Math Bridge: Indoor BLE calibration

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Math BridgeUX DesignStruggle-friendly runway

How can -72 dBm mean 2.55 or 3.31 metres?

Make the building exponent visible instead of treating RSSI as a ruler.

UX Uma, the guideUX Uma guides
The one targetCalculate BLE distance as the path exponent changes.
The chapter case-59 dBm reference; -65/-72/-68 dBm beacons; n=2.5 and 3.2.
What it buys youAn indoor-location claim tied to site calibration.

A field team faces an unresolved physical question: How can -72 dBm mean 2.55 or 3.31 metres? They must answer it before changing path exponent on the real device. Predict the direction first.

See the relationship before changing it

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

The retained audit below checks several chapter fixtures. This added model holds every other chapter fixture fixed, so the numeric fixture does not switch without explanation.

Path exponent changes beacon a An input card leads through the page relationship to the beacon a result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. The path exponent sits in the denominator of the logarithmic exponent, so site calibration controls every converted distance.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for path exponent is 2.5.

  2. 2

    Name the relationship. at n=2.5: -65→1.74 m; -72→3.31 m; -68→2.29 m at hospital n=3.2: -72→10^(13/32)=2.55 m ±3 dB at n=2.5 multiplies by 0.759 and 1.318 3.31 m therefore spans about 2.51-4.36 m before geometry error

  3. 3

    Substitute the chapter fixture. Set path exponent to 2.5. The page ledger gives beacon a as 1.74 m.

  4. 4

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

Predict, then change path exponent

Try Predict the direction of beacon a. Move one control, calculate, then check your prediction.

2.5
Chapter baseline
Beacon A

Observe The path exponent sits in the denominator of the logarithmic exponent, so site calibration controls every converted distance. Reset the control to 2.5 and compare beacon a.

Explain Only path exponent moves here. The other chapter fixtures remain fixed.

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 exponent moves. Field effects named in the page's technical boundary stay fixed.

Honesty note. This teaching model changes only path exponent. It replays the chapter fixture; it is not a field prediction.

1. Treat n as a summary of the building

The path-loss exponent describes how quickly received power falls with distance in a calibrated environment. A larger n says the building absorbs or scatters power faster, so the same dB drop maps to a shorter distance.

UX Uma: RSSI does not carry metres; calibration supplies the missing environment.

2. Name every algebra move

1

Find the power dropΔR=TxPower−RSSI.

2

Divide by site losse=ΔR/(10n).

3

Undo log base tend=10e metres for a 1 m reference.

4

Turn noise into a multiplierm±=10^(±ΔRnoise/(10n)).

5

Apply the banddlow=d m− and dhigh=d m+.

3. Reproduce the three beacon estimates

at n=2.5: -65→1.74 m; -72→3.31 m; -68→2.29 m
at hospital n=3.2: -72→10^(13/32)=2.55 m
±3 dB at n=2.5 multiplies by 0.759 and 1.318
3.31 m therefore spans about 2.51-4.36 m before geometry error

The two values for -72 dBm do not contradict each other. They answer the equation with two different site models.

4. Try the path exponent

TryChange n while keeping all three readings and the -59 dBm reference fixed.

Path exponent
Beacon A
Beacon B
Beacon C
Hospital B at n=3.2
+3 dB multiplier
-3 dB multiplier
Noisy low
Noisy high

ObserveRaising n pulls all three estimates inward because the model assigns more loss to each metre.

ExplainThe path exponent sits in the denominator of the logarithmic exponent, so site calibration controls every converted distance.

Technical boundaries.

One path exponent is a coarse site summary, not a complete radio map.

Space
Walls, shelves, bodies, doors, and orientations vary by position
Time
Occupancy and interference change distributions after calibration
Geometry
Three range estimates still need anchor positions and a robust solver

Calibrate and validate zones on site with representative devices and changes.

5. Test the building model

Collect ground-truth readings across rooms, corridors, doors, crowds, phone orientations, device lots, and time. Compare zone decisions, not only mean metre error.

6. Record the indoor state

Store beacon power, reference, n, anchors, site map, sample distribution, device, firmware, solver, confidence, fallback, recalibration date, and change triggers.

7. Check yourself

Why does larger n produce a shorter estimate?
Answer: The model expects power to fall faster with distance, so the observed drop needs fewer metres.
Why can ±3 dB create an asymmetric metre band?
Answer: dB is inverted through a power of ten, producing reciprocal multiplicative factors.
Does fitting n=3.2 qualify a hospital system?
Answer: No. Spatial variation, time drift, device differences, anchor geometry, solver behaviour, and product risk remain.
Honesty boundary.

The reference, three beacon readings, n values, and ±3 dB case reproduce the chapter's worked region.

n=2.5
Illustrative calibrated indoor environment
n=3.2
Illustrative concrete-hospital comparison
±3 dB
Noise sensitivity example, not a confidence guarantee

Correct, not complete: a log-distance calibration does not qualify indoor positioning.

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