Math Bridge: Audio Aliasing and dBFS

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Where does a 30 kHz tone go at 48 kHz?

One thread from Nyquist folding to relative dBFS and the limits of an ideal 16-bit noise number.

Phoebe, the physics guidePhoebe guides
The one targetRead a sampled spectrum without inventing evidence.
The chapter case48 kHz, 30→18 kHz, 16-bit 98.1 dB.
What it buys youSeparate digital level from calibrated sound.

A field team faces an unresolved physical question: Where does a 30 kHz tone go at 48 kHz? They must answer it before changing input audio tone frequency on the real device. Predict the direction first.

See the relationship before changing it

The figure reads from left to right. The blue card is input audio tone frequency. The middle card applies this page's relationship. The green card is observed alias. 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 added model holds every other chapter fixture fixed, so the numeric fixture does not switch without explanation.

Input audio tone frequency changes observed alias An input card leads through the page relationship to the observed alias result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. The sample rate fixes where a tone folds. dBFS describes its digital amplitude, while anti-alias rejection and real noise belong to the device path and must be tested.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for input audio tone frequency is 30000.

  2. 2

    Name the relationship. fN=fs/2; falias=|f-fs·round(f/fs)|; L=20log10(A/AFS); SNR=6.02N+1.76

  3. 3

    Substitute the chapter fixture. Set input audio tone frequency to 30000. The page ledger gives observed alias as 18000 Hz.

  4. 4

    Read the result. Keep Hz beside the value. Use it only inside the technical boundary on this page.

Predict, then change input audio tone frequency

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

30000
Chapter baseline
Observed alias

Observe The sample rate fixes where a tone folds. dBFS describes its digital amplitude, while anti-alias rejection and real noise belong to the device path and must be tested. Reset the control to 30000 and compare observed alias.

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

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 input audio tone frequency moves. Field effects named in the page's technical boundary stay fixed.

Honesty note. This teaching model changes only input audio tone frequency. It replays the chapter fixture; it is not a field prediction.

1. Sampling repeats the spectrum

A sampler can represent frequencies only inside one Nyquist band. Frequencies outside it are mirrored into that band unless the analogue input path attenuates them first.

Phoebe: A tone does not disappear just because it is ultrasonic; it can return under a false lower-frequency name.

2. Fold to the nearest sample-rate copy

1

Find the ceiling48,000/2 = 24,000 Hz.

2

Choose the nearest copyk = round(f/fs).

3

Take the absolute differencefalias = |f − kfs|.

|30,000 − 48,000| = 18,000 Hz

3. Keep level scales separate

LdBFS=20log10(amplitude/full-scale amplitude)

dBFS is a relative digital ratio. It becomes absolute sound pressure level only after microphone-path calibration. The ideal converter result SNR ≈ 6.02N + 1.76 is another bounded ratio, not a measured phone noise floor.

4. Try the input tone

fN=fs/2; falias=|f−fs·round(f/fs)|; L=20log10(A/AFS); SNR=6.02N+1.76

TryMove an ideal input tone above the chapter's 24 kHz ceiling while the 48 kHz rate stays fixed. The half-scale amplitude is labelled interaction-only.

Nyquist ceiling
Observed alias
Ideal 16-bit SNR
Half-scale interactionFS

ObserveAt 30 kHz the sampled record shows 18 kHz; at 26 kHz it shows 22 kHz. Both values are below 24 kHz and can look like genuine in-band tones.

ExplainThe sample rate fixes where a tone folds. dBFS describes its digital amplitude, while anti-alias rejection and real noise belong to the device path and must be tested.

Technical boundaries.

The folding model assumes an ideal unfiltered sampler.

measured evidence
Needs separate evidence

Use field evidence or a deeper model before release.

5. Reproduce the chapter checks

30 kHz → 18 kHz; 26 kHz → 22 kHz at fs=48 kHz
Ideal N=16 ⇒ 6.02(16)+1.76=98.1 dB

The 98.1 dB figure is total ideal quantisation-noise power under a full-scale-sine assumption, not a spectral-bin floor.

6. Write an honest lab claim

Report sample rate, FFT size, window, gain, processing, and relative dBFS. Use a controlled quiet-input or reference measurement for the actual floor, and avoid claiming SPL without calibration.

7. Check yourself

Why does 30 kHz appear at 18 kHz?
Answer: The nearest 48 kHz spectrum copy is one sample rate away, so |30−48|=18 kHz.
What does −6.02 dBFS mean in the interaction?
Answer: The amplitude is one half of digital full scale; it does not state acoustic SPL.
Does 98.1 dB prove the phone's noise floor?
Answer: No. It is an ideal quantisation result under specific assumptions.
Honesty boundary.

These are the chapter inputs, worked results, and named teaching assumptions.

48 kHz rate
Frequency, sample rate, or event rate
24 kHz ceiling
Frequency, sample rate, or event rate
30→18 kHz
Frequency, sample rate, or event rate
26→22 kHz folds
Frequency, sample rate, or event rate
ideal 16-bit 98.1 dB result reproduce the chapter
Gain, loss, margin, or level ratio
The half-scale −6.02 dBFS readout is labelled as an interaction value
Named teaching assumption

This page does not certify a phone's anti-alias filter, noise floor, or SPL.

Return to the chapter with the maths unpacked →