Math Bridge: UWB PSD Link Margin

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Math BridgeUWB channel 5Margin screen

How much first-path margin survives the PSD ceiling?

Turn a per-megahertz limit into total EIRP, free-space receive power, and one-wall margin.

Eddie, the electronics guideEddie guides
The one targetConnect occupied bandwidth to a bounded first-path link screen.
The chapter caseChannel 5 at 6,489.6 MHz, 499.2 MHz bandwidth, 10 m, and one 6 dB wall.
What it buys youA test distance and obstruction hypothesis for ranging evidence.

A field team has a real problem to settle: How much first-path margin survives the PSD ceiling? They must decide what happens before they change distance on the device. Predict the direction first.

See the relationship first

The figure reads from left to right. The blue card is distance. The middle card uses this page's rule. The green card is bandwidth correction. 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.

Distance changes bandwidth correction An input card leads through the page rule to the bandwidth correction result. SET INPUT ONE CONTROL USE RULE predict calculate check units READ RESULT
Follow the arrows. More bandwidth raises integrated total power only according to the PSD cap; distance and obstruction still consume the resulting link budget.

Derive the baseline in four moves

  1. 1

    Name the input. The chapter baseline for distance is 10.

  2. 2

    Name the rule. EIRPtotal = -41.3 + 10 log10(499.2) = -14.32 dBm λ = 3x10⁸ / 6.4896x10⁹ = 0.0462 m FSPL10m = 68.70 dB; Pr = -83.02 dBm Pr,wall = -89.02 dBm

  3. 3

    Put in the chapter value. Set distance to 10. The page rule gives bandwidth correction as 26.98 dB.

  4. 4

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

Predict, then change distance

Try Predict what happens to bandwidth correction. Move one control, calculate, then check your idea.

10
Chapter baseline
Bandwidth correction

Observe More bandwidth raises integrated total power only according to the PSD cap; distance and obstruction still consume the resulting link budget. Reset to 10 and compare bandwidth correction.

Explain Only distance 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 distance moves. Field effects named in the page limits stay fixed.

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

1. Start with the physical story

UWB gains timing resolution from wide bandwidth, but a spectral-density ceiling keeps total transmit power unusually small.

Eddie: A decoded reflection does not prove the first path had enough margin for trustworthy ranging.

2. Name every algebra move

1

Integrate the ceilingAdd 10 log10 B to the dBm/MHz limit.

2

Find wavelengthUse λ = c/f at channel centre.

3

Spread powerUse FSPL = 20 log10(4πd/λ).

4

Reserve marginSubtract path and wall loss, then compare with sensitivity.

3. Reproduce the chapter case

EIRPtotal = −41.3 + 10 log10(499.2) = −14.32 dBm
λ = 3×10⁸ / 6.4896×10⁹ = 0.0462 m
FSPL10m = 68.70 dB; Pr = −83.02 dBm
Pr,wall = −89.02 dBm

The clear ideal screen barely clears −85 dBm, while the illustrative wall moves it below sensitivity.

4. Try one real input

TryMove anchor distance while the channel, bandwidth, PSD ceiling, wall loss, and sensitivity stay fixed.

Distance
Bandwidth correction
Total EIRP
Total EIRP (mW)
20 dBm / UWB power
Wavelength
Free-space loss
Clear received level
Clear margin
One-wall received level
One-wall margin

ObserveAt 10 m the ideal clear margin is about 2 dB, while the one-wall screen is about 4 dB below sensitivity.

ExplainMore bandwidth raises integrated total power only according to the PSD cap; distance and obstruction still consume the resulting link budget.

Technical boundaries.

This is an average-EIRP free-space screen, not a UWB ranging channel model.

Regulation
Use the applicable regional mask, measurement bandwidth, indoor/outdoor rules, peak limit, and certified antenna.
Receiver
Implementation loss, antenna gains, noise, preamble, data rate, integration, and detection thresholds remain separate.
Ranging
NLOS bias, first-path detection, clock error, multipath, geometry, and quality metrics decide position trust.

Correct, not complete: this ledger does not approve a UWB link or ranging result.

5. Use the result in the lab

Place anchors at distances around the screened margin, then add the real obstruction and compare first-path, receive-power, NLOS, and range-error traces.

6. Record the evidence state

Keep region and mask, channel, antenna, power settings, distance, obstruction, first-path metrics, sensitivity basis, error, and stale-data rule.

7. Check yourself

Does 499.2 MHz bandwidth permit 499.2 times the dBm value?
Answer: No. Integrate in linear power, which becomes +10 log10 B in dB.
Does a decoded packet prove a trustworthy first path?
Answer: No. A later reflection may decode while the first path is weak or biased.
Is the 6 dB wall value universal?
Answer: No. Material, thickness, angle, moisture, frequency, and geometry must be measured.
Honesty boundary.

The ledger turns an explicit PSD ceiling into a reproducible distance-and-wall hypothesis.

Computed
Integrated EIRP, wavelength, free-space loss, receive level, and margins are reproducible.
Specified
Certified mask, antenna, receiver mode, sensitivity, and material loss replace the examples.
Observed
First-path metrics, NLOS flags, range error, packet state, and repeatability decide trust.

Correct, not complete: retain channel-quality and geometry evidence with every range.

Return to UWB Fundamentals with the PSD maths unpacked →