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Excellent answer and I am sure you are aware of this, but like to point out: "Mathematically, the Fourier transform is "simply" a way of representing time sign

Excellent answer and I am sure you are aware of this, but like to point out: "Mathematically, the Fourier transform is "simply" a way of representing time signals in a certain orthogonal vectorial basis." Not just time signals but any piecewise continuous and differentiable as well as Dirichlet integrable function. This has many applications, just a few examples from the top of my head: image processing, solving differential equations, fast multiplication. I'd also like to add that from a mathematical point of view these transforms are "lossless" in the sense that the transformed function has the exact same information as the original and you can get back the exact original even if all you have is the transform. I feel this often gets lost when people approach the Fourier transform from a more engineering perspective, not at least because we often do the transform to throw away unwanted information, like certain frequency components. In the end it really is just one of many perspectives to look at a function.

quibono 3y ago
> I feel this often gets lost when people approach the Fourier transform from a more engineering perspective, not at least because we often do the transform to throw away unwanted information, like certain frequency components. That was my problem as well. My first introduction to Fourier transforms was through more of an engineering lens. I remember having trouble with the _inverse_ Fourier transform. I was OK with a Fourier inverse of an already transformed function but I wasn't quite sure what that would mean when applied to a non-transformed, "regular" function.
- takd 3y ago
  As a related aside, the terms "cepstrum" and the "quefrencies" [1] (c.f. spectrum and frequencies) sound so hilarious that when I first heard about them I was convinced it was some kind of prank. [1] https://en.wikipedia.org/wiki/Cepstrum https://en.wikipedia.org/wiki/Cepstrum
  - cmehdy 3y ago
    This does read like a joke, I had never heard of it either and I'm wondering if many people do use this at all.. Operations on cepstra are labelled quefrency analysis (or quefrency alanysis[1]), liftering, or cepstral analysis. It may be pronounced in the two ways given, the second having the advantage of avoiding confusion with kepstrum.
    - aswanson 3y ago
      It's used in speech signal processing & seismic signal analysis.
    - ljosifov 3y ago
      Almost all speech recognizers until this latest crop (of end-to-end DL NN ASR) operated on cepstral coefficients (and their delta-s and delta-delta-s) as their feature vector.
- pas 3y ago
  > [...] I wasn't quite sure what that would mean when applied to a non-transformed, "regular" function. Have you gained some intuition/understanding for this? I tried a few inputs in WolframAlpha, but unless I manually type in the integral for the inverse transform there's not even a graph :) (and I have no idea whether it's even the same thing without putting a `t` in the exponent and wrapping it in an f(t) = ... ) https://www.wolframalpha.com/input?i=integral+%28sin%28x%29+%2F+x%29*e%5E%28i+2pi+x%29+ https://www.wolframalpha.com/input?i=integral+%28sin%28x%29+...
  - weinzierl 3y ago
    Not parent (but GP) and intuition can mean many things but what helped me was keeping in mind: Every continuous periodic function turns into a discrete aperiodic one when transformed. Works both ways. Continuous aperiodic stays continuous aperiodic. Discrete periodic stays discrete periodic.
- sharpneli 3y ago
  Inverse fourier transform of a non transformed signal gives you basically the fourier transform with some changes (I can't remember which, were the numbers conjugates or something?). Applying it the second time gives you same result as if you'd do the forward direction transform twice. If you apply fourier transform 4 times you get your original function back. You can think of it as 90 degree rotation. Inverse transform just rotates it in the opposite direction. The rotation analog is not even too far fetched as fractional fourier transform allows you to do an arbitrary angle rotation.
