> 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.
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.
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].
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.
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.
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.
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.
> [...] 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) = ... )
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).
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.