Math Bridge: The Indoor Path Exponent

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Math BridgeBluetooth and BLEStruggle-friendly runway

Why does the same −74 dBm mean 3.98 m or 5.62 m?

Derive the free-space exponent, recompute the chapter’s −59/−74 dBm beacon, and test the NEAR/FAR fade margin.

Radio Remi, the guideRadio Remi guides
The one targetSee how n changes an RSSI distance.
The chapter case−59 dBm, −74 dBm: 3.98 m at n=2.5.
What it buys youTreat zone boundaries as measured margins, not exact metres.

A technician must decide whether indoor distance estimate is safe before changing measured rssi on the real device. The result is unresolved until the rule and units are checked. Predict the direction first.

See the relationship before changing it

The figure reads from left to right. The blue card is measured rssi. The middle card applies this page's rule. The green card is indoor distance estimate. 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 model keeps those stated values fixed and changes only measured rssi, so the numeric fixture does not switch without explanation.

Measured RSSI changes indoor distance estimate An input card leads through the rule implementation distance = 10^((-59 dBm - RSSI) / 25) to the indoor distance estimate result. INPUT PAGE INPUT APPLY THE RULE predict calculate check units OUTPUT RESULT
Walk the arrows. A weaker measured RSSI maps to a longer distance under the fixed indoor exponent.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline is -74 dBm.

  2. 2

    Name the relationship. implementation distance = 10^((-59 dBm - RSSI) / 25)

  3. 3

    Substitute with units. 10^((-59 - -74) / 25) = 3.98 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 measured rssi

Try Predict the direction of implementation distance = 10^((-59 dBm - RSSI) / 25). Test another measured rssi, then compare indoor distance estimate.

-74 dBm
Chapter baseline
Indoor distance estimate

Observe A weaker measured RSSI maps to a longer distance under the fixed indoor exponent. Reset measured rssi to -74 and compare indoor distance estimate.

Explain A weaker measured RSSI maps to a longer distance under the fixed indoor exponent.

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 measured rssi moves here. Field effects named in the technical boundary stay fixed.

Honesty note. This teaching model changes only measured rssi. It is not a field prediction.

1. The exponent describes the room

Free-space power spreads over a sphere, so it falls with distance squared and n starts at 2. Walls, bodies, reflections, and diffraction change the measured slope; a fitted indoor n summarises that environment over a stated range.

Radio Remi: Keep units and assumptions beside every number.

2. Name every algebra move

1

Write the room modelRSSI=TxPower−10n log10(d).

2

Move and divide(TxPower−RSSI)/(10n)=log10(d).

3

Raise tend=10^((TxPower−RSSI)/(10n)).

3. A zone boundary needs margin

FM=RSSI_predicted(d)−RSSI_threshold

A positive margin stays on the intended side of the threshold before shadowing. A negative margin means the nominal model has already crossed the classifier boundary.

4. Try one controlled change

FM=RSSI_predicted(d)−RSSI_threshold

TryMove the measured RSSI while both candidate exponents stay fixed.

RSSI
Distance at n=2.5
Distance at n=2
Exponent-only gap
−84 dBm example
3 m prediction
Boundary margin
Body-swing strong
Body-swing weak

ObserveAt −74 dBm the indoor model gives 3.98 m and free space gives 5.62 m, a 41.3% gap caused only by the assumed environment.

ExplainThe formula is not wrong; the model choice is unverified. At 3 m the nominal prediction is −70.9 dBm, so ±8 dB shadowing crosses both sides of the −70 dBm rule.

Technical boundaries.

One path-loss exponent is a teaching model, not a room map.

Obstruction and fading
Vary across rooms and channels
Calibration and orientation
Change the measured RSSI reference
Zone decisions
Need distributions and hysteresis

Do not assign a zone from one RSSI sample or one fitted exponent.

5. Reproduce the chapter values

With TxPower=−59 dBm, RSSI=−74 dBm, and n=2.5, d=10^(15/25)=3.98 m. Using n=2 gives 10^(15/20)=5.62 m. At 3 m the indoor model gives −59−25log10(3)=−70.9 dBm; applying ±8 dB gives about −62.9 to −78.9 dBm.

6. Carry the evidence forward

Record device and firmware, calibrated one-metre power, fitted n and range, channel, orientation, room state, raw RSSI distribution, zone hysteresis, false transitions, and labelled test positions.

7. Check yourself

Where does free-space n=2 come from?
Answer: Power density falls as 1/d² when energy spreads over a sphere.
Is −74 dBm an exact distance?
Answer: No. It becomes a distance only after choosing and validating TxPower and n.
What does a −0.9 dB margin mean?
Answer: The nominal 3 m prediction is already on the FAR side of the −70 dBm threshold.
Honesty boundary.

The chapter values compare two stated environments.

n=2.5
Illustrative indoor path
n=2.0
Illustrative open path
±8 dB
Sensitivity study, not a confidence interval

Correct model outputs do not prove real location accuracy.

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