THD Measurement: Do it represent the actual nonlinear harmonic distortion?

[Pinox67]

As part of a study of the effects of non-linear distortions in amplifiers, I came across an aspect that left me a little perplexed. This is the way in which the THD value (and also the IMD and TD+N) of the data collected by the standard measurements is calculated.
​

Measurement of distortions in static nonlinear systems​

To introduce the discussion, let's start by summarizing what THD is: it represents the measure of harmonic distortion present in a signal. It is obtained by introducing a pure tone into the device under test, generally a sine wave at f1=1KHz, to then reveal the overall level of harmonic distortions at frequencies of 2f1, 3f1, etc. in the output signal. It is then calculated by the ratio between the RMS (Root Mean Square) value of all these harmonic distortions and the RMS value of the tones in the original signal.

With another indicator, the IMD, we can measure the overall level of intermodulation products that arise when using test signals typically consisting of two tones. If their frequencies are f1 and f2 we will have the new tones at frequencies if1±jf2, with i and j positive integers. These tones are in general non-harmonic. IMD is then calculated by the ratio between the RMS value of the overall intermodulation products and the RMS value of the tones in the original signal. It should be noted that in some measurement standards such as CCIF and SMPTE, for simplicity of calculation, some tones are neglected in the calculation.
​

Criticisms have been raised in the literature of the THD and IMD indicators as being too simple to be representative of the amount of distortion experienced by real musical signal. To bring the measurements closer to the real world, it can be adopted multitone test signals, where there will be a proliferation of harmonic distortions and above all of intermodulation products, which together form a sort of "carpet". Since it is very complex to separate them, the indicator that is used for this signal, the TD+N, does not distinguish them: it is the ratio between the RMS value of all distortions (including any noise) and the RMS value of all tones in the original signal. Generally, for all indicators the DC component is not considered in the calculations.

But let's take a step back to identify the cause of the nonlinear distortion. This is due to the non-constancy of the g gain of the device for each level of the input signal. In other words, the input/output transfer curve f(x) of our device is not represented by a straight line (of slope g) in the working interval but has "imperfections". If we model this curve and calculate the output values for each value of the input signal we will be able to numerically simulate the behavior of our device for any signal. If we want to simulate a system that has 2nd and 3rd harmonic distortions, the transfer function f(x) can be modeled by a third degree polynomial:​

In fact, by inserting a sinusoidal x(t) signal in this polynomial and carrying out the appropriate trigonometric transformations, we will have that the different addends control:​

• a0: component in DC, normally null.
• a1x: amplification (of gain g) of the fundamental component (i.e. H1).
• a2x²: 2nd order distortion, consisting of two components:
  • 2nd harmonic (HD2), out of phase by +90° or -90°;
  • DC, of the same entity as HD2.
• a3x³: 3rd order distortion, consisting of two components:
  • 3rd harmonic (HD3), 0° or 180° out of phase;
  • contribution to the fundamental component (HD1), 9.5dB 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. To better establish the concepts, let's make an example. The following figure shows:​

• The transfer function (left), relating to a mixed distortion of 2nd and 3rd order, with HD2 = -70dB and HD3 = -60dB, having a phase of 90° and 0° respectively; the gain g is unity.
• The non linear distortion undergone by a single tone (right): in blue the resulting curve, where a “compressive” effect can be seen, and in red the isolated distortion component.

In both graphs the curves are amplified by 40 times to better show the shape of the distortion. As explained above on the transfer function, the distortion curve in red is here composed of four harmonics: HD2 + DC, due to 2nd order distortion, and HD1 + HD3, due to 3rd order distortion.​

Fig. 1 - Transfer function (left) and distortion on a single tone (right).​

The one described is a simplified approach, capable of modeling non linear systems called static, without memory, i.e. the output value depends only on the instantaneous value of the input signal (except for the crossing time of the system), which constitutes the point starting point for the following considerations.

In the common calculation of the THD in the numerator here are considered only the values of HD2 and HD3, determining a value of -59.6dB, and the DC and HD1 are omitted. Now, if the DC component can be neglected, since it does not reach our speakers due to the decoupling capacitors between the electronic devices that block it, the same discourse is not applicable to the HD1 component, which instead crosses the entire audio chain downstream, and it is mainly responsible in this case for the compressive effect on the signal. Therefore, the THD value calculated in this way is not really representative of the entity of the distortion related to the red curve shown in the previous figure (actually, it is relative to the curve in green), although the definition of THD speaks of "all" harmonic distortion components, and not only "new". If HD1 were included in the calculation, we will have a THD of -50dB, almost 10dB higher.

