Frequency Modulated Continuous Wave (FMCW) Radar

Unit Objectives

By the end of this unit, the student should be able to:

1. Define FMCW radar and explain how frequency modulation endows a CW radar with ranging capability.
2. Describe the common modulation waveforms (sawtooth/ramp, triangular, square-wave FSK, stepped chirp) and their trade-offs.
3. Derive the beat-frequency equations for sawtooth and triangular FMCW, obtaining range and range-rate expressions.
4. Determine range resolution, maximum unambiguous range, and beat-frequency limits for a given waveform design.
5. Explain range–Doppler coupling and how the two-ramp (triangular) technique decouples range and velocity.
6. Outline the digital signal processing chain: mixing, low-pass filtering, sampling, and 2-D FFT range–Doppler processing.
7. Analyse the effects of sweep nonlinearity and describe correction techniques.
8. Describe chirp generation methods (VCO, DDS, PLL) and the isolation requirements of FMCW front-ends.
9. Evaluate FMCW applications: automotive radar, altimetry, level sensing, and satellite/spaceborne ranging.
10. Solve complete design problems combining bandwidth, sweep time, beat frequency, range and velocity.

Prerequisites: Unit on Continuous Wave (CW) radar; signals & systems (Fourier analysis, sampling); communication systems (mixing, PLLs).

1. Introduction: Why Modulate the Frequency?

A plain CW radar transmits a pure tone and measures only the Doppler shift — it has no way to tell how far the target is, because the echo delay hides inside an unobservable constant phase. Frequency-modulated CW (FMCW) radar solves this by sweeping the transmitted frequency in a known pattern. The echo, delayed by t_d = 2R/c, is then a time-shifted copy of the sweep; at any instant the receiver sees a frequency difference between what is being transmitted now and what was transmitted when the echo left. This difference — the beat frequency f_b — is proportional to delay, and hence to range.

1.1 FMCW vs. Other Radar Architectures

2. FMCW Modulation Waveforms

The choice of frequency modulation pattern determines what can be measured and how cleanly. Four waveforms dominate practice:

3. Sawtooth FMCW: Beat Frequency Derivation

3.1 Transmitted and Received Signals

Let the transmitted frequency ramp linearly from f₀ to f₀+B over sweep time T_m. The instantaneous transmitted frequency and phase are:

The echo from a target at range R arrives after delay t_d = 2R/c, and (if the target moves) carries a Doppler shift f_d = 2v_r/λ:

3.2 Mixing to the Beat Note

The receiver mixes the echo with a sample of the transmitted signal; the difference term of the product cos(φ_TX)·cos(φ_RX) contains the phase difference Δφ = φ_TX − φ_RX. A low-pass filter removes the sum term, leaving:

For stationary or slow targets, the Doppler term is negligible or lumped into a range error, giving the working equation:

4. Triangular FMCW: Simultaneous Range and Velocity

With a triangular waveform, the ramp slope reverses on the down-leg. The Doppler shift f_d keeps the same sign on both legs (it depends on motion, not on the sweep direction), while the range-induced beat changes sign with the slope:

Adding and subtracting the two measured beats recovers both quantities:

Equivalently, f_d = (f_down − f_up)/2 and v_r = f_dλ/2.

4.1 The Multiple-Ramp Alternative (Modern Automotive Practice)

Rather than one up/down pair, automotive radars transmit a frame of N chirps with slightly different slopes (a chirp sequence). Range comes from the beat frequency (fast-time FFT); velocity comes from the phase progression of the beat across the chirp sequence (slow-time FFT — the same Doppler-FFT idea as pulsed radar, with chirps playing the role of pulses). This 2-D processing cleanly separates range and Doppler and resolves multiple targets.

5. Resolution, Ambiguity and Waveform Design

5.1 Range Resolution

Set purely by sweep bandwidth — identical to the pulsed-radar result (B ≈ 1/τ). A 4 GHz sweep gives 3.75 cm resolution; this is why automotive short-range radar uses multi-GHz chirps.

5.2 Maximum Unambiguous Range and Beat-Frequency Limits

Beat frequencies above half the sampling rate f_s alias (Nyquist). The maximum measurable beat, and hence range, is:

The fast-time observation window T_m sets the beat-frequency resolution Δf_b = 1/T_m, which maps back to a range resolution — consistency with ΔR = c/2B requires B·T_m ≥ 1 (the time–bandwidth product must exceed unity).

5.3 Velocity Resolution and Maximum Unambiguous Velocity (Chirp-Sequence Waveforms)

where T_c is the chirp repetition interval and N the number of chirps in the frame. Short chirps give wide unambiguous velocity; long frames give fine velocity resolution — a direct echo of pulsed-radar PRF trade-offs.

