FMCW Radar Explained – Chirps, Range and Doppler

CHIRP SIGNAL THEORY

What is FMCW radar?

FMCW stands for Frequency Modulated Continuous Wave. Unlike pulse radar systems, which transmit short bursts of energy and wait for their echo, an FMCW radar transmits continuously a signal whose frequency changes linearly over time. This frequency-modulated signal is called a chirp.

Pulse radar (classical)

Transmits short, high-power pulses (kW–MW). Measures the echo's round-trip delay Δt directly. Advantage: simple signal processing. Disadvantage: high peak power, expensive transmitters (magnetron, klystron), blind during the transmit phase.

FMCW radar (modern)

Transmits continuously at low power (mW). Measures range from the frequency difference between the transmit and receive signal. Advantage: low-cost (CMOS technology), high resolution, simultaneous range + velocity. Disadvantage: requires transmitter–receiver isolation.

The chirp signal is mathematically a linear frequency sweep. The instantaneous frequency rises linearly from f₀ (start frequency) to f₀ + B (end frequency) over the chirp duration T_c. The continuous-time transmit signal is:

Chirp signal (transmit signal):

s(t) = A · cos(2πf₀t + πμt²)

Here μ = B / T_c is the chirp rate (frequency change per unit time in Hz/s). The instantaneous frequency follows from the derivative of the phase: f(t) = f₀ + μ·t.

Parameter	Symbol	Typical value (77 GHz)	Description
Start frequency	f₀	77 GHz	Carrier frequency; determines the wavelength λ = c/f₀
Bandwidth	B	1–4 GHz	Frequency sweep of the chirp; determines the range resolution
Chirp duration	T_c	20–200 μs	Duration of a single chirp; affects max. range and Doppler ambiguity
Chirp rate	μ = B/T_c	20–80 MHz/μs	Slope of the frequency ramp in the time–frequency diagram

In the time–frequency diagram the chirp appears as a rising line: the x-axis shows time, the y-axis the instantaneous frequency. The received signal (echo) is a time-delayed copy of the transmit signal – the vertical difference between the two lines corresponds to the beat frequency, which is proportional to the target range.

BEAT FREQUENCY

Deriving the range measurement

A target at range R reflects the transmit signal. The echo reaches the receiver with a round-trip delay τ. Because the transmit frequency has kept rising during this delay, there is a constant frequency difference between the current transmit signal and the received echo – the beat frequency f_b.

Step 1: round-trip delay

τ = 2R / c

The signal travels radar → target → radar (there and back = 2R). With c = 3×10⁸ m/s and R = 100 m, τ = 667 ns.

Step 2: mixing (dechirping)

In the receiver, the echo signal is mixed (multiplied) with the current transmit signal. The mixer produces a signal at the difference frequency – the beat frequency:

f_b = μ · τ = (B / T_c) · (2R / c) = 2BR / (c · T_c)

Step 3: computing range

Solving for R gives the central FMCW range formula:

R = (f_b · c · T_c) / (2B)

Range is directly proportional to the beat frequency. The farther away the target, the higher f_b.

Numerical example:

                    // Given parameters

                    B  = 4 GHz   = 4×109 Hz

                    Tc = 100 μs  = 100×10-6 s

                    fb = 1 MHz   = 1×106 Hz

                    c  = 3×108 m/s

                    // Calculation

                    R = (fb · c · Tc) / (2 · B)

                    R = (1×106 · 3×108 · 100×10-6) / (2 · 4×109)

                    R = 30000 / 8000

                    R = 3.75 m

Important: the beat frequency typically lies in the range of kHz to a few MHz – independent of the carrier frequency (77 GHz). This down-conversion to baseband is the key advantage of the FMCW principle: the entire signal chain (ADC, FFT) can be built with inexpensive, low-frequency electronics.

RANGE–DOPPLER MATRIX

From chirps to a 2D map

A single chirp only provides range. To additionally measure a target's velocity, several chirps are grouped into a frame. A two-stage FFT process produces the range–Doppler matrix (RDM) – the central data product of an FMCW radar.

1. Range FFT (fast time)

For each individual chirp, an FFT is computed over the sampled beat-signal samples. Each frequency bin corresponds to a specific range (range bin). The spacing between bins is ΔR = c/(2B).

FFT along the axis: samples per chirp

2. Doppler FFT (slow time)

Across N chirps, the phase of the signal in a given range bin changes systematically if the target is moving. A second FFT over this phase change extracts the Doppler frequency f_d and thus the radial velocity.

FFT along the axis: chirps per frame

Velocity from the Doppler frequency:

v = λ · f_d / 2

Here λ = c/f₀ is the wavelength and f_d the Doppler frequency. An approaching target produces a positive Doppler shift (higher frequency), a receding target a negative one.

The result is a 2D matrix: the x-axis represents range (range bins), the y-axis velocity (Doppler bins). Each target appears as a peak in this matrix. Peak height grows with the target's RCS, but depends just as strongly on range: by the radar equation, received power P_r ∝ σ/R⁴, so amplitude ∝ √σ/R². Peak height alone therefore cannot be read as RCS.

                # Range–Doppler processing pipeline (pseudocode)

                # 1. Data: N chirps with M samples each

                data = matrix[N_chirps, M_samples]

                # 2. Range FFT (along each row), usually windowed

                for chirp in range(N):

                  range_profile[chirp] = FFT(data[chirp, :] * w_range)  # M-point FFT

                # 3. Doppler FFT (along each column)

                for bin in range(M):

                  rdm[:, bin] = FFT(range_profile[:, bin] * w_dopp)  # N-point FFT

                # 4. Result: range–Doppler matrix (N × M)

                plot_heatmap(rdm, x="Range", y="Velocity")

The window functions w_range and w_dopp (e.g. Hann or Blackman) suppress the sidelobes that otherwise arise from the finite observation window and can mask weak targets next to strong ones. In exchange, they slightly widen the main lobe. Which window is appropriate depends on the scenario.

