DSP Toolbox · Page 39

Harmonics hit the mirror

Nonlinear processing creates harmonics above the original frequency. Some may sit above human hearing — and some may also cross the digital system's Nyquist boundary. In sampled audio they do not simply continue upward forever: they fold back into the representable spectrum as aliases.

48 kHz sample rate → Nyquist = 24 kHz audible region ≈ 0–20 kHz 20–24 kHz = representable, but above the usual nominal hearing range above 24 kHz = cannot be represented at 48 kHz → FOLD / ALIAS

1. Drag the fundamental and watch the reflections

The upper half of the ruler shows where the harmonics mathematically want to go. Once a harmonic passes 24 kHz, its sampled version appears reflected back below Nyquist.

5.00 kHz
10
48.0 kHz
Representable harmonic Harmonic above Nyquist Alias reflected into output Nominal hearing limit / Nyquist
Nyquist24.0 kHz
First harmonic beyond Nyquist5th · 25.0 kHz
Alias returns at23.0 kHz
Audible aliases—
The grey line is not an extra sound stored above Nyquist. It is a teaching view of the frequency the nonlinear maths attempted to create. The red line is where that component appears after sampling.

2. The reflection rule

Example at 48 kHz: fundamental = 10 kHz 2nd harmonic = 20 kHz ✓ 3rd harmonic = 30 kHz ✗ above 24 kHz 30 kHz folds around Nyquist: 24 − (30 − 24) = 18 kHz So a wanted 30 kHz harmonic appears as an 18 kHz alias.

Higher components can fold more than once. The interactive ruler calculates the final alias inside the 0 → Nyquist interval.

3. Follow individual harmonics

HarmonicWanted frequencyStatusSampled result

Red entries are especially interesting when the final alias lands below roughly 20 kHz: an ultrasonic harmonic has returned as audible inharmonic content.

Why this matters

4. Distortion can create its own aliasing problem

A clean 8 kHz sine is perfectly representable at 48 kHz. But a nonlinear function may generate 16, 24, 32, 40 kHz and beyond. The source signal was legal; the processor created the troublesome frequencies.

Next step: run the nonlinear stage at a higher internal sample rate, filter the newly created ultrasonic content, then return to the host sample rate. That is the reason for oversampling.