Hello, Electrician.

Yes your transfer function is periodically repetitive and continuous.
But it is an idealisation and further we are talking about continuity of the output waveform, not the transfer function.
Further still your transfer function cannot lead to the output wavefoirm shown because your output shows curvature near/in the crossover region.

For real amplifiers we are discussing distortion ie deviation from the ideal. So an idealisation is not really suitable.

The actual output at very low levels is indeterminate ie neither predictable nor periodically reproducible.

It is also the case that the open loop gain drops dramatically at very low levels of signal current or voltage for many real components.

 

For real amplifiers we are discussing distortion ie deviation from the ideal. So an idealisation is not really suitable.

No real amplifier can ever produce an actual discontinuity; rise and fall times are always greater than zero. A true discontinuity is an idealization, so to explain what you said, "In crossover distortion there are an infinite number of discontinuities.", one must consider idealities.

I still don't see where I will find an infinite number of discontinuities in crossover distortion, which can't happen in the real world, but I would be interested to know how this can happen in an ideal case. Can you explain further?

 

I still don't see where I will find an infinite number of discontinuities in crossover distortion, which can't happen in the real world, but I would be interested to know how this can happen in an ideal case. Can you explain further?

To find a discontinuity in an ideal system with ideal noise.

1) Set the ouput to zero.

2) Observe the actual output. This will not be precisely zero. every measurement will be different in a random manner, there will be no smooth transition from one meausrement to the next.

3) Let the measurement interval tend to zero, the measurments will take on the appearance of Cantors dust.

 

So then are you in effect saying that a amplifier with crossover distortion and output level greater than zero will not exhibit an infinite number of disconituities?

 

No, you asked about an ideal amplifier.

A real amplifier will have that characteristic in the crossover or indeteminate zone, not only due to noise.

The point with indeterminacy is that it is not going to be the same in repetive 'cycles', and periodic is one other criterion of Fourier analysis.

 

No, you asked about an ideal amplifier.

A real amplifier will have that characteristic in the crossover or indeteminate zone, not only due to noise.

The point with indeterminacy is that it is not going to be the same in repetive 'cycles', and periodic is one other criterion of Fourier analysis.

And an ideal amplifier with crossover distortion, having the transfer characteristic shown in the image I posted in post #23, has zero output when the input is in the region between breakpoints. That image is just something I found on the web to illustrate crossover distortion. Whoever made the image shows a blue output waveform which is rounded in the crossover region; it should have sharp corners and zero output when the input is in the crossover region. But the transfer characteristic shown is ideal, and that is what I am considering.

There is no indeterminacy in this ideal case. The output in the crossover region is zero.

In a real amplifier the output in the crossover zone isn't indeterminate either except for input indeterminacy such as noise at the input or noise added by the amplifier. Its transfer characteristic is just not a linear continuation of the transfer characteristic outside the crossover zone, and this causes distortion.

So for the process you describe, and the ideal transfer characteristic shown:

"To find a discontinuity in an ideal system with ideal noise.

1) Set the ouput to zero.

2) Observe the actual output. This will not be precisely zero. every measurement will be different in a random manner, there will be no smooth transition from one meausrement to the next.

3) Let the measurement interval tend to zero, the measurments will take on the appearance of Cantors dust. "

the actual output will be precisely zero. There may be ideal noise at the input, but given the ideal transfer characteristic shown in post #23, there will be no output. The transfer characteristic has a noise free output. There will be no "Cantor's dust" at the output.

Furthermore, even if the part of the transfer characteristic between the breakpoints were not perfectly horizontal (but adding no additional noise to the output), "ideal noise" at the input would not produce a discontinuity at the output unless there were a discontinuity at the input. To get an infinite number of discontinuities at the output we must have an infinite number of discontinuities at the input.

To say that "...every measurement will be different in a random manner, there will be no smooth transition from one meausrement to the next." even when the measurement interval approaches zero implies that the signal being measured has infinite bandwidth.

A signal with a true discontinuity must have infinite bandwidth; a signal with an infinite number of discontinuities (in a finite interval of time) even more so. A noise signal at the input with infinite bandwidth must have its power at increasing frequencies decrease at a rate sufficient to avoid an Ultraviolet Catastrophe--infinite power. This means that the power which is contributed by the infinite number of discontinuities at the output will be vanishingly small, and of no consequence in the Fourier description of the output with ordinary input waveforms.

This notion that "In crossover distortion there are an infinite number of discontinuities." only has meaning in some ideal way not relevant for real amplifiers. A real amplifier can't produce a true output discontinuity even if there were a discontinuity at the input, and any "Cantor's dust" at the output (which there can't be with a real amplifier) will have negligible power.

You further say "periodic is one other criterion of Fourier analysis." If we let the fact that a signal is not strictly periodic because of some small contamination with noise prevent us using Fourier analysis, then in the real world, no signal can be so analyzed. So why should the output from amplifiers with crossover distortion be less amenable to Fourier analysis than any other amplifier? Any occurrence of an infinite number of (very small) discontinuities in the output of an amplifier with crossover will contribute negligible power to the output, and will not interfere with a practical Fourier analysis.

This is akin to the thread with an ideal short on the output of an ideal voltage source.

 

This is akin to the thread with an ideal short on the output of an ideal voltage source.

We will have to agree to disagree.