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Physics and perception of low-order nonlinear distortions

Pinox67

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Introduction​

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 amplitude and phase of the signal. 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 this study is to investigate the cause of these effects on perception, looking for a correlation between the physical aspects of nonlinear distortions on signals and some results in the field of psychoacoustics and neuroscience. Beyond the measurements and above all the listening tests in support of the above that are not detailed here, the main investigation tool used will be the computer simulation of the behavior of some families of non-linear systems for music-like signals. This will lead us to define numerical indicators useful for carrying out qualitative speculations with the effects on perception.

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 extent of the former with the THD and the latter with the IMD. Both indicators are calculated as the ratio between the RMS (Root Mean Square) value of the harmonics introduced by the device with the RMS value of the harmonics in the source signal. For example, on top of in figure 1 it is shown the distortions introduced by a real tube preamp (6H30 in a Mu-follower stage) for a 1KHz sinusoidal signal.​

Fig. 1 - Measure of non-linear distortions: Two tones @19KHz+20KHz (CCIF test, top) and a single tone of at 1KHz (bottom).
fig 1+2 sin+ccif.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, below in the same figure, we see the intermodulation products, at 1KHz and in the side bands. 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 underlying the generation of non-linear distortion in the amplifiers: the non-constancy of the gain for each level of the input signal. In other words, the input / output transfer curve f(x) is not a straight line in the working range but has "imperfections" around 0 or towards the extremes. By modeling this curve and calculating the output values for each value of the input signal, a simulator of our non-linear system can be obtained. This simple approach determines a static type model, for which the value of the output signal depends only on the instantaneous value of the input signal. In real systems, including common audio amplifiers, there is instead a dependence also on the values assumed in the past: it is said that the nonlinear system is “with memory” and the relative model, much more complex, is of the dynamic type.

The static model, which we will adopt in the study presented, however, constitutes a good approximation of the behavior of amplifiers that show very limited memory effects when they remain in their normal working area. This class is recognizable by the following characteristics:​
  • Flat frequency response, with extension at least up to 100KHz.
  • Low level of harmonic distortions (< 1%) and frequency independence in the audio band.
  • Phases of harmonic distortions independent of both frequency (always in audio band) and signal level.
In these hypotheses we will be able to model the transfer function f(x) by means of a polynomial. For the 2nd and 3rd order distortions we are interested in studying we can limit ourselves to considering third degree polynomials:​

Capture f.PNG

We will also assume that the coefficients are all non negative; we will remove both of these restrictions later. If we now insert in this polynomial a simple sinusoidal signal (x(t) = sin(ωt)) we will have that the different addends "control":​
  • a0 : component in DC, normally null.
  • a1x : amplification (gain g) of the fundamental component.
  • a2x²: 2nd order distortion, consisting of two components:
    • 2nd harmonic (HD2), out of phase by -90 degrees;
    • DC, of the same entity as HD2.
  • a3x³: 3rd order distortion, consisting of two components:
    • 3rd harmonic (HD3), 180 degrees out of phase;
    • contribution to the fundamental component (HD1), 10dB higher than HD3.
Therefore, it is appropriate to distinguish the distortion of order i due to the i-th power addition of the transfer function, from the harmonic component of order i. In formulas, the first is represented by the i-th power by the function sin(ωt) and the second by the harmonic sin(iωt).

To perform the discrete simulation of these non-linear distortions for a preamplifier the function f(x) is inserted in a flow of operations illustrated in the following logic diagram.​

Fig. 2 - Scheme of the non-linear distortion simulator.
fig 3 - Scheme NEW.png

The discrete input signal x[m] is subjected to the following operations in cascade (chain in the center):
  1. Upsampling of x[m] by an integer factor M, so that all subsequent processing works with a signal at a sampling frequency of at least 192KHz. This avoids potential aliasing phenomena due to the subsequent application of the transfer function f(x) which can create harmonics beyond the maximum frequency that can be represented with the original sampling rate.
  2. Attenuation of the gx value, which simulates the input volume control.
  3. Application of the non-linear distortion curve f(x), of gain a1. Optionally you can here:
    • Apply a negative global linear feedback (chain down) with factor β, resulting in a gain G = a1 / (1 + βa1).
    • Apply the complementary function g(x) in place of f(x) such as to cancel the effect of f(x), i.e. g(f(x)) = x.
    • Eliminate the DC component created by the distortion.
  4. Addition of Gaussian white noise N[n] to simulate the contribution of thermal noise, of a configurable entity.
  5. Attenuation of the gy value, which simulates the output volume control.
The signal thus obtained, y[n], represents the distorted signal. The subsequent statistical analyses however require the separation of y[n] into two parts: only the distortion component d[n] and the signal s[n], equal to the amplified input signal of the product of the reported values (gx, G and gy). The separation is obtained by subtracting from y[n] the same amplified input signal in a perfectly linear (top chain) and synchronous way.

