Nonlinear distortion: measurements with complex signals, audibility questions, model of music

Modeling polyphonic music with complex tones for worst-case testing.

July 18, 2023

When we want to assess the audibility of nonlinear distortion we are facing with an infinite possibilities of signals. The most important criteria for choosing the 'right' signals is the harmonic structure since the audibility of nonlinear distortion largely determined by the harmonic structure of the signal. Harmonic structure also determines the relation to real-life signals. Measurements and tests with pure tones and two-tones provide high sensitivity, but they are far from real-life signals. On the other hand, 1/3 octave spaced tones are too 'dense' and easily hide distortion components.

Another challenge is modeling polyphonic music, synthesize the harmonic structure of polyphonic music in a way that is suitable for worst-case nonlinear distortion testing/measurement.

For the following measurements I used an old (but excellent) Toshiba Satellite laptop. The laptop has a Realtek audio chip and it came with a "Certified for Windows" logo, which means that the audio hardware meets some quality requirements.

Measurements are loop measurements, which means that the headphone out was connected to microphone input. FFT size was 32768, which is close to NSD (Noise Spectral Density) at a sample rate of 44.1 kHz and 48 kHz (only affects the displayed noise floor, not important now). Microphone input has higher noise than the headphone out.

Harmonic distortion measurements

Distortion at 1 kHz / 0 dBFS:

Distortion components below audibility thresholds. SINAD is 77 dB, THD is 0.015%.

Distortion at 1 kHz / -6 dBFS:

Distortion at 5 kHz / 0 dBFS:

SINAD is now 64 dB (3rd harmonic distortion has a ~6 dB/octave rise from 1 kHz).

So how does it perform with muzik?

Here comes the multitones

Nonlinear behavior of amplifiers and DACs can be perfectly characterized by harmonic distortion measurements. We only need to take two measurements: a high level sine sweep measurement and a distortion vs. power (or distortion vs. output voltage) measurement at 1 kHz. From this point of view, measuring amplifiers and DACs with complex tones is redundant. (loudspeakers are different...)

Measuring nonlinear distortion with multitones have the following benefits:

we can see the effect of different nonlinearities in one signal;
20 kHz bandwidth is sufficient for all distortion measurements;
assessment of audibility of nonlinear distortion at high frequencies;
modeling polyphonic music is only possible with multitones (in our case the harmonic structure is important).

Four tone magic

First, let's meet a four-tone signal with a unique harmonic structure. The signal is actually a combination of two two-tone signal with different base frequencies. Intermodulation between different frequency ranges and intermodulation within frequency ranges can be detected. The large gap is intentional since it gives the signal a clean sound.

Components	Crest factor, dBre	Level of one tone, dBFS
1f₀, 1.33f₀, 20f₀, 23.33f₀	8.4	-11.4
1f₀, 1.33f₀, 20f₀, 26.66f₀	8.4	-11.4

Level of one tone is interpreted for 0 dBFS (digital full scale sine wave, digital peak).

With 500 Hz base frequency we get: 500 Hz, 666.6 Hz, 10 kHz, 11.66 kHz and 500 Hz, 666.6 Hz, 10 kHz, 13.33 kHz.

Measurement data for 500 Hz + 666.6 Hz + 10 kHz + 11.66 kHz. Signal level was 0 dBFS:

Measurement data for 500 Hz + 666.6 Hz + 10 kHz + 13.33 kHz. Again, signal level was 0 dBFS:

The latter measurement with hearing threshold (combination of masking threshold and Absolute Threshold of Hearing). 0 dBFS corresponds to 100 dBSPL and masking threshold was calculated for 80 dBSPL tonal masker:

Minor issues at 4 kHz, but nothing serious. (Note: distortion components should be summed on the ERB scale (~1/9 octave above 1 kHz), but it doesn't affect our results so we can ignore this step.)

Six tone magic

The six tone version is a really good worst-case model of polyphonic music. It covers the full audio range, complex, but still clean sounding, not too strict, not too permissive. Frequencies: 60Hz, 80 Hz, 500 Hz, 666.6 Hz, 10 kHz, 13.33 kHz. Crest factor is 10.35 dBre, level of harmonics in 0 dBFS signal is -15.1 dBFS.

Six tone complex set to peak level (0 dBFS):

With hearing threshold (0 dBFS = 100 dBSPL):

Distortion components are well below the masking threshold. With this DAC nonlinear distortion is only audible with high frequency two-tones (such as 14 kHz + 15 kHz), or with complex signals that lack frequency components below 10 kHz.

Notes: My original intention was to write about multitones - even the working title of the article was "Playing with multitones". It turned out - as one step followed another - that it could also be about modeling polyphonic music.

Csaba Horváth

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