Audibility thresholds for SINAD / THD+N measurements


Audibility thresholds for SINAD / THD+N measurements from auditory masking curves and hard clipped signal harmonics (pure-tone & two-tone).


November 23, 2023

Audibility thresholds for SINAD / THD+N measurements is a recurring question in audio magazines, forums. Although amplifier/DAC nonlinear distortion measurements were developed decades ago, interpretation of the measurements is still not clear and open to debate. So it's not surprise that exaggerated claims about SINAD/THD are still very common, arguments are based on fantasy drawn distortion threshold curves with little connection to reality. The spread of misconceptions is supported with the multitude of flawed measurement methods or incorrect interpretation of the results.

A typical mistake is to ignore the properties of hearing or real-life audio signals. As a general rule, any problems related to audio fidelity requires a three-level analysis:

Applying the previous concept to nonlinear distortion, only three questions need to be answered:


Basic concepts

Nonlinear distortion adds new components to the original signal. In the case of sinusoidal signals the resulting components are usually harmonic (there are of course exceptions, e.g. distortion from the power supply is not harmonic), in the case of "standard" complex periodic signals (e.g. triangular waveform) and solo instruments the distortion is also harmonic, while in the case of certain complex signals (several instruments playing in different pitches at the same time) both harmonic and non-harmonic components are created.

What we can hear is determined by the Absolute Threshold Of Hearing (threshold in silence) and auditory masking. Auditory masking is a change in hearing threshold due to the presence of a loud sound. Masking is most significant near the frequency of the component (shown in the figures below). This means that if the level of the distortion components falls below the absolute hearing threshold or the masking threshold, then the distorted signal is indistinguishable to the ear from the original (undistorted) version.

In audio devices, the generated distortion depends on the frequency and intensity of the signal, and in the case of complex signals, it also depends on the harmonic structure. Similarly, audibility of distortion depends on the frequency and intensity of the signal, and in the case of complex signals, on the harmonic structure. The duration of the signal is also an important factor: in transients nonlinear distortion is less audible. In fact, during transients the distortion - if audible - is only perceived as a subtle change in timbre.

Measurement of nonlinear distortion at a fixed frequency and level (THD+N, SINAD) may not be indicative of audible performance. Nonlinear distortion should be measured in the entire audio frequency range, with a signal close to maximum level (e.g. at half power or quarter power). The distortion should also be measured in a wide dynamic range at a single frequency (e.g. at 1 kHz, between maximum level and -60dBre).

An overview of nonlinear distortion measurement methods can be found in the following technical papers (pdf):
Audio Specifications (RANE)
Comparison Of Nonlinear Distortion Measurement Methods, Richard C. Cabot



Method

Since SINAD/THD+N doesn't apply weighting to harmonics, the only way to define audibility thresholds for SINAD/THD+N is to perform tests with hard clipped pure tones and two-tones. Hard clipping (limiting) is the worst kind of "normal" distortion (polarity change or signal drop-out during amplifier overdrive are worse, but they can't be considered "normal"). Fortunately, creating hard clipped signals is very simple with audio editors, though the process requires some measurements (calculating THD for every amplification factor).

Since nonlinear distortion is most audible with pure tones and two-tones, a detection threshold analysis should be based on these types of signals. Two-tones are needed because it is not possible to properly test the audibility of distortion above 1 kHz with a sine signal.

Level of the largest harmonics in music is 10-15 dB below the peak level. If the sound pressure level of a full-scale sine (0 dBFS) is 100 dBSPL and the peak level in the recording is around 0 dBFS, the level of largest harmonics will be at about 85-90 dBSPL. If we also consider that our hearing is most sensitive to high order distortion at 80-85 dBSPL (masking), then listening tests and generation of masking curves should be performed at 75 dBSPL, 80 dBSPL and 85 dBSPL.

Audibility threshold graphs were generated by mapping distortion harmonics to the SPL domain and comparing the level of harmonics with masking curves.

Masking curves were generated with the masking patterns created by Zwicker & Fastl (Psychoacoustics: Facts and Models). This masking patterns are referred as "normalized masking patterns" in their book. A combined outer and middle ear response were added to the model (result is a slightly sharper roll-off between 2 and 5 kHz).