To focus on the main concepts of the study in the first simulations reported we will make the following assumptions:​
  • Absence of feedback (β = 0).
  • Absence of thermal noise (N[n] = 0).
  • Attenuation and unity gains (gx = a1 = gy = 1), so that the input signal x[m] and output signal s[n], except for oversampling, coincide.
The preamplifier described in the previous paragraph (moreover, without global feedback) is well represented by this model. As a check we can generate test signals and compare the output of the amplifier with that of the simulator. As a sample, the first diagram in figure 3 shows the preamplifier output for a series of 8 equally spaced tones in frequency from 10KHz to 17KHz, level at -18dB and fs = 192KHz. The graph clearly shows the intermodulation products in the side bands of the original tones. The second diagram shows the result of the simulator for the same source signal with a 2nd and 3rd order distortion level obtained from measurements with single tones, together with the ‘noise carpet’, distinguishing the undistorted signal s[n] in red from the distortion component d[n] in green.​

Fig. 3 - Measure (top) and Simulation (below) with 8 tones @ 10KHz, 11KHz, …, 17KHz; input at 1Vrms; 0dB Gain.
fig 4 - MSin MS.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, which we will return to later.​
As a last note, it should be pointed out that the use of y[n] in listening tests to simulate the distortions of a real amplifier should be taken with precautions, the details on this post.

Analysis of sinusoidal and impulsive signals for single order distortions

Let's start by analyzing the shape of the distortions in the time domain related to transfer curves that have only one of the two orders, 2nd or 3rd. It is not a very realistic situation, but it allows us to understand how the two orders act separately on the signal. The first two graphs at the top of Figure 4 show these transfer curves in the normalized working range [-1, +1], with a0 = 0 (DC) and a1 = 1 (unit gain); the one on the left with only HD2 = -60 (a2 = 0.002, a3 = 0) and the one on the right with only HD3 = -60dB (a2 = 0, a3 = 0.004). For both, the THD is equal to 0.1%. The curves are amplified by 100 times with respect to the 45 degree straight line to show its trend.​

Fig. 4 - Source signal (top) and its 2nd order (middle) and 3rd order (bottom) distortions, last two amplified. Detail, Time simulation.
fig 5 - fx 2 3.png

The lower part shows the deformation undergone by the classic sine wave: the source curve in orange; in blue the distorted curve and in red the distortion alone, always amplified by 100 times. The transfer functions act on the signal as follows:
  • 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.
We also note how the distortions are curves composed of several harmonics, as already mentioned: the 2nd order from DC+HD2 and the 3rd from HD1+HD3.

We now use the simulator to study more complex signals: we build a source signal composed of several equally spaced tones in frequency at the same level and phase: 100 tones spaced 200Hz from each other, 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 summed up in a few short time windows: the signal takes the form of a sinc pulse train, given the band limitation at 20KHz, with alternating phases of 0, -90, -180 , -270, 0, ... degrees, and spaced 1/200 sec.

Let’s now simulate the passage of this signal in two amplifiers, without memory effects and 0dB of gain, where the first is affected by distortion of only 2nd order and the second only of 3rd. So, in the measurements, we will observe for a single tone only the 2nd harmonic and only the 3rd harmonic respectively. In the simulation we will set these two values at -60dB, which determine a THD of 0.1%. Figure 5 reports the detail for a single pulse out of phase by -90 degrees in channel 1; channels 2 and 3 report only the resulting distortion components, both amplified by 50dB to highlight the trend. The signal is sampled with fs = 192KHz.​

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.
Therefore, the expansive and compressive effects behave as in the simple sinusoidal case. Suppose now that these alterations are examples of what happens more generally 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 some notions of psychoacoustics.

From the point of view of frequency impacts, it is now well understood that even non-negligible levels of low-order harmonic distortions are not audible. Simplifying, this effect is mainly due to the fact that the ear, when stimulated by a pure tone, generates impulses towards our brain that identify a set of frequencies in the "surrounding" of the fundamental, more extended towards the high frequencies than the low ones and dependent on the signal level. This generates the so-called masking effect, which raises the audibility threshold of frequencies close to the tone of the highest masking level. The amount of masking also depends on the relative phase between the tones involved, and also occurs when the signals are separated in time. Thus the so-called "critical bands" are determined, which cause different stimuli for tones that fall in the same band or in different bands (the MP3 format is based on this phenomenon for compression). It should be added that the effect is milder for the intermodulation products, generally present at frequencies theoretically much more distant from the main ones, non-harmonic, therefore "dissonant" and potentially able to confuse other tones present in the signal.

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 auditory system is able to detect frequencies up to around 20KHz, the temporal resolution, that is the ability to locate transient events over time, is between 6μs and 10μs. The math tells us that this value corresponds to a frequency of around 125KHz, well 7 times higher than our audible audio band. It is therefore hypothesized that these two aspects, contained in frequency and arrival times of a signal, are processed 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 MQA format has among the key aspects of its design that of "time precision" (further details in this article).

Returning to the physical effects of the distortions reported in the previous diagram, we have seen how these act on transients, with effects of an expansive or compressive type, and consequent advance or delay in time. It is therefore reasonable to hypothesize that non-linear distortions may have effects on our perception also for the temporal aspects of sounds, which are added to the well-known ones relating to the presence of additional harmonic components. It is on the temporal aspect that we will focus in this study to find a possible justification for the effects described in the introduction.

