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Physics and perception of low-order nonlinear distortions


Apr 8, 2020


In the engineering of electronic devices for music production or reproduction all designers are faced with the problem of minimizing the different forms of distortion inexorably introduced by their creations on the audio signal. They are commonly classified into two categories: linear and nonlinear. With reference to the representation of the signal in the frequency domain, the former are characterized by alterations of the modulus and phase of the signal; the tone controls are a typical example. The latter instead add new frequency components not present in the original signal, due to the nonlinear interaction of the frequencies present in the input signal. Both forms of distortion are often created voluntarily in production to give particular characters to voices or musical instruments, or to manage loudness and dynamics, but they are to be avoided in musical playback, especially nonlinear ones, since the effects they cause on our perception may be slightly "euphonic", not reversible and are still the subject of studies. However, it should be noted that some audiophiles prefer, even unconsciously, the presence of some types of nonlinear distortion in playback. More specifically, common experiences show that modest amounts of low-order nonlinear distortion are able to give sound properties such as:
  • 2nd order distortion: “warmer” and “softer” sound;
  • 3rd order distortion: sound with more “dynamic contrast”.
The aim of the following study is to examine in detail the physical properties of the alterations caused by nonlinear distortions on musical-like signals and, based on advances in psychoacoustics and neuroscience, to look for correlations with the effects on perception mentioned. As we will see, the reported speculations can only be of a qualitative type, being based, more than on listening tests, on the execution of simulations using ad-hoc written programs that model the nonlinear behavior of some families of amplifiers.

Nonlinear distortion measurement

The commonly adopted approach to measuring nonlinear distortions on audio devices such as an amplifier is based on assuming that this is linear and permanent, and then looking for deviations from this ideal model. We then measure the amount of tones generated by our device by introducing single or multiple tones (synusoidal components at fixed frequencies). In the first case we will have new harmonic components at multiple frequencies of the fundamental; in the second we will have intermodulation products resulting from the interaction of two or more tones at frequencies identified by their linear combination. We then numerically quantify the magnitude of the former with THD; the latter with IMD. For example, the graph in Figure 1 shows distortions introduced by a real tube preamplifier (6H30 in a Mu-follower stage) for a 1KHz sine signal.

Fig. 1 - Measure of nonlinear distortions for a tone of 1Vrms@1KHz, 0dB gain.
fig 1 - L sin.png

The 2nd order harmonic level is here of [email protected] while that of 3rd is very small, [email protected]; the THD detected is -72dB. The higher orders all have a lower level and not detectable here; the other tones present at lower frequencies are signal independent and are due to power supply. With two tones at 19KHz and 20KHz in a 1:1 ratio of Figure 2, we see the intermodulation products, at 1KHz and in the side bands.

Fig. 2 - Measure of nonlinear distortions for two tones @19KHz+20KHz (CCIF test).
fig 2 - CCIF.png

The detected IMD is -76dB. The increase in the number of tones will cause a considerable proliferation of intermodulation products, in this case at linear combination frequencies of all pairs or triplets of original tones, which form a sort of “carpet” at non-harmonic frequencies. Finally, other measurements can detect THD and IMD levels for each frequency or different levels of the input signal.

Listening experiences of this preamp reveal soft sound, with a slight emphasis on mid-low frequencies and a modest soundstage, at least when compared to a neutral solid state preamplifier like the Threshold FET Ten/e used as a reference. Which is in line with common experiences on the perception of distortions, as the preamp has predominantly 2nd order distortions. Now, none of the measurements or parameters listed provide clear clues to physical alterations on the signal that can be related to this perceived character for 2nd-order distortion, different from that of the 3rd. So let's try to understand what may be the determining aspects hidden in the measurements made.

Mathematical model of nonlinearity

One way to study in detail the effects of nonlinear distortions is to mathematically model the input/output behavior of amplifiers for these aspects. We will then be able to build a simulator (a program) that replicates its behavior for any input signal and thus perform all our investigations on any type of signal.

The definition of the model starts from identifying the cause behind the generation of nonlinear distortion in amplifiers: the non-constant gain for each level of the input signal. In other words, the input/output transfer curve f(x) is not a perfect line in the working interval but has imperfections around 0 or towards extremes. By modeling this curve and calculating the output values for each input signal value we can get a simulator of our amplifier. This approach determines a static model, since the curve is constant over time; if instead this curve is dependent on it, it is said that the amplifier “has memory” and it is necessary to resort to more complex models, called dynamic. In a previous work I dealt with these issues in more detail.

