The Noise Budget
The Noise Budget
Ada re-derives this chapter’s own numbers step by step, at full precision
ADA · CALCULATION AUDIT
The Noise Budget
Thermal noise, averaging gain, and ADC resolution, ~4 minutes
Averaging is only defensible when the noise model and units agree. Recalculate the chapter’s own constants before deciding whether thermal noise, quantization, or sensor accuracy is the limiting factor.
A 10 kΩ source at room temperature (~300 K) sets a thermal-noise density near 13 nV per sqrt(Hz), and the chapter’s worked example claims averaging 16 samples buys +12 dB of SNR while a 12-bit ADC over 3.3 V resolves about 0.081 degree C per count. Its own noise-limited-resolution example then works out to 0.002 degree C against a sensor accuracy of only ±0.5 degree C. This audit asks the question those numbers invite: does the arithmetic really support the claim that more averaging cannot turn a ±0.5 degree C sensor into a 0.002 degree C instrument?
Companion to the chapter Sensor Noise and Averaging Limits — every number here comes from that chapter.
See the relationship before changing it
The figure reads from left to right. The blue card is noise-ledger samples. The middle card applies this page's rule. The green card is white-noise snr gain. 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 noise-ledger samples, so the numeric fixture does not switch without explanation.
Derive the baseline in four named moves
- 1
Name the input. The chapter baseline is 16 samples.
- 2
Name the relationship. gain = 20 log10(sqrt(N)) = 10 log10(N)
- 3
Substitute with units. 10 log10(16) = 12.04 dB
- 4
Read the result. Keep the unit beside the value. Use it only inside the technical boundary on this page.
Predict, then change noise-ledger samples
Try Predict the direction of gain = 20 log10(sqrt(N)) = 10 log10(N). Test another noise-ledger samples, then compare white-noise snr gain.
Observe Independent averaging improves SNR by the square-root law but cannot remove fixed accuracy error. Reset noise-ledger samples to 16 and compare white-noise snr gain.
Explain Independent averaging improves SNR by the square-root law but cannot remove fixed accuracy error.
Check yourself
What should you do before trusting a moved-control result?
What does this small model leave out?
Treat T = 300 K, R = 10 kOhm, bandwidth = 1000 Hz, and N = 16 as fixed noise-ledger inputs; predict the trend before Check audit.
Thermal density reads 12.9 nV/sqrt(Hz), integrated front-end noise maps to about 0.002 C, yet the sensor accuracy remains +/-0.5 C.
Averaging N = 16 samples quarters independent white noise, but fixed accuracy error and drift remain correlated and do not average away.
The chapter’s own constants, rechecked
Use only the values already on this page: k = 1.38e-23 J/K, T = 300 K, R = 10000 ohms, bandwidth = 1000 Hz, N = 16, a 12-bit ADC over 3.3 V, sensor sensitivity 10 mV per degree C, and integrated input noise 20 uV RMS.
| Check | Arithmetic | Review meaning |
|---|---|---|
| Thermal-noise density | sqrt(4 x 1.38e-23 x 300 x 10000) = 1.286e-8 V/sqrt(Hz) = 12.9 nV/sqrt(Hz) |
The resistor’s physical noise floor is tiny, but it scales with bandwidth. |
| Integrated RMS noise | 12.9 nV x sqrt(1000) = 407 nV = 0.407 uV RMS |
At this bandwidth, the thermal noise is far below the later 20 uV input-noise example. |
| Averaging gain for 16 samples | sqrt(16) = 4; 20 x log10(4) = 12.04 dB |
The 12 dB claim is the same as four times less white-noise RMS. |
| ADC temperature step | 3.3 V / 4096 = 0.000805664 V = 0.805664 mV; 0.805664 mV / 10 mV per degree C = 0.0806 degree C |
One ADC count is about 0.081 degree C before calibration and sensor accuracy are considered. |
| Noise-limited temperature resolution | 20 uV / (10000 uV per degree C) = 0.002 degree C |
The electrical noise estimate is much smaller than the example sensor accuracy limit of +/-0.5 degree C. |
Audit conclusion: the arithmetic supports the chapter’s warning. More averaging can improve white-noise-limited readings, but it does not make a +/-0.5 degree C sensor into a 0.002 degree C instrument, and it cannot remove drift or fixed bias.
Every number above is taken from the chapter’s own noise and averaging examples and re-derived step by step.
Technical boundaries: The RMS budget assumes stationary uncorrelated white sources and omits 1/f noise, interference tones, ADC reference noise, aliasing, sensor drift, calibration bias, self-heating, and temperature-dependent bandwidth.