Math Bridge: Antenna detuning in an RSSI log

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Math BridgePrototypingStruggle-friendly runway

Can an RSSI log really see enclosure detuning?

Turn antenna gain loss into a received-power ratio, then compare that change with the radio's own reporting step.

Voltage Vera, the prototyping guideVoltage Vera guides
The one targetDecide whether an RSSI shift is larger than reporting noise.
The chapter caseA 4 dB enclosure loss and 1 dB RSSI steps.
What it buys youAn enclosure comparison based on resolvable evidence.

A field team faces an unresolved physical question: Can an RSSI log really see enclosure detuning? They must answer it before changing gain loss 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 gain loss. The middle card applies this page's relationship. The green card is aperture retained. 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.

Gain loss changes aperture retained An input card leads through the page relationship to the aperture retained result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. The ratio tells you whether the proposed change is large relative to this narrow noise model; it does not prove what caused the change.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for gain loss is 4.

  2. 2

    Name the relationship. aperture ratio=10 -4/10 =0.398 aperture retained=39.8% σ q =1/√12=0.289 dB evidence scale=4/0.289=13.9

  3. 3

    Substitute the chapter fixture. Set gain loss to 4. The page ledger gives aperture retained as 39.8%.

  4. 4

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

Predict, then change gain loss

Try Predict the direction of aperture retained. Move one control, calculate, then check your prediction.

4
Chapter baseline
Aperture retained

Observe The ratio tells you whether the proposed change is large relative to this narrow noise model; it does not prove what caused the change. Reset the control to 4 and compare aperture retained.

Explain Only gain loss 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 gain loss moves. Field effects named in the page's technical boundary stay fixed.

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

1. Start with gain as redistribution

Antenna gain does not create extra radio power. It concentrates available power in some directions and gives up coverage in others. For a signal arriving from one fixed direction, effective aperture describes how much power the antenna can collect. A loss in gain causes the same loss in aperture and received power.

Voltage Vera: The enclosure test needs two scales: how much the antenna changed and how coarsely the radio reports that change.

2. Name every algebra move

1

Write the aperture relationAe=Gλ²/(4π).

2

Cancel the unchanged termsAt one frequency, the aperture ratio is the gain ratio.

3

Turn dB into a power ratioUse 10−L/10 for a loss L.

4

Estimate one-step quantisation noiseσq=q/√12.

5

Compare the scalesDivide the observed dB change by σq.

3. Reproduce the chapter case

aperture ratio=10−4/10=0.398
aperture retained=39.8%
σq=1/√12=0.289 dB
evidence scale=4/0.289=13.9

A 4 dB loss leaves 39.8% of the former effective aperture. Against 1 dB RSSI steps, the simple quantisation model makes that change nearly fourteen noise standard deviations. That is far larger than ordinary one-step wobble in this model.

4. Try the enclosure loss

TryMove the measured gain loss while the RSSI reporting step stays at 1 dB.

Gain loss
Aperture retained
Aperture lost
RSSI quantisation noise
Evidence scale

ObserveA 1 dB loss keeps 79.4% of the aperture and sits much closer to the radio's reporting scale. A 4 dB loss is much harder to dismiss as one quantised step.

ExplainThe ratio tells you whether the proposed change is large relative to this narrow noise model; it does not prove what caused the change.

Technical boundaries.

This is an aperture-ratio and uniform-quantisation comparison, not a complete radio measurement model.

RSSI error
Calibration error, fading, packet variation, AGC, orientation, and interference can exceed quantisation noise
Aperture
The same-direction, same-frequency ratio cancels wavelength but not the environment
Causation
An RSSI shift alone cannot isolate the enclosure from placement or channel changes

Use repeated, paired measurements with the same nodes, channel, position, and traffic.

5. Build the paired test

Measure the bare board and enclosed board at fixed separation, orientation, channel, transmit power, packet count, and surroundings. Swap only the enclosure state, then repeat the pair. Keep the distribution of RSSI readings, not only one average.

6. Record the evidence state

Store the board, antenna, enclosure, battery position, channel, transmit setting, distance, orientation, packet count, RSSI resolution, radio calibration note, room layout, and raw logs. A moved cable or person can matter at 2.4 GHz.

7. Check yourself

Why does a 4 dB loss leave about 39.8%?
Answer: A power loss uses 10−L/10, so 10−0.4=0.398.
Why is a 1 dB step modelled as 0.289 dB of noise?
Answer: Uniform quantisation noise has standard deviation q/√12, so 1/√12=0.289 dB.
Does a 13.9-noise-step shift prove the enclosure caused it?
Answer: No. It says the shift is large relative to quantisation noise; the paired test must control other radio changes.
Honesty boundary.

The arithmetic reproduces the chapter's illustrative 4 dB loss and catalog-typical 1 dB RSSI step.

39.8%
An ideal same-frequency aperture ratio
0.289 dB
Quantisation noise only, not total RSSI uncertainty
13.9×
A comparison scale, not causal proof

Correct, not complete: this ledger does not certify antenna performance or explain a field RSSI shift by itself.

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