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.

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.