Chirp Signal
Turning Duration into Bandwidth
The instantaneous frequency of a linear chirp ramps from f0 − B/2 to f0 + B/2 over the pulse duration T, where f0 is the center frequency and B is the sweep bandwidth. The chirp rate is k = B/T in Hz per second. At the receiver, a matched filter (or its equivalent, a dechirp mixer) correlates the received echo with a replica of the transmitted waveform, compressing the long pulse into a peak with a width of approximately 1/B.
s(t) = A · cos(2πf0t + πkt²) for −T/2 ≤ t ≤ T/2
where k = B/T (chirp rate, Hz/s)
Compressed pulse width: τc = 1/B
Range resolution: ΔR = c/(2B)
Compression ratio: CR = T · B = TBP
Processing gain: Gp = 10 · log10(TBP) dB
Chirp Parameters for Common Radar Types
| Radar Type | Bandwidth | Pulse Width | TBP | Range Resolution | Processing Gain |
|---|---|---|---|---|---|
| Air surveillance (L-band) | 5 MHz | 100 μs | 500 | 30 m | 27 dB |
| Weather radar (S-band) | 2 MHz | 50 μs | 100 | 75 m | 20 dB |
| Fire control (X-band) | 500 MHz | 20 μs | 10,000 | 0.3 m | 40 dB |
| Automotive FMCW (77 GHz) | 1 GHz | 40 μs | 40,000 | 0.15 m | 46 dB |
| SAR imaging (X-band) | 600 MHz | 10 μs | 6,000 | 0.25 m | 38 dB |
Sidelobe Control After Compression
The matched filter output has sinc-function sidelobes at −13.2 dB. For radars detecting small targets near large ones, these sidelobes must be suppressed. Amplitude weighting (windowing) the received signal before compression trades main lobe width for sidelobe reduction:
- Hamming: −43 dB sidelobes, 1.4× wider main lobe, −1.3 dB processing gain loss
- Taylor (−40 dB, n̄=5): −40 dB sidelobes, 1.2× wider, −0.8 dB loss
- Chebyshev (−50 dB): −50 dB sidelobes, 1.5× wider, −1.8 dB loss
Frequently Asked Questions
What is the time-bandwidth product?
TBP = T × B, the compression ratio. A 10 μs chirp with 100 MHz BW has TBP = 1000: the compressed pulse is 1000× shorter than the transmitted pulse, giving 30 dB of processing gain.
What causes range sidelobes?
The matched filter output is sinc-like with −13.2 dB sidelobes. These mask weak targets near strong ones. Windowing (Hamming, Taylor) reduces sidelobes to −40 to −50 dB at the cost of slightly wider main lobe and 0.8 to 1.8 dB processing gain loss.
How does FMCW differ from pulsed chirp?
FMCW transmits continuously while receiving, using separate TX/RX antennas. The received signal is mixed with the live transmit signal, producing a beat frequency proportional to range. FMCW avoids high peak power and is used in automotive radar, altimeters, and level gauges. Pulsed chirp uses a single antenna with T/R switching.