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.
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.