FMCW & Pulsed Radar — Waveforms & Resolution
The radar equation gives the received signal power, but the waveform determines how accurately range and velocity can be measured. The three main radar waveform families — pulsed, CW, and FMCW — each make a different tradeoff between range resolution, velocity measurement capability, and hardware complexity.
Pulsed Radar
A pulsed radar transmits short bursts of RF energy and measures the round-trip travel time to determine range. The transmitter is silent while the receiver listens for echoes:
where τ is the pulse round-trip time, and ΔR is the range resolution set by the signal bandwidth B (not the pulse duration, for pulse-compressed systems). A 1 µs pulse → 150 m range resolution. A 10 MHz bandwidth (pulse-compressed or stepped frequency) → 15 m resolution.
| Parameter | Formula | Notes |
|---|---|---|
| Max unambiguous range | R_max = c/(2·PRF) | PRF = pulse repetition frequency |
| Range resolution | ΔR = c/(2B) | B = bandwidth; 150 m/µs for CW pulse |
| Duty cycle | DC = τ·PRF | Typically 0.001–0.1 for pulsed systems |
| Average power | P_avg = P_peak · DC | Sets detection range via SNR |
CW Radar
Continuous wave radar transmits a single frequency continuously. It cannot measure range (no time reference) but measures velocity extremely well via the Doppler shift:
where v_r is the radial velocity (positive = approaching). A 10 GHz radar and 100 km/h target: f_D = 2 × 27.8 / 0.03 = 1.85 kHz. Police speed guns and simple motion detectors use CW radar.
FMCW Radar
Frequency-modulated CW (FMCW) combines range and velocity measurement by sweeping the transmit frequency linearly. The echo from a target at range R arrives with a time delay τ = 2R/c, creating a constant "beat frequency" f_b between the current transmit frequency and the received echo:
where S is the frequency sweep rate (Hz/s), B the sweep bandwidth, and T_sweep the sweep duration. By measuring f_b with an FFT, range is determined directly. Modern automotive radar (77 GHz, 4 GHz bandwidth) achieves ΔR = c/(2×4GHz) = 3.75 cm range resolution.
Range-Doppler Matrix
A moving target creates both a range displacement and a Doppler frequency. In FMCW, velocity causes a phase rotation of the beat signal between sweeps. By taking a 2D FFT (range FFT per sweep, then Doppler FFT across sweeps), the range-Doppler matrix simultaneously shows all targets' range and velocity:
where N is the number of sweeps in the coherent processing interval (CPI).
Ambiguity Function
The radar ambiguity function χ(τ, f_D) describes the range-Doppler response of a waveform — how much a target at (range τ₀, Doppler f_D₀) leaks into adjacent range-Doppler cells. A perfect ambiguity function has a single narrow spike at the origin and zero everywhere else (impossible). The uncertainty principle of radar states: ΔR × Δv ≥ λc/4. Short pulses give good range resolution but poor velocity resolution; long CPI gives good velocity but requires coherence across many sweeps.
FMCW Applications
- Automotive radar (77 GHz): 4 GHz sweep, 3.75 cm range resolution, ±200 km/h velocity, 200 m range. Used for adaptive cruise control, AEB, blind-spot detection.
- Level measurement: Industrial FMCW sensors measure tank fill level to mm accuracy over tens of metres.
- Ground-penetrating radar (GPR): 100 MHz–3 GHz sweep through soil to detect buried utilities, unexploded ordnance, or archaeological features.
- Indoor positioning: 60 GHz UWB FMCW can locate people to < 10 cm accuracy.