Timing Evidence: Time of Flight, Round Trips, and Proximity Thresholds
Timing Evidence: Time of Flight, Round Trips, and Proximity Thresholds
Ada re-derives the chapter’s own time-of-flight distances, round-trip range error, and proximity timing thresholds
ADA · CALCULATION AUDIT
Timing Evidence: Time of Flight, Round Trips, and Proximity Thresholds
UWB range evidence is timing evidence with physics attached. Keep the speed-of-light conversion visible before a centimetre display becomes an access, tracking, or safety decision.
UWB matters when a tag near a door, tool, forklift, phone, or robot needs more than a vague nearby claim: radio waves travel at roughly 300,000,000 m/s, making 1 ns of one-way timing error about 0.30 m, and a 2 ns round-trip timing error still about 0.30 m of range error once the two-way exchange divides it. A displayed reading of 42 cm can look precise even when the underlying evidence is only a few nanoseconds wide, and a phone-to-lock check may rely on a rule as simple as inside 1.2 m with fresh confidence. This audit asks the question that precision invites: does the timing evidence behind a number like 42 cm actually justify a 1.2 m proximity decision, or does the display just look more certain than the physics supports?
Companion to the chapter How UWB Measures Distance — every number here comes from that chapter.
See the relationship before changing it
The figure reads from left to right. The blue card is round-trip timing error. The middle card applies this page's rule. The green card is two-way range error. 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 round-trip timing error, so the numeric fixture does not switch without explanation.
Derive the baseline in four named moves
- 1
Name the input. The chapter baseline is 2 ns.
- 2
Name the relationship. range error = 0.30 m/ns x timing error / 2
- 3
Substitute with units. 0.30 x 2 / 2 = 0.30 m
- 4
Read the result. Keep the unit beside the value. Use it only inside the technical boundary on this page.
Predict, then change round-trip timing error
Try Predict the direction of range error = 0.30 m/ns x timing error / 2. Test another round-trip timing error, then compare two-way range error.
Observe Two-way ranging halves travelled error, but nanoseconds still become visible distance. Reset round-trip timing error to 2 and compare two-way range error.
Explain Two-way ranging halves travelled error, but nanoseconds still become visible distance.
Check yourself
What should you do before trusting a moved-control result?
What does this small model leave out?
Use c = 300000000 m/s to convert 1 ns one-way and derive the round-trip timing boundary for a 1.2 m proximity rule.
One nanosecond corresponds to 0.30 m of one-way travel, and the 1.2 m rule occupies only an 8 ns round-trip interval.
Radio flight time scales distance directly by light speed; a two-way exchange divides elapsed propagation by two because the signal traverses the path outward and back.
From Time of Flight to Range and Proximity
| Audit question | Arithmetic | Review consequence |
|---|---|---|
| How much distance is one nanosecond? | 300,000,000 m/s x 0.000000001 s = 0.300000000 m | One nanosecond of one-way timing error is 0.30 m, so antenna delay and timestamp calibration are not cosmetic details. |
| Why does a two-way exchange divide by two? | 300,000,000 m/s x 0.000000002 s / 2 = 0.300000000 m | A 2 ns round-trip error still becomes 0.30 m of range error after the outbound and return path are averaged. |
| What timing boundary supports a 1.2 m proximity rule? | 2 x 1.2 m / 300,000,000 m/s = 0.000000008 s = 8 ns round trip | The phone-to-lock policy should record that its 1.2 m decision depends on about 8 ns of round-trip timing evidence plus channel confidence. |
| What does a displayed 42 cm imply? | 2 x 0.42 m / 300,000,000 m/s = 0.0000000028 s = 2.8 ns round trip | The display looks precise, but the evidence claim is only a few nanoseconds wide; rounding is final, not an excuse to skip channel review. |
Use the full timing conversion during review, then round the final distance for presentation. If channel impulse response, calibration, geometry, or confidence does not support the timing claim, the application should recheck or fall back instead of consuming the number.
Every number above is taken from the chapter’s own UWB ranging example and re-derived step by step.
Technical boundaries: This ideal symmetric-flight calculation excludes antenna and cable delays, clock offset and drift, timestamp quantisation, NLOS bias, multipath, device geometry, calibration temperature, and confidence estimation.