For the following analysis we will adopt the static model, in which the transfer function f(x) is modeled by a third degree polynomial:

Capture f.PNG

By introducing a sinusoidal x(t) signal into this polynomial we will have that the different addendums control:
  • a0 : Contributed to DC, normally 0.
  • a1x : Gain of the device.
  • a2x² : 2nd order distortion amount, -90 degrees phase shift + DC input.
  • a3x³ : 3rd order distortion amount, 180 degrees phase shift + original frequency contribution.
Having to normalize the result to 0dBFS working with discrete signals, the polynomial will then be divided by a1; then, completely eliminating the x addendum will give only the distortion component. Furthermore, we will assume for the moment that the coefficients are all positive; we will see later the implications of this hypothesis.

The preamplifier described in the previous paragraph (without global feedback) is well represented by this model. As a verification we can generate test signals and compare the output of the amplifier with that of the simulator. As a sample, the following graph shows the output of the preamplifier for a series of 8 balanced tones in frequency from 10KHz to 17Hz, level at -18dB and fs = 192KHz.

Fig. 3 - Measure with 8 tones @ 10KHz, 11KHz, …, 17KHz; input at 1Vrms; 0dB Gain.
fig 3 - MSin meas.jpg

The graph clearly shows the intermodulation products in the side bands of the original tones. The result of the simulator for the same source signal with distortion level of 2nd and 3rd order derived from measurements in single tones is in the following diagram. The input signal is depicted in orange; distortion component alone is in blue.

Fig. 4 - Simulation with 8 tones @ 10KHz, 11KHz, …, 17KHz, in a static nonlinear system.
fig 4 - Msin sim.png

The agreement is very good; the small differences are due to neglecting linear distortions (the frequency response of the preamp not perfect for module and phase) and interactions with the nonlinear distortions of the DAC/ADC converters of the measuring instrument. Note the distortion components coinciding with the original tones not detectable in the measurements.

Analysis of impulsive signals for single order distortions

Let’s start by analyzing distortion behavior in the time domain with a band limited impulsive signal. We then build a source signal composed of multiple equispaced tones in frequency at the same level and phase: 100 tones spaced apart by 200Hz, from 50Hz to 20KHz, level at -40dB to avoid clipping; for the phase we choose the same constant value, for example -90 degrees, to move the signal more. In the time domain all these tones are constructively added into a few short time windows: the signal takes the form of a train of sinc pulses, given the 20KHz band limitation, with alternating phases of 0, -90, -180, -270, 0, ... degrees, and spaced by 1/200 sec..

Let's now simulate the transit of this sampled signal to fs = 192KHz into two amplifiers (without memory effects): the first introduces a distortion of only of 2nd order and the second only of 3rd order, both at -60dB (THD = 0.1%) on a single source tone. Figure 5 shows the detail for a single pulse (phased by -90 degrees) in channel 1; channels 2 and 3 report only the resulting distortion components, amplified by 50dB to highlight its trend.

Fig. 5 - Source signal (top) and its 2nd order (middle) and 3rd order (bottom) distortions, last two amplified. Detail, Time simulation.
fig 5 - 2-3 time.png

We observe that:
  • 2nd order distortion always assumes positive values, proportional to the absolute value of the source signal, compressing the negative half-wave and expanding the positive half-wave (asymmetrical distortion). As a consequence we will have simultaneously effects of attenuation and enhancement of the transients.
  • 3rd order distortion instead enhances both positive and negative half-waves, thus increasing the steepness of all transients; the signal extension also increases symmetrically.
Suppose now that these alterations are examples of what also happens on musical signals: what are the characteristics of our auditory system that come into play determining the way we perceive them?

To try to answer this question, we must resort to notions of psychoacoustics. It is now well understood that, from the point of view of frequency impacts, low-order harmonic distortions (Figure 1) are not audible as distinct tones from the main tone. This is due to the masking effect of our auditory system which, by raising the audibility threshold of frequencies close to the main one, makes them undetectable as such. It should be added that the musical signals are rich in harmonic components, and therefore the harmonic distortions are superimposed on those in the original signal, making them less "invasive" in practice. This discourse does not apply to intermodulation products (Figure 2), present at frequencies theoretically very distant from the main, non-harmonic ones, potentially capable of confusing the weakest signal levels.

