Track Correlation and Bandwidth

Track Correlation and Bandwidth

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

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Ada 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.

Try

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. Calculate this case.

Observe

This audit checks track correlation and bandwidth together, because each holds only under the independence it assumes. Check shows this.

Explain

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. Check confirms it.

See the relationship before changing it

The figure reads from left to right. The blue input is rooms. The middle card names the page’s rule. The green output is traffic reduction. The arrow matters: change the input, apply the rule once, then read the result with its unit.

Rooms changes traffic reduction A three-part teaching diagram connects rooms, the rule saving = 1 - (rooms x 64/30 + 128/30) / (rooms x 4 x 32/10), and traffic reduction. INPUT Rooms APPLY THE RULE predict calculate check units OUTPUT RESULT
Walk the arrow. The fixed building summary matters most in a small estate.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline is 50 rooms.

  2. 2

    Name the relationship. saving = 1 - (rooms x 64/30 + 128/30) / (rooms x 4 x 32/10)

  3. 3

    Substitute with units. 50 rooms gives 82.67% less traffic

  4. 4

    Read the result. Keep the unit beside the value, then use the result only inside the technical boundary below.

Predict, then change rooms

Try Predict how traffic reduction responds when rooms moves. Calculate rooms; compare traffic reduction with that prediction.

50 rooms
Chapter baseline
Traffic reduction

Observe Return to 50 rooms. Recheck traffic reduction with rooms at its chapter value.

Explain The fixed building summary matters most in a small estate.

Check yourself

What should you do before trusting a moved-slider result?
Answer: Predict its direction, apply the displayed relationship, keep the units, and compare the reset value with the chapter’s worked baseline.
What does this small model leave out?
Answer: Only rooms moves here. The traffic reduction calculation excludes field effects listed below.

Honesty note. The traffic reduction result holds rooms as the only changing input. It is not a field prediction.

Technical boundaries

Outside the fixed “Track Correlation and Bandwidth” arithmetic are track-association mistakes, packet loss, timestamp skew, burst traffic, compression overhead, or compute and queueing latency; “Track Correlation and Bandwidth” therefore reports only its named fixtures.

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.