How Much the Path-Loss Exponent Moves the Fix

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.

See the relationship before changing it

The figure reads from left to right. The blue card is path-loss exponent. The middle card applies the page rule. The green card is beacon a distance. Walk the arrows once: set the input, apply the rule, then read the result with its unit.

Path-loss exponent changes beacon a distance An input card leads through the rule distance = 10^(6 / (10 x path-loss exponent)) to the beacon a distance result. INPUT PAGE INPUT APPLY THE RULE predict calculate check units OUTPUT RESULT
Walk the arrows. A wrong site exponent moves every RSSI circle even when the radio reading is unchanged.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline is 2.5.

  2. 2

    Name the relationship. distance = 10^(6 / (10 x path-loss exponent))

  3. 3

    Substitute with units. 10^(6 / (10 x 2.5)) = 1.74 m

  4. 4

    Read the result. Keep the unit beside the value. Use it only inside the technical boundary on this page.

Predict, then change path-loss exponent

Try Predict the direction of distance = 10^(6 / (10 x path-loss exponent)). Test another path-loss exponent, then compare beacon a distance.

2.5
Chapter baseline
Beacon A distance

Observe A wrong site exponent moves every RSSI circle even when the radio reading is unchanged. Reset path-loss exponent to 2.5 and compare beacon a distance.

Explain A wrong site exponent moves every RSSI circle even when the radio reading is unchanged.

Check yourself

What should you do before trusting a moved-control result?
Answer: Predict its direction, apply the shown relationship, keep the units, and reset to the worked baseline.
What does this small model leave out?
Answer: Only path-loss exponent moves here. Field effects named in the technical boundary stay fixed.

Honesty note. This teaching model changes only path-loss exponent. It is not a field prediction.

TryChoose Check derivation with TxPower -59 dBm, RSSI -65 dBm, and calibrated path-loss exponent n = 2.5.
ObserveThe first range reads 1.74 m at n = 2.5 but drifts to about 2.00 m when n is mis-set to 2.0.
ExplainDistance is 10 raised to the RSSI difference divided by 10n, so exponent calibration turns the same -65 dBm reading into different ranges.

Ready: use the stated baseline inputs, then compare each displayed result.

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.

Technical boundaries. Walls, antenna orientation, multipath, RSSI noise, body shadowing, and time-varying calibration are excluded from the fixed-reference log-distance conversion.