Eikonal Equation: Coverage Wavefront
Equation: |∇T(x)|² = 1/c(x)² is the Eikonal equation, where T(x) is the arrival time of the signal wavefront at position x, and c(x) is the local propagation speed.
Fast Marching Method: A Dijkstra-like algorithm that computes T(x) efficiently. Towers initialize T=0. The front expands outward, with arrival time increasing with distance and decreasing with speed.
c(x) field: Free space: c = c₀. Inside buildings/obstacles: c = c₀ × slowdown factor. This models diffraction and penetration loss — signals propagate slowly through buildings, creating coverage shadows.
Isochrones: Contour lines at equal T values show the wavefront shape — like ripples in a pond, but distorted by obstacles. Closely spaced contours → rapid arrival (strong coverage gradient).
LTE relevance: T(x) predicts the signal propagation delay and coverage "shadow zones" behind buildings. Used in Radio Environment Map (REM) construction. Minimum T over all towers gives the actual best-server coverage map. This directly informs antenna placement optimization.
Colors: Dark red = near tower (early arrival). Yellow-orange = mid-range. White = far from all towers (late/poor coverage). Gray = obstacle cells.
Try: Increase obstacle density to see complex shadow patterns. Add more towers to fill coverage gaps. Adjust c₀ to simulate different frequency bands (higher freq = more obstruction).