  - araes 3y ago
    Having never heard of this for the Fourier Transform, needed to read. F0: original signal F1: frequency domain signal F2: reverse time signal F3: inverse fourier signal F4: original signal Also, has further weird applications I've never heard of with "Fractional Fourier Transforms" [1] which can apparently result in smooth smears of time -> frequency domain [2]. [1] https://en.wikipedia.org/wiki/Fractional_Fourier_transform https://en.wikipedia.org/wiki/Fractional_Fourier_transform [2] https://en.wikipedia.org/wiki/File:FracFT_Rec_by_stevencys.jpg https://en.wikipedia.org/wiki/File:FracFT_Rec_by_stevencys.j...
    - laszlokorte 3y ago
      See [1] for my visualization of this 4-cycle and the fractional fourier transform [1]: https://static.laszlokorte.de/frft-cube/ https://static.laszlokorte.de/frft-cube/
      - araes 3y ago
        Thanks. It's neat being able to visualize them, and the 3D display's actually pretty cool looking for all the different functions. The Fractional DFT part though, doesn't seem to do anything no matter the function chosen. Firefox 125. Edit: Nvm, figured it out. Have to visualize from the top down to see the Fractional DFT portion. Haven't seen many visualization systems where each orientation shows a different type of data. Actually a pretty neat idea from a UI perspective.
- Salgat 3y ago
  A fourier transform basically gives you an infinite number of sine waves with different amplitudes/phases at every frequency. If you add them all back together (the inverse fourier transform), you get back your original signal. Audio compression in this case would just be excluding the sine waves that are too high frequency too hear when you add them all back. I always hate how people try to make the fourier transform sound more complex than it actually is (and yes there is more nuance to compression than this, but this is just the basic idea).
computerfriend 3y ago
I wonder how Fourier transformations are taught in engineering courses, because the idea that it could be "lossy" is strange and not obvious to me. It has an inverse after all.
- aswanson 3y ago
  A square wave would take an infinite number of sinusoidal waves to perfectly reconstruct. To approximate it, one truncates the coefficients. This is done in any engineering application where memory isn't infinite. Which is all of them.
  - mkaic 3y ago
    Of course, a discrete, finite sampling of a square wave at a set of points in time only requires a finite number of coefficients to perfectly reconstruct.
    - bisby 3y ago
      Which is the point. A discrete fourier transform can never recreate a square wave unless you have infinite samples. It can only recreate the finite sampled signal (which is only an approximation of the square wave, and not a real square wave). This means that sure, the Fourier transform itself isn't lossy (garbage in, garbage out) but Fourier transforms would be used in contexts where loss are introduced. If I have a real perfect square wave, and I want to a take a fourier transform of it, the sampling is going to introduce loss, so to associate sampling losses with the transform itself is fair. Real square wave ran through a DFT program on my computer is going to spit out an approximation of a square wave -- loss.
      - davidgay 3y ago
        The good news of course is that if you were sampling a real signal, then that signal was not actually a perfect square wave. So the fact that you can't (re)construct a perfact square wave is somewhat moot...
aj7 3y ago
The DFT has quite severe limitations that do not appear in the Fourier transform. In particular, the Nyquist criteria that there be zero signal energy for ALL frequencies above half the sampling frequency can only be approximated, and must be accomplished BEFORE sampling, i.e. in the time domain.

