Radio Link Lab: Link Budget, Fresnel Zones, Knife-Edge Diffraction and Terrain Viewshed

Blender 5.2 + Three.js

Loading Blender-built valley, masts and vehicle...

P_rx - margin - d - Fresnel -

RSSI at the receiver (dBm)

-130-90-50

Scenarios

Live numbers

Graph

Relay siting

Search every hilltop for the site that covers the most village area while keeping hop 1 alive.

Experiments

Press "Run all checks" to compare the page's own numbers with closed-form theory and a brute-force reference.

Scope. Civil radio-propagation physics: emergency communications, rescue relays, rural links, amateur radio. The valley is a procedurally generated fictional landscape, not a real elevation model. The page contains no jamming, interception or direction-finding tools, and no military network design.

The science

Teaching model. Textbook radio engineering with the simplifications stated below; real planning uses calibrated tools and measured clutter data (ITU-R P.526, P.530, P.833, P.838).

Free-space loss. FSPL = 20 log10(4 pi d / lambda) dB, equivalent to 32.44 + 20 log10 d(km) + 20 log10 f(MHz). Doubling the distance or the frequency adds 6 dB.
Link budget. P_rx = P_tx + G_tx + G_rx - L_fs - L_diff - L_foliage - L_rain - L_cable - L_other (dBm). The fade margin is P_rx - sensitivity; 10 to 20 dB is a usual design margin. Directional antennas use a quadratic-in-dB pattern G(theta) = G0 - 3 (theta / (BW/2))^2 (capped at -30 dB). Dish gain 10 log10(0.55 (pi D / lambda)^2), beamwidth about 70 lambda / D degrees.
Radio horizon and Earth bulge. With effective Earth radius k R (k = 4/3, standard atmosphere): d = 4.12 (sqrt(h1) + sqrt(h2)) km (h in m) and the bulge in a path is h_b = d1 d2 / (12.74 k) m (d in km). The profile graph adds this bulge to the terrain.
Fresnel zones. First-zone radius r1 = sqrt(lambda d1 d2 / (d1 + d2)). If at least 60 % of r1 is clear everywhere, the path behaves like free space (rule of thumb). The 3D ellipsoid is drawn enlarged by the chosen factor and coloured by local clearance (green, amber, red).
Knife-edge diffraction. Fresnel-Kirchhoff parameter nu = h sqrt(2 (d1 + d2) / (lambda d1 d2)) (h above the line of sight; positive = obstructed). Lee approximation J(nu) = 6.9 + 20 log10( sqrt((nu - 0.1)^2 + 1) + nu - 0.1 ) dB for nu > -0.78; 6 dB at grazing (nu = 0). The check below compares it with the exact Fresnel integral |F|^2 = 1/2 [ (0.5 - C(nu))^2 + (0.5 - S(nu))^2 ] evaluated by numerical quadrature.
Several edges. Deygout method: the edge with the largest nu is the principal edge; the sub-paths on each side are treated the same way (principal + two secondary edges, secondary edges at least 240 m from their neighbours). Deygout over-estimates for closely spaced edges. The coverage map uses only the dominant edge (maximum nu along each ray).
Foliage and rain. Foliage: modified exponential decay in the style of Weissberger / ITU-R P.833 for in-leaf trees, L = 1.33 f^0.284 d^0.588 dB (f in GHz, d metres of path inside forest below the 18 m canopy, 14 to 400 m; 0.45 f^0.284 d below 14 m), scaled by the density slider. Rain: gamma = k R^alpha dB/km with table values in the style of ITU-R P.838 (negligible below 1 GHz) and the P.530 distance factor 1 / (1 + d / d0). Both are rough, illustrative numbers.
Two-ray ground reflection. Direct and reflected rays with path difference delta = sqrt(d^2 + (ht + hr)^2) - sqrt(d^2 + (ht - hr)^2), about 2 ht hr / d interfere; the reflection coefficient is the horizontal-polarisation Fresnel coefficient of damp ground (er = 15, sigma = 5 mS/m). The last maximum is at the break distance d_b = 4 ht hr / lambda; beyond it the loss follows 40 log10 d - 20 log10(ht hr) (40 dB per decade).
Viewshed and coverage. Horizon-angle algorithm: subtract the tangent-plane drop s^2 / (2 k R) from every terrain sample on the ray from the transmitter; a cell is visible when its elevation angle is not below the running maximum. Each cell also gets P_rx = EIRP - L_fs - J(nu_max) - L_foliage - L_rain + G_rx. The map assumes the receive antenna is aimed at the transmitter and the transmit pattern is aimed at the receiver (or the relay).
Relay. Two hops: the end-to-end margin is the smaller of the two hop margins (regenerative repeater, no extra delay or frequency-offset modelling). The optimiser tries every local hilltop and scores the village area covered, with hop 1 required to close.
Simplifications. 62.5 m terrain cells with bilinear heights, no ducting or tropospheric scatter, no polarisation or building clutter, flat-Earth 3D drawing (the bulge is in the maths and in the profile graph), one sensitivity number instead of a noise budget, models enlarged when the camera is far away (physics always uses the true heights).