Math Bridge: Accelerometer aliasing and quantisation

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Math BridgeDesign MethodologyStruggle-friendly runway

How can a 240 Hz fault look like a 40 Hz wobble?

Keep amplitude rounding and time sampling separate, then use the datasheet conditions to predict what firmware will see.

Blueprint Bina, the design guideBlueprint Bina guides
The one targetPredict code step and frequency alias.
The chapter case±16 g; 13 bits; 200 Hz ODR; 1,800 RPM motor.
What it buys youA datasheet comparison tied to the fault signal.

A field team has a real problem to settle: How can a 240 Hz fault look like a 40 Hz wobble? They must decide what happens before they change real signal on the device. Predict the direction first.

See the relationship first

The figure reads from left to right. The blue card is real signal. The middle card uses this page's rule. The green card is motor rotation. Follow the arrows: set the input, use the rule, then read the result and its unit.

The audit later on checks more than one number. Here, the added model uses the baseline named below and holds every other chapter value fixed. That sentence bridges the fixtures, so the numbers do not change without a reason.

Real signal changes motor rotation An input card leads through the page rule to the motor rotation result. SET INPUT ONE CONTROL USE RULE predict calculate check units READ RESULT
Follow the arrows. Sampling repeats the spectrum every 200 Hz. Firmware receives the nearest folded copy, not the original frequency label.

Derive the baseline in four moves

  1. 1

    Name the input. The chapter baseline for real signal is 240.

  2. 2

    Name the rule. q=32 g/8192=3.906 mg/LSB SNRideal=6.02x13+1.76=80.02 dB 1800 RPM/60=30.0 Hz; Nyquist=200/2=100 Hz 240 Hz is the eighth rotation harmonic and aliases to |200-240|=40.0 Hz

  3. 3

    Put in the chapter value. Set real signal to 240. The page rule gives motor rotation as 30.0 Hz.

  4. 4

    Read the result. Keep Hz next to the value. Use it only within the limits on this page.

Predict, then change real signal

Try Predict what happens to motor rotation. Move one control, calculate, then check your idea.

240
Chapter baseline
Motor rotation

Observe Sampling repeats the spectrum every 200 Hz. Firmware receives the nearest folded copy, not the original frequency label. Reset to 240 and compare motor rotation.

Explain Only real signal moves here. The other chapter values stay fixed.

Check yourself

What should you do before you trust the result?
Answer: Predict its direction, use the shown rule, keep the units, and reset to the worked baseline.
What does this small model leave out?
Answer: Only real signal moves. Field effects named in the page limits stay fixed.

Honesty note. This teaching model changes only real signal. It replays the chapter baseline; it is not a field prediction.

1. Two grids act on one signal

Quantisation rounds acceleration onto vertical amplitude steps. Sampling records the signal only at horizontal time instants. More bits make smaller vertical steps; higher sample rate moves the Nyquist boundary.

Blueprint Bina: A good amplitude code cannot rescue a frequency that was sampled too slowly.

2. Name the algebra moves

1

Use the full spanA ±16 g range spans 32 g.

2

Count levels13 bits gives 2¹³=8,192 codes.

3

Divideq=2FS/2ᴺ.

4

Set NyquistfN=fs/2.

5

Fold the signalfalias=|nfs−f| using the nearest repeated spectrum.

3. Reproduce the motor case

q=32 g/8192=3.906 mg/LSB
SNRideal=6.02×13+1.76=80.02 dB
1800 RPM/60=30.0 Hz; Nyquist=200/2=100 Hz
240 Hz is the eighth rotation harmonic and aliases to |200−240|=40.0 Hz

The 40 Hz output is not random noise. It is the deterministic low-frequency disguise created by sampling a 240 Hz component at 200 samples per second.

4. Try the real vibration frequency

TryMove the real component across the 90–300 Hz bearing-defect band while ODR stays 200 Hz.

Real signal
Motor rotation
Nyquist
Code step
Ideal SNR ceiling
Recorded alias
Rotation harmonic

ObserveCode step and ideal SNR stay fixed because bits and range stay fixed. The alias moves as the real frequency moves.

ExplainSampling repeats the spectrum every 200 Hz. Firmware receives the nearest folded copy, not the original frequency label.

Technical boundaries.

This is ideal sampling and quantisation arithmetic, not a complete accelerometer response.

SNR
80.02 dB is a quantisation ceiling before sensor, circuit, and reference noise
Aliasing
Real antialias filters attenuate frequencies gradually rather than at a perfect wall
Mechanics
Mounting and MEMS resonance shape which vibration reaches the digital path

Measure the installed spectrum and verify ODR, filter, range, and noise settings.

5. Test the datasheet row

Inject or measure known vibration across the target band. Compare raw spectra at several ODR and filter settings while watching for clipping and noise.

6. Record the sampling state

Store range, bits, ODR, filter bandwidth, mount, motor speed, expected harmonics, raw samples, analysis window, and the conditions attached to each datasheet value.

7. Check yourself

What does 13-bit resolution set here?
Answer: A 3.906 mg/LSB ideal amplitude step across ±16 g.
Why does 240 Hz appear at 40 Hz?
Answer: At 200 samples/s, the nearest repeated spectrum gives |200−240|=40 Hz.
Does 80.02 dB describe the real sensor?
Answer: No. It is only the ideal quantisation ceiling.
Honesty boundary.

The ±16 g, 13-bit, 200 Hz, and 1,800 RPM values reproduce the chapter's teaching example.

3.906 mg/LSB
Ideal code spacing
80.02 dB
Ideal quantisation ceiling
40.0 Hz
Ideal alias of one 240 Hz component

Correct, not complete: ideal sampling arithmetic does not qualify vibration monitoring.

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