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 NEW.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. 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.​

DSA (Derivative Sign Agreement)
With the first indicator, the DSA, we characterize the type of alteration suffered by the transients, where by “type” we mean the two effects encountered, which we will call expansive and compressive. Mathematically, the DSA is expressed by the fraction of the overall agreement between the transients of the source signal and those of the distortion. The transients are qualified through the derivative of each signal, which expresses the speed with which the signals vary: it is positive when the signal grows; negative when it decreases. Therefore:​
  • Where there is agreement between the sign of the derivative of the source signal and that of the distortion we will have an increase in the speed of the transient, increasing or decreasing; therefore, an expansive effect.
  • Where there is discord, the distortion will reduce the speed and level reached of the transient; hence, a compressive effect.
If we have upsampled the signal over time x4 or more, we can get a good approximation of the derivatives by simply calculating the difference between the consecutive pairs of samples at this higher frequency. Let's see what has been described with an example.​

Fig. 8 – DSA Domains identification - Example.
fig 8 - dsa details.png

Figure 8 shows at top a segment of the original signal s[n] in blue ever-increasing; in purple the distorted signal s[n]+d[n] (heavy way to show the details). The difference between the two signals, in red, is therefore the distortion component d[n]: it grows fast in a first stretch and then decreases more slowly. In the center is shown the s'[n] derivative of the original signal that always assumes positive values, since s[n] is always increasing. At the bottom the derivative d'[n] of the distortion component: up to n=5 it is positive, like s’[n]: so the distortion increases the slope of s[n], as is evident in the curve in purple. Subsequently, d'[n] becomes negative, a sign opposite to s'[n], indicating that the distortion decreases the slope of s[n]. Here the transient agreement is 5/(5+9) = 0.36.

We will indicate below the set (or domain, generically indicated with D) of the indices of the signal samples where there is expansion with '+' (positive); with '-' (negative) where there is compression. As defined, we will indicate with DSA the fraction of the only expanded transients; unless otherwise specified, the negative ones will be equal to 1 - DSA.​

PSD (Partialized Signal Distortion)
With this second indicator we quantify the amount of distortion suffered by the signal limited to each of the two types of alteration, expansive or compressive. By borrowing the definition of THD or IMD, we will define the PSD as the ratio between the RMS value of the distortion component d[i] and the RMS value of the undistorted signal component s[i], calculated on the samples related to one of the two domains, '+' and '-'. In formulas:
Cap PSD 2.PNG

Compared to THD, or rather IMD, we note that:
  • PSD is calculated in the time domain instead of frequency. However, this is not a determining aspect: for Parseval's Theorem, the two methods are equivalent.
  • The PSD can be evaluated on any signal, and covers all the distortions suffered, even the noise.
  • Only segments of the overall signal are considered.
Also for the PSD, higher values (expressed in dB or as a percentage) imply higher distortion, which therefore is potentially more audible. In the following we will indicate for brevity the two values of the PSD with s+ and s-.

Returning to the signal in question, we will have:
⁃ 2nd order: DSA = 50%, s+ = -61.2dB, s- = -61.2dB IMD = -63.1dB
⁃ 3rd order: DSA = 100%, s- = -60.3dB, s- = N.D. IMD = -60.3dB

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 (also here the DSA assumes the same values). PSD values tell us that the amount of distortion is slightly greater for the 3rd order.

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, where the reference is with respect to the level of the source signal, decreasing in frequency to -2dB/KHz. Figure 9a shows the 2nd order distortion and the same without the components at the same frequencies coinciding with those in the source signal; Figure 9b that of 3rd order.​

Fig. 9a - 2nd order distortion (in dB, blue line) and average (red line); reference to input signal.
fig 8 - 2 rnd freq L NEW.png

Fig. 9b - 3rd order distortion (in dB, blue line) and average (red line); reference to input signal.
fig 9 - 3 rnd freq L NEW.png

On the qualitative trend of the distortions we can say that:​
  1. The level of the "carpet" of distortion is almost constant for both orders.
  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 (phase synchronized, not shown).
These physical characteristics, combined with those detected in the time domain, allow us to speculate a correlation with the effects on perception: the 2nd order distortion emphasizes the medium-low frequencies (moreover not detected by the normal measures of frequency responses) and "softens" the signal (points 1 and 2), while that of 3rd increases the dynamic contrast (point 3). It must be said that these considerations are derived from the analysis of the trends of the distortions on a hypothetical musical signal; carrying out other simulations with signals with different characteristics, a change is observed for the quantitative aspects, i.e. more slope of the carpet, variations in the density of the peaks etc., but the behaviors described, in the working hypotheses set out, remain substantially unchanged.​

Analysis of music-like signals with distortions of more orders

Let's now analyze how DSA and PSD behave in more realistic situations where both orders of distortion are present. Let's start by setting a reference level for the 2nd harmonic and vary the level of the 3rd. We will briefly express this variation as the ratio between the 3rd/2nd levels in dB. Figure 10a shows two graphs; the first, as a reference for a single tone, reports the curves:​
  • The level of harmonics related to 2nd order distortion (in red) and 3rd order (in blue). For the 2nd order we have the 2nd harmonic (HD2) equal to the reference value set at -90dB (DC is omitted). For the 3rd order, we have two components: the 3rd harmonic (HD3) and the contribution to the fundamental frequency (HD1), which is always higher than 10dB. These increase progressively, from -48dB to + 48dB compared to the HD2 level.
  • The True-THD (in purple), with which we indicate the value of the Total Harmonic Distortion in which we also consider the distortion of the fundamental (HD1), normally neglected. It coincides with the “classic” THD when the distortion is mainly due to the 2nd order (on the left in the graph); it is higher than 10dB when it is due to the 3rd (right in the graph).
  • The DSA (in gray, with an ordered reference on the right) which shows the percentage trend of the expanded transients; compressed ones are here equal to 1 - DSA, i.e. symmetrical. In line with what has already been seen, where the 2nd order prevails, there is a parity between expanded and compressed transients (“warm” effect); where 3rd order prevails there are only expanded transients (“dynamic” effect). Between -12dB and +3dB there is the transition from the first to the second situation, with a very pronounced slope.
The second graph shows the DSA and PSD curves for the simulated music-like signal, parameterized on three reference values of the 2nd harmonic: -110dB, -90dB and -70dB.​

