In cellular network positioning, we measure Time of Flight (ToF) or Signal Strength (RSSI) to estimate distance from each base station. Each distance defines a circle (or sphere in 3D) around that tower. The intersection of three circles collapses the Area of Uncertainty into a single coordinate. However, four cells are needed for a robust real-world solution.

Pseudo-Range and Clock Bias

The UE's clock is not synchronized with the towers. The measured pseudo-range is:

p_i = √((x−x_i)² + (y−y_i)² + (z−z_i)²) + c·Δt

where Δt is the clock bias (time offset). In high-speed signals, time is distance: since radio travels at c ≈ 300,000 km/s, a 1 µs error yields a 300 m positioning error. The UE uses a cheap quartz oscillator, not an atomic clock, so its timer drifts. The 4th tower is mathematically required to solve for Δt.

3D and the Z-Axis

With only 3 towers, the intersection of three spheres yields two potential points (one above ground, one below). The 4th tower disambiguates the true altitude. In urban environments (e.g., 10th floor), height matters.

Dilution of Precision (DOP)

If towers are collinear, the matrix becomes singular and vertical precision is lost. Good geometry requires towers spread around the UE.

Noise Types

1. Thermal Jitter: Electronics noise causes ToA to fluctuate; the position "jitters" even when the UE is still.

2. Multipath/NLOS: Signals bouncing off buildings (Non-Line-of-Sight) travel farther; the tower thinks you are further away than you are.

3. GDOP: Poor tower geometry amplifies small measurement errors into large positioning errors. A 1 m noise can become 50 m when towers are collinear.

Centroid LLS: The 3-tower solver uses a centroid-based approach (not Tower 1-anchored), distributing error across all towers for a fair, unbiased estimate.

Weighted 4D Solver

The 4-tower solver uses weighted least squares to prioritize 2D (map) accuracy. When tower heights are similar (vertical geometry weak), the solver dampens altitude updates and weights X,Y more heavily than Z. This mimics real GPS: HDOP (horizontal) is typically better than VDOP (vertical). Without weighting, the 4-cell solution can sometimes show a larger 2D error than the 3-cell case because the solver "trades off" horizontal accuracy to fit altitude or clock bias.

Usage

Defaults: 3 towers, UE Height 0, Clock Drift 0, Thermal Jitter 0.

Drag UE: Move the black circle. The search rings (sphere projections) expand/contract.

Drag Towers: Reposition T1–T4. T4 appears only when "4 Towers" is selected.

Towers: Switch between 3 and 4 towers (default: 3). With 4 towers, the system solves for clock bias Δt; the weighted solver prioritizes 2D accuracy when clock bias is present.

UE Height: Simulates altitude (e.g., building floor). Affects 3D distance and projected circle radii.

UE Clock Drift: Simulates UE clock offset (1 µs = 300 m). With 3 towers the circles form a "buffer zone" triangle; with 4 towers the solved position stays stable. Dashed circles = true range R; solid = pseudo-range R′ = R + c·Δt.

Residual (RMS): Red = 3 towers (assume perfect clock); Green = 4 towers (clock bias corrected).

Thermal Jitter: Gaussian-like ToA fluctuation. Position "jitters" even when UE is still.

Multipath (Building): Drag the building between a tower and UE. Blocked LOS adds +50 px "ghost" distance (NLOS).

Satellite View: Overlays a map background (NYC area). Canvas pixels map to GPS: top-left (0,0) = 40.7128°N, 74.0060°W. Drag UE to see GPS fix in degrees/minutes/seconds. Heatmap alpha = 0.1 so the map stays visible. Lines are drawn thicker for visibility. Use "Satellite (NYC)" preset to position the building over a visible structure; multipath triggers when UE is behind it.

GDOP: Poor tower geometry (collinear) amplifies noise. Red uncertainty = unreliable fix.

Math Sidebar: Shows symbolic pseudo-range equation, plugged equations with (x,y,z) coordinates per tower, final solution, and Error = √((x_true−x_est)²+(y_true−y_est)²) in pixels. Residual (RMS) measures how well the circles meet at the estimate.

Key Concepts

• 4th tower: Solves for clock bias and disambiguates altitude. The weighted solver prioritizes 2D accuracy when vertical geometry is weak.
• Thermal jitter: Electronics noise causes position to dance.
• Multipath/NLOS: Reflected signal makes you appear farther than you are.
• GDOP: Poor geometry amplifies small errors into large ones.
• 3 vs 4 towers: Set UE Clock Drift to 50 µs to see the 3-tower error explode; 4-tower weighted solver corrects for clock bias and reduces 2D error.