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How do you sum up distortion in the sound chain?

EdW

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Except in the rather unlikely case of both pieces of equipment producing identical distortion characteristic (same harmonic in the same phase) they do not add arithmetically - just as you correctly surmised. Generally non linearities are added on an ‘rms’ basis with the assumption that they are not correlated in the absence of any information to the contrary.
So this addition is done as follows:
Distortion = SQRT(0.3^2 + 0.03^2) = 0.301%
 
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DanielT

DanielT

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Thanks EdW for the explanation and clarification.:)

Edit:
In other words, it is rather pointless to chase low distortion in amplifiers if it is like this fictitious example:

Scenario 1
Speaker: 0.7% THD
Amplifier: 0.06% THD

Scenario 2
Speaker: 0.7% THD
Amplifier: 0.02% THD

Not to mention the low distortion found in today's cheap high-performance DACs.:)

As usual then. The old mantra. Put the gunpowder on the speakers.:)
 
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charleski

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with the assumption that they are not correlated
I may be wrong, but isn't this really just the equation for adding non-correlated (random) noise figures? I wondered about this, and surely the problem is that harmonic distortion is correlated, with the nonlinearity adding up at specific tones (2nd, 3rd, 4th harmonics, etc)?

The definition of THD is 100*(P2+P3+P4 ...)/P1, where P1 is the power of the fundamental and P2, etc are the powers of the harmonics. Using power rather than voltage makes the calculations easier. If device A has 0.3%THD, then its nonlinearities (P2+P3+P4 ...) = 0.003*P1. If device B has 0.03%THD then it adds nonlinearities that amount to 0.0003*P1 at the same tones as device A. So surely the THD of a system consisting of devices A+B should sum up all these tones: 100*(P2A+P3A+P4A ... +P2B+P3B+P4B ...)/P1.
But it doesn't end there. Device A is receiving a single tone, but device B is receiving a signal with multiple tones (the harmonics from device A), and each of these will produce additional nonlinearites which will all lie on the harmonics of the original tone. So you end up with a series like:
[P2A +P3A +P4A ...
+ P2B + P3B +P4B ...
+ (P2B)*(P2A + P3A +P4A ...)
+ (P3B)*(P2A + P3A +P4A ...)
... ]
/P1

So the end result will be slightly higher than simply adding the two distortion figures together.
But as I said, I might be wrong about this. I couldn't find an authoritative source on the question with a quick search.
 

EdW

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I may be wrong, but isn't this really just the equation for adding non-correlated (random) noise figures? I wondered about this, and surely the problem is that harmonic distortion is correlated, with the nonlinearity adding up at specific tones (2nd, 3rd, 4th harmonics, etc)?

The definition of THD is 100*(P2+P3+P4 ...)/P1, where P1 is the power of the fundamental and P2, etc are the powers of the harmonics. Using power rather than voltage makes the calculations easier. If device A has 0.3%THD, then its nonlinearities (P2+P3+P4 ...) = 0.003*P1. If device B has 0.03%THD then it adds nonlinearities that amount to 0.0003*P1 at the same tones as device A. So surely the THD of a system consisting of devices A+B should sum up all these tones: 100*(P2A+P3A+P4A ... +P2B+P3B+P4B ...)/P1.
But it doesn't end there. Device A is receiving a single tone, but device B is receiving a signal with multiple tones (the harmonics from device A), and each of these will produce additional nonlinearites which will all lie on the harmonics of the original tone. So you end up with a series like:
[P2A +P3A +P4A ...
+ P2B + P3B +P4B ...
+ (P2B)*(P2A + P3A +P4A ...)
+ (P3B)*(P2A + P3A +P4A ...)
... ]
/P1

So the end result will be slightly higher than simply adding the two distortion figures together.
But as I said, I might be wrong about this. I couldn't find an authoritative source on the question with a quick search.
To take a perhaps unlikely case we could find that the distortion of one piece of equipment in the chain is anti phase with the distortion generated in the second piece of equipment thus cancellation occurs resulting in lower overall distortion. So as usual what appears as a simple question has a slightly more complex answer . . .
 

charleski

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Yes, differences in the phase of the distortion products would add a load of complication (my example assumed everything was in phase), which might explain why there isn't an easy equation floating around for this.
 