Fig. 10a - HD level, True-THD, DSA (top) and PSD, DSA (below) per 3rd/2nd harmonic ratio (dB), with more 2nd harmonic references.
fig 10a - dsa L NEW3.png

Here we observe that:
  • The DSA curve (in gray, with ordinate reference on the right), similar in trend to that obtained for a single tone, less steep in the initial climb and shifted to the right. This highlights that the presence of more tones delays and makes the emergence of 3rd order dynamic effects more gradual. Furthermore, the curve is invariant with respect to the different reference levels: therefore, the mix of the two types of distortion on the transients depends only on the ratio of the levels of the two harmonics.
  • The PSD curves are affected, as expected, by the variation of the 2nd order reference level which vertically translates the pairs of PSD+ (in red) and PSD- (in blue) curves by the same amount as the variation of the reference, 20dB. For a given reference level, where the 2nd harmonic prevails, the PSD curves remain constant and coincide with the value of the same reference; after that:
    • The PSD+, associated with the expanded transients begins to increase around the 3rd/2nd harmonic ratio of -12dB, reaching a constant slope in addition to the 6dB ratio, similar to the trend of True-THD. Therefore, the dynamic effect due to 3rd harmonic distortion begins to act before the DSA increases, around -3dB.
    • The PSD-, associated with the compressed transients, has a symmetrical trend to the PSD+, with an advance of the descent of about 10dB on the 3rd/2nd ratio. This indicates that, as the 3rd order increases, the amount of distortion on the same transients decreases, faster than the contributions to the expanded transients increase.
We now trace the same curves, always as a function of the value ratio between 3rd/2nd harmonic, keeping constant the value of the overall distortion, that is the True-THD, constant on three reference values: -80dB, -70dB and -60dB.​

Fig. 10b - HD level, True-THD, DSA (top) and PSD, DSA (below) per 3rd/2nd harmonic ratio (dB), with more True-THD references values.
fig 10b - dsa T NEW3.png

The harmonic distortion graph refers for simplicity to a single True-THD value, -80dB. As expected, for music-like signals the DSA remains invariant, while the PSD curves translate vertically by the same value relative to the variation of the True-THD. Distortion is equally distributed between expanded and compressed transients where 2nd order prevails. When the level of the 3rd harmonic approaches that of the 2nd, the expanded transients (PSD+) undergo a decrease of 10dB. This effect is due to the fact that the True-THD reference value also considers the distortion on the fundamental (HD1). Compressed transients (PSD-) suffer more prominently, decreasing quickly in the same situation.

Let's now change the perspective, analyzing the distortions as a function of the level of the source signal. The graph in Figure 11 shows the True-THD, DSA and PSD curves at the input levels of 0dB, -10dB, -20dB and -30dB, setting the 2nd harmonic reference level at -70dB.​

Fig. 11 - True-THD, DSA (top) and PSD, DSA (below) per 3rd/2nd harmonic ratio (dB), for more input levels, ref. 2nd harmonic.
fig 12a  dsa2 NEW3.png

The curves in the first True-THD and DSA graph are always relative to the distortion experienced by a single tone for different input levels. The True-THD curve is translated vertically downwards and to the right by the same amount of attenuation, 10dB, while the DSA curve is only shifted to the right by the same amount. These effects are due to the fact that the 3rd order distortion decreases faster than the 2nd order distortion, respectively with the cube and square of the signal level. Therefore, with the same ratio between the two harmonics, the characteristic of the 3rd order distortion (dynamic effect) will “struggle” more to manifest itself as the input level decreases. The second graph shows the PSD and DSA curves for the music-like signal, translated in the same way as the True-THD and DSA of the previous graph for the same reasons. For the rest, the trend is similar to that already described for the curves of fig. 10a.

It should be noted that these effects are obtained by acting on the volume of the reproduction if the amplifier has this control at the signal input or within the same piece of music when passing from parts to high to other lower levels and vice versa. This last aspect is more evident in the graph in fig. 12 which reports the DSA and PSD values as a function of the input level. The curves refer to different 3rd/2nd harmonic ratios (from 0dB to 30dB in 10dB steps), with the 2nd reference at -70dB.​

Fig. 12 - True-THD, DSA (top) and PSD, DSA (below) per 3rd/2nd harmonic ratio (dB), for more input levels, ref. 2nd harmonic.
fig 12b -dsa2 lev NEW4.png

For low input levels the distortion is always dominated by the 2nd order characteristics, as evidenced by all the curves; for higher values of the input level the expansive effects of the 3rd order are felt, both on the DSA and on the PSD, so much stronger the higher the 3rd/2nd harmonic ratio. It should be added that the curves do not take into account the level of background noise which in practice hides the lowest levels of distortion. With the noise, the previous graph would be a straight line that, from high values at low input levels, drops progressively until the reported curves are intercepted.​

Preliminary Conclusions

The above helps us to bridge the apparent gap between the subjective experiences of listening to music and the measures of non-linear distortions. The study shows that the "dynamism" effect is probably caused by the expansive contributions of 3rd order distortion to the tones in the main signal, which strengthen its energy content and therefore the transients, to which we have a high sensitivity. These contributions are 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 of distortion produces intermediate effects, also dependent on the level of the input signal. With the DSA and PSD parameters we are able to numerically qualify these effects. And here we stop for now 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.​
 
