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RE: Long-term DBT example

This is the point I think Rick is trying to make, that this circuit simulates the "less worse" kind of distortion, meaning primarily even-order, while not addressing what is generally considered the worst kind of distortion, namely odd and/or crossover distortion, which plagues nearly all Push/pull (or complementary if you prefer) Class A/B amplifiers.

I do understand what Rick's point was, although it might not have been obvious from what I said :-). My point is that harmonics are a symptom of nonlinearity, and that it's not sufficient to use harmonics to describe a nonlinearity or determine the potential subjective effects of it.

Here's an example. Keith Howard synthesized his nonlinear characteristics in software using polynomial functions. These functions are continuous, and all their derivatives are continuous. They are "smooth" (analytic) functions. They have the property that when measuring or computing distortion with a sine wave input, the distortion approaches zero as the amplitude of the sine wave approaches zero. There are some quirks relating to the quantization of the digital domain, but let's neglect that for the moment, as that's a complex mathematical problem involving dither and such.

But when analyzing the "grunge circuit" assuming ideal op-amps and diodes, its nonlinear relationship between input Vin and output Vout looks like this:

Vout = K1 * Vin for Vin > = 0
Vout = K2 * Vin for Vin < 0

and K1 is not equal to K2.

You can see that because K1 is different from K2, there is a discontinuity in the first derivative of this function at the origin, or a "kink" as it were. It's not a smooth function. As I mentioned in my other post to Rick, if we call this function f(Vin), it has the property:

f(a * Vin) = a * f(Vin) where "a" is a constant.

That is, if you scale the input, the output scales by the same amount, retaining its shape as the amplitude of the input changes. This gives it the strange property that the percent distortion with a sine wave input is independent of the amplitude of the sine wave. The distortion does not go to zero as the amplitude of the sine wave becomes small. This is totally unlike any nonlinearity that's present in real-world amplifiers.

Because the input-output relationship has a "kink" at the origin (discontinuity of the first derivative), this should raise concern that the nature of the distortion could be more subjectively nasty than one might initially think if one only considers that it produces only even-order harmonics.

In his article on distortion in the AES, Gene Czerwinski made the amusing comment, "If you get hit in the head by a rock, it is not the velocity of the rock that hurts you, it is the rock". The "velocity of the rock" in this case is the harmonic distortion, and the "rock" is the nonlinear input-output characteristic. His comment was in reference to the mixing up of concepts that sometimes occurs in the discussion of distortion. My view is that a similar confusion has come up here. It's not sufficient to look at harmonics. We really need to examine the input-output characteristic itself to obtain the most insight.

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