[Dan Banquer]

There are dozens of different measurements that can be used to characterize a componet and when the meaning of these is not fully understood it gives people the impression that maybe there are new types of measurement yet to be discovered that show things that all of the current ones don't show.

The truth is that there will be new types of measurements but they will show the same thing the already existing types of measurement show.

Translation - We already know everything - the earth is flat.

My point was not that there might be new types of measurements - but new things TO measure.  What if at some point we find something that dismisses the need to assume systems are linear (IMO a bad assumption since NO system is perfectly linear - much less when having to interact with other dynamic, non-linear systems)?  What if the proof turns out to be different than the assumption?  Yes, what we know and assume now gets us pretty close, but not exact - and there is no 'e' to define that error as there is in calculus.  Is it even a constant 'e'?

Bryan

Well Bryan; I think there's a difference between "knowing everything" and implementing what is already known.
Personally, my take on most of audio is that there is a very real difference between the two.
            d.b.

[ctviggen]

It is very common to hear someone commenting yeah the frequency response is flat but what about the time domain, you need to look at it's square wave resonse to see how it handles transients.

For any element with a nonlinear characteristic, this is true.  Remember that a function has to be linear for one to transfer something from the frequency domain to the time domain (or vice versa).  For instance, what's the Fourier transform of x^2 or log(x)?  There isn't one, because these are nonlinear functions and the rule of supposition does not work for nonlinear functions (in other words, no matter how many sine/cosine functions you use, you cannot replicate a nonlinear function through the use a sine/cosine functions).

[Daryl]

It is very common to hear someone commenting yeah the frequency response is flat but what about the time domain, you need to look at it's square wave resonse to see how it handles transients.

For any element with a nonlinear characteristic, this is true.  Remember that a function has to be linear for one to transfer something from the frequency domain to the time domain (or vice versa).  For instance, what's the Fourier transform of x^2 or log(x)?  There isn't one, because these are nonlinear functions and the rule of supposition does not work for nonlinear functions (in other words, no matter how many sine/cosine functions you use, you cannot replicate a nonlinear function through the use a sine/cosine functions).

That point has been covered repeatedly in the last few posts and not just by me.

Again the frequency domain ALWAYS is equivilant to time domain.

Once a system becomes non-linear to a certain extent trying to quantize linear transfer function is just a waste of time (it won't be stable).

No system is completely linear but any system we would use for hi-fi must be darn close and it's linear transfer function will be stable.

You will consider the non-linear characteristics separately, no one is saying linear a linear transfer function describes the  non-linear characteristics of a system.

You can also see above where I mentioned that when an amplifier is subject to a condition that causes it to become unstable it's linear transfer function will become meaningless until it recovers.

[ctviggen]

Then you're wrong.  The system HAS TO BE linear and time invariant for a Fourier transform to apply. 

[Daryl]

Then you're wrong.  The system HAS TO BE linear and time invariant for a Fourier transform to apply. 

Yikes Ctviggen,

Your going to extremes!

As I keep repeating a linear transfer function will NOT describe non-linear characteristics.

A fourrier transform can only describe a transfer function Fully and Exactly when the system is PERFECTLY linear and TIME INVARIANT.

As I said before audio systems are nearly linear and their linear transfer function is stable for all practical purposes.

When a frequency response is smooth you know there are no signifigant time domain anomalies.

You can transform a componets frequency response into it's step response and it will be valid for the devices linear characteristics and not include it's non-linear characteristics or noise characteristics.

You can consider a devices non-linear characteristics separately from it's linear characteristics.

Thus the step response you calculate will not include the deviations caused by non-linearity or noise.

If the systems non-linearity is -60db peak then you can expect a deviation from your fourrier transform of 1/1000 the full scale.

More importantly your ears don't function like an oscilloscope so the frequency response is the perspective you want to concern yourself with when looking at devices with smooth frequency response.

When considering non-linear characteristics you still want to use spectral analysis because your ears operate the way an RTA does.

The simplest form of measurement is simply a direct comparison of input vs. output taking gain into consideration.

This one measurement shows all forms of reproduction errors without the possibility of missing anything.

There is no phsycic signal that accompanies the electrical signal that cannot be measured but is easily heard.

That is the hardest thing for some to come to terms with, an audio signal is a simple one dimensional data set.

Because of the way your ears function you can have errors in reproduction that exceed 100% that are inaudible and then have different reproduction errors that cause a deviation of less than .1% and have them be perfectly audible.

Because if this you will separate reproduction errors into different categories so they can be placed into context considering how your ears percieve them.

You can separate a systems transfer function into three main categories....

Noise
Linear
Non-Linear

With no possibility of having an anomaly that isn't quantized by the three and be better able to investigate reproduction errors within separate categories.