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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. 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. 13 - Top: Transfer function for 2nd order distortion (left) and 3rd order (right), negative. Bottom: related effect on a sine wave.
fig 14 - fx 2 3 neg.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 14, and therefore also in musical ones.​

Fig. 14 - 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 15 and 16.​

Fig. 15 - Simulation of 2nd order distortion: positive at left; negative at right; original RAW image at center.
fig 16a 2nd.jpg

Fig. 16 - Simulation of 3rd order distortion: positive at left; negative at 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 have DSA different from the case of positive coefficients only for the 3rd order; the PSD remains unchanged:​
  • 2nd order: DSA = 50% , s+ = -61.2dB, s- = -61.2dB
  • 3rd order: DSA = 0%, s- = N.D . , s- = -60.3dB
In the frequency domain the distortions do not show any difference in the modulus with respect to those reported in figures 8 and 9. The phases of the 3rd order peaks will in any case have opposite phase (180 degrees) with respect to the source tones. For a mix of 2nd and 3rd order distortions both negative (-/-) we will have the situation in Figure 17.​

Fig. 17 - HD level, True-THD, DSA (top) and PSD, DSA (below) per 3rd/2nd harmonic ratio (dB), negative coeff., with more 2nd harmonic references.
fig 17 - dsa all NEW3.png

The trend of harmonic distortions and True-THD are identical to those in Figure 10a for distortions with positive coefficients. The DSA curve (always relative to the fraction of the expanded transients; the compressed ones are 1 - DSA) is perfectly symmetrical: it starts from the usual 50% with a distortion of only 2nd harmonic to decrease to 0% (all transients compressed) as the 3rd harmonic prevails. The PSD+ and PSD- curves are exchanged with each other, indicating the greater alteration of the compressed transients as the 3rd harmonic increases. We add that these trends remain unchanged even when the 2nd order distortion is positive (+/-); the same applies to the graph of Figure 10a for the 2nd negative order (-/+).

By way of illustration, Figure 18 shows the transfer curves (top) 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 corresponding effect on a sine wave, where the intervals in which the distortion increases (red background) or decreases (blue background) the transients of the original curve are highlighted. For this “simple” signal, the DSA and the PSD values are.​
  • +/+: DSA = 88%, s + = -52.2dB, s- = -72.6dB
  • +/-: DSA = 13%, s + = -72.6dB, s- = -52.2dB
Fig. 18 - Top: Transfer function for 2nd and 3rd order distortions both at -60dB. Bottom: related effect on a sine wave.
fig 18 - fx mix.png

In practice, the case in which we are for a given amplifier is detectable by measuring the phase value of the 2nd and 3rd harmonic distortion produced by a single tone:​

Harmonic​
Phase [degree]​
Transfer Function Coeff.​
2nd​
-90​
a2 > 0​
+90​
a2 < 0​
3rd​
+/-180​
a3 > 0​
0​
a3 < 0​

From some basic tests and experiences of people who have deepened these aspects such as Nelson Pass, all this translates into the following effects on our perception:​
  • 2nd order distortion: in addition to a ‘swelling’ of medium-low frequencies due to the density of the intermodulation products of type |fi–fj|, transients cause a softening effect. The differences due to the phase are in an approach to the sound listener and greater detail when the 2nd harmonic has 90 degree phase; deeper soundstage and better localization when the phase is -90 degrees.
  • 3rd order distortion: the main effect is of greater dynamic contrast when the 3rd harmonic phase is 180 degrees; on the contrary, reduction when it is 0 degrees.
  • Distortion mix: you have intermediate effects, with a predominance of the 2nd order ones for the lowest signal levels.
To be fair, it must be said that often the phases of the harmonics 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.​
 
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manisandher

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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!

Mani.
 

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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.
 

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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.
 

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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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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 comments.

Often times, a study is not known exactly in which direction it will lead you. In the one presented, the further insights in the next post have also resulted in changes to some of the parts exposed in the previous posts, necessary to make the entire thread consistent and not to create confusion. The changes are from the beginning of September 2022, they are not substantial, but rather refine the aspects already exposed.
 
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After examining in detail the 2nd and 3rd order distortions, usually of the most important level, let's explore the physical characteristics of the higher order ones. This in-depth study will give us the opportunity to better fix some of the concepts exposed and highlight further characteristics of the distortions.

Mathematical model of n-order distortions

We have seen how with a third degree polynomial we can model the transfer function for the distortions introduced by non-linear systems without memory up to the 3rd harmonic. It is easy to predict that, by extending the degree of the polynomial, distortions of any harmonic order n can be modeled:​

cap fn.PNG

By introducing a sinusoidal signal into the polynomial and applying the appropriate trigonometric transformations we will have that the different addends "control":​
  • a0: component in DC, normally null.
  • a1x: fundamental component, amplified by the gain g of the device.
  • aix^i: i-th order distortion, consisting of the set of harmonics of order j = i, i-2, i-4, i-6, ..., with phase equal to: 90·(-1)^(j/2) degrees if i is even; 180·(j-1)/2 if i is odd.
Therefore, the distortion of order i is made up of i/2+1 harmonic components (including DC): all the even ones are linked together, determined by the value of the coefficients a2, a4, a6, … and independent from the odd ones; in a similar way, the latter are independent from the even ones, identified by the coefficients a3, a5, a7,…. The following Tables allow us to calculate the modulus and phase of harmonics related to distortions up to the 11th order.​

tab 1-2.png

The values contained in Tables 1a or 1b show the coefficients of the harmonics sin(jx) (in the columns) whose sum constitutes the expansion of sin^i(x), the order of distortion (in the rows); Tables 2a and 2b show the phases of each harmonic. Let's see how to use them.

