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I Fourier transform -- what physical variables am I allowed to transform between?

  1. Jul 5, 2