Math Bridge: Audio sampling and aliasing

A field team faces an unresolved physical question: Why can a 16 kHz audio pipeline turn ultrasound into an audible feature? They must answer it before changing tone on the real device. Predict the direction first.

See the relationship before changing it

Set the input, apply the rule, then read the result with its unit.

The retained audit below checks several chapter fixtures. This added model holds every other chapter fixture fixed, so the numeric fixture does not switch without explanation.

Derive the baseline in four named moves

1
Name the input. The chapter baseline for tone is 20000.
2
Name the relationship. 16,000/2 = 8,000 Hz 16,000x0.025 = 400 samples; 16,000x0.010 = 160 samples floor((16,000-400)/160)+1 = 98 frames/s 20,000 Hz → |20,000-16,000| = 4,000 Hz; 16 bit → 98.1 dB
3
Substitute the chapter fixture. Set tone to 20000. The page ledger gives alias as 4.0 kHz.
4
Read the result. Keep kHz beside the value. Use it only inside the technical boundary on this page.

Predict, then change tone

Try Predict the direction of alias. Move one control, calculate, then check your prediction.

Observe Predict an increase in alias as the input increases over this range. Reset to the chapter baseline and compare.

Explain Only tone moves here. The other chapter fixtures remain fixed.

Source: content/analytics-ml/modeling-audio-features.qmd — Phoebe's Field Notes: Why 16 kHz Audio Cannot Hear An 18-22 kHz Tone Honestly. Use the chapter equation and fixed ledger above; only the named input varies.

Check yourself

What should you do before trusting a moved-control result?

Answer: Predict its direction, apply the shown relationship, keep the units, and reset to the worked baseline.

What does this small model leave out?

Answer: Only tone moves. Field effects named in the page's technical boundary stay fixed.

1. Start with the clock

A sample is one timed measurement. The sample clock decides which frequencies can be distinguished before an FFT or MFCC exists.

Data Dora: A larger buffer stores more samples; it does not raise the Nyquist ceiling.

2. Name every algebra move

Halve the rateNyquist ceiling = fs/2.

Multiply by timesamples = fs × seconds.

Fold to the nearest copyfalias = |f − round(f/fs)fs|.

Count ideal converter rangeSNR = 6.02N + 1.76.

3. Reproduce the chapter

16,000/2 = 8,000 Hz
16,000×0.025 = 400 samples; 16,000×0.010 = 160 samples
floor((16,000−400)/160)+1 = 98 frames/s
20,000 Hz → |20,000−16,000| = 4,000 Hz; 16 bit → 98.1 dB

The 18, 20, and 22 kHz examples fold to 2, 4, and 6 kHz.

4. Try an emitted tone

TryMove the sensing tone through 18–22 kHz.

ObserveThe feature schedule stays fixed while the false in-band alias moves.

ExplainAliasing creates plausible evidence at the wrong frequency, so metadata must bind sample rate to the model.

Technical boundaries.

This engine treats tones and an ideal converter.

Clock: No jitter or analogue filter response
Signal: No broadband noise or microphone response
MFCC: No window leakage or mel-filter loss

Measure the complete microphone and feature pipeline.

5. Separate the use cases

Speech and ultrasonic interaction need different acquisition contracts even when both end in compact features.

6. Record what travels

Version sample rate, frame, hop, bit depth, filter, window, FFT, mel bank, model, and threshold together.

7. Check yourself

What is the 16 kHz Nyquist ceiling?

Answer: 8 kHz.

Where does 20 kHz fold?

Answer: 4 kHz.

Does 98.1 dB prove the microphone is that quiet?

Answer: No; it is an ideal 16-bit converter ceiling.

Honesty boundary.

The chapter owns the digital schedule; the tone is its separate sensing example.

16 kHz, 25 ms, 10 ms: Chapter keyword pipeline
18–22 kHz: Chapter ultrasonic interaction band
98.1 dB: Ideal PCM relation

Correct, not complete: this arithmetic does not qualify an audio detector.