Let us take into account the cij element of Table 1a or 1b relating to the distortion of order i and harmonic j. The product ai·cij identifies the amount of the contribution of the distortion of order i to the harmonic of order j for a sinusoidal signal of 0dB level. The harmonic phase is identified by the corresponding element in Table 2a or 2b, to be increased by 180 degrees if ai is negative. Adding the products relative to the harmonic column of order j we will have the total value of the distortion for this harmonic due to all the distortion orders. The phase of the resulting distortion is reported in the same column of Table 2a or 2b, to which 180 degrees must be added if the result of the summation is negative. The following graphs give us an idea of the trend of the contributions in dB of the different distortion order, broken down by even and odd orders.​

Fig. 19 - Harmonic components trend for even and odd orders distortion.
fig 19 - dist 3D.png

Qualitatively we note that:
  • The contributions to the highest harmonics gradually decrease as the order of distortion increases, with decreasing speed as the order of distortion increases.
  • The contributions to the lower order harmonics differ by a few dB for the different distortion orders.
These curves are relative to a sinusoidal signal of unitary amplitude (0dB). For lower amplitudes the contribution ai·cij must be multiplied by the i-th power of the amplitude of the sinusoid. This implies that the level of the i-th harmonic will decrease the faster ("slightly less" than the i-th power) the higher its order is as the signal level decreases. Figure 20 graphically shows the value of this factor in dB.​

Fig. 20 - Even and odd Harmonics reduction factor per input level.
fig 20 - x xpower.png

We observe that in the measurements of real systems we can detect the modulus and phase of the different harmonics, while the coefficients ai of the transfer function are to be determined. To obtain these values it is necessary to write a system of linear equations in the unknowns ai, where it equals:​
  • On the one hand, the expressions of the different harmonics resulting from the sum of the values in the columns of Table 1a and 1b, referring to the level of the fundamental.
  • On the other hand, the modules of the harmonics measured, taken with a positive sign if the phase of the harmonic is close to the corresponding values in Table 2a and 2b; negative if close to the same values increased by 180 degrees.
Solving the system with the classical methods we will obtain the values of ai sought.​

Analysis of n-order distortions on sinusoidal signals

In the previous paragraph we have explored the relationships between the distortions of n-order and the related harmonic contents. We continue the study by analyzing these distortions in the time domain. We have already seen the waveforms relating to distortions of only 2nd or 3rd order (see Figure 13 and 14), composed respectively of the 2nd harmonic+DC and 3rd harmonic+fundamental. For higher orders we will have more harmonics, with the ratios indicated in Tables 1a and 1b. The graphs in Figure 21 show the time trend of the distortion for increasing "pure" distortion orders, normalized to the maximum levels to compare the shapes. The relative derivatives used for the analysis of transients are also shown.​

Fig. 21 - Even and Odd Distortion orders for a sine wave (top) and their derivatives (bottom).
fig 21 - Dist Sin 2-11.png

Even visually, it is easy to deduce that for all even-order distortion curves the DSA is always equal to 50%; for those of odd order, 100%. Hence, the relative weight of the expanded and compressed transients does not change. But it is also evident that as the order n increases the distortions become more concentrated, with steeper edges, less and less "similar" to the original signal. Wanting to characterize this aspect numerically, regardless of the amount of distortion expressed by the PSD, we can define a new statistical indicator.​

PCD (Partialized Correlation on the Derivative)
In the theory of signals, a measure of the similarity of two signals is given by the cross-correlation operation, commonly used to search in a signal for the position of a shorter and more known section. For two real discrete signals x[i] and y[i] is defined by the relation:​

cap Rxy.PNG

Suppose we have two x and y signals that differ from each other only by an offset and possibly by a scale factor: the formula "scrolls" the y signal along the time axis, calculating the discrete integral (summation) of the product with the x signal in each position (value n). When the signals match, the value of Rxy(n) will be maximized, given that when the "peaks" and "valleys" of the two signals are aligned, we’ll have the highest contributions and all of the same sign to the integral. Then dividing Rxy(n) by the product of the square roots of the auto-correlations Rxx(0) and Ryy(0) of each signal (cross-correlation of the signals with themselves, proportional to the RMS value), we will normalize the resulting value in the interval [-1,1], obtaining what is called the correlation coefficient, c(n). The different values will indicate:​

1: The two signals are identical, except for a scale factor.
0: Complete decorrelation (orthogonality) of the two signals.
-1: The two signals are the opposite of each other, except for a scale factor.

Intermediate values in the interval [-1,1] indicate similarity levels. Since in this context we are interested in studying the similarity of transients on already aligned signals, we will get the PDC by calculating c(n) in the following way:​
  • The x and y signals will be the derivative of the source signal and of the distortion component only: therefore x[i] = s'[i] and y[i] = d'[i].
  • Given the alignment, we will consider only the value of the coefficient for n = 0.
  • We will partition the domain of the indices on which c(0) is calculated into two parts: positive where s'[i] and d'[i] have the same sign; negative where the signs are discordant.
In formulas:​
cap pcd.PNG

Therefore we will also have two values for the PCD: the first, indicated with c'+ (positive correlation), which expresses the degree of similarity of the expanded transients; the second, c’- (negative correlation) relating to compressed transients. With reference to the last two curves at the bottom of figure 7, c'+ will give us information about the similarity of the parts in white background; c'- for those in gray background.

