I was gonna ask how the heck you knew that, then saw your tag line was one of his quotes..😀
Jn
You mean the zero-crossings line up?
If so, zero crossing times are dependent on the amplitudes of the two original signals. What if you change the amplitude of one signal?
If not zero-crossings, looked to me like the so-called 20kHz sine wave didn't always line up at some other times with the superimposed 20kHz waveform you added for comparison. If they don't line up at any point at all, then they don't line up 'exactly.'
Just saying, something appears to be wrong with your analysis/conclusions related to the drawing.
All the above based on the appearance of the drawing (given possible inaccuracies not yet accounted for).
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I've understood everything posted here so far, as well as all the frequency-domain-only attempts being used to try and prove that a trig identity is fiction.
Jn
I assume you did not see my edits, I really hate the IPad.
Look up the trig identity sin(a)+ sin(b), plug in 17.5 for a and 22.5 for b.
Jn
Oh, sorry...where is it again?
And, why is it important to listen, the trig identity says it all.
Also, that link Richard posted a while ago for me had some verbage on humans ability to distinguish individual tones, some kind of relation between difference, duration, whatever..I glanced by it as I was not looking for that.
Jn
You also must look up the trig identity I mentioned.
I do not know how to link on this infernal device, clark university summary of trig identities is the web page I found most useful.
Scroll down to truly obscure identities. It's the first one there in the product-sum identities.
Jn
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Okay, but conversely it is still only 17.5 and 22.5 from the point of view of filtering or sampling.
From the perspective of hearing it depends on whether the ear/brain perceives the time domain wiggle at 20kHz as a tone, or say, as something essentially tone-like ('tone' is the perceptual experience associated with physical frequency).
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I don't know, I'll save you the trouble, there is no 200Hz tone to be heard, same as there is no 20kHz tone, go figure.
Okay but....... Really?😀
And your second point is exactly what I have been saying all along. That composite waveform, whether it was created by two frequencies beating from addition, or one modulating another, are still only 17.5 and 22.5 from the point of view of filtering. And if the filter removes only the 22.5, what is left?
Oh, the argument of not hearing the 20k? Who cares? The concern is a filter removing the 22.5 and leaving the 17.5 unscathed.
Jn
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Sigh, it's taught in seventh grade in the USA. We've all had plenty of time to forget it.
Jn
Really, you're going with that?
I detail exactly how a modulated sine and a pair of sines added are identical mathematically, detailed exactly what Han's waveforms were showing, and your going with "time domain viewing can be misleading"?
Really?
Jn
No.
If the frequencies are added, then the process is reversible by filtering out one of the original frequencies. Only the other original frequency is left. No beat pattern remains.
If the frequencies are modulated that is nonlinear processing that can't be undone by a filter. The sum and difference frequencies would continue to exist if one or both of the original frequencies was/were selectively filtered out.
You do not understand the concept of mathematical identity.
If I hand you a waveform created by adding 17.5k to 22.5 k, you cannot tell if it was made by addition or modulation. That is what an identity is.
Jn
Yes, you can tell. The waveforms will be different in both cases (addition or modulation).
You are arguing against a mathematical identity equation.
I really think you should find a math professor and argue with him/her.
I am not the one who created that trig identity.
Jn
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