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Signal Processing

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  • $\begingroup$ hm but there's a theorem that if you take a discontinuous function, apply Fourier (or alike) transform on it, and the inverse of that, you get a function where the value at the discontinuity is in the middle of the left- and right-side limit of the original function. Gibb's phenomenon is a result of exactly that. $\endgroup$ Commented Dec 19, 2021 at 18:49
  • $\begingroup$ @MarcusMüller: That's right! The inversion formula is only valid if we assume the function value at jump discontinuities is the average of the two one-sided limits. But that's just one way of defining a function value at a discontinuity. I mean, why should we apply a Fourier transform and its inverse in the first place? The problem has nothing to do with the way we need to define function values for the inverse Fourier transform to hold true. $\endgroup$ Commented Dec 19, 2021 at 18:56
  • $\begingroup$ point is that when we wonder about what happens at "cutoff frequency", we're asking ourselves what the result of the point-wise multiplication of the signal's Fourier transform (i.e. the $\pm f_c$ Diracs) and the system's Fourier transform is, transformed back to time – and this consideration says it's multiplication with a factor $\frac12$ in freq domain, so we'll arrive at a sine of half the original amplitude in time. $\endgroup$ Commented Dec 19, 2021 at 19:04
  • $\begingroup$ @MarcusMüller: That's a reasonable definition, but it remains a definition in my opinion. $\endgroup$ Commented Dec 19, 2021 at 19:08
  • $\begingroup$ Hm, I think (my proof on paper here isn't airtight, so I'd love to say "I know", but really can't) this is actually the answer to question "what happens when you convolve sin x with (sin x)/x", and I don't think that's really something that should be dependent on the definition of what the value of the rect function is – because that's just an "intuition thing" for the human who likes to thing in the frequency domain, but "the ideal brickwall filter with cutoff frequency $f_c$ convolved with the signal $\sin(f_ct)$"should be independent of that - the first is unambigously a scaled sinc with $\endgroup$ Commented Dec 19, 2021 at 19:14