Range Resolution

A pulse-compression sonar (or radar) can only tell two reflectors apart if they are separated by more than c / (2B), where c is the wave speed and B is the transmitted bandwidth. Drag the sliders below: as you shrink B, the matched-filter peaks fatten until two distinct targets merge into a single blob.

Controls

2000 Hz
1.00 m
1.20 m

Chirp: 8 ms linear sweep starting at 500 Hz, sweeping a bandwidth B. Wave speed c = 343 m/s. Both targets reflect with equal amplitude.

Predicted resolution

Target separation — |R₂ − R₁|
c / (2B) — smallest separation that resolves
Status — —

Matched-filter envelope · distance axis

|matched filter|²

Yellow dashed lines mark the true target positions R₁, R₂. The yellow bar at the top is c / (2B) drawn to scale — if it is narrower than the gap between the dashed lines, you should see two peaks; otherwise the peaks merge.

What the receiver sees, before pulse compression

received signal r(t)

Sum of two delayed copies of the transmitted chirp. As you bring the targets close, the two echoes overlap into a single complicated waveform — raw inspection cannot count the targets.

Transmitted chirp

s(t)
Why c / (2B)?
Step 1 · the matched filter compresses to width 1/B

A chirp of duration T and bandwidth B spreads its energy across both time and frequency. After matched filtering, all of that energy concentrates into a single peak whose width in time is approximately 1/B — a sinc-shaped main lobe set by the frequency support, not by T. (This is the time–bandwidth uncertainty principle: the narrowest pulse you can synthesize from frequencies in a band of width B is about 1/B wide.)

Step 2 · round trip divides delay by 2

A target at one-way distance R produces a round-trip delay τ = 2R/c. Two targets separated in range by ΔR are separated in delay by 2ΔR/c. To distinguish them after matched filtering, this delay gap must exceed the peak width 1/B:

2 · ΔR / c  >  1 / B   ⇔   ΔR  >  c / (2B)

Notice what is not in the answer: pulse duration T, transmit power, or the carrier frequency. T only buys signal-to-noise ratio (more energy on target). The resolution is set by how much spectrum you sweep through, full stop.

Implication

Range resolution is bought with bandwidth, not with time or power. A long, narrow-band tone (B → 0) has zero resolution no matter how loud or how long. A short broadband click resolves down to cT/2 ≈ c/(2B). Wideband chirps give you both: the SNR of a long pulse and the resolution of a short one. That trick is called pulse compression.