5.4 Design Rules of Thumb

6. Digital Signal Processing Chain

6.1 The Two-Dimensional FFT

1. Fast-time FFT (per chirp): each chirp's beat signal is windowed and FFT'd → range profile with bins ΔR = c/2B.
2. Slow-time FFT (across chirps): for each range bin, the sequence of complex samples across N chirps is FFT'd → Doppler/velocity bins Δv = λ/(2NT_c).
3. Result: a range–Doppler map where every target appears as a peak at (range bin, Doppler bin) — targets at the same range but different velocities are separated.
4. Detection & tracking: CFAR thresholding declares targets; an α-β or Kalman filter tracker maintains target histories.

7. Sweep Nonlinearity and Its Correction

The derivation in Section 3 assumes a perfectly linear frequency ramp. Real VCOs exhibit nonlinear tuning (S-curves, drift), which broadens and distorts the beat spectrum: instead of a single clean tone per target, the beat becomes a chirp-like smear, degrading resolution and detection.

7.1 Correction Techniques

• PLL-locked VCO: the VCO is disciplined by a phase-locked loop referenced to a linear ramp from a DDS — the loop forces the VCO to follow the ideal slope.
• Direct digital chirp synthesis (DDS): generate the ramp numerically and upconvert; inherently linear to DAC resolution.
• Reference-delay calibration: route a delayed copy of the TX signal through a known delay line into the mixer; the residual beat reveals the instantaneous nonlinearity, which is applied as a digital correction.
• Post-processing re-sampling: measure the actual frequency trajectory (e.g., via a frequency discriminator) and resample the beat signal on a uniform-frequency grid before the FFT.

8. FMCW Front-End and Chirp Generation

8.1 Chirp Generation Methods

8.2 Isolation and Leakage (Revisited from the CW Unit)

Because TX is always on, FMCW inherits the CW isolation problem — worsened by the fact that the near-field leakage from the TX antenna directly enters the mixer, producing a huge "zero-beat" DC component that can saturate the ADC. Mitigations: antenna separation, absorptive shielding, TX-leakage cancellation networks, and digital DC-offset removal. Monostatic (single-antenna) automotive radars rely on circulators/isolators with typically 20–30 dB of inherent isolation plus calibration.

9. Applications of FMCW Radar

9.1 Automotive Radar (Largest Modern Market)

• 77 GHz long-range radar (LRR): ~250 m range, adaptive cruise control (ACC); bandwidth ~0.8–1 GHz.
• 77/79 GHz short/medium-range (SRR/MRR): blind-spot detection, lane-change assist, cross-traffic alert, autonomous emergency braking; bandwidth up to 4 GHz for ~4 cm resolution.
• Chirp-sequence waveforms with 2-D FFT processing separate vehicles at the same range by relative velocity — essential in dense traffic.

9.2 Radar Altimeters

Aircraft FMCW altimeters (C-band, ~4.3 GHz, B ≈ 100–165 MHz) give 0.5–1 m resolution from ~1,500 ft down to the runway — no minimum-range dead zone, low peak power. Spaceborne and planetary-lander altimetry uses related FM/chirp techniques for precision descent ranging.

9.3 Industrial Level and Distance Sensing

26/60/80 GHz FMCW level radars measure tank and silo contents with millimetre accuracy, immune to dust, vapor and temperature — replacing ultrasonic and guided-wave devices in harsh process plants.

9.4 Security, Surveillance and Through-Wall Sensing

Low-power FMCW at 24/60 GHz supports perimeter intrusion detection, people counting, and vital-signs monitoring (chest-motion micro-Doppler at a few tens of cm).

9.5 Missile Seekers and Proximity Sensing

Triangular-FMCW seekers provide passive-appearing ranging in air-to-air engagements; artillery proximity fuzes use FM ranging derivatives for height-of-burst control.

9.6 Geoscience

Ground-penetrating radar (GPR) and snow/ice thickness sounders frequently use stepped-FMCW or chirp waveforms to build high-resolution subsurface profiles.

10. FMCW in Satellite & Space Systems (Course Link)

1. Planetary landing radars: FMCW/chirp altimeters provide continuous altitude and velocity during descent (e.g., lunar/Mars lander radar demonstrators) — low peak power suits power-limited landers, and the absence of a minimum range is critical in the final metres.
2. Satellite-to-satellite / formation flying ranging: cross-link chirp ranging measures inter-satellite distances to centimetre level (optical/microwave metrology complements transponder-based two-way ranging).
3. Spaceborne altimetry & scatterometry heritage: ocean-topography altimeters use chirp-like pulse-compression waveforms — the pulsed equivalent of the FMCW principle (same c/2B resolution law).
4. Ground-station test & measurement: FMCW techniques calibrate antenna range-delay and feed chains at satellite earth stations.

