A physics teaching model. Everything here is textbook radar physics with generic round numbers; it is not a model of any real system, and the echo objects (airliner, light aircraft, bird flock, ship, rain) are inanimate civil examples. Nothing here optimises detection, evasion or targeting of anything.
1. The radar range equation
A transmitter of peak power Pt and antenna gain G illuminates a target of radar cross-section σ; the echo collected by the same antenna is
Pr = Pt G² λ² σ / ((4π)³ R&sup4;), G = η (π D / λ)² (η = 0.6)
Receiver noise energy is kT0FL (k = 1.38e-23 J/K, T0 = 290 K, noise figure F = 3 dB, losses L = 4 dB, fixed here). A matched filter turns the pulse into a single sample with
SNR1 = Pt τ G² λ² σ / ((4π)³ R&sup4; kT0FL) → Rmax&sup4; = Pt τ G² λ² σ / ((4π)³ kT0FL · SNRmin)
The R-4 law: doubling the range costs 12 dB; to double Rmax you need 16 times the energy Ptτ. Coherent integration of M pulses adds 10 log M dB.
2. Detection, false alarms and the ROC
With complex Gaussian noise (std σn per quadrature) the envelope is Rayleigh. A threshold T gives Pfa = exp(-T²/2σn²). For a steady (Swerling 0) target of power SNR, Pd = Q1(√(2 SNR), T/σn) (Marcum Q function, evaluated numerically). Pd = 0.9 at Pfa = 1e-6 needs about 13.2 dB. Real targets fluctuate (Swerling 1-4) and need several dB more; not modelled.
3. Pulse timing, ambiguities and Doppler
R = c Δt / 2, ΔR = c / 2B, Runamb = c / (2 PRF)
fd = 2 vr / λ, vunamb = ± λ PRF / 4, blind speeds n λ PRF / 2
The two ambiguities pull in opposite directions: Runamb · vunamb = cλ/8 is fixed by the wavelength, so a low PRF gives unambiguous range but aliased speeds and a high PRF the reverse. Echoes beyond Runamb fold back to R mod Runamb (the dashed amber ring on the PPI).
4. The range-Doppler map and MTI
A coherent processing interval of M pulses is windowed and Fourier-transformed along slow time for each range gate; one Doppler cell is λ PRF / (2M) in speed and the target peak gains M (rectangular window), or about 0.67 M with Hann in exchange for much lower sidelobes. Ground clutter sits at zero Doppler with a small Gaussian spread σv (0.8 m/s here, wind-blown vegetation) and falls roughly as R-3. A moving-target indicator is a slow-time FIR filter: 2-pulse |H|² = 4 sin²(π fdT), 3-pulse 16 sin4(π fdT), with noise gains 2 and 6. For Gaussian clutter the improvement factor of the 2-pulse canceller is 1/(1 - ρ(T)), ρ = exp(-2π²σf²T²), σf = 2σv/λ. The price is the notch itself: any target whose Doppler is a multiple of the PRF (a blind speed), or that moves tangentially (vr = 0), is cancelled along with the clutter. Try the "Doppler demo" preset and set the probe speed to the first blind speed.
5. Pulse compression
A plain pulse ties resolution to length (ΔR = cτ/2): finer resolution needs a shorter pulse and therefore less energy. A linear-FM chirp s(t) = exp(jπ(B/τ)t²) spreads bandwidth B over a long pulse τ; the matched filter (a correlation, done here by FFT) compresses it to about 0.886/B, a compression ratio of Bτ, while keeping the energy Ptτ (so the SNR is that of the long pulse). The unweighted output is a sinc with a -13.3 dB first sidelobe. A weighting window lowers the sidelobes (Hamming about -41 dB here) at the cost of a wider main lobe (x1.3) and a mismatch loss (1.35 dB). Range-Doppler coupling of the chirp is not modelled.
6. Antenna beam and scan
Half-power beamwidth of a uniformly lit aperture: θ ≈ 0.886 λ/D. Hits per beam = PRF · θ / ω (ω = rotation rate); these are integrated coherently on the PPI. The scope shows threshold crossings (set by Pfa) with persistence, so noise false alarms appear at the expected rate.
7. The radio horizon
With standard atmospheric refraction, rays are straight on an Earth of effective radius kRE, k = 4/3. The horizon distance for height h is d = √(2kREh) = 4.12 √h km (h in metres), and a radar at hr sees a target at ht out to d = 4.12(√hr + √ht) km. A target below that line is invisible no matter how much power you add. The ridge adds an obstacle to the line-of-sight test (ray height versus the bulge x(d-x)/(2kRE) plus terrain).
8. Weather radar
Reflectivity Z = Σ D6 per unit volume (mm6/m3), quoted as dBZ = 10 log Z. The operational relation is Z = 200 R1.6 (Marshall and Palmer; WSR-57 convention); integrating the exponential Marshall-Palmer drop-size distribution N(D) = 8000 e-ΛD, Λ = 4.1 R-0.21, gives about Z = 296 R1.47 (dashed). A beam-filling rain volume follows the Probert-Jones equation Pr = π³ Pt G² θφ h |K|² Z / (1024 ln2 λ² R²) (|K|² = 0.93, h = pulse length): the power falls only as R-2, so the weakest detectable reflectivity rises 20 dB per decade of range. Attenuation by rain and the melting-layer bright band are not modelled.
9. What the experiments check
"SNR vs R^-4" runs the whole simulated signal chain (noise, tone, Doppler FFT, normalisation) and compares the measured peak with the closed form, so it tests the processing and bookkeeping rather than re-deriving the equation. The Monte-Carlo ROC, the MTI clutter attenuation, the radio-horizon ray trace, the drop-size integrals and the pulse-compression width, sidelobe and gain are independent numerical results compared with their analytic values. Tolerances are printed with each line.
Simplifications
Single fan beam with a Gaussian azimuth pattern, no elevation pattern, multipath or lobing; free-space propagation (no atmospheric or rain attenuation); steady targets; white Gaussian noise; a fixed receiver chain; range gates sampled at half a resolution cell with independent noise; no range-Doppler coupling; clutter from a fixed procedural map. The targets move along simple lines so that the PPI stays readable; the pink probe is held at your chosen range for repeatable measurements. Models in the 3D view are enlarged (1 unit = 5 km); the beam width, Earth curvature and ranges are true to scale.
References
M. Skolnik, Introduction to Radar Systems; M. Richards, Fundamentals of Radar Signal Processing; J. Marcum (1960) on the probability of detection; J. Marshall and W. Palmer (1948) on the distribution of raindrops; P. Probert-Jones (1962) on the weather radar equation; R. Doviak and D. Zrnic, Doppler Radar and Weather Observations.