The graphs in Figure 22 show in the case of the sinusoidal signals of Figure 21 the trend of correlations. These start from high values for the 2nd and 3rd harmonic (0.85 and 0.71), to reach very low values on the 10th and 11th harmonic (-0.34 and -0.32) to indicate, as expected, the less similarity of the transients.​

Fig. 22 - PCD for odd and even distortion orders for a sine wave, 0dB.
fig 22 - Hist corr 2-11.png

We add that as explained so far, the distortion of order n has been considered "pure", that is, in the transfer function only an is different from 0, which also determines the value of all harmonics (even or odd) of lower order, as reported in Figure 19. Typically, the other coefficients also assume values other than 0, causing constructive or destructive interference between the harmonics that vary depending on the signal level.​

Correlation for mix of 2nd and 3rd order distortions on music-like signals

Let's take the DSA graphs on a mix of 2nd and 3rd order distortions for a music-like signal and enrich it by plotting the PCD curves, for all the sign combinations of a2 and a3. We will have the curves shown in Figure 23.​

Fig. 23 - PCD and DSA per 3rd/2nd order ratio (dB) for all combinations of a2 and a3 signs.
fig 23 - corr 2-3.png

The curves relating to the positive sign of a3 are in continuous lines; those with a negative sign are dashed. In details:​

Correlation with positive a3
Both c'+ (red curve) and c’- (blue curve) start on the left from high values, respectively 0.8 (expansion) and -0.8 (compression), where 2nd order distortion prevails. As the 3rd order component (and DSA, gray curve, from 50% to 100%) increases, c'+ decreases to 0.6, indicating that expanded transients, even if they increase in percentage and magnitude, are altered in a more "abrupt" way. The negative correlation, c’-, increases (-0.9), with the expansive effects of the 3rd order which therefore compensate for the compression effects of the 2nd order. As with the PSD, these variations begin before the DSA begins to grow. The final part of the curve, given the small amount of compressed transients, is not significant.​

Correlation with negative a3
Both values c'+ and c’- always start from the same values 0.8 and -0.8. But here, as the 3rd order component increases, c’+ rises to 0.9, indicating greater agreement in the expanded transients, well before the DSA begins to descend to 0%, with only compressed transients. Instead, the negative correlation c’- decreases (-0.6), where the more abrupt compression effects of the 3rd order are added to those of the 2nd order.

We add that the sign of a2 has no influence on the curves, as well as the reference level chosen for the 2nd harmonic.

Now let's see what happens to the PCD curves when the input level changes, from 0dB to -30dB in 10dB steps (input volume control). The resulting graph is illustrated in Figure 24 for distortions with coefficients a2 and a3 positive.​

Fig. 24 - PCD and DSA per 3rd/2nd order ratio (dB), for more input levels.
fig 24 - corr L 2-3.png

We observe that, as the input level decreases, all curves translate to the right by the same amount of attenuation, 10dB. This effect is always due to the fact that the 3rd order distortion decreases faster than the 2nd order one: similarly to the PSD, the curves detect the “longer” maintenance of the 2nd order characteristics. The other aspects are similar to those of Figure 23. Figure 25 finally shows the PCD as a function of the input level for different values of the 3rd/2nd order ratio.​

Fig. 25 - PCD and DSA per input levels, for more 3rd/2nd distortion ratios.
fig 25 - corr L2 2-3.png

Here too, it is evident that for low input levels the distortion is always dominated by the 2nd order characteristics, which turn into 3rd order the faster the higher the 3rd/2nd harmonic ratio.​

Distortion analysis for exponential transfer functions

Let's now analyze what happens for transfer curves with exponential trend, typical of active components such as transistors. Figure 26 reports:​
  • The transfer function, as usual amplified to show the details.
  • The level of the harmonics modeled up to the 10th order for a starting level of -60dB on the 2nd harmonic; it is also shown how each decreases as the input level decreases from 0dB to -15dB in -3dB steps.
  • The shape of the distortion on a sinusoidal (amplified) signal for the input levels 0dB, -3dB, -6dB and -9dB.
  • The shape of the distortion derivative (amplified) for the same input levels.
Fig. 26 - Distortions for an exponential transfer function.
fig 26 - Sig exp.png

The transfer curve and the distortion graphs show an expansion of the positive values of the signal and a compression of the negative ones, therefore a "2nd harmonic" behavior, as is also evident from the following figure which shows the frequency trend of the distortion, with and without the frequency components coinciding with the source signal.​

Fig. 27 - Distortion in frequency for an exponential transfer function, with and without the original signal components; ref. to the input signal.
fig 27 - Freq exp.png

From the previous figures it appears that the DSA is 50%; for the other indicators we will have, for a level of 0dB in input:​
  • Expanded transients: s + = -60.3dB; c’+ = +0.77
  • Compressed transients: s- = -63.4dB; c'- = -0.84
As expected, these values show that the expansive distortions act more consistently than the compressive ones (3dB), but the latter are a little more similar in shape to the original transients. With the decrease in the input level these differences are progressively reduced, as is evident in Figure 28, which shows PSD and PCD as a function of the input level; The reference is on HD2 at -60dB.​

Fig. 28 - PSD, PCD and DSA for an exponential transfer function.
fig 28 - Sig L exp.png

These curves have a similar trend to those of Figure 12b and 25, obtained for 2nd and 3rd order distortions. The main difference is in a greater symmetry of the positive and negative PSD and PCD curves, with the DSA always remaining at 50%.

