Track Correlation and Bandwidth

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

ADA · CALCULATION AUDIT

Track Correlation and Bandwidth

The chapter fuses two trackers reporting 10.2 m and 10.4 m, each with 0.4 m uncertainty, into a combined 0.28 m — and separately cuts a 50-room sensor stream from 640 bytes/s to about 111 bytes/s, an 83% reduction. Both results are architecture choices about how much evidence moves and how confident the answer may be. This audit checks track correlation and bandwidth together, because each holds only under the independence it assumes.

Companion to the chapter Sensor Fusion Architectures — every number here comes from that chapter.

Ada: The architecture choices in this chapter are really arithmetic choices about how much evidence moves and how confident the fused state is allowed to be. Both worked examples hold up, but only while the independence assumption is kept in view.

Take the track-to-track fusion above: tracker A reports 10.2 m and tracker B reports 10.4 m, each with sigma 0.4 m. Treating them as independent:

• Each precision: 1 / 0.4^2 = 1 / 0.16 = 6.250000
• Combined precision: 6.250000 + 6.250000 = 12.500000
• Combined position: (10.2 x 6.25 + 10.4 x 6.25) / 12.5 = (63.75 + 65.0) / 12.5 = 128.75 / 12.5 = 10.300000 m
• Combined sigma: sqrt(1 / 12.5) = sqrt(0.08) = 0.282843 m, rounded to 0.28 m.

That 0.28 m is smaller than either tracker’s 0.4 m, but only if the two tracks did not both lean on the same upstream anchor. If they did, the true uncertainty stays near 0.4 m and the 0.28 m is optimistic by roughly 0.400 - 0.283 = 0.117 m.

The hierarchical example is the same discipline applied to bytes. Fifty rooms, four sensors each, 32-byte readings every 10 seconds:

• Centralised raw stream: 50 x 4 x 32 / 10 = 6400 / 10 = 640.000000 bytes/s
• Room-level state, one 64-byte message every 30 s: 50 x 64 / 30 = 3200 / 30 = 106.666667 bytes/s, rounded to 106.7.
• Building summary, one 128-byte message every 30 s: 128 / 30 = 4.266667 bytes/s, rounded to 4.3.
• Upstream total: 106.666667 + 4.266667 = 110.933333 bytes/s, rounded to 111.0.
• Reduction: 1 - 110.933333 / 640 = 1 - 0.173333 = 0.826667, about 83% less traffic.

The audit conclusion is narrow: an 83% bandwidth cut and a 0.28 m fused sigma are both real, but each is safe only when the summary still carries the time span, sensor health, and common-source flags that prove the numbers were not double-counted.

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