Scenario
Aircraft on the scope (truth vs what the radar reports)
| Target | Range km
true / scope | Radial m/s
true / meas. | SNR dB | Pd
theory / meas. |
|---|
Radial velocity: + approaching, - receding. "Scope" range and "meas." velocity are the folded (ambiguous) values. Click the scope to list the ranges consistent with an echo.
Click the PPI to measure a point.
Detection vs range (analytic curves, test points)
Press the button: 3 000 looks at each of 40 ranges, measured Pd vs the closed-form prediction.
The science
Radar range equation. A pulse of peak power Pt from an antenna of gain G illuminates a target of radar cross-section σ at range R. The echo power back at the antenna is
Pr = Pt G² λ² σ / ((4π)³ R⁴ L).
Doubling the range costs 12 dB (R-4). Gain follows from the beam: G ≈ 0.5 · 41253 / (θaz θel), with θ ≈ 70λ/D.
Noise and threshold. Receiver noise N = kT0BF with B ≈ 1/τ, so SNR = Pr/N. A noise-only envelope is Rayleigh distributed; a threshold VT gives false-alarm probability Pfa = exp(-VT²/2σ²). A steady target adds a Rician envelope, so Pd = Q1(√(2 SNR), √(-2 ln Pfa)) (Marcum Q); for a Swerling-1 target Pd = Pfa1/(1+SNR). Check: Pfa = 10-6, Pd = 0.9 needs 13.2 dB for a steady target.
Integration. About n = PRF · θaz / (6 · rpm) pulses hit the target per scan; this lab integrates them ideally (SNR × n, coherent).
Resolution. Range resolution ΔR = cτ/2; the receiver is blind inside cτ/2. Azimuth resolution is the beamwidth, which is why a point target smears into an arc on the PPI.
Ambiguity. Pulses are sent every 1/PRF, so an echo is placed at R mod Ru with Ru = c/(2 PRF) (second-trip echoes fold back). The Doppler shift fd = 2vr/λ is only measured modulo PRF, giving vu = λ PRF/4. Hence Ru · vu = cλ/8: raising the PRF to fix velocity shortens the range, and vice versa. Two or more PRFs resolve both (Chinese-remainder idea). An MTI canceller multiplies the echo power by sin²(2πvr/(λPRF)), so speeds at multiples of λPRF/2 are "blind".
Weather. Rain is a volume target; the Probert-Jones equation gives Pr = π³ Pt G² θφ cτ |K|² Z / (1024 ln2 λ² R² L) (R-2, not R-4), with Z = 200 R1.6 (Marshall-Palmer).
Simplifications: flat-earth slant range with a 4/3-earth horizon cut-off, no atmospheric attenuation, ideal integration, truth-assisted plot extraction (a plot is made from threshold-crossing cells that belong to the target), traffic is time-lapsed (the Time-lapse slider speeds the aircraft, not the antenna). This is a physics teaching model of civil surveillance radar, not a model of any real system; numbers are illustrative.
Controls
Scope: click to probe a point (apparent range and the true ranges it could be). 3D site: drag to orbit, wheel / pinch to zoom, buttons switch camera. On a phone use the tabs above the stage to flip between scope and 3D site.