Let's investigate the phenomenon better by verifying what happens for different mixes of distortion.​

Fig. 2 - HD, “True-THD”, THD [dB] for more ratio of 3rd/2nd harmonic.
​

The figure illustrates the trend of distortions in dB as a function of the ratio between the levels of the 3rd and 2nd harmonic, with the latter set at -90dB. The red curve relates to the 2nd harmonic (HD2), constant, while the light blue curve shows the value of the 3rd harmonic (HD3), increasing. The resulting THD value for these 2 harmonics for each value of their ratio is represented by the dashed purple curve. Including in the calculation also HD1, shown in blue (parallel to HD3 and 10dB higher), we will have the “True-THD” curve shown in purple. This coincides with the “classic” THD value as long as the distortion is mainly 2nd order, while it is higher up to 10dB when 3rd order prevails; moreover, it begins to grow much earlier.

Figure 2b shows what happens if we observe the same quantities as a function of the input signal level for the transfer curve in Figure 1. Hence, HD2 = -70dB and HD3 = -60dB. The reference here is H1; if instead we considered H1+HD1 as a reference we would not have any “visible” impact in the curves.​

Fig. 2b - HD, “True-THD”, THD for input levels from -60dB to 0dB.
​

From the figure it is evident that the classic THD coincides with the True-THD for low input values, below -30dB; for higher levels the True-THD deviates up to 10dB, as in the previous graph. This is due to the fact that the effect of 3rd order distortion (and therefore both HD1 and HD3) at low levels is not felt very much, since it depends on the cube of the signal level, leaving "the field" at the 2nd order (HD2) which depends on the square of the signal level. It should be added that by increasing or decreasing the HD2 and HD3 values by the same amount (in dB), all curves translate respectively upwards or downwards. Therefore, the discrepancies between the classic THD and the True-THD depend on the relationships between the harmonics, not on the absolute values.

From what has been described it follows that not considering the harmonic HD1 in the calculation of the THD distorts its "representativeness" as an indicator for the overall amount of distortion suffered by the signal when the 3rd harmonic appears. More generally, we can say that it is not correct to compare THD values close to each other when the harmonics involved in its determination are of a different nature, i.e. mainly of even or odd order: there is a risk of committing gross errors of evaluation, since not considering HD1 can lead to differences of 10dB.

For the IMD there are no discrepancies given that HD1 is to be considered "harmonic" and therefore not included in the calculation of this indicator. Instead for the TD+N, which contemplates all forms of distortion, the discrepancy reappears. To get an idea, the following graph shows the simulation of the distortion suffered by a signal composed of 8 tones equally spaced in frequency from 5KHz to 12KHz, level at -18dB, when passing through a system with only 2nd order distortion (HD2 = -60dB) and in another with only 3rd order (HD3 = -60dB).​

Fig. 3 - Signal and distortion (HD2 = -60dB) in the frequency domain for 8 tones @ 10KHz, 11KHz, …, 17KHz - Level = -18dB.

Fig. 4 - Signal and distortion (HD3 = -60dB) in the frequency domain for 8 tones @ 10KHz, 11KHz, …, 17KHz - Level = -18dB.​

In both graphs all non linear distortion components are shown in orange, while the source signal in blue. The distortion components coinciding with the tones of the source signal, of different levels and of not exactly negligible entity, are evident. These tones would not be detected directly by any measurement. The comparison between the classic and real TD+N values is as follows:
​

- 2nd order (Fig. 3): Classic / True TD+N = -57.8dB / -57.5dB
- 3rd order (Fig. 4): Classic / True TD+N = -58.5dB / -52.5dB

Therefore, even in the case of the TD+N calculation, the differences are not always negligible, the value is underestimated and the discrepancy is higher when there are odd-order distortions.