Typical frame parameters (77 GHz, drone detection):

N = 128 chirps per frame, M = 256 samples per chirp, T_c = 60 μs, B = 4 GHz.
Frame duration: 128 × 60 μs = 7.68 ms → theoretical upper bound on frame rate ≈ 130 Hz. Real systems fall well below this because of idle time between chirps and frames and processing latency.
Range bins: 256, Doppler bins: 128.

RESOLUTION

Bandwidth, resolution and ambiguity

The fundamental limits of an FMCW radar are set by the chirp parameters. Bandwidth B defines the range resolution, chirp duration T_c and count N determine the velocity resolution, and the maximum values follow from the sampling limits.

Range resolution

ΔR = c / (2B)

Two targets can only be distinguished if their separation exceeds ΔR. Only the bandwidth B determines the range resolution – not the carrier frequency.

Velocity resolution

Δv = λ / (2 · N · T_c)

Depends on the total observation duration N·T_c (frame duration). More chirps per frame = finer velocity resolution.

Bandwidth B	ΔR (range res.)	T_c	N (chirps)	Δv (vel. res.)	Application
250 MHz	60 cm	200 μs	64	0.15 m/s	Long-range surveillance
1 GHz	15 cm	100 μs	128	0.15 m/s	Automotive (ACC)
2 GHz	7.5 cm	60 μs	128	0.25 m/s	Person detection
4 GHz	3.75 cm	60 μs	256	0.13 m/s	Drone detection / gesture

Maximum unambiguous range

R_max = f_s · c / (2μ)

Determined by the ADC sampling rate f_s. This form applies to complex (I/Q) sampling, where the beat frequency may reach up to f_s. Under real-valued sampling, the Nyquist criterion f_b,max = f_s/2 applies instead, giving R_max = f_s·c/(4μ). For the same reason, an M-point FFT under real-valued sampling yields only M/2 usable range bins. A second, usually non-critical bound is R_max = c·T_c/2, if the echo must arrive within one chirp.

Maximum unambiguous velocity

v_max = λ / (4 · T_c)

Determined by the chirp repetition rate 1/T_c. Faster chirps allow higher unambiguous velocities but reduce velocity resolution.

Design trade-off:

There is a fundamental conflict: short chirps (small T_c) increase v_max but worsen Δv. Many chirps (large N) improve Δv but lengthen the frame duration. High bandwidth B improves ΔR but requires a faster ADC. These trade-offs are the core challenge of FMCW radar design.

SAMPLING THEOREM & ALIASING

Nyquist criterion and ambiguities

Both in range and in Doppler, FMCW radar is subject to the sampling theorem. If it is violated, aliasing occurs – signals are “folded” into the wrong bins and produce ghost targets.

Range aliasing (ADC sampling):

f_s ≥ 2 · f_b,max

The ADC must sample the beat signal at at least twice the maximum beat frequency. If f_s is too low, distant targets (high f_b) fold into nearer range bins. The maximum f_b follows from R_max: f_b,max = 2·B·R_max/(c·T_c).

Doppler aliasing (chirp repetition rate):

v_max = λ / (4 · T_c)

The chirps sample the Doppler phase. The sampling frequency in the Doppler domain is 1/T_c. Targets with |v| > v_max fold into the range [-v_max, +v_max] and appear with the wrong velocity. The apparent velocity is: v_alias = v − 2·n·v_max (n an integer).

Example: Doppler aliasing

At f₀ = 77 GHz (λ = 3.9 mm) and T_c = 100 μs:
v_max = 0.0039 / (4 × 100×10^-6) = 9.74 m/s (≈ 35 km/h).

A car at v = 50 km/h (13.9 m/s) gets folded:
v_alias = 13.9 − 2×9.74 = −5.6 m/s
It appears as an oncoming target!

Countermeasures

1. Shorten T_c → higher v_max, but worse Δv.
2. Multiple PRF / multi-slope: two chirp configurations with different T_c → CRT unwrapping resolves the ambiguity.
3. Chirp sequence with multiple slopes (up+down chirp) to resolve the range–Doppler coupling.

Chirp parameter design against aliasing:

A good FMCW design requires simultaneously accounting for R_max, v_max, ΔR, Δv and f_s. The typical workflow:
1. Fix ΔR → determine B.
2. Fix v_max → determine T_c.
3. Fix R_max → determine f_s.
4. Fix Δv → determine N (chirps per frame).
5. Check whether the frame rate is sufficient for the application.

INTERACTIVE

FMCW explorer: chirp & beat frequency

Move the sliders to change the target range and the bandwidth. Watch how the delay of the RX chirp and the resulting beat frequency change in the FFT spectrum.

Target range: 50 m Simulation parameter – not a demonstrated Ozense range. Bandwidth: 2.0 GHz

Computed values:

f_b = 6.67 MHz

ΔR = 7.50 cm

τ = 333.3 ns

Time–frequency diagram – TX chirp (yellow) and RX chirp (blue)

Beat-signal FFT – spectrum with range peak

FMCW Basics – Frequency-Modulated Continuous-Wave Radar

Academy Modules

What is FMCW radar?

Deriving the range measurement

From chirps to a 2D map

Bandwidth, resolution and ambiguity

Nyquist criterion and ambiguities

FMCW explorer: chirp & beat frequency

Scope and limits

All Academy Modules