Stacking Noise Reduction and Sizing the Deadband

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

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Ada ADA · CALCULATION AUDIT

Stacking Noise Reduction and Sizing the Deadband

Three DS18B20 sensors, each accurate to only ±0.5°C, sit behind a filter-and-fusion calculator that promises averaging 10 readings shrinks that noise to 0.158°C, and fusing all three sensors together gets to 0.289°C. The same panel sizes a hysteresis deadband so a relay turns ON at 28°C and OFF at 26.5°C, a 1.5°C band. This audit asks the question those numbers invite: is that 1.5°C deadband really tied to the sensors’ own noise, or just a round number that happens to look safe?

Companion to the chapter Lab: Sensor Best Practices — every number here comes from that chapter.

Ada: The “Putting Numbers to It” panel reduces noise three ways – filtering, fusion, and a hysteresis band. Two of those combine in a way worth making explicit, and the third is pinned to the sensor’s own noise. Let me check each with the panel’s sigma = 0.5 C.

  • Filtering N samples: 0.5 / sqrt(5) = 0.224 C, 0.5 / sqrt(10) = 0.158 C, 0.5 / sqrt(20) = 0.112 C – the panel’s 2.24x, 3.16x, 4.47x.
  • Fusing M = 3 independent sensors: 0.5 / sqrt(3) = 0.289 C, a 1.73x gain.
  • Do both – fuse 3 sensors, then average 10 samples: 0.5 / sqrt(3 x 10) = 0.5 / sqrt(30) = 0.0913 C, a sqrt(30) = 5.48x total. The two gains multiply because independent variances add, so sqrt(3) x sqrt(10) = sqrt(30) – the calculator’s combined figure.
  • Hysteresis deadband: 2 x 0.5 + 0.5 = 1.5 C, giving ON at 28 C and OFF at 26.5 C.

The deadband is not an arbitrary 1.5 C – it is 2 sigma of measurement noise plus a control margin, so random jitter alone (roughly +/-2 sigma) cannot carry a reading across both thresholds and chatter the relay. That links the two halves of the chapter: drive the noise down first – fusion then filtering leaves 0.0913 C – and the same rule would permit a far tighter band near 2 x 0.0913 + 0.5 = 0.68 C. The stability you can hold is set by the measurement uncertainty you are willing to pay for.

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