From the point of view of the temporal impacts of the distortions we can refer to recent advances in the field of neuroscience. These confirmed that our auditory system under certain circumstances exhibits a much higher temporal resolution than that in frequency. In fact, if it is now widely established that our hearing system is able to detect frequencies up to 18KHz (a limit that can reach 24KHz), the temporal resolution, that is the ability to localize transient events over time, is between 6 and 10μs. The math tells us that this value corresponds to a frequency of around 125KHz, well above 18KHz. It is therefore hypothesized that these two aspects, frequency content and arrival times of a signal, are managed by different parts of our auditory system, probably in a similar way to what our eye does, in which the cells of the retina are differentiated into two types, rods and cones, to reveal brightness and color separately. Not surprisingly, the new MQA format has among the key aspects of its design that of "time precision" (further details in this article).

Returning to the distortions of the previous diagram, we have that 8μs (the average) correspond to 1.5 times the distance between one sample and another (fs = 192KHz). It is therefore reasonable to hypothesize that the nonlinear distortions, acting on the transients, can have an impact on the temporal precision. Again an analogy with our visual system regarding transient alterations is curious. If we look at the 3rd order distortion curve in Figure 5, we note that this is attributable to the effect caused by a filter known as "unsharp mask". The filter (due to Laplace) does nothing but detect the edges in an image: subtracting the result of this filter (possibly attenuated) from the original image, a sort of double border is added to each element contained in the image, where the darker part is further darkened and the lighter part is lightened. This effect is clearly visible at very close range, but by observing the image from a distance it induces an effect of greater sharpness on our visual system. The filter relating to the 2nd order, on the other hand, is not used. An example of applying these filters in a non-aggressive way is in the following figure (to be observed at 100% of the dimensions).

Fig. 6 - Simulation of 3rd order distortion al left; 2nd at right; original RAW image at center.
fig 6 - pic.jpg

The image on the left (equivalent to the 3rd order) appears sharper, more "dynamic", enhancing both transients; in the one on the right (equivalent to the 2nd order) this effect is still present but less pronounced, enhancing the transients on the light tones and compressing those relating to the dark tones, appearing overall more “soft”. Both modified versions seem to improve on the original image: of course it is not exactly the same processing that our hearing system does, but it seems a nice coincidence for the equivalent effect on the sound!

Analysis of music-like signals for single order distortions

Now let's try to understand how much of the considerations in the previous paragraph applies to a signal more similar to a musical one. We build a harmonic signal, with different phases and decreasing intensity with frequency: 64 tones from 30Hz to 48KHz in 1/6 octave steps, starting from -25dB, decreasing on the frequency level linearly to -2dB/KHz, each with random phase. This signal will appear erratic over time, given the random phases and the relationships between frequencies, which is difficult to analyze directly over time. Also the 2nd and 3rd order distortions, always at -60dB on single tone, will follow the same trend as shown in the following figure.

Fig. 7 - Source signal (top) and its 2nd order (middle) and 3rd order (bottom) distortions, last two amplified of 55dB. Time simulation, detail.
fig 7 - rnd 2-3 time L.png

Upon visual inspection, both distortions seem to be attributable to the curves of Figure 5, applied to each micro-transient: here too the 3rd order distortion enhances all transients; that of 2nd enhances the transients where the signal is positive and attenuates the negative ones. Furthermore, the 2nd order distortion values also appear negative. At this point it is appropriate to use a couple of statistical indicators that give quantitative evidence, based on the availability of the source signal and that relating only to the distortion perfectly time-aligned.

TSD (Total Signal Distortion)
With this indicator we quantify the overall distortion suffered by the signal, calculated as the ratio between the RMS level of the distortion signal d[i] and that of the source signal s[i]:


The higher this value, expressed in dB or percentage, the more distorted the signal is, and therefore the distortion is potentially more audible. If the DC component is null (typically so) the TSD is proportional to the standard deviation of the distortion, i.e. its variance, unless a multiplicative constant equal to the RMS value of the source signal.