A fourrier transform for a nearly linear system is perfectly valid so long as you understand that the device will also have noise and non-linear characteristics which will be in addition to whatever deviations are due to it's linear transfer function and must also be considered.

[opaqueice]

For any element with a nonlinear characteristic, this is true.  Remember that a function has to be linear for one to transfer something from the frequency domain to the time domain (or vice versa).  For instance, what's the Fourier transform of x^2 or log(x)?  There isn't one, because these are nonlinear functions and the rule of supposition does not work for nonlinear functions (in other words, no matter how many sine/cosine functions you use, you cannot replicate a nonlinear function through the use a sine/cosine functions).

Of course one can take the FT of a non-linear function - otherwise it would be a rather uninteresting procedure!  I'm not sure what you mean by time-invariant - the only function invariant under time translation is a constant, which again isn't very interesting.

To answer your specific questions, the FT of x^2 is the second derivative of a delta function, just as the FT of x (a linear function) is the first derivative of a delta, and the FT of a constant is a delta.  The FT of log is something like 1/f.  Any reasonable function has a unique Fourier transform (reasonable means smooth and bounded).

There seem to be some very basic misconceptions here concerning why frequency domain analysis is useful.

[JohnR]

This is so tedious. ctviggen is talking about the relationship between the value of the output and the input of a system at a given time. Not a function from time to a value.

I think the basic problem here is that you don't understand the difference between a signal and a system. Sorry if that's blunt, don't really have time to do better at this point...

BTW thanks Daryl for the synopsys, that captures it well

[opaqueice]

This is so tedious. ctviggen is talking about the relationship between the value of the output and the input of a system at a given time. Not a function from time to a value.

What ctviggen said simply didn't make sense, as far as I can tell. 

You take the FT of the output and compare it to FT of the input; it's as simple as that.  Obviously you can do that no matter what the relationship is between the input and output - linear, non-linear, it doesn't matter.  If the two FT are related linearly exactly for all inputs, then the system is exactly linear, with exactly zero distortion, period.  If they are not, then the system is non-linear, but the FT (of all possible inputs) fully characterizes that non-linear response.  If you take the FT only of some limited set of inputs, like a sweep tone or something, that doesn't fully characterize the behavior unless you also assume something.

Again - there is nothing wrong with or inapplicable about Fourier analysis no matter how non-linear the system is.  As has been emphasized repeatedly the FT is completely equivalent to the time domain, period.  The only issue is exactly what you measure.  I think people here are confusing the limitations of a frequency response measured with a particular set of pure tones, which is a real limitation, with some kind of imagined limitation on the general technique of using frequency domain analysis.

[JohnR]

You take the FT of the output and compare it to FT of the input; it's as simple as that.

Well, more specifically, you divide the FT of the output by the FT of the input. So suppose I input a 1V sinewave at 1 khz, and the output is 1v at 1 kHz, and 0.00001V at 2kHz, what is the result of this operation at 2 kHz?

Again - there is nothing wrong with or inapplicable about Fourier analysis no matter how non-linear the system is.  As has been emphasized repeatedly the FT is completely equivalent to the time domain, period.

Yes. For signals. But you are attempting to apply this fact to the frequency RESPONSE of a SYSTEM which is something different. (Hopefully the above example might give you pause to think about why.)

I'm sorry but I don't have much more to say on this. Not understanding is one thing but insulting other people as a result is completely another.

JohnR

[opaqueice]

Well, more specifically, you divide the FT of the output by the FT of the input. So suppose I input a 1V sinewave at 1 khz, and the output is 1v at 1 kHz, and 0.00001V at 2kHz, what is the result of this operation at 2 kHz?

Well, dividing is only one of the many techniques you might use to compare the two FTs.  In this case it would give you a function of frequency which is 1 at 1 kHz, big at 2kHz, and determined by noise everywhere else.  I don't get the point - why do you ask about that? 

Another technique that might be useful here is to do this same measurement, and then divide the sum of the powers of all the harmonics by the power at the fundamental (which in your example will give 10^{-10}).  That's called Total Harmonic Distortion, or THD, and it's an example of using FT and frequency domain analysis to measure non-linear effects.

Yes. For signals. But you are attempting to apply this fact to the frequency RESPONSE of a SYSTEM which is something different. (Hopefully the above example might give you pause to think about why.)

I guess I'm missing something here, because I honestly don't know what you're getting at.  Is it just that the impulse response, or some particular limited set of measurements like that, doesn't fully characterize the behavior of a non-linear system?  If so we're agreeing but not communicating, because I keep repeating that. 

I'm sorry but I don't have much more to say on this. Not understanding is one thing but insulting other people as a result is completely another.

I apologize if I insulted anyone.  Could you please tell me where and how, so I can try not to do it again?