Signal Processing Lab Calculation Audit
Signal Processing Lab Calculation Audit
Ada checks sample-rate margin, ADC step size, and spectral resolution before a lab trace is release evidence
ADA · CALCULATION AUDIT
Signal Processing Lab Calculation Audit
A clean lab trace is only trustworthy when the arithmetic proves the capture could preserve the physical event before any smoothing or downsampling makes the display look calm.
A vibration lab expects useful energy up to 1 kHz and needs to see amplitude changes as small as 1 mV, so the team tests sample rates of 3.2 kHz and 4 kHz against the Nyquist minimum and checks an 8-bit ADC against a 12-bit ADC at a 3.3 V reference. This audit asks the question that setup invites: does the chosen sample-rate margin and ADC bit depth actually preserve a 1 mV event, or does the 8-bit plan fail on arithmetic alone?
Companion to the chapter Lab: Signal Processing — every number here comes from that chapter.
See the relationship before changing it
The figure reads from left to right. The blue card is highest useful vibration frequency. The middle card applies this page's rule. The green card is nyquist minimum sample rate. 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 highest useful vibration frequency, so the numeric fixture does not switch without explanation.
Derive the baseline in four named moves
- 1
Name the input. The chapter baseline is 1000 Hz.
- 2
Name the relationship. minimum sample rate = 2 x highest frequency
- 3
Substitute with units. 2 x 1,000 Hz = 2,000 samples/s
- 4
Read the result. Keep the unit beside the value. Use it only inside the technical boundary on this page.
Predict, then change highest useful vibration frequency
Try Predict the direction of minimum sample rate = 2 x highest frequency. Test another highest useful vibration frequency, then compare nyquist minimum sample rate.
Observe A wider trusted signal band demands a proportionally faster sample rate. Reset highest useful vibration frequency to 1000 and compare nyquist minimum sample rate.
Explain A wider trusted signal band demands a proportionally faster sample rate.
Check yourself
What should you do before trusting a moved-control result?
What does this small model leave out?
Treat the 1 kHz vibration band, 4.0 kHz sampling, 8-, 12-, and 16-bit code steps, and 1 mV shift as the fixed comparison; predict the result.
The 4.0 kHz plan is 2× the Nyquist minimum; at 3.3 V, one code shrinks from 12.9 mV to 0.806 mV to 0.050 mV as resolution rises.
Sample rate determines whether time variation is representable, while the ADC code step determines whether a voltage change spans enough digital levels to be distinguished.
1. Timing evidence starts from Nyquist, then adds margin
The chapter's vibration example names useful energy up to 1 kHz. The lower mathematical boundary is just the starting line:
3.2 kHz / 2 kHz = 1.6 x minimum
4.0 kHz / 2 kHz = 2.0 x minimum
2. ADC resolution is a voltage-step test, not a bit-count boast
With a 3.3 V reference, the code step is the reference divided by the available codes:
12-bit step = 3.3 V / 4096 = 0.000805664 V = 0.806 mV
16-bit step = 3.3 V / 65536 = 0.000050354 V = 0.050 mV
3. A 1 mV requirement rejects the 8-bit plan on arithmetic alone
The smallest meaningful change is smaller than one 8-bit code, but larger than one 12-bit code:
1 mV / 0.806 mV = 1.24 codes
1 mV / 0.050 mV = 19.9 codes
4. Frequency evidence needs the record length
The chapter's FFT rule stays symbolic until the lab records both sample rate and sample count:
At fs = 4 kHz: bin spacing = 4000 / N Hz
What the audit buys you: the lab can defend a sample-rate choice, reject a converter that cannot resolve the physical change, and explain why frequency claims need the captured record length before the result is release evidence.
Every number above is taken from this chapter's own worked example and re-derived step by step.
Technical boundaries: This arithmetic assumes an ideal converter and brick-wall bandwidth; it omits anti-alias filter roll-off, reference noise, ENOB, aperture jitter, record-window leakage, sensor noise, and front-end saturation.