The Absolute Threshold Of Hearing (ATH) is the ISO 226-2003 threshold in quiet curve up to 16 kHz. A short analysis of ATH and age groups can be found at the end of the article (link).

An alternative method would be using just-noticeable levels of a combined 3rd, 5th and 7th order (harmonic) distortion.

Polarity inversion (or polarity reversal). The ear is insensitive to polarity inversion. However, we can detect polarity indirectly by distortion byproducts. If we send an asymmetrical signal through an asymmetrical nonlinear stage, the generated distortion products change according to the polarity of the signal. Cone displacement in loudspeakers is rarely symmetrical, which result in 2nd order distortion at low frequencies, ear has its own asymmetrical nonlinearity (increasing 2nd order distortion from 80 dBSPL). Therefore, a "hidden" polarity inversion may generate false positive test results.
A similar problem occurs when the audibility of phase shift is tested with complex periodical tones. Even large phase shift is not audible with complex periodical signals. However, nonlinear phase shift changes the waveform and phase shifted version of complex signals can be detected "indirectly" when nonlinear distortion is not masked properly (e.g. in two-tones, three-tones). Nonlinear distortion can falsify phase shift listening tests with complex periodical signals.


Masking curves

Some notable frequencies and levels. In pure tone graphs only harmonics up to the 15th harmonic are shown (since clipping is symmetrical, even order harmonics are missing). Green line is the Absolute Threshold Of Hearing, blue line is the masking threshold.

Pure tone at 80 dBSPL / 500 Hz. THD = 0.1%:

Pure tone at 80 dBSPL / 500 Hz. THD = 0.1%:

Pure tone at 80 dBSPL / 500 Hz. THD = 0.02%:

Two-tone distortion. 80dBSPL per tone, Frequencies: 10kHz & 12.5kHz. THD = 0.1%:

It's clear that high-order distortion is more audible at low frequencies than at high frequencies.

At 1 kHz we have to investigate a special worst case: when the distortion spectrum contains only the 3rd and 5th harmonics at equal level. This lowers the threshold from 0.1% to 0.05%. Below and above 1 kHz this type of distortion is not important.


Audibility thresholds

Just-noticeable hard clipping vs. frequency in a pure tone:

Combined threshold: below 1.5 kHz detection threshold with pure tone, above 1.5 kHz detection threshold with two-tone. Detection threshold for two-tone is measured with a pure tone with the same amplitude, and not IMD:

Same combined threshold, just in THD:

Threshold for THD vs level at 1 kHz. Calculated for 0 dB = 100 dBSPL. If 0dB = 110 dBSPL, then the curve has to be shifted to the left with 10 dB and the right side kept unchanged.

What can we say about "hum" from the power supply? Since Absolute Threshold of Hearing is about 25 dBSPL at 100 Hz, the distortion component at 100Hz / 120 Hz should be 75 dB below the clipping level measured at least at quarter power (-6dB), or maximum at half power (-3dB). If SINAD is greater than 75 dB, then this distortion component is below the -75 dBre limit.

And what if the distortion is mainly 3rd order? The just-noticeable third harmonic distortion with pure tones is 55-60 dB, so even 60 dB SINAD (0.1% THD+N) is sufficient in the full audio range.


An alternative "worst-case"

2024.01.26.

We can determine just-noticeable distortion levels for equal level 3rd+5th+7th harmonics and equal level 3rd+5th harmonics (required for threshold at 1 kHz). The advantage of this method is simplicity: since the distortion spectrum is reduced to three harmonics, bandlimiting has practically no effect on signal-to-distortion ratio. In addition, we can compare the results obtained with two different methods.

Below 1500 Hz the curve is calculated from the minimum of the 3rd+5th threshold and the 3rd+5th+7th threshold. At 1500 Hz the curve is merged with the pure-tone equivalent of the hard-clipped two-tone threshold (probably the difference is minor).

Good agreement with the just-noticeable hard-clipping [dB] vs. frequency curve between 300 Hz and 1500 Hz.


Appendix

Some details of the measurement methods are beyond general interest, so I have separated them from the main article.

Importance of measurement bandwidth

Hard clipping is the best choice for defining audibility threshold for SINAD/THD due to the large amount of high order harmonics. However, high order harmonics mess up traditional THD measurements.

