LFM Chirp
Understanding the LFM Chirp
Early radar systems used simple unmodulated pulses. Range resolution was determined by pulse width: a 1 μs pulse gives 150 m resolution (cτ/2). To improve resolution to 1 m requires a 6.7 ns pulse, but such a short pulse contains very little energy, limiting detection range. The LFM chirp resolves this dilemma by encoding bandwidth within a long pulse. A 10 μs chirp with 500 MHz bandwidth has the detection range of a 10 μs pulse but, after matched filter compression, the range resolution of a 2 ns pulse (0.3 m).
The chirp signal is described mathematically as s(t) = rect(t/τ) × exp(j2π(f0t + ½μt²)), where μ = BW/τ is the chirp rate (Hz/s), f0 is the center frequency, and τ is the pulse duration. The instantaneous frequency f(t) = f0 + μt increases linearly from f0 − BW/2 to f0 + BW/2 over the pulse duration. At the receiver, a matched filter (the conjugate time-reversed replica of the chirp) compresses the echo into a sinc-like pulse with mainlobe width 1/BW.
LFM Chirp Equations
s(t) = A × exp(j2π(f0t + ½μt²)), |t| ≤ τ/2
Chirp Rate:
μ = BW / τ (Hz/s)
Time-Bandwidth Product (Compression Ratio):
TBP = τ × BW
Processing Gain = 10 log10(TBP) dB
Range Resolution:
ΔR = c / (2 × BW)
500 MHz → 0.3 m; 4 GHz → 3.75 cm
Compressed Pulse Width:
τcomp = 1/BW
LFM Chirp Parameters by Application
| Application | Pulse Duration (τ) | Bandwidth (BW) | TBP | Range Resolution |
|---|---|---|---|---|
| Automotive FMCW (77 GHz) | 60 μs | 4 GHz | 240,000 | 3.75 cm |
| Weather Radar (S-band) | 1 ms | 1 MHz | 1,000 | 150 m |
| Air Surveillance (L-band) | 100 μs | 10 MHz | 1,000 | 15 m |
| SAR Imaging (X-band) | 20 μs | 500 MHz | 10,000 | 0.3 m |
| Fire Control (Ka-band) | 10 μs | 200 MHz | 2,000 | 0.75 m |
Frequently Asked Questions
What is pulse compression and how does an LFM chirp achieve it?
Pulse compression transmits a long pulse for high energy while achieving the range resolution of a much shorter pulse. The LFM chirp sweeps bandwidth BW over duration τ. The matched filter correlates the echo with the known chirp, compressing it into a spike of width 1/BW. A 10 μs chirp with 500 MHz bandwidth has TBP = 5,000: the compressed pulse is 5,000 times narrower than the transmitted pulse, giving 0.3 m range resolution while maintaining the detection range of a 10 μs pulse.
What is the time-bandwidth product?
TBP = τ × BW equals the pulse compression ratio and determines the processing gain: 10 log10(TBP) dB. A TBP of 1,000 provides 30 dB processing gain, meaning the compressed peak is 30 dB above noise relative to an uncompressed pulse of equal energy. Modern radar chirps range from TBP = 100 for simple systems to over 1,000,000 for SAR and high-resolution imaging where both long dwell times and extreme bandwidth are used simultaneously.
How is an LFM chirp generated in a modern radar?
Modern systems use Direct Digital Synthesis (DDS) or Arbitrary Waveform Generators running at 1 to 10 GSPS to produce the chirp at an intermediate frequency, then upconvert to the RF carrier via a mixer and local oscillator. Digital generation provides precise control over chirp linearity, start/stop frequencies, and phase continuity. Older systems used SAW dispersive delay lines or voltage-controlled oscillators with linearization feedback, but these analog methods have been largely replaced by digital waveform generation.