In conclusion, beyond the difficulty of detecting directly the values of the distortions at the same frequencies of the test tones in the measurements, I cannot see any valid reason to neglect these harmonics in the calculation of the THD (and TD+N). These are in fact the main responsible for the compressive or expansive effects on the signal (in this thread more details). However, it would be possible to develop a more sophisticated dynamic model, (i.e., with memory) which contemplates arbitrary phase values of the distorsion, to estimate the entity of these harmonics from the values of the other harmonics detected and then correct the calculation of these indicators, at least for the simplest cases. Perhaps also providing additional test signals to facilitate this calculation.

[EDIT] Analysis updated here also for the dynamic model.​

 

[JohnPM]

Why should a component at the same frequency as the input be considered a distortion?

 

[Pinox67]

Why should a component at the same frequency as the input be considered a distortion?

Well, if it depended linearly on the signal, it might not be considered distortion, yes. But here we are talking about a cubic dependence on the signal level (see fig. 2b). If higher order harmonics are present we will have contributions to HD1 also with the fifth powers, seventh etc. And in the presence of more tones, as illustrated in figs. 3 and 4, there are much more complex situations...

In short, it has all the characteristics of a distortion, the only thing that differentiates it is that it is "hidden" in the harmonics of the source signal, but it is there, it alters the signal in a non-linear way, it is not negligible and, if not considered, adds no small margin of uncertainty when comparing THDs or IMDs of different devices.

 

[fpitas]

Just from the viewpoint of audibility, I could argue that intermod is the better metric. Especially multitone IMD. A lot of those new tones are not harmonically related, so they are inherently unmusical.

 

[JohnPM]

The THD figure summarises the energy in the harmonics relative to the fundamental. It is not a complete characterisation of the system, nor is it presented as such. The effect of the fundamental contributions is to change the gain at the fundamental from a constant to a level-dependent curve, the linearity plot of a stepped THD measurement shows that characteristic but why should THD include it?

 

[pkane]

Well, if it depended linearly on the signal, it might not be considered distortion, yes. But here we are talking about a cubic dependence on the signal (see fig. 2). If higher order harmonics are present we will have contributions to HD1 also with the fifth powers, seventh etc. And in the presence of more tones, as illustrated in figs. 3 and 4, there are much more complex situations...

In short, it has all the characteristics of a distortion, the only thing that differentiates it is that it is "hidden" in the harmonics of the source signal, but it is there, it alters the signal in a non-linear way, it is not negligible and, if not considered, adds no small margin of uncertainty when comparing THDs or IMDs of different devices.

THD or IMD is a single number that doesn't reveal the details where the energy is coming from, from which harmonic. IMD calculation frequently only uses 2nd and 3rd order products, for example. I see @JohnPM just posted a similar response while I was typing this.

 

[DonH56]

HD1 is the fundamental. Nonlinear compression creates distortion terms as well as reducing the amplitude of the fundamental. The THD calculation RSS's the distortion terms and ratios the result to the fundamental, so if the fundamental is compressed, the overall THD will reflect that by being higher (worse) due to the lower fundamental (HD1) amplitude.

Unless I am completely missing something...

One thing (OK, among many) that THD will not show are non-harmonic distortion terms that may arise, e.g. in physical (speaker) systems. It also won't show things like power supply noise, clock or other coupled noise (EMI/RFI, 60 kHz spurs from fluorescent lights, etc.), and so forth.

 

[Pinox67]

The THD figure summarises the energy in the harmonics relative to the fundamental. It is not a complete characterisation of the system, nor is it presented as such. The effect of the fundamental contributions is to change the gain at the fundamental from a constant to a level-dependent curve, the linearity plot of a stepped THD measurement shows that characteristic but why should THD include it?

THD, by definition, is the measure of harmonic distortion in a signal. HD1 is a component of the distortion, hidden in the fundamental. If I neglect it, I am not giving evidence of all (and not only “new”) harmonic distortions produced by my system. I am well aware of the difficulties in calculations in considering HD1, "immersed" in the fundamental measured on systems that modify the fundamental input by means of a gain factor, I struggle with it too...

We can also see the situation from another point of view. When we calculate the RMS value of a signal you know very well that we can proceed in two ways:

• Time-Domain: we perform the square root of the integral (sum in the discrete case) of the square of the signal, on the value of the time interval.
• Frequency-Domain: we perform the Fourier transform of the signal and the integral (sum) of the square of the modules of all the frequency components.

