Definitive measurement of nonlinear distortion with a simple harmonic spectrum multitone
Can we measure all nonlinear effects with one signal with high sensitivity? A multitone measurement that combines the sensitivity of sine signal measurement with the allrevealing nature of multitone measurements.
November 3, 2023
Nonlinear distortion measurements with sine signals and closely spaced twotones show the local effects of nonlinear distortion only. Measurements with multitone signals, on the other hand, reveal all frequencydependent effects in a single measurement. However, traditional multitone measurements cannot be considered worstcase measurement, since nonlinear distortion is most audible heard with a sine signal and twotones (e.g. 13 kHz + 14 kHz).
The most important criteria for choosing the 'right' measurement signal(s) 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 reallife signals. Measurements and tests with pure tones and twotones provide high sensitivity, but they are far from reallife signals. On the other hand, 1/3 octave spaced tones are too 'dense' and easily hide distortion components.
My goal was, therefore, to create a multitone measurement that combines the sensitivity of the sine signal measurement with the allrevealing nature of multitone measurement. Such a measurement signal should contain three sinusoidal components. A lowfrequency, a mediumfrequency and a highfrequency component. In order for a bandwidth of 20 kHz to be sufficient for distortion measurement, the highfrequency component must be split into two components and their level reduced by 3 decibels compared to the others. The optimal distance between the two components is about 1.5 kHz  2 kHz.
An overview of nonlinear distortion measurement methods can be found in the following technical papers (pdf):
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Audio Specifications (RANE)
•
Comparison Of Nonlinear Distortion Measurement Methods, Richard C. Cabot
The benefits of multitone measurements
Nonlinear behavior of amplifiers and DACs can be perfectly characterized by harmonic distortion measurements. We need only two measurements: a distortion vs frequency measurement usually performed at high level 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 is easier;
 modeling polyphonic music is only possible with multitones (in our case the harmonic structure is important).
Using multitone signals in measurements has another benefit: the relative level of harmonics (their level relative to the amplitude and their level relative to the maximum value) is similar to the relative level of harmonics in reallife audio signals. In general, the level of the largest harmonics in recordings is 1015 dB below the amplitude and the frequency range of the largest harmonics lies between 40 Hz and 2 kHz. Harmonic distortion measurement or a two tone measurement at "justbelowclipping" signal level above 10 kHz can be misleading. (Just a side note: In a loudspeaker box designer software, when cone displacement analysis is performed with a sine signal, the peak SPL can be corrected by +10 dB. )
Requirements of a worstcase multitone distortion measurement
Back to our goals, we want to measure all nonlinear effects with one signal and also apply the concept of worstcase analysis. In order to find the best multitone measurement, we need to consider the following critical points:

Measurement of all nonlinear effects with one signal
Problem: We need a measurement with high sensitivity to all effects of nonlinearity. Low, mid and high frequency distortion, including slew rate induced distortion.
Solution: multitone signal that covers the audio range. 
Harmonic structure, masking and sensitivity of the measurement
Problem: Sensitivity of a distortion measurement is a function harmonic structure. Nonlinear distortion is the most audible with signals with simple harmonic spectrum. On the contrary, signals with too dense spectrum hide the distortion components.
Solution: Large separation (at least three octave) between the components. 
Measurement of high frequency distortion with inband components and not with harmonics
Problem: In high frequency harmonic distortion measurements the distortion components fall into the ultrasonic region. Testing the audibility of distortion above ~5 kHz with pure tones is pointless. However, the audibility of distortion still matters.
Solution: The only way to assess the audibility of distortion above 5 kHz is to use twotones as a test signal (CCIF type, closely spaced tones) with correct level and spacing. 
Measurement has to take into account auditory masking (at least in a form of a weighting function)
Problem: highorder distortion is more audible than low order distortion
Solution: Masking threshold calculation or applying a weighting function (more simple). 
Condition of using the signal in listening tests
As far as possible, the signal should be pleasant sounding, clean and nondissonant. The purest intervals are powers of two (2F_{0}, 4F_{0}, 8F_{0}...). E.g. listening to a mix of 100 Hz and 800 Hz (2*2*2*100 = 800) is more pleasant than listening to a mix of 100Hz and 700 Hz (7* 100 = 700).
So the concept is: one harmonic for the bass, one for the mid and two for the hightreble. The treble part is split into two signals so that the energy of highfrequency distortion is concentrated mostly in inband distortion components. The bass and treble harmonics have equal level, the level of the treble harmonics is reduced by 3 dB.
The following measurements use these harmonics: 80Hz, 640Hz, 10240 Hz and 11520 Hz. (100Hz, 800Hz, 12800Hz, 14400Hz also a good choice).
Measurement of a DAC (laptop headphone out)
For the following measurements an old (but excellent) Toshiba Satellite laptop was used. 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).
Multitone measurements
Multitone is set to 0 dBFS. Harmonics at equal level:
It's interesting that SINAD is only 62.18 dB in this measurement and 62.47 dB in the following measurement (no Aweighting).
Same harmonics, but the two highest component is lowered with 3 dB:
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:
Distortion components are well below the masking threshold. With this DAC nonlinear distortion is only audible with high frequency twotones (such as 14 kHz + 15 kHz), or with complex signals that lack frequency components below 10 kHz. (Note: distortion components should be summed on the ERB scale (~1/9 octave above 1 kHz). Since it doesn't affect the result in this case, this step was ignored.)
An alternative method:
If the application of weighting or masking seems too complicated, we can calculate a segmented signaltodistortion ratio in each main frequency range (bass, mid, treble). Much better than an overall signaltodistortion ratio.
Csaba Horváth
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