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.

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.