DSA (Derivative Sign Agreement)
With the second indicator, the DSA, we quantify the effects on transients. This expresses the overall percentage of agreement between the transients of the source signal and those of the distortion. Transients are qualified by calculating the derivative of each signal, which expresses the rate at which signals vary: when it is positive the signal grows; when it is negative it decreases. So:
  • Where there will be agreement between the sign of the source signal derivative and that of distortion, we will have an increase in the speed of the transient, positive or negative;
  • Where there is discrepancy, distortion will reduce the speed and level reached of the transient.
If we have sampled the signals over time with a high frequency we can get a good approximation of the derivatives by calculating the difference between the consecutive pairs of samples of each signal.

Returning to the signal in question, we will have:
  • 2nd order: TSD = -62.2dB (0.077%), DSA = 50.0%
  • 3rd order: TSD = -62.3dB (0.077%), DSA = 100.0%
TSD values tell us that there is no difference between the overall amounts of distortion suffered by the signal. The DSA values confirm the visual inspection of the transients: both orders act on the micro-transients according to when seen for the impulsive signal (again the DSA assumes the values 50% and 100% respectively).

To complete the picture, we perform the analysis in the frequency domain of the distortions. The following diagrams show the spectral trend of the distortions in Figure 7, normalized to the source signal level, decreasing in frequency to -2dB/KHz. Figure 8 shows the 2nd order distortion and the same without the components at the same frequencies of the source tones; Figure 9 that of 3rd order.

Fig. 8 - Normalised 2nd order distortions (in dB, blue line) and average (red line) - Simulation.
fig 8 - 2 rnd freq L NEW.png

Fig. 9 - Normalised 3rd order distortions (in dB, blue line) and average (red line) - Simulation.
fig 9 - 3 rnd freq L NEW.png

Both distortions have an uneven frequency distribution. On their qualitative trend we can say that:
  1. The level of the "carpet" of distortion is almost constant for both orders, higher for the 2nd order.
  2. The peaks of 2nd order distortion are modest, almost at the same level at each frequency, denser in the medium-low frequencies and generally not coincident with the tones of the original signal.
  3. The peaks of 3rd order distortion are very pronounced, almost at the same level at each frequency and coinciding with the tones of the original signal.
These physical characteristics, also considering what is reported in the previous paragraph, allow us to speculate on the correlation with the effects on perception: the 2nd order distortion emphasizes the medium-low frequencies and "softens" the signal (points 1 and 2), while the of 3rd makes the sound more precise and dynamic (point 3). It must be said that these trends are related to a signal that simulates a hypothetical musical signal; with other types of signals experimented what is observed is a change for the quantitative aspects, that is more slope of the carpet, variations in the density of the peaks etc., but the behaviors described, for the specific simulated non-linear system, remain substantially unchanged.

Analysis of music-like signals with more distortion orders

Let's analyze now what happens in the presence of both distortion orders, where we expect a mix of the two effects. The graphs in Figure 10 show TSD and DSA values as a function of the ratio of distortion levels of 3rd and 2nd order (expressed in dB) to single tone. The different curves refer to three references for the 2nd order: -110dB, -90dB, and -70dB.

Fig. 10 - Left: TSD per 3rd/2nd order ratio (dB). Right: DSA per 3rd/2nd order ratio (dB). Change of 2nd order reference.
fig 10 - dsa L NEW.png

On the left side of the graphs 2nd order distortion prevails, at a constant value; on the right side the 3rd order prevails, which determines the overall increase in distortion. In detail we have:

TSD Chart
The graph shows the magnitude of the overall distortion, coupled with the variability which is about 12dB lower. Changing the 2nd order reference level logically translates the curves upwards by the same quantity. These remain constant where 2nd order distortion (constant) prevails up to a ratio of -6dB; then they grow with constant slope on the right side where the 3rd order prevails over the ratio 6dB.

DSA Chart
The graph shows how the character of the distortion progressively shifts from that of the 2nd to 3rd order, therefore of the perception of the “warm” effect (DSA = 50%) towards the "dynamic" effect (DSA = 100%). Since the curve is unique for all the reference levels we will have that the character of the distortion depends only on the relationship between the two types of distortion, while the different reference levels make them more or less audible (TSD). With a ratio of 6dB we have the intermediate situation between the two orders for transient effects, with DSA = 74%. Finally, we note that the most important variations occur when the ratio between the two distortions is between -3dB and 20dB.