By Parseval's Theorem, the two methods are equivalent, they give the same result. Now, excluding the DC component, the RMS value of the distortion component only (red curve in fig. 1, which represents the real distortion injected into the signal) calculated in the time-domain coincides with the RMS value calculated on the same component in the frequency-domain only if we consider all harmonics, even HD1. If excluded, the RMS values calculated in the two domains coincide only if the signal has no odd harmonics (I have experienced it directly in the simulations). This highlights once again that by performing the calculations as today in the frequency domain (only “new” vs “all” harmonics) we are losing energy components. And the most serious thing in my opinion is that this loss occurs only in some cases.

 

[Pinox67]

THD or IMD is a single number that doesn't reveal the details where the energy is coming from, from which harmonic. IMD calculation frequently only uses 2nd and 3rd order products, for example. I see @JohnPM just posted a similar response while I was typing this.

Yes, the THD and the IMD are single numbers, which do not “carry around” information on the nature of the harmonics that determine them. But precisely for this reason it must allow to compare THD value of different systems having distortions of different order. If in the case of 2nd order distortions it correctly reflects the relative energy levels, but in the case of 3rd order distortions it "loses pieces" that are not negligible, as the mathematical models reveal, it loses a bit of validity as a tool for making comparisons.

 

[Pinox67]

HD1 is the fundamental. Nonlinear compression creates distortion terms as well as reducing the amplitude of the fundamental. The THD calculation RSS's the distortion terms and ratios the result to the fundamental, so if the fundamental is compressed, the overall THD will reflect that by being higher (worse) due to the lower fundamental (HD1) amplitude.

In my description by HD1 I mean the 1st order Harmonic Distortion, at the same frequency as the fundamental. The fundamental is indicated with H1, without HD1. In the calculations of the THD we perform the ratio of the RMS value of the distortions with the RMS value of H1+HD1, without distinguishing them, yes.
But generally, in the Hi-Fi field, the levels of harmonic distortions, including the value of HD1, are much smaller than those of H1. Therefore the error in considering H1+HD1 or H1 in the calculations is small: from the simulations, to approach 1dB of error, we would have to overcome levels of distortion in harmonics of -30dB...

One thing (OK, among many) that THD will not show are non-harmonic distortion terms that may arise, e.g. in physical (speaker) systems. It also won't show things like power supply noise, clock or other coupled noise (EMI/RFI, 60 kHz spurs from fluorescent lights, etc.), and so forth.

By definition, THD only accounts for harmonic distortion. All other aspects, however important, are not taken into consideration, yes, other measures, graphs, etc. are needed.

 

[pkane]

Yes, the THD and the IMD are single numbers, which do not “carry around” information on the nature of the harmonics that determine them. But precisely for this reason it must allow to compare THD and IMD values of different systems having distortions of different order. If in the case of 2nd order distortions it correctly reflects the relative energy levels, but in the case of 3rd order distortions it "loses pieces" that are not negligible, as the mathematical models reveal, it loses a bit of validity as a tool for making comparisons.

It's valid for what it's designed for. You want THD to be a true measure of non-linearity. That it is not. It is an indication of energy in harmonic (and IMD) components relative to the fundamental caused by the nonlinearity. In fact, changes to the fundamental are not considered distortions in audio and signal processing -- distortion is reserved for new frequency components that didn't exist in the original signal.

What you seem to want to do is to introduce a new metric, call it F+THD or FHD, or some-such. Let's see if that proves to be any more useful than THD

 

[JohnPM]

the RMS value of the distortion component only (red curve in fig. 1, which represents the real distortion injected into the signal) calculated in the time-domain coincides with the RMS value calculated on the same component in the frequency-domain only if we consider all harmonics, even HD1.

That's only "real distortion" in the context of your polynomial model because that model causes you to define a separate model-based fundamental component of the output. From a black box perspective the distortion on the output is the content which is not at the frequency of the input and what you wish to call HD1 is part of the fundamental.