quibono 3y ago

> I feel this often gets lost when people approach the Fourier transform from a more engineering perspective, not at least because we often do the transform to throw away unwanted information, like certain frequency components. That was my problem as well. My first introduction to Fourier transforms was through more of an engineering lens. I remember having trouble with the _inverse_ Fourier transform. I was OK with a Fourier inverse of an already transformed function but I wasn't quite sure what that would mean when applied to a non-transformed, "regular" function.

takd 3y ago
As a related aside, the terms "cepstrum" and the "quefrencies" [1] (c.f. spectrum and frequencies) sound so hilarious that when I first heard about them I was convinced it was some kind of prank. [1] https://en.wikipedia.org/wiki/Cepstrum https://en.wikipedia.org/wiki/Cepstrum
- cmehdy 3y ago
  This does read like a joke, I had never heard of it either and I'm wondering if many people do use this at all.. Operations on cepstra are labelled quefrency analysis (or quefrency alanysis[1]), liftering, or cepstral analysis. It may be pronounced in the two ways given, the second having the advantage of avoiding confusion with kepstrum.
  - aswanson 3y ago
    It's used in speech signal processing & seismic signal analysis.
  - ljosifov 3y ago
    Almost all speech recognizers until this latest crop (of end-to-end DL NN ASR) operated on cepstral coefficients (and their delta-s and delta-delta-s) as their feature vector.
pas 3y ago
> [...] I wasn't quite sure what that would mean when applied to a non-transformed, "regular" function. Have you gained some intuition/understanding for this? I tried a few inputs in WolframAlpha, but unless I manually type in the integral for the inverse transform there's not even a graph :) (and I have no idea whether it's even the same thing without putting a `t` in the exponent and wrapping it in an f(t) = ... ) https://www.wolframalpha.com/input?i=integral+%28sin%28x%29+%2F+x%29*e%5E%28i+2pi+x%29+ https://www.wolframalpha.com/input?i=integral+%28sin%28x%29+...
- weinzierl 3y ago
  Not parent (but GP) and intuition can mean many things but what helped me was keeping in mind: Every continuous periodic function turns into a discrete aperiodic one when transformed. Works both ways. Continuous aperiodic stays continuous aperiodic. Discrete periodic stays discrete periodic.
sharpneli 3y ago
Inverse fourier transform of a non transformed signal gives you basically the fourier transform with some changes (I can't remember which, were the numbers conjugates or something?). Applying it the second time gives you same result as if you'd do the forward direction transform twice. If you apply fourier transform 4 times you get your original function back. You can think of it as 90 degree rotation. Inverse transform just rotates it in the opposite direction. The rotation analog is not even too far fetched as fractional fourier transform allows you to do an arbitrary angle rotation.
- araes 3y ago
  Having never heard of this for the Fourier Transform, needed to read. F0: original signal F1: frequency domain signal F2: reverse time signal F3: inverse fourier signal F4: original signal Also, has further weird applications I've never heard of with "Fractional Fourier Transforms" [1] which can apparently result in smooth smears of time -> frequency domain [2]. [1] https://en.wikipedia.org/wiki/Fractional_Fourier_transform https://en.wikipedia.org/wiki/Fractional_Fourier_transform [2] https://en.wikipedia.org/wiki/File:FracFT_Rec_by_stevencys.jpg https://en.wikipedia.org/wiki/File:FracFT_Rec_by_stevencys.j...
  - laszlokorte 3y ago
    See [1] for my visualization of this 4-cycle and the fractional fourier transform [1]: https://static.laszlokorte.de/frft-cube/ https://static.laszlokorte.de/frft-cube/
    - araes 3y ago
      Thanks. It's neat being able to visualize them, and the 3D display's actually pretty cool looking for all the different functions. The Fractional DFT part though, doesn't seem to do anything no matter the function chosen. Firefox 125. Edit: Nvm, figured it out. Have to visualize from the top down to see the Fractional DFT portion. Haven't seen many visualization systems where each orientation shows a different type of data. Actually a pretty neat idea from a UI perspective.
Salgat 3y ago
A fourier transform basically gives you an infinite number of sine waves with different amplitudes/phases at every frequency. If you add them all back together (the inverse fourier transform), you get back your original signal. Audio compression in this case would just be excluding the sine waves that are too high frequency too hear when you add them all back. I always hate how people try to make the fourier transform sound more complex than it actually is (and yes there is more nuance to compression than this, but this is just the basic idea).

computerfriend 3y ago

I wonder how Fourier transformations are taught in engineering courses, because the idea that it could be "lossy" is strange and not obvious to me. It has an inverse after all.

aswanson 3y ago
A square wave would take an infinite number of sinusoidal waves to perfectly reconstruct. To approximate it, one truncates the coefficients. This is done in any engineering application where memory isn't infinite. Which is all of them.
- mkaic 3y ago
  Of course, a discrete, finite sampling of a square wave at a set of points in time only requires a finite number of coefficients to perfectly reconstruct.
  - bisby 3y ago
    Which is the point. A discrete fourier transform can never recreate a square wave unless you have infinite samples. It can only recreate the finite sampled signal (which is only an approximation of the square wave, and not a real square wave). This means that sure, the Fourier transform itself isn't lossy (garbage in, garbage out) but Fourier transforms would be used in contexts where loss are introduced. If I have a real perfect square wave, and I want to a take a fourier transform of it, the sampling is going to introduce loss, so to associate sampling losses with the transform itself is fair. Real square wave ran through a DFT program on my computer is going to spit out an approximation of a square wave -- loss.
    - davidgay 3y ago
      The good news of course is that if you were sampling a real signal, then that signal was not actually a perfect square wave. So the fact that you can't (re)construct a perfact square wave is somewhat moot...

aj7 3y ago

The DFT has quite severe limitations that do not appear in the Fourier transform. In particular, the Nyquist criteria that there be zero signal energy for ALL frequencies above half the sampling frequency can only be approximated, and must be accomplished BEFORE sampling, i.e. in the time domain.