	Freq	BW	ΔR	Range
Auto long-range	77 GHz	1 GHz	15 cm	250 m
Auto short-range	77 GHz	4 GHz	3.75 cm	30 m
Level sensing	24/80 GHz	2 GHz	7.5 cm	100 m
Vital signs	60 GHz	4 GHz	3.75 cm	2 m
Gesture	60 GHz	7 GHz	2.1 cm	1 m

Common Questions

Frequently Asked Questions

Range measurement?

Chirp: freq sweeps f_start to f_start+BW over time T. Return delayed by τ=2R/c. Mixer: beat freq f_b = BW/T × τ = 2R×BW/(cT). FFT: each peak = one target. ΔR = c/(2BW). 4 GHz BW: 3.75 cm. Low peak power, slow ADC sufficient.

Velocity?

Phase change between consecutive chirps = Doppler. 2nd FFT across N chirps at each range bin. v = λf_d/2. v_max = λ/(4T_chirp). 77GHz, 40μs: v_max=24.4 m/s. Δv = λ/(2NT_chirp). Result: 2D range-Doppler map. Every pixel has range + velocity.

Why automotive?

Low peak power (mW-W, not kW). Single-chip CMOS/SiGe (<$10). 4 GHz BW = 3.75 cm resolution. Simultaneous range + velocity. Simple HW: mixer + slow ADC. MIMO: 3TX×4RX = 12 virtual elements for angular resolution. Credit-card-sized module. Every new car has 5+ FMCW sensors.

Related Terms

← FM FFT →

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