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I'm found of seeing the Fourier transform as a fancy way of changing basis / coordinates, not sure how mathematically correct is that. In high school physics s

by pid-1 4y ago

I'm found of seeing the Fourier transform as a fancy way of changing basis / coordinates, not sure how mathematically correct is that. In high school physics some problems got way easier by redefining coordinates as x' and y', solving for those, them going back to x and y. That's what the Fourier transform does, but for functions. Looking at its formula, we can see it looks like we are projecting a function into a series of complex exponentials, just like we need to project a vector in x' and y' for a change of basis in the euclidian space. Then, the Fourier coefficients can be thought as "how much my function looks like a complex exponential of this given frequency". Compare the formulas for the Pearson Correlation and Convolution, they are almost the same. Since the Fourier transform is just a particular case of changing basis, there are many, many other integral transforms that might be useful in different contexts. We can even create our own. But unis teach about the Fourier transform because it is really good for solving differential equations, which turns out to be a popular way of modeling problems in engineering.

necroforest 4y ago
>I'm found of seeing the Fourier transform as a fancy way of changing basis / coordinates, not sure how mathematically correct is that. It's exactly a change of basis
walnutclosefarm 4y ago
It's a mathematically correct way of looking at it as long as you're clear that that the bases are functions, and you're considering how a particular mathematical artifact (the signal function) is represented as a sum of basis functions. In the naive signal trace, the basis functions are one-hot functions (zero everywhere except at a single point, where they have value 1), in the Fourier basis, the basis functions are sine waves of amplitude 1. So the signal is the sum of the basis functions multiplied by a unique amplitude for each basis function. Sine waves aren't the only basis with which you can make this transformation. Your basis can be an arbitrary set of periodic functions as long as it meets certain requirements. Decomposition into wavelet functions is commonly used in seismic signal analysis, for example.
dimatura 4y ago
Yes, that totally makes sense - you can think of a function as an "infinite-dimensional" vector, which you can express in the (orthogonal) basis of a bunch of sines and cosines of different frequencies.
lathyrus_long 4y ago
As others note, it's exactly a change in basis. In particular, it's an orthonormal basis, meaning that the dot product (correlation) between a basis function and itself is one, and between any two different basis functions is zero. This gives some intuition for why the forward and inverse transforms look so similar, in the same way that the inverse of an orthonormal matrix (like a rotation matrix) is just its transpose.

by pid-1 4y ago

necroforest 4y ago

>I'm found of seeing the Fourier transform as a fancy way of changing basis / coordinates, not sure how mathematically correct is that. It's exactly a change of basis

walnutclosefarm 4y ago

It's a mathematically correct way of looking at it as long as you're clear that that the bases are functions, and you're considering how a particular mathematical artifact (the signal function) is represented as a sum of basis functions. In the naive signal trace, the basis functions are one-hot functions (zero everywhere except at a single point, where they have value 1), in the Fourier basis, the basis functions are sine waves of amplitude 1. So the signal is the sum of the basis functions multiplied by a unique amplitude for each basis function. Sine waves aren't the only basis with which you can make this transformation. Your basis can be an arbitrary set of periodic functions as long as it meets certain requirements. Decomposition into wavelet functions is commonly used in seismic signal analysis, for example.

dimatura 4y ago

Yes, that totally makes sense - you can think of a function as an "infinite-dimensional" vector, which you can express in the (orthogonal) basis of a bunch of sines and cosines of different frequencies.

lathyrus_long 4y ago

As others note, it's exactly a change in basis. In particular, it's an orthonormal basis, meaning that the dot product (correlation) between a basis function and itself is one, and between any two different basis functions is zero. This gives some intuition for why the forward and inverse transforms look so similar, in the same way that the inverse of an orthonormal matrix (like a rotation matrix) is just its transpose.