Stacking Noise Reduction and Sizing the Deadband
Stacking Noise Reduction and Sizing the Deadband
Ada re-derives this chapter’s own numbers step by step, at full precision
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.
See the relationship before changing it
The figure reads from left to right. The blue card is samples per sensor. The middle card applies this page's rule. The green card is three-sensor random spread. Walk the arrows once: set the input, apply the rule, then read the result with its unit.
The retained audit below checks several chapter fixtures. This model keeps those stated values fixed and changes only samples per sensor, so the numeric fixture does not switch without explanation.
Derive the baseline in four named moves
- 1
Name the input. The chapter baseline is 10 samples.
- 2
Name the relationship. spread = 0.5 deg C / sqrt(3 sensors x samples)
- 3
Substitute with units. 0.5 / sqrt(3 x 10) = 0.0913 deg C
- 4
Read the result. Keep the unit beside the value. Use it only inside the technical boundary on this page.
Predict, then change samples per sensor
Try Predict the direction of spread = 0.5 deg C / sqrt(3 sensors x samples). Test another samples per sensor, then compare three-sensor random spread.
Observe Fusion and averaging multiply independent sample counts before the square root is taken. Reset samples per sensor to 10 and compare three-sensor random spread.
Explain Fusion and averaging multiply independent sample counts before the square root is taken.
Check yourself
What should you do before trusting a moved-control result?
What does this small model leave out?
Treat 0.5 C sensor noise, M = 3 fused devices, and N = 10 filtered samples as the fixed audit case; predict the combined uncertainty before choosing Check audit.
The combined-noise readout falls to 0.0913 C; the original hysteresis rule yields a 1.5 C deadband and the reduced-noise case permits about 0.68 C.
Fusion M = 3 and averaging N = 10 multiply independent sample counts, while a deadband above twice the remaining noise prevents relay chatter.
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, asqrt(30) = 5.48xtotal. The two gains multiply because independent variances add, sosqrt(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.
Technical boundaries: The sqrt(MN) estimate presumes independent Gaussian noise and ignores common-mode bias, sensor correlation, filter delay, process dynamics, actuator lag, asymmetric noise, and relay wear.