How Much the Path-Loss Exponent Moves the Fix

Ada re-derives this chapter’s own numbers step by step, at full precision

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Ada ADA · CALCULATION AUDIT

How Much the Path-Loss Exponent Moves the Fix

A retail store drops BLE beacons for shopper navigation and turns three RSSI readings into three distances, every one leaning on a single calibrated number: the path-loss exponent n. With TxPower = −59 dBm and a calibrated n = 2.5, the chapter’s own formula turns a −65 dBm beacon into 1.74 m — but leave n at the free-space 2.0 and the same reading drifts. This audit re-runs the trilateration to ask how much a mis-set path-loss exponent really moves the fix.

Companion to the chapter Indoor Positioning Technologies — every number here comes from that chapter.

Ada: The retail trilateration example turns three RSSI readings into three distances, and every one of them leans on a single calibrated number: the path-loss exponent n. Let me re-run the chapter’s own conversion d = 10^((TxPower - RSSI) / (10n)) with TxPower = -59 dBm and n = 2.5, then show what a mis-set n does.

  • Beacon A, RSSI -65: exponent (-59 - (-65)) / (10 x 2.5) = 6 / 25 = 0.24, so d = 10^0.24 = 1.7378 m -> 1.74 m.
  • Beacon B, RSSI -72: exponent 13 / 25 = 0.52, so d = 10^0.52 = 3.3113 m -> 3.31 m.
  • Beacon C, RSSI -68: exponent 9 / 25 = 0.36, so d = 10^0.36 = 2.2909 m -> 2.29 m.

All three match the chapter. Now the sensitivity. Take Beacon A’s identical -65 dBm reading but assume the installer left n at the free-space 2.0 the chapter warns against, instead of the calibrated 2.5:

  • d = 10^((-59 - (-65)) / (10 x 2.0)) = 10^(6/20) = 10^0.30 = 1.9953 m -> 2.00 m.

Same radio, same beacon, but the estimate jumps from 1.74 m to 2.00 m – a 14.8% distance error from one wrong exponent, and it compounds across all three circles in the trilateration. The design meaning is why the chapter insists on on-site calibration: n is not a datasheet default but the single most consequential number in the fix, so the honest indoor record stores the calibrated exponent right beside every coordinate it produces.

Every number above is taken from the chapter’s own material and re-derived step by step.