Math Bridge: UWB Link Margin Before GDOP

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Math BridgeUWBPositioning

Is the far-corner error weak geometry, or no detectable first path?

Close the anchor link budget before asking geometry to amplify a measurement.

Eddie, the electronics guideEddie guides
The one targetCompute whether the chapter's far corner reaches the receiver floor.
The chapter case30 m by 20 m, 4.62 cm wavelength, -14.3 dBm EIRP, and -85 dBm sensitivity.
What it buys youA clean split between a missing range and an imprecise geometric solution.

A field team has a real problem to settle: Is the far-corner error weak geometry, or no detectable first path? They must decide what happens before they change path exponent on the device. Predict the direction first.

See the relationship first

The figure reads from left to right. The blue card is path exponent. The middle card uses this page's rule. The green card is floor diagonal. Follow the arrows: set the input, use the rule, then read the result and its unit.

The audit later on checks more than one number. Here, the added model uses the baseline named below and holds every other chapter value fixed. That sentence bridges the fixtures, so the numbers do not change without a reason.

Path exponent changes floor diagonal An input card leads through the page rule to the floor diagonal result. SET INPUT ONE CONTROL USE RULE predict calculate check units READ RESULT
Follow the arrows. A negative link margin is a detectability failure. GDOP describes what geometry does after valid range observations exist.

Derive the baseline in four moves

  1. 1

    Name the input. The chapter baseline for path exponent is 2.

  2. 2

    Name the rule. d = √(30² + 20²) = 36.06 m; PL(1 m) = 48.69 dB For n = 2: PL(36.06 m) = 79.83 dB Pr = -14.3 - 79.83 = -94.13 dBm; margin = -9.13 dB

  3. 3

    Put in the chapter value. Set path exponent to 2. The page rule gives floor diagonal as 36.06 m.

  4. 4

    Read the result. Keep m next to the value. Use it only within the limits on this page.

Predict, then change path exponent

Try Predict what happens to floor diagonal. Move one control, calculate, then check your idea.

2
Chapter baseline
Floor diagonal

Observe A negative link margin is a detectability failure. GDOP describes what geometry does after valid range observations exist. Reset to 2 and compare floor diagonal.

Explain Only path exponent moves here. The other chapter values stay fixed.

Check yourself

What should you do before you trust the result?
Answer: Predict its direction, use the shown rule, 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 limits stay fixed.

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

1. Start with the physical story

GDOP can only amplify ranges that exist. If the first path falls below the receiver floor, changing anchor angles does not turn that absent observation into a valid range.

Eddie: Name the physical limit first; the algebra then has one honest job.

2. Name every algebra move

1

Find the diagonalSquare both floor sides, add them, then take the square root.

2

Anchor the lossCompute free-space loss at one metre from wavelength.

3

Scale by environmentAdd 10n log10(d) for the selected path exponent.

4

Close marginSubtract path loss from EIRP, then subtract receiver sensitivity.

3. Reproduce the chapter case

d = √(30² + 20²) = 36.06 m; PL(1 m) = 48.69 dB
For n = 2: PL(36.06 m) = 79.83 dB
Pr = −14.3 − 79.83 = −94.13 dBm; margin = −9.13 dB

This reproduces the chapter's far-corner shortfall before any GDOP multiplication is applied.

4. Try one real input

TryMove the path exponent from free space toward clutter while floor size, EIRP, wavelength, and receiver floor stay fixed.

Path exponent
Floor diagonal
Loss at 1 m
Loss at 10 m
Power at 10 m
Margin at 10 m
Loss at diagonal
Power at diagonal
Margin at diagonal
Ideal zero-margin range

ObserveAt n = 2, 10 m keeps about 2.01 dB while the 36.06 m diagonal is 9.13 dB below the named receiver floor.

ExplainA negative link margin is a detectability failure. GDOP describes what geometry does after valid range observations exist.

Technical boundaries.

This is a bounded formula screen, not a deployment approval.

EIRP
The -14.3 dBm total value is the chapter's regulatory screen, not every radio's measured output.
Exponent
One exponent cannot describe every aisle, shelf, body, and obstruction.
Positioning
Positive margin still does not certify timestamp bias or position accuracy.

Correct, not complete: use the measured state named above before release.

5. Use the result in the lab

Map first-path detectability and margin before tuning anchor geometry; add anchors where signal reach fails, not only where GDOP looks weak.

6. Record the evidence state

Keep anchor and tag parts, EIRP, channel, receiver configuration, floor geometry, obstruction state, first-path quality, ranges, and position residuals.

7. Check yourself

Can good GDOP rescue a -9.13 dB link?
Answer: No. Geometry cannot use a range that was not detected reliably.
What does n = 2 mean?
Answer: It is the ideal free-space distance exponent, used here as the most generous screen.
Does positive 10 m margin certify 10 m position accuracy?
Answer: No. Detection, timestamp bias, multipath, calibration, and geometry remain separate evidence.
Honesty boundary.

The bridge keeps calculation, chosen inputs, and field evidence separate.

Computed
Diagonal, one-metre anchor loss, distance loss, received powers, margins, and ideal zero-margin range.
Specified
Wavelength, EIRP, sensitivity, floor dimensions, and selected path exponent.
Observed
First-path detection, range bias, obstruction state, residuals, and final position error.

Correct, not complete: this page does not certify hardware, coverage, safety, capacity, or compliance.

Return to UWB Positioning Systems with detectability checked before GDOP →