EdW

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Thanks EdW for the explanation and clarification.:)

Edit:
In other words, it is rather pointless to chase low distortion in amplifiers if it is like this fictitious example:

Scenario 1
Speaker: 0.7% THD
Amplifier: 0.06% THD

Scenario 2
Speaker: 0.7% THD
Amplifier: 0.02% THD

Not to mention the low distortion found in today's cheap high-performance DACs.:)

As usual then. The old mantra. Put the gunpowder on the speakers.:)
You make a good point. Speakers distort way more than amps and DACs. Particularly at high levels and low frequencies. But speakers are pretty clean at modest levels where amps might suffer from crossover distortion and DAC linearity is (on low level signals) not be as good as when the signal level is nearer maximum modulation (0dB). This might be of slight concern if you are using the digital volume control on the DAC rather than a precision resistor volume control in a premium grade audio preamp.
Amps, DACs are substantially cheaper than speakers though so it makes sense to have a DAC and amp which outperform the speaker, where the amp never appreciably distorts even driving the impedance troughs of the speaker at levels a little beyond the nominal power rating of the speaker. An amplifier running into clip could potentially generate a slew of harmonics which could burn out the tweeter.
 
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DanielT

DanielT

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Yes, differences in the phase of the distortion products would add a load of complication (my example assumed everything was in phase), which might explain why there isn't an easy equation floating around for this.
But in any case then more interesting and challenging than I thought.:)
 

EEE272

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[P2A +P3A +P4A ...
+ P2B + P3B +P4B ...
+ (P2B)*(P2A + P3A +P4A ...)
+ (P3B)*(P2A + P3A +P4A ...)
... ]
/P1

So the end result will be slightly higher than simply adding the two distortion figures together.
But as I said, I might be wrong about this. I couldn't find an authoritative source on the question with a quick search.
I do not claim to be an authority on this topic. I would think you are right. Nevertheless, let's take two devices that create a lot of harmonic distortion, with -60dB peaks. Then going through both devices, the harmonic of the harmonic is at -120dB, which is below the noise floor of most devices.
Hence, it might be OK to assume that the sum is a good upper bound.
I believe Ethan Winer had a video, where he showed that in practice, because of all the above reasons mentioned by others, the noise does not get to this bound.

Speakers are the real limitation, just like you said.
 

ZolaIII

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Simply by measuring it at end of the chain (analog reproduction on speakers or hedaphones) and comparing it to the master source input signal. And don't worry everything will be there.
 

EEE272

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Simply by measuring it at end of the chain (analog reproduction on speakers or hedaphones) and comparing it to the master source input signal. And don't worry everything will be there.
True but I think the question, at least for me, was to estimate how much "trouble" a device could cause, which I do not own yet, based on its measurements.
 

EdTice

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True but I think the question, at least for me, was to estimate how much "trouble" a device could cause, which I do not own yet, based on its measurements.

This is a valid question because one wants to estimate a priori (before buying equipment) if it's going to have the characteristics you want. For measuring *distortion*, the RMS equation is probably fine since two devices aren't going to have the same distortion pattern.

But as you point out, if there is high *noise* (or distortion) very early in the chain, it's going to get multiplied in later gain stages. For a simple example of this (if you are old like me), many of the first classical music recordings on CDs sounded awful because they were mastered at a level where consumer DACs were noisy vs pop/rock that was mastered "hot" and sounded much better on the same format. These were the same DACs, same amps, and same speakers. But getting rid or early stage noise made the difference between awful and great.

You can't make up for a weak preamp with later gain stages! So I still don't have a formula for you but, early gain stages definitely are the most problematic (until you get to the speakers)
 
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