As part of the insights into the behavior of distortions through measurements and simulations, I came across an aspect, actually secondary to my study but which intrigued me enough. This is the calculation of the THD and IMD in non-linear systems (in simpler words, our audio amplifiers…).
In short, the modeling of these systems shows that among the new harmonic components that are introduced in the output signal (HD2, HD3, HD4 ...) there is also a component, HD1, "hidden" in the fundamental (H1). This component is naturally not detected in the measurements, as the instruments read H1+HD1, they cannot be distinguished, and refer the level of the other harmonics to this value. But the computational models say that there is…
As quantity, HD1 is not exactly negligible: it can be up to 10dB higher than the third harmonic (HD3). And it must be considered distortion, since it depends on the cube of the signal level (or higher powers in the presence of other harmonics). This causes dynamic compressive or expansive effects on the signal and phase shifts.
Now, beyond the questions on the audibility of these effects, this aspect also causes errors in the calculation of the THD which, "losing" this component in the numerator of the classic calculation formula (HD1 is not there...), is underestimated by up to 10dB in the presence of odd order distortions (taking HD1 into account or not in the denominator causes negligible errors, at least in the "normal" case of audio device distortions). In other words, the THD is not representative of all the nonlinear distortions introduced by the system. Therefore, even the comparison of the THD (and also of IMD, SINAD etc.) of different devices loses significance when these values are close to each other.
I reported more details in these two posts:
In short, the modeling of these systems shows that among the new harmonic components that are introduced in the output signal (HD2, HD3, HD4 ...) there is also a component, HD1, "hidden" in the fundamental (H1). This component is naturally not detected in the measurements, as the instruments read H1+HD1, they cannot be distinguished, and refer the level of the other harmonics to this value. But the computational models say that there is…
As quantity, HD1 is not exactly negligible: it can be up to 10dB higher than the third harmonic (HD3). And it must be considered distortion, since it depends on the cube of the signal level (or higher powers in the presence of other harmonics). This causes dynamic compressive or expansive effects on the signal and phase shifts.
Now, beyond the questions on the audibility of these effects, this aspect also causes errors in the calculation of the THD which, "losing" this component in the numerator of the classic calculation formula (HD1 is not there...), is underestimated by up to 10dB in the presence of odd order distortions (taking HD1 into account or not in the denominator causes negligible errors, at least in the "normal" case of audio device distortions). In other words, the THD is not representative of all the nonlinear distortions introduced by the system. Therefore, even the comparison of the THD (and also of IMD, SINAD etc.) of different devices loses significance when these values are close to each other.
I reported more details in these two posts:
- Static nonlinear model: basic case, useful to understand "where HD1 comes from".
- Dynamic nonlinear model: situation more representative of what happens on real systems.
Has anyone come across these aspects before?
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Yes, I've noticed that also when writing out equations.
Suppose you have 1 % of first harmonic distortion. The first harmonic level then changes by +/- 1 % if the distortion is in phase or in antiphase with the desired signal, corresponding to a level error of about +/- 0.08 dB.
It is a non-linear effect, but the distortion is very well masked because it is spectrally right on top of the desired signal, and the amount of instantaneous compression or expansion is also quite small.
Suppose you have 1 % of first harmonic distortion. The first harmonic level then changes by +/- 1 % if the distortion is in phase or in antiphase with the desired signal, corresponding to a level error of about +/- 0.08 dB.
It is a non-linear effect, but the distortion is very well masked because it is spectrally right on top of the desired signal, and the amount of instantaneous compression or expansion is also quite small.
The point is that the same non-linearity that causes the second and higher harmonics to appear, also causes the fundamental to be not exactly proportional to the input signal. The deviation from proportionality could be regarded as a kind of distortion, first harmonic distortion. In the case of a third power term, the resulting first harmonic distortion is three times as strong as the third harmonic distortion. It makes perfect mathematical sense, but still, no one ever bothers to look at it, and it is not even included in the normal definition of total harmonic distortion. The same holds for even powers and zeroth harmonic distortion, by the way.
I think there are some very pragmatic reasons for this. Measuring a very small deviation at the first harmonic will be very difficult to do, the effect on the sound is quite benign compared to the other distortion products and when everyone measures THD without including the zeroth and first harmonic distortion, you still get a rough impression of which device is the closest and which the furthest from perfect linearity.
I think there are some very pragmatic reasons for this. Measuring a very small deviation at the first harmonic will be very difficult to do, the effect on the sound is quite benign compared to the other distortion products and when everyone measures THD without including the zeroth and first harmonic distortion, you still get a rough impression of which device is the closest and which the furthest from perfect linearity.
I haven't thought of this in audio, but in RF (which in many cases operates in Class C and is not concerned about keeping the waveshape, as harmonics are filtered out by further circuitry and the antenna) there's the 1dB and 3dB compression points, measurements to show when an RF amp is beyond the point where output level is linearly proportional to input. Of course, an equivalent signal in audio would be well into clipping.
Such RF amplifiers are still concerned with linearity. Third order IMD is the most concerning because it can be high and typically falls in the amplifier’s pass band. It degrades the signal to noise ratio and can fall into an adjacent channel. Even a single tone, such as a pulsed radar application still feels this to some degree as it can cause distortion of the phase of even a single tone. That can be corrected digitally at the receiver, but any correction requires CPU cycles And therefore ultimately power.