Therefore, we would expect a "warmth" effect, with almost level-independent characteristics. However, it must be said that for higher order harmonics the masking effect of our ear is lower, so these harmonics could become audible "individually", adding sensations of harshness. To complicate things there is a further psychoacoustic effect to be taken into consideration, which is that of the aural harmonic distortion. In short, our ear, when stimulated by a pure tone, autonomously generates harmonics (of a compressive type), similar to the non-linearities described so far for electronic devices. To get a more precise idea, the following figure shows the analytical curve, obtained by D. H. Cheever from the data of a work by H. F. Olson in 1967 in the RCA laboratories, which approximates the level of the harmonics generated by our ears. Only integer values are significant here.​

Fig. 28b - Aural Harmonic Distortion for some SPL (dBA).
fig 29 - Aural Dist.png

Now, if it has been verified that the ear sends impulses towards our brain both for the fundamental and for each of the self-generated harmonics, it is also true that the brain ignores the latter: we perceive only the fundamental tone. This aspect, which adds to that of the acoustic masking already described, is used by some as a distortion model for an ideal amplifier: if its distortions conform to this pattern, then it is likely that our brain does not detect them. The exponential trend of Fig. 26 is very close to these curves, so these distortions could be viewed not so negatively. At present, I do not know of any in-depth studies on this aspect.

Conclusions

In this study we explored many characteristics of the non-linear distortions introduced by audio devices with negligible memory effects, in order to derive qualitative correlations with the effects on our perception. The main attention was paid to the study of the relationships between the transients of the original signal and the alterations introduced by the distortions, working mainly on the time domain. In fact, several studies have shown a high sensitivity of our auditory system to temporal resolution, far superior to that in frequency and with no apparent connection with this.

On the basis of simulations on music-like signals, it was natural to classify the alterations into two types: expansive and compressive. Each causes clearly identifiable physical effects, correlated to perceptual aspects that we commonly refer to with the terms of micro and macro-dynamics. The mix of the two types of alterations, defined numerically by the DSA indicator, seems to be a good guide to frame the effects of “warm”, "dynamic expansion" or "compression" that we perceive, at least for low orders of distortion and at modest levels. Given the infinite nuances of these effects, two other physical indicators help us in the analysis: the PSD, which determines the amount of distortion injected into the signal in the two types of transients, thus revealing their audibility, and the PCD, which quantifies the degree of similarity to the original transients.

All the indicators can be obtained by performing module and phase measurements of the harmonic distortions of real devices, assuming that these are sufficiently devoid of memory effects. It is clear that the presence of these effects, as well as noise can make the distortion shape more "tortuous" and consequently more difficult to translate into effects on our perception. Also in this cases, the defined quantities can be useful investigation tools to frame the alterations suffered by the signal. How much a more or less wide variation of these physical quantities is perceived as more or less important must then be experimented with listening or other tests.​
 
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You spent all this time doing numerical analysis of signals and allowed yourself to make layman comments and interpretations of the audible effects? Unless you introduce a loudness model any such discussion is moot.
 

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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.
Yes, phase is very important. Nelson Pass discusses the effects of the 2nd harmonic-to-fundamental phase relationship here:

 
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You spent all this time doing numerical analysis of signals and allowed yourself to make layman comments and interpretations of the audible effects? Unless you introduce a loudness model any such discussion is moot.

The study delves into the physical aspects related to non-linear distortions in the time-domain, to which our ear has a much higher sensitivity than in frequency. I have not found much documentation on these aspects, hence the insights and the attempt to define some useful indicators to qualify them. The same is for correlations with effects/hearing levels (those classics work in frequency): if you are aware of them, share!
 

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Curvature

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The study delves into the physical aspects related to non-linear distortions in the time-domain, to which our ear has a much higher sensitivity than in frequency. I have not found much documentation on these aspects, hence the insights and the attempt to define some useful indicators to qualify them. The same is for correlations with effects/hearing levels (those classics work in frequency): if you are aware of them, share!
I'll work on some sources if you wouldn't mind sharing the basis for your comment on frequency vs. timing.

Are you familiar with binaural unmasking?
 

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Oh yessir. I'm certainly sorry if I offended!
Sarcasm engaged, I see.


Aren't you an engineer? Why would you continue to disseminate false info? Plainly false. Circuit diagrams aside.
 

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Since "perception" is in the thread title, is it the intention to derive perception from pure mathematics here?
 
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Pinox67

Pinox67

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I'll work on some sources if you wouldn't mind sharing the basis for your comment on frequency vs. timing.

Are you familiar with binaural unmasking?
I can't rescue the main articles now (they are from works of Von Békésy and Nordmark about inter-aural resolution), but there are also more recent books on psychoacoustic that describes these aspects. If I remember well, also MQA format (albeit questionable) takes care of this aspect.
 
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Hi, a small question - would you try to investigate an effect of crossover distortion rather than polynomial frequency independent (static) distortion? Thanks for the answer.
 
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Pinox67

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Since "perception" is in the thread title, is it the intention to derive perception from pure mathematics here?
The intention is described clearly at the beginning of thread...and also in the conclusions: investigation on physical alterations of distortions, to help with speculations with the effects on perception. How much a more or less wide variation of these physical quantities is perceived as more or less important must be experimented with listening tests (running, albeit more difficult and longer than simulations).
 
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Pinox67

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Hi, a small question - would you try to investigate an effect of crossover distortion rather than polynomial frequency independent (static) distortion? Thanks for the answer.

Good question! I actually thought about it a few days ago... I have to update the simulator or model this distortion with a high-degree polynomial. I will try.
 
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