 

[DonH56]

In my description by HD1 I mean the 1st order Harmonic Distortion, at the same frequency as the fundamental. The fundamental is indicated with H1, without HD1. In the calculations of the THD (and IMD) we perform the ratio of the RMS value of the distortions with the RMS value of H1+HD1, without distinguishing them, yes.
But generally, in the Hi-Fi field, the levels of harmonic distortions, including the value of HD1, are much smaller than those of H1. Therefore the error in considering H1+HD1 or H1 in the calculations is small: from the simulations, to approach 1dB of error, we would have to overcome levels of distortion in harmonics of -30dB...

By definition, THD only accounts for harmonic distortion. All other aspects, however important, are not taken into consideration, yes, other measures, graphs, etc. are needed.

OK, I misunderstood, or more likely did not read thoroughly. It would not be "harmonic" distortion, but rather distortion at the fundamental frequency. Harmonic means at an integer multiple of the fundamental. In the RF world fundamental compression is considered when assessing performance, typically the -1 dB power compression point, but I have never seen that used in audio. Instead, for audio clipping is defined as occurring at the "knee" of the THD curve, such as when distortion has reached 1% with respect to the magnitude of the fundamental (which is compressed at that point as you say). Similarly, the concept of intercept points commonly used in the RF world (e.g. IP2, IP3) could also be applied to the audio world, but the onset of clipping for typical audio circuits is so fast that I am not sure it is useful. At one point I presented a program that would generate -1 dB compression and the intercept points (2, 3, 4, 5...) for an amplifier (any amplifier), but nobody in the audio world knew what I was talking about, and the RF-based metrics did not seem to offer anything beyond "normal" THD and IMD measurements.

As @pkane and @JohnPM have said, it sounds like you are arguing for a new metric, perhaps to apply something like the power compression metric used for RF amplifiers to the audio world. An interesting idea, technically accurate, but may not add much value in the real world? My intuition says it may be more useful for circuits using very low feedback that exhibit relatively soft saturation. Typical designs using significant feedback to achieve low distortion tend to transition very sharply from low distortion to clipping (high distortion) so the fundamental compression point is very sharply defined. That implies little difference between classic THD measurements and your new approach that includes the effect of fundamental power (or voltage) compression.

Unless I have misunderstood again... - Don

 

[Pinox67]

It's valid for what it's designed for. You want THD to be a true measure of non-linearity. That it is not. It is an indication of energy in harmonic (and IMD) components relative to the fundamental caused by the nonlinearity. In fact, changes to the fundamental are not considered distortions in audio and signal processing -- distortion is reserved for new frequency components that didn't exist in the original signal.

What you seem to want to do is to introduce a new metric, call it F+THD or FHD, or some-such. Let's see if that proves to be any more useful than THD

So, the answer to the question in the thread title, if THD represents the nonlinear distortion of a system, would be a nice NO.

[EDIT] You must admit that this difference is very fine. The non-linearities of a system generate harmonic distortions. I measure the energy level of these distortions but, since in some cases some of them escape, I cannot attribute this measure as a indicator of the non-linearities present in my device. But this association is what people commonly do and expect… The difference should be explained to them, since it would be necessary to correct the calculation of the THD to make them representative, or to prepare new, more sophisticated metrics.

 

[Pinox67]

That's only "real distortion" in the context of your polynomial model because that model causes you to define a separate model-based fundamental component of the output. From a black box perspective the distortion on the output is the content which is not at the frequency of the input and what you wish to call HD1 is part of the fundamental.

[EDIT] I agree with you, in a "black box" model I don't know a priori what the non-linear distortion model of my device is, and therefore I can't know if there is an HD1 component or not. But already with a very simple distortion model like the one presented it is evident that the current way of calculating THD can potentially lose energy components that are not of small entity... and therefore does not correctly represent the non-linearities in my device, simple or sophisticated that are. So, the answer to the thread title is NO, and it is not sound so good in my opinion...

I would like to add that together with my electronic technical friends who are very skilled in the construction of preamps and power amps, we have experienced that the described model predicts very well the behavior of (non-linear) circuits with negligible memory effects. This essentially means systems with a wide bandwidth (about 100KHz), harmonic distortions with little frequency dependence (at least in the audio band they do not vary more than 6dB) and above all phases of harmonic distortions that approximate the reported values, independent of the signal level. So, at least for this preamp class, through simulations we can understand a lot of what happens with the harmonic distortions, which is what I reported. It is sufficient to measure the distortions for a single tone (in module and phase) to predict with good precision the result for any signal. In the initial part of this thread some comparisons between measurements and simulations. To add that the circuits are tube based and without negative feedback.

