Math Bridge: UWB Bandwidth and Timing Scales

A field team has a real problem to settle: What does bandwidth tell us about separating UWB signal paths? They must decide what happens before they change bandwidth on the device. Predict the direction first.

See the relationship first

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.

Derive the baseline in four moves

1
Name the input. The chapter baseline for bandwidth is 499.2.
2
Name the rule. Δt = 1/(499.2x10⁶) = 2.003 ns d = (3x10⁸ x 2.003x10⁻⁹)/2 = 0.3005 m At 20 MHz: Δt = 50 ns and d = 7.50 m
3
Put in the chapter value. Set bandwidth to 499.2. The page rule gives an inverse-bandwidth time scale of 2.0 ns.
4
Read the result. Keep ns next to the value. Use it only within the limits on this page.

Predict, then change bandwidth

Try Predict what happens to the inverse-bandwidth time scale. Move one control, calculate, then check your idea.

Observe Inverse bandwidth is a path-separation scale, not a hard timestamp-precision or ranging-error floor. Signal processing can estimate an isolated arrival more precisely; multipath and calibration can increase actual error. Reset to 499.2 and compare the time scale.

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

1. Start with the physical story

Inverse bandwidth gives a rough time scale for separating overlapping signal features. It does not set a hard limit on timestamp precision or range error: an isolated arrival can be estimated more precisely when signal quality and processing permit, while multipath and calibration can make error larger.

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

2. Name every algebra move

Invert bandwidthInvert bandwidth to estimate a rough time scale for separating signal features.

Convert unitsMultiply seconds by one billion to report nanoseconds.

Convert the time scaleMultiply by wave speed and divide by two to make a two-way distance comparison.

Compare scalesDivide bandwidths and compare the distance scale with the robot boundary.

3. Reproduce the chapter case

Δt = 1/(499.2×10⁶) = 2.003 ns
d = (3×10⁸ × 2.003×10⁻⁹)/2 = 0.3005 m
At 20 MHz: Δt = 50 ns and d = 7.50 m

The 0.3005 m value is a two-way distance comparison scale derived from 1/B; it is not a hard floor on timestamp precision or range-estimation error.

4. Try one real input

TryMove occupied bandwidth and watch the inverse relation connect a time scale, pulse length, and distance comparison.

ObserveAt 499.2 MHz, 1/B is about 2.00 ns and the two-way distance comparison is 0.300 m, 24.96x smaller than the 20 MHz comparison scale.

ExplainInverse bandwidth is a path-separation scale, not a hard timestamp-precision or ranging-error floor. Signal processing can estimate an isolated arrival more precisely; multipath and calibration can increase actual error.

Technical boundaries.

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

Approximation: 1/B is a rough path-separation time scale, not a timestamp-precision or range-error limit.
Propagation: Multipath and non-line-of-sight bias can dominate measured range error.
Safety: A 0.300 m comparison scale does not certify a 0.25 m exclusion boundary.

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

5. Use the result in the lab

Compare the screen with the required boundary, then measure bias and spread across line-of-sight and obstructed positions.

6. Record the evidence state

Keep channel, bandwidth, preamble and correlator settings, antenna delay calibration, geometry, reference distance, bias, and spread.

7. Check yourself

Does 1/B set a hard floor on timestamp precision or ranging error?

Answer: No. It is a rough path-separation time scale; an isolated arrival can be estimated more precisely, while multipath and calibration can make error larger.

Why divide by two in the distance equation?

Answer: The named time is a round-trip interval, while range is one-way distance.

Does a 499.2 MHz channel guarantee 30 cm accuracy?

Answer: No. The 0.300 m value is a comparison scale, not a hard error floor or an accuracy guarantee; signal quality, processing, multipath, and calibration affect the estimate.

Honesty boundary.

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

Computed: Inverse-bandwidth time scale, pulse length, two-way distance comparison, and comparison ratios.
Specified: Occupied bandwidth, wave speed, reference bandwidth, and robot boundary.
Observed: Bias, repeatability, first-path quality, and errors under real geometry and obstruction.

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