FMCW radar

Up to ten point targets, each with a range, a range rate (negative means closing), a radar cross section (how much power it reflects: roughly 10 dBsm for a car, −10 to 0 dBsm for a person) and an angle off boresight. Press Run and they move. The chirp settings on the left fix what the radar can measure: the numbers below follow from them. Moving targets turn around at 5 m and at 80% of the maximum range, so they stay where the radar measures them unambiguously; a target placed beyond that stays beyond it.

The transmitter sweeps up in frequency by the bandwidth B over the ramp time, rests for the idle time, and repeats. Each echo is the same ramp arriving a round-trip delay τ = 2R/c later, shifted by the Doppler frequency 2v/λ. The delay is tiny next to the ramp, so the left plot looks like one line; the zoom on the right shows the gap. At any instant the received ramp sits a fixed frequency below the transmitted one: that gap is the beat frequency.

The mixer multiplies the echo by the transmitted chirp. What comes out is the beat (IF) signal: one tone per target at f_b = 2SR/c + 2v/λ, where S = B/T_c is the chirp slope. It is sampled as I and Q (a complex signal) or, in cheaper designs, as I only. How strong each tone is follows from the radar equation, and it is usually below the thermal noise per sample. The FFTs that follow add up every sample of every chirp, which lifts the target above the noise.

An FFT of one chirp's samples turns each beat tone into a peak. Bin k corresponds to a range of about k·c/2B (a moving target is offset slightly by its Doppler shift), so the bandwidth alone sets the spacing between range bins. The sample rate sets the farthest range that fits: beyond it, a tone wraps around and shows up at the wrong range. The window trades the width of each peak against how far its sidelobes spread.

A second FFT runs across the chirps in each range bin. A moving target's phase advances by 4πvT/λ per chirp, where T is the chirp period, so this FFT sorts targets by velocity. Speeds beyond ±λ/4T wrap around: the phase step has passed half a turn and looks like the opposite direction. The velocity bins are λ/2NT apart, so a longer frame (more chirps) separates closer speeds. Click the map to choose the cuts for stage 6.

Constant false-alarm rate (CFAR) detection compares each cell with the noise around it rather than with a fixed level. Training cells in a ring around the cell under test estimate the local noise; guard cells next to it are skipped so a target does not raise its own threshold. Cell averaging (CA) uses the mean of the training cells; ordered statistic (OS) uses a rank, such as the 75th percentile, which copes better when another target or clutter sits in the ring. The threshold factor is set from the design false-alarm rate. Cells above the threshold are then thinned to local peaks and interpolated between bins.

The design rate assumes the training cells hold independent noise. Hann and Blackman-Harris windows make neighbouring cells correlated, so the measured rate runs above the design value (about 1.2× for Hann and 1.5× for Blackman-Harris at 10⁻⁴); with the rectangular window it matches.

Two equal targets closer together than about one bin blur into one peak. In range the bin is c/2B, so only more bandwidth separates them; in velocity it is λ/2NT, so only a longer frame does. Windows widen the peaks, so a windowed FFT needs more than one bin. The line is the profile averaged over the targets' relative phase; the band shows how far it swings between in-phase and opposite-phase echoes.

With four receive antennas half a wavelength apart, an echo from angle θ reaches each one with a phase step of π·sin θ. A small FFT across the four channels, at each detection's range–Doppler cell, turns that step into an angle. Four elements give a wide beam (about 26° between the −3 dB points at boresight), so two targets at the same range and speed are hard to split by angle; that is why they are separated in range and velocity first.