The following figure shows the detail of the frequency trend of distortions for the 1:2, 1:1 and 2:1 ratios, always normalised to the level of the source signal by frequency.

Fig. 11 - Normalised 2nd+3rd order distortion (in dB) for 1:2, 1:1 and 2:1 ratios - Simulation.
fig 11 - 2-3 rnd mix freq L NEW.png

The graphs show how the progressive increase of the 3rd order distortion marginally increases the level of the carpet, acting mainly on the peaks.

Finally, let's verify how to change the distortions in function of the level of the source signal. The graphs in Figure 12a show the graphs of the TSD and DSA values in correspondence with the reduction of input levels of 10dB, 20dB, 30dB and 40dB. The 2nd order reference level is -70dB.

Fig. 12a - TSD (dB, left) and DSA (%, right) per 3rd/2nd distortion ratio, for more input levels.
fig 12 - dsa2 NEW.png

Both graphs show that 3rd order distortion decreases faster than 2nd order with the reduction of the level (the 3rd decreases with the cube of the signal level; the 2nd with the square). This implies that all curves translate to the right by the same amount as the reduction in the input level (10dB). Furthermore:

TSD Chart
The level of distortions decreases with the level of the source signal. Where the 2nd order prevails, the level of distortion decreases by the same amount as the entry level; where the 3rd prevails, the level of distortion decreases by double the level of the input.

DSA Chart
From the curves it can be seen that with the same ratio between the two orders, the characterization of the 3rd order distortion (i.e. dynamic) decreases very quickly as the entry level decreases. For example, if for a level of 0dB we have a DSA = 74.4% for the 3rd/2nd ratio = 6dB, for a 10dB reduction of the level we have DSA = 51.0%, i.e. the disappearance of the dynamic effect.

It should be added that these effects are obtained by acting on the volume of the reproduction, if the amplifier has this control at the signal input (in this study more details), or within the same music track when passing from parts at high level to other lower ones and vice versa. This latter aspect is more evident from the graphs in Figure 12b, where the TSD and DSA values are shown as a function of the input level (dB) for different 3rd/2nd order ratios (from 0dB to 30dB, in steps of 6dB), with a reference for 2nd order = -70dB.

Fig. 12b - TSD (dB, left) and DSA (%, right) per input level, for more 3rd/2nd distortion ratio.
fig 12b - dsa2 lev NEW.png

Finally, the curves do not take into account the level of background noise which in practice hides the lower levels of distortion.


The above helps us to bridge the apparent gap between the subjective experiences of listening to music and the measures of nonlinear distortions. The study shows that the "dynamic" effect is likely caused by the contributions of 3rd order distortion to the tones in the main signal, which strengthen the energy content of the musical signal and the transients, to which we are very sensitive. These contributions are much milder for the 2nd order, which distributes energy in the form of a "carpet", more pronounced in the medium-low frequencies. The presence of both orders produces intermediate effects, as well as the variation of the input signal level. And here we stop with speculations: to what extent a more or less large variation in physical quantities is perceived as more or less important must be experimented with listening tests or other types. There is also the most popular world of “memory” amplifiers to explore that have different behaviors about distortions.
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Apr 8, 2020
After examining in detail the physics behind the distortions in the case that we can define as "base", with the consequent characterizations of “warm” and "dynamics", we explore other cases that determine different impacts on the signal and therefore on perception.

Effects of negative coefficients in the transfer curve

In the previous section we examined the characteristics of the 2nd and 3rd order nonlinear distortions through a simulation based on the modeling of the input/output transfer curve, represented by a third degree polynomial:

Capture f.PNG
In this equation the coefficient a2 controls the amount of the 2nd order distortion while a3 that of the 3rd. We can trace the trend of this curve to have a more complete view. The first two graphs at the top of Figure 13 show the curves in the normalized working interval [-1,+1], with the a0 = 0 (DC) and a1 = 1 (unity gain), separately for the distortions of 2nd and 3rd order at -60dB; the curves are amplified by 100 times compared to the ideal straight line at 45 degrees to show the trend. The lower part shows the deformation undergone by the classic sine wave: the source curve in orange; in blue the distorted curve; in red the distortion alone, always amplified by 100 times.

