Timing Precision Sets the Accuracy Floor

Timing Precision Sets the Accuracy Floor

Ada re-derives this chapter’s own numbers step by step, at full precision

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Ada ADA · CALCULATION AUDIT

Timing Precision Sets the Accuracy Floor

This chapter lines up GPS, Wi-Fi, BLE, and UWB and claims each one’s accuracy falls straight out of how precisely it can time a signal. GPS times to about 10 ns and lands near 3 m of ranging error; UWB’s wider 500 MHz+ bandwidth resolves under a nanosecond and reaches 7.5 cm; BLE, with no fine timing, infers distance from signal strength and drifts about 0.46d for a 4 dB fade. This audit re-runs those figures to test whether timing precision, not branding, is what really sets each technology’s accuracy floor.

Companion to the chapter Location Awareness Fundamentals — every number here comes from that chapter.

See the relationship before changing it

The figure reads from left to right. The blue card is timing error. The middle card applies the page rule. The green card is range error. Walk the arrows once: set the input, apply the rule, then read the result with its unit.

Timing error changes range error An input card leads through the rule range error = 3 x 10^8 m/s x timing error in seconds to the range error result. INPUT PAGE INPUT APPLY THE RULE predict calculate check units OUTPUT RESULT
Walk the arrows. Finer timing directly tightens range before geometry multiplies it.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline is 10 ns.

  2. 2

    Name the relationship. range error = 3 x 10^8 m/s x timing error in seconds

  3. 3

    Substitute with units. 3 x 10^8 x 10 x 10^-9 = 3.00 m

  4. 4

    Read the result. Keep the unit beside the value. Use it only inside the technical boundary on this page.

Predict, then change timing error

Try Predict the direction of range error = 3 x 10^8 m/s x timing error in seconds. Test another timing error, then compare range error.

10 ns
Chapter baseline
Range error

Observe Finer timing directly tightens range before geometry multiplies it. Reset timing error to 10 and compare range error.

Explain Finer timing directly tightens range before geometry multiplies it.

Check yourself

What should you do before trusting a moved-control result?
Answer: Predict its direction, apply the shown relationship, keep the units, and reset to the worked baseline.
What does this small model leave out?
Answer: Only timing error moves here. Field effects named in the technical boundary stay fixed.

Honesty note. This teaching model changes only timing error. It is not a field prediction.

TryRun Check derivation for GPS at 10 ns, BLE with a 4 dB fade, and UWB at 0.5 ns timing precision.
ObserveThe accuracy-floor results separate into 3 m for GPS, about 0.46d for BLE, and 7.5 cm for UWB.
ExplainPropagation speed converts timing error directly into range error; UWB's 0.5 ns resolution beats GPS, while BLE substitutes a noisier RSSI-distance inference.

Ready: use the stated baseline inputs, then compare each displayed result.

Ada: The trade-off section claims each technology’s accuracy falls straight out of how precisely it can time a signal. That is checkable from c = 3 x 10^8 m/s and each system’s timing resolution. Start with GPS at about 10 ns:

  • Range error: Delta d = c x Delta t = 3 x 10^8 x 10 x 10^-9 = 3 m
  • Position error with GDOP 1.5: 1.5 x 3 = 4.5 m

BLE has no fine timing at all – it infers distance from signal strength, so a 4 dB fade propagates through Delta d = d x ln(10) x (Delta RSSI) / (10 n) with path-loss exponent n = 2: ln(10) x 4 / 20 = 2.3026 x 0.2 = 0.4605, i.e. about 0.46 d. At 5 m that is 0.46 x 5 = 2.3 m. UWB flips the advantage with sub-nanosecond timing on its 500 MHz-plus bandwidth, using two-way flight d = c x t / 2:

  • With 0.5 ns: Delta d = 3 x 10^8 x 0.5 x 10^-9 / 2 = 0.15 / 2 = 0.075 m = 7.5 cm

Three technologies, three timing floors: 3 m, a strength-limited 0.46 d, and 7.5 cm. The design-meaningful reading is that indoor-versus-outdoor accuracy is not a branding difference but a bandwidth-and-timing consequence – UWB is about 40x tighter than GPS because wider bandwidth buys finer time resolution, which is why you match the positioning technology to the decision’s clearance rather than to the number that sounds most precise.

Every number above is taken from the chapter’s own material and re-derived step by step.

Technical boundaries. Receiver acquisition, oscillator drift, multipath, estimator bias, and antenna delay are absent; fixed timing precision is converted to one-way range using propagation speed.