THD/SINAD measurements require a specified bandwidth. In traditional THD/SINAD measurements the bandwidth is either 20 kHz (for a single frequency analysis at 1 kHz) or approximately 100 kHz (for THD vs. frequency or a single frequency analysis at 20 kHz).

When we increase the bandwidth (high-frequency cut-off) during a distortion measurement, once a limit is reached, increasing the bandwidth further does not increase THD significantly. This is valid for all kinds of distortion. The only difference between hard clipping and traditional amp/DAC distortion is that the limit for traditional amp/DAC distortion is approx. five times the frequency of the fundamental, while for hard clipping the limit can be twenty times the frequency of the fundamental.

In this study THD is calculated with a moving filter. The cut-off frequency is 20 times the frequency of the signal.

The following table compares different moving filters. First column is calculated with a filter with a cut-off frequency of 10 times the frequency of the signal (LP-10F0), the second column is calculated with a filter with a cut-off frequency of 20 times the frequency of the signal (LP-20F0). The comparison is valid only for hard clipping.

LP-20F0
THD [%]
LP-10F0
THD [%]
0.10.07
0.040.028
0.0140.01

How to convert two-tone audibility thresholds to pure tone thresholds?

We can't test the audibility of distortion close to the top of the human hearing range with pure tones. The only way to assess the audibility of distortion above ~2 kHz is using two-tones as a test signal with correct level and spacing. However, in this case the signal-to-distortion ratio is not directly comparable with a harmonic distortion measurement.

We can solve this problem if the listening test is performed with a two-tone signal, but distortion is measured with a sine signal of the same amplitude. First, we send a sine wave through a clipping stage and measure THD+N. Then we send a two-tone with the same amplitude through the clipping stage. We perform the test with the distorted two-tone signal, repeat the process with different distortion levels until we find the just-noticeable distortion level. The just-noticeable distortion level is the measured THD+N for the equivalent sinusoidal signal.

With this method we can define audibility threshold as THD+N at those frequencies where tests with pure tone fail.

What is the criterion for low-level audibility thresholds?

At low signal levels, the audibility of nonlinear distortion is determined by the Absolute Threshold of Hearing (ATH). Distortion components below the ATH are not audible. However, even close to the Absolute Threshold of Hearing THD (and not THD+N) should be kept below 10% to avoid linearity issues (audible amplitude changes in the signal).

THD and THD+N measurements may give quite different results at very low and very high signal levels. Since THD measures distortion without noise, THD is more suitable at low signal levels (noise is included in the measurement of Signal-To-Noise Ratio). On the contrary, since THD+N measures noise and every kind of distortion, including hum from the power supply, THD+N is the better choice at high signal levels.

In practice THD+N is measured. The 10% rule can be applied to THD+N and SINAD with the following restriction: if our upper limit for THD+N is 10% (min. 20 dB for SINAD) at very low signal levels, then the signal level in SINAD/THD measurement should be higher than - (SNR - 20dB) [dB].



2024.01.26.

How does the ISO ATH curve relate to age groups?

The ISO absolute threshold is measured between the ages of 18 and 25. Therefore, the ISO curve is NOT a threshold of an average person: it falls below the "global" average hearing threshold. I compared the ISO curve with the threshold values (mean, P25, P5, minimum) reported in other studies(1, 2, 3) to estimate how many people might have significantly lower threshold (meaning of significantly lower in this case: difference is more than 5 dB).

A more accurate percentile for each age group could be determined from higher resolution data.

Just-audible hard clipping vs. frequency with the lowest Absolute Threshold of Hearing (ATH is lower than the ISO-ATH by 10 decibels):

Note: the curve is only two decibels lower than the normal curve between 400 Hz and 1 kHz.

Csaba Horváth

Reference:
[1] Extended high-frequency (9-20 kHz) audiometry reference thresholds in 645 healthy subjects, A. Rodríguez Valiente , A. Trinidad , J.R. García Berrocal , C. Górriz & R. Ramírez (2014)
[2] Threshold of hearing for pure tone under free-field listening conditions, H. Takeshima, Y. Suzuki, M. Kumagai, T. Sone, T. Fujimori and H. Miura (1994)
[3] Threshold of hearing in free field for high-frequency tones from 1 to 20 kHz, Kenji Kurakata, Kaoru Ashihara, Kazuma Matsushita, Hideki Tamai and Yoshiyuki Ihara (2003)

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