11. Worked Examples

Interactive FMCW Range Calculator

Practice Problems (unsolved)

1. A 24 GHz FMCW level radar sweeps 250 MHz in 4 ms and measures f_b = 32 kHz. Find the range and resolution.
2. For the Example-2 radar, what beat frequency would a target at 100 m produce on the up-ramp (ignore Doppler)?
3. Show that with chirp-sequence processing, two targets at the same range but with velocities differing by more than Δv are resolved in the Doppler FFT.

12. Summary of Key Points

1. FMCW imprints a known frequency pattern on a CW carrier; the echo's delay converts into a beat frequency, giving range with no minimum range and low peak power.
2. Sawtooth FMCW: R = c·T_m·f_b/2B. Range–Doppler coupling prevents simultaneous range+velocity from a single ramp.
3. Triangular FMCW decouples them: R from the sum of up/down beats, v_r from the difference (via f_d).
4. Range resolution ΔR = c/2B; unambiguous range limited by ADC sampling of the beat; chirp-sequence waveforms add v_max = λ/4T_c and Δv = λ/2NT_c.
5. Processing is a 2-D FFT (fast-time range, slow-time Doppler) — mathematically the dual of pulsed Doppler radar.
6. Sweep linearity is critical: PLL/DDS synthesis or reference-delay correction is required for high resolution.
7. Applications: automotive ACC/AEB, altimeters, industrial level radar, security sensing, seekers, GPR, and satellite landing/formation-flying ranging.

13. Self-Test Questions

Q1. State the two measurable quantities provided by a triangular-FMCW radar and the beat-frequency combinations that yield each.

Answer: Range: R = c·T_m(f_up+f_down)/4B; Velocity: from f_d = (f_down−f_up)/2, v_r = f_dλ/2.

Q2. An FMCW radar sweeps 500 MHz in 2 ms and measures a beat of 25 kHz. Find the range.

Answer: R = 3×10⁸×2×10⁻³×2.5×10⁴/(10⁹) = 15 m.

Q3. Explain range–Doppler coupling in a single-ramp FMCW waveform.

Answer: The measured beat f_b = (B/T_m)(2R/c) + f_d mixes range and velocity information; a moving target appears at an apparent range offset ΔR = c·T_m·f_d/(2B) — indistinguishable from a range change without a second measurement.

Q4. What bandwidth gives 5 cm range resolution, and why?

Answer: B = c/(2ΔR) = 3×10⁸/0.1 = 3 GHz; resolution depends only on bandwidth, not carrier.

Q5. Why does sweep nonlinearity degrade FMCW performance, and name two correction methods.

Answer: Nonlinear frequency-vs-time smears the beat tone into a chirp, broadening the spectral peak and degrading resolution/detection. Corrections: PLL-locked VCO, DDS chirp synthesis, reference-delay calibration, frequency-domain resampling (any two).

Q6. In chirp-sequence processing, which FFT gives range and which gives velocity?

Answer: Fast-time FFT (within each chirp) → range; slow-time FFT (across the N chirps per range bin) → Doppler/velocity.

Q7. A 79 GHz (λ = 3.8 mm) chirp-sequence radar uses T_c = 25 μs. Find v_max.

Answer: v_max = λ/(4T_c) = 3.8×10⁻³/10⁻⁴ = 38 m/s.

Q8. Give two reasons FMCW suits planetary landing radar compared with pulsed radar.

Answer: Low peak power (power-limited lander) and no minimum-range dead zone (needed in final descent metres); also continuous measurement of altitude/rate (any two).

Q9. What sets the maximum unambiguous range in an FMCW system?

Answer: The ADC sampling rate: f_b must stay below f_s/2 (Nyquist), so R_max = c·T_m·f_s/(4B).

Q10. A triangular-FMCW radar with λ = 3 cm, B = 150 MHz, T_m = 1 ms measures f_up = 45 kHz, f_down = 55 kHz. Find R and v_r.

Answer: R = 3×10⁸×10⁻³×10⁵/(6×10⁸) = 50 m; f_d = 5 kHz; v_r = 5000×0.03/2 = 75 m/s.

14. References and Further Reading

1. Skolnik, M. I., Introduction to Radar Systems, 3rd ed., McGraw-Hill, 2001 — CW and FMCW chapter.
2. Skolnik, M. I. (ed.), Radar Handbook, 3rd ed., McGraw-Hill, 2008 — altimeter and FMCW sections.
3. Stove, A. G., "Linear FMCW radar techniques," IEE Proceedings F — Radar and Signal Processing, Vol. 139, No. 5, 1992.
4. Richards, M. A., Fundamentals of Radar Signal Processing, 2nd ed., McGraw-Hill, 2014 — 2-D Doppler processing.
5. Hasch, J. et al., "Millimeter-wave technology for automotive radar sensors in the 77 GHz frequency band," IEEE Transactions on Microwave Theory and Techniques, 2012.
6. Fortescue, P., Swinerd, G. & Stark, J. (eds.), Spacecraft Systems Engineering, 4th ed., Wiley — landing and TT&C chapters.