Please explain diffraction in lenses

[paulr]

Let's listen to two sinusoidal sound waves at a slightly different frequency.
Now consider that out ears+brains consist only in a linear detector.
And propose a model where the difference in frequencies could appear in this 100% linear detection scheme.
The difference in frequencies comes out immediately if the detector performs some non linear operations with respect to the amplitude of sound vibrations. But a purely linear detector, by no means can yield different Fourier frequencies in output.

Maybe you're trying to say something different from what I'm understanding.

But as I see, you have two slightly out-of-tune sine waves, you add them together, and you get a very low frequency undertone. It's what we hear, it's what a microphone pics up, and it's what an oscilloscope shows. It looks like this:

[Jac@stafford.net]

[...] Let's listen to two sinusoidal sound waves at a slightly different frequency.
Now consider that out ears+brains consist only in a linear detector [...]

But do our ears sense in only a linear manner? We know our eye-brain does not.
.

[paulr]

But do our ears sense in only a linear manner? We know our eye-brain does not.
.

You really have to define your terms here. We respond to some aspects of sound and light in a linear manner, others not. In this case, what we hear is pretty well predicted by what you see in those waveforms, so I'm not sure what's being questioned ...

My question all along is the relevance of this to MTF curves. Non-linearity was brought up in objection to the notion that you can model the sharpness of a system by multiplying MTF curves. This criticism strikes me as both illogical and vague, and moreover I'm certain it's wrong. It's what optical scientists do, and it's one of the things they value about the measurement.

[Emmanuel BIGLER]

Paul.
Sorry to insist, and I apologize to our readers, this is not one of those controversies we are familiar with on discussion forums, but a fundamental scientific question.
And to put it frankly: your are plain wrong regarding the beatnote for sound.
Or you do not understand what Fourier analysis means, which would be surprising.

If the Fourier spectrum for a certain sound is composed only of two elementary components, i.e. two different sinusoidal vibrations at different frequencies, any linear and time-invariant (non-sampling) detector cannot do anything but filter the two components with a certain coefficient for each frequency. Hence the spectrum of the output signal contains only the two original frequencies. And nothing else.
In the diagram that you've posted with the sum of two sine waves, the Fourier spectrum contains only the two original frequencies. No sum and no difference of frequencies.
Of course we can see the beatnote as plotted as the envelope of the wave. But no difference in frequencies exist in the Fourier spectrum.

Paul, we really need to peacefully agree on this before continuing the discussion.

A sampling detector will create a spectrum for the sampled signal which is different from the original spectrum. Aliasing can result from such a detector when the sampled signal is re-filtered with a linear low-pass filter. But our ears are not a sampling detector.

If you do not agree with what I just stated, then I'm afraid that you a missing some of the fundamental signification of Fourier analysis.
And I am not vague stating this: I am very precise regarding what Fourier analysis means.
So you need to answer the question relative to sound beatnotes before we continue the discussion.

The situation is simple.
If you believe that the difference of frequencies can appear in the Fourier spectrum detected by a linear and time-invariant detector, you are wrong.

Now we apply Fourier analysis to vision and images, and my objection is: our vision system is non linear, hence MTF analysis, valid only for a linear and space-invariant, non sampling image detector, cannot explain everything.

[Nodda Duma]

My question all along is the relevance of this to MTF curves. Non-linearity was brought up in objection to the notion that you can model the sharpness of a system by multiplying MTF curves. This criticism strikes me as both illogical and vague, and moreover I'm certain it's wrong. It's what optical scientists do, and it's one of the things they value about the measurement.

It's because there is confusion between the idea of a *linear output* and *linear systems*. I gave up trying to explain the difference and checked out earlier in the thread.

Make no mistake: imaging systems are entirely linear and predictable. The proof is in the fact that precision optics exist.