Fig. 13 - Top: Transfer function for 2nd order distortion (left) and 3rd order (right). Bottom: related effect on a sine wave.
fig 13 - Transf pos NEW.png

According to what has already been described, the curves act on signal in the following way:
  • 2nd order: Expansion of the positive values of signal (f(x) > x for x > 0) and compression of the negative ones (f(x) < x for x < 0) - asymmetric distortion.
  • 3rd order: Expansion of both positive and negative values of signal (|f(x)| > |x| for |x| > 0) - symmetrical distortion.
The shape of the curves, and consequently the value of a2 and a3 coefficients, is determined by the type of circuit and the working point of the active components in the amplifier. In the above we have assumed that these coefficients assume positive values, but this is by no means the rule. There are also circuit solutions that determine negative values for a2 and/or a3 (we will leave a1 positive, to indicate the non-inverting effect of the device). In this situation the characteristics of the distortions change, and not a little. Based on the survey tools we have seen, it is relatively easy to understand their impacts. Let's start with the analysis of the effects on the transfer curve, illustrated in the following figure.

Fig. 14 - Top: Transfer function for 2nd order distortion (left) and 3rd order (right), negative. Bottom: related effect on a sine wave.
fig 14 - Transf neg NEW.png

Here the curves act in the following way on signal:
  • 2nd order: Compression of the positive values of signal and expansion of the negative ones.
  • 3rd order: Compression of both positive and negative values of signal.
Therefore, the 2nd order distortion reverses the action on transients; that of 3rd order changes its nature, transforming itself from an action of strengthening the transients to that of weakening. The same effects are found in more complex signals, such as the impulsive signal in figure 15, and therefore also in musical ones.

Fig. 15 - Source signal (top) and its 2nd order (middle) and 3rd order (bottom) distortions, last two amplified. Detail, Time simulation.
fig 15 - imp neg.png

The percentage of agreement of transients (DSA) always remains at 50% for the 2nd order, while for the 3rd order it tends to 0%, indicating precisely “compression”. This effect is evident in the simulation of the analogous effect in the photographic field shown in Figures 16a and 16b.

Fig. 16a - Simulation of 2nd order distortion: positive at left; negative and right; original RAW image at center.
fig 16a 2nd.jpg

Fig. 16b - Simulation of 3rd order distortion: positive at left; negative and right; original RAW image at center.
fig 16b 3rd.jpg

For the 3rd order, the effect is analogous to the blur effect: the transitions from high tones to low tones and vice versa are both attenuated. For the 2nd order, the transitions to high tones are attenuated and those to dark tones are increased.

For the “music-like” signal the situation is no different: we will only have different DSA for the 3rd order; the TSD remains unchanged:
  • 2nd order: TSD = -62.2dB (0.077%), DSA = 50.0%
  • 3rd order: TSD = -62.3dB (0.077%), DSA = 0.0%
In the frequency domain the distortions do not show any difference in the module from those shown in figures 8 and 9. The phases of the 3rd order peaks will however have the opposite phase (180 degrees) with respect to the source tones.

For 2nd and 3rd order distortions mix we will have the TSD curve invariant with respect to the sign change of a2 and a3. As expected, the DSA curves shown in figure 17 differ when a3 is negative: they start from the usual DSA = 50% when the 2nd order prevails and arrive to 0 when the 3rd order distortion prevails, indicating the complete discordance of the transients.

Fig. 17 - DSA per 3rd/2nd order ratio (dB) for all combinations of a2 and a3 signs.
fig 17 - dsa all NEW.png

As an illustration, Figure 18 shows the transfer curves for both orders at -60dB, in the cases of both coefficients a2 and a3 positive (top left) and with only a3 negative (top right), always amplified by 100 times. Below, the usual corresponding effect on a sine wave, highlighting where the distortion increases (red background) or decreases (blue background) more or less strongly the transients of the original curve. For this “simple” signal, the DSA values are 89.2% and 10.8% respectively.