For systems with more conspicuous (non-linear) memory effects, detectable by verifying the phases of the harmonic distortions, very different from the values reported, more sophisticated models are needed, such as the Volterra Kernel. I am working on them, they are much more difficult, both in the measurements and calculations. Maybe it takes someone much better than me!

Therefore, from the measurements, it should be possible to trace a representative "black-box" model that allows to derive the value of HD1. An alternative way could be to prepare ad-hoc test signals, similar to those used for IMD, multitone, to measure the differences between the levels (and phases?) of the fundamentals. All to be explored…

 

[mansr]

That's only "real distortion" in the context of your polynomial model because that model causes you to define a separate model-based fundamental component of the output. From a black box perspective the distortion on the output is the content which is not at the frequency of the input and what you wish to call HD1 is part of the fundamental.

Put differently, any change to the fundamental is a linear distortion that will show up in a standard frequency response measurement.

 

[pma]

How about 3D to catch fluctuations of H1 and harmonic and non-harmonic distortion components?

 

[syn08]

Put differently, any change to the fundamental is a linear distortion that will show up in a standard frequency response measurement.

Are you sure? Unless we have different understandings of "linear distortion", a system transfer function that somehow modulates (in time) the fundamental and its harmonics is no longer a time invariant system (trivial to prove mathematically, think of the time variant definition, F(t)=f(x(t), t), that is having an explicit dependency on time). As such, it does not satisfy the Fourier theorem conditions *) and, as a result, the output may have nonharmonic components. Everything we assume (in a standard model) about THD crumbles, since the energy transformed by the nonlinearities is no longer distributed in the fundamental frequency multiple bins only.

*) There are methods developed to extend the Fourier theorem to time variant systems, the most commonly used is a Fourier transform that has kernel of non constant frequency (goodbye Dirichlet!); the immediate engineering application is when the time variant effect is small, then its effect can be separated and handled using some sort of perturbation approach. Check this if interested https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.493.9952&rep=rep1&type=pdf, but to me this approach is, for audio, a useless band aid on an non existing wound. For all purposes, audio systems/circuits are time invariant, of minimum phase (that is, the system and its inverse are causal and stable).

 

[pkane]

Are you sure? Unless we have different understandings of "linear distortion", a system transfer function that somehow modulates (in time) the fundamental and its harmonics is no longer a time invariant system (trivial to prove mathematically, think of the time variant definition, F(t)=f(x(t), t), that is having an explicit dependency on time). As such, it does not satisfy the Fourier theorem conditions *) and, as a result, the output may have nonharmonic components. Everything we assume (in a standard model) about THD crumbles, since the energy transformed by the nonlinearities is no longer distributed in the fundamental frequency multiple bins only.

*) There are methods developed to extend the Fourier theorem to time variant systems, the most commonly used is a Fourier transform that has kernel of non constant frequency (goodbye Dirichlet!); the immediate engineering application is when the time variant effect is small, then its effect can be separated and handled using some sort of perturbation approach. Check this if interested https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.493.9952&rep=rep1&type=pdf to me this approach is, for audio, a useless band aid. For all purposes, audio systems/circuits are time invariant, of minimum phase (that is, the system and its inverse are causal and stable).

How did this just become a time-variable transfer function discussion when from the very first OP it was about a static (memory-less) nonlinearity? Things become much more complex when memory is added, and the simple polynomial mentioned in the OP will not suffice to model it.

 

[syn08]

I was strictly commenting on @mansr statement: "any change to the fundamental is a linear distortion" which is, in general, IMO incorrect (or wrongly formulated).

For the rest, IMO the OP premises are incorrect; while some sort of integral transform using a kernel of polynomials can be developed, it would not render any other results than the standard Fourier transform. I believe what the OP considers as "contribution to the fundamental component" is simply a reflection of the obvious effect of energy transfer from the fundamental to the harmonics. Otherwise, I may not understand what's the fuss here. Dear Fourier is good enough, we don't need anything else, unless the OP wants to model a time variant system (like dielectric memory, or thermal behavior). For which no polynomials will ever help.