BLE Beacon Indoor Positioning

Radio Environment

Path-loss exponent n 2.5

RSSI noise σ (dB) 3.0

Position smoothing (EMA) 0.25

Phone walk speed 1.0×

Live Beacons (min 3 needed)

Position Estimate

Position error

0.00 m

Running RMSE

0.00 m

True (x, z)

0.0, 0.0

Estimated (x, z)

0.0, 0.0

How it works

Each fixed BLE beacon broadcasts at a known reference power. The phone measures Received Signal Strength Indicator (RSSI) and inverts the log-distance path-loss model to estimate its range to that beacon:

RSSI(d) = TxPower − 10·n·log10(d) + noise
d_est   = 10 ^ ((TxPower − RSSI) / (10·n))

TxPower is the calibrated RSSI at 1 m (−59 dBm here). n is the path-loss exponent — 2.0 in free space, rising toward 4 in a room with walls and furniture that absorb and scatter the 2.4 GHz signal. Real hardware adds thermal noise and multipath fading, modelled here as Gaussian noise on the dB reading before the inversion, which is why RSSI-based ranging is famously noisy even though the underlying formula is simple.

With three or more distance estimates d₁…d_k to beacons at known positions, the phone's 2D position (x, z) is recovered by linearizing the circle-intersection equations and solving the resulting normal equations with weighted least squares (Gauss–Newton, one iteration per frame is enough since the phone moves slowly relative to the update rate):

‖(x,z) − beacon_i‖ = d_i   for i = 1..k
→ linearize around last estimate, solve Aᵀ·A·Δ = Aᵀ·b

An exponential moving average (EMA) smooths the raw per-frame solve, trading responsiveness for stability — exactly the tradeoff a real indoor-positioning SDK makes when fusing BLE ranges with accelerometer/gyroscope dead-reckoning. Toggle beacons off to watch the estimate degrade below three active anchors (multilateration becomes under-determined), raise the path-loss exponent to see a cluttered room blow up the error, and drop noise to zero to confirm the solver converges to the exact true position.

BLE beacon Phone (true) Estimated position True path Estimated path

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