Fig. 18 - Top: Transfer function for 2nd and 3rd order distortions both at -60dB. Bottom: related effect on a sine wave.
fig 18 - Transf mix NEW2.png

In conclusion, in the cases examined the complete picture for the effects on perception can be summarized as follows:
  • The 3rd order has an obvious effect: when the phase is positive it adds dynamic contrast; on the contrary, it reduces it when negative.
  • The 2nd order has finer effects: there is a general softening effect. The differences due to the phase, from the experiences of people who have explored these aspects such as Nelson Pass, are in an approach to the listener of sound and greater detail when positive; deeper soundstage and better localization when negative.
In practice, the case in which we are for a given amplifier can be detected by measuring the phase value of the 2nd and 3rd harmonic distortion produced by a single tone:
  • 2nd harmonic: -90 degree phase —> a2 positive; +90 degree —> a2 negative;
  • 3rd harmonic: +/-180 degree phase —> a3 positive; 0 degree —> a3 negative.
To be fair, it must be said that often the phases of the distortions resulting from the measurements of real devices differ from these values. This highlights the presence of memory effects in the device and therefore the non-applicability of the static model used here. From the above it should be clear that the memory effects make the distortion form more “tortuous” and consequently more difficult to translate into effects on our perception.
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Senior Member
Nov 6, 2016
Royal Leamington Spa, UK

When I first quickly scanned through your post and glanced at these images, I assumed the first was the original. I actually preferred the second, as it looked more natural and less 'sharp'. The third was simply too 'soft'. My conclusion was that 2nd order distortion might have a positive effect after all, and that increasing orders of distortion beyond 2nd simply soften things too much.

But on reading correctly, the deleterious effects of both 2nd and 3rd order distortions are clear to see - the former having a 'softening' effect, the latter a 'sharpening' effect. If your system needs either of these effects to 'sound better', then it's likely there's something fundamentally wrong with it in the first place.

Thanks so much for sharing. Fascinating stuff!



Addicted to Fun and Learning
Jan 9, 2022
Exploring perception of different distortion components is interesting, thank you for the elaboration. It goes some way toward theoretical explanation of perceived sonics of different components, which isn't as so easy to infer from the standard measurement suites. Relating/illustrating the frequency and time domain influences is good to see.

The images were interesting, assuming they are usefully analogous. I can see (for example) highlights in the fabric weave that read as dynamic but blow out some of the pixels. Also, colour saturation is affected. I expect the former from unsharp mask, but the latter isn't something I considered previously.


Grand Contributor
Central Scrutinizer
Mar 9, 2016
Riverview FL
I'd like to see examples other than what appears to be a calculated result.

I'd also note that the harmonics may not be in phase with the fundamental (as reported by REW with speakers), which would change the values of the sum of the waves.


Active Member
Nov 21, 2021
Thanks for the interesting read and the effort put into the post. The analogy with image processing filters makes a lot of sense to me.

Just a couple of comments...

You may wish to remove the following repeated passage of text.
Furthermore, the 2nd order distortion values also appear negative. At this point it is appropriate to use a couple of statistical indicators that give quantitative evidence, based on the availability of the source signal and that relating only to the distortion perfectly time-aligned.

At this point it is appropriate to use a couple of statistical indicators that give quantitative evidence, based on the availability of the source signal and that relating to the distortion component only perfectly time aligned.

Your statement on the effects of second order distortion don't seem quite right to me...
  • 2nd order distortion always assumes positive values, proportional to the absolute value of the source signal, reducing the extension of the negative half-wave and increasing the positive one (asymmetric distortion). As a result we will also have an accentuation of the slopes of positive transients and a reduction in negative ones.
  • 3rd order distortion instead enhances both positive and negative half-waves, thus increasing the steepness of all transients; the signal extension also increases symmetrically.

To me it looks more like...
  • 2nd order distortion always assumes positive values, proportional to the absolute value of the source signal, but reduces the extension of both the negative and positive half-waves and increasing whilst inverting (rectifying) the positive negative one (asymmetric distortion). As a result we will also have an accentuation of the slopes of positive transients and a reduction in negative ones. As a result we will have an attenuation and a broadening of transients (essentially converting a single impulse into a time delayed double pulse of lower amplitude). In other words a smoothing effect.
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Apr 8, 2020
Thanks for the interesting read and the effort put into the post. The analogy with image processing filters makes a lot of sense to me.

Thanks for your useful clarifications.
I've updated the graphs and some considerations after simulations with higher calculation precision on July 20 and August 9, 2022.
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