Math Bridge: BLE RSSI Distance Zones

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Math BridgeBluetooth and BLEStruggle-friendly runway

Why can −67.8 dBm mean anywhere from 1.29 to 3.91 m?

Undo the log-distance equation and carry a ±6 dB swing through the same formula instead of drawing a symmetric error bar.

Radio Remi, the guideRadio Remi guides
The one targetTurn BLE RSSI into an honest distance zone.
The chapter case−59 dBm at 1 m, n=2.5, measured RSSI −67.8 dBm.
What it buys youUse zones without pretending RSSI is a tape measure.

A technician must decide whether nominal indoor distance is safe before changing observed beacon 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 observed beacon rssi. The middle card applies this page's rule. The green card is nominal indoor distance. 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 observed beacon rssi, so the numeric fixture does not switch without explanation.

Observed beacon RSSI changes nominal indoor distance An input card leads through the rule Python estimate = 10^((-59 dBm - RSSI) / 25) to the nominal indoor distance result. INPUT PAGE INPUT APPLY THE RULE predict calculate check units OUTPUT RESULT
Walk the arrows. Weaker observed RSSI maps to a greater distance under the fixed calibration.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline is -67.8 dBm.

  2. 2

    Name the relationship. Python estimate = 10^((-59 dBm - RSSI) / 25)

  3. 3

    Substitute with units. 10^((-59 - -67.8) / 25) = 2.25 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 observed beacon rssi

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

-67.8 dBm
Chapter baseline
Nominal indoor distance

Observe Weaker observed RSSI maps to a greater distance under the fixed calibration. Reset observed beacon rssi to -67.8 and compare nominal indoor distance.

Explain Weaker observed RSSI maps to a greater distance under the fixed calibration.

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

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

1. RSSI is a logarithmic power reading

The scanner compares measured power with a one-metre calibration. Subtracting dBm values gives a dB loss. Dividing by 10n and raising 10 to that power turns the loss back into a distance multiplier.

Radio Remi: A fixed dB error multiplies distance; it does not add a fixed number of metres.

2. Name the rearrangement

1

Start with log distanceRSSI=RSSI1m−10n log10(d).

2

Isolate the logarithmlog10(d)=(RSSI1m−RSSI)/(10n).

3

Undo log base 10d=10^((RSSI1m−RSSI)/(10n)).

3. Carry uncertainty through the exponent

dnom=10^((P1m−RSSI)/(10n)); dlow=dnom/10^(Δ/(10n)); dhigh=dnom×10^(Δ/(10n))

With n=2.5 and Δ=6 dB, the multiplier is 10^(6/25)=1.738. That produces asymmetric metre differences around the nominal estimate.

4. Try one controlled change

d=10^((RSSI1m−RSSI)/(10n))

TryMove only the measured RSSI. Calibration, path exponent, ±6 dB field swing, and zone thresholds stay fixed.

Nominal distance
−6 dB-side distance
+6 dB-side distance
Uncertainty factor
−70 dBm boundary
−55 dBm boundary
2.4 GHz wavelength
1 m free-space term

ObserveAt −67.8 dBm, the estimate is 2.25 m. The ±6 dB swing spans 1.29–3.91 m; −70 dBm maps to 2.75 m and −55 dBm to 0.692 m.

ExplainThe logarithm makes equal dB changes equal distance ratios. A zone boundary can be useful, but the calculated metres are not centimetre-accurate positions.

Technical boundaries.

The interaction holds one fitted path exponent and one-metre calibration constant.

RSSI variation
Bodies, orientation, multipath, antennas, and receiver filtering move it
Path exponent
Must be fitted for the environment and may change across the route
Distance zone
Supports coarse proximity decisions, not safety-critical position

Calibrate at the site and validate confusion rates around every chosen threshold.

5. Reproduce the worked sample

d=10^((−59−(−67.8))/25)=10^0.352=2.25 m. The 6 dB factor is 10^(6/25)=1.738, so 2.25/1.738=1.29 m and 2.25×1.738=3.91 m. This corrects the tempting but incompatible symmetric ±0.8 m shortcut.

6. Carry the evidence forward

Record beacon calibration, receiver model, antenna and enclosure, height, orientation, smoothing window, raw RSSI distribution, fitted n, threshold confusion matrix, obstruction states, and site retest date.

7. Check yourself

Why does weaker RSSI produce a larger distance?
Answer: RSSI1m−RSSI grows, so the exponent and distance multiplier grow.
Why are the metre errors asymmetric?
Answer: The same dB swing multiplies and divides distance by 1.738 rather than adding a fixed amount.
What does the −70 dBm boundary prove?
Answer: Only a modelled zone threshold under the stated calibration, not exact physical range.
Honesty boundary.

The page reproduces the chapter’s equation and corrects its uncertainty statement.

2.25 m
Exact model output for −67.8 dBm with n=2.5
1.29–3.91 m
Same model under a full ±6 dB swing
Zone labels
Operational guardrails requiring site validation

RSSI measures received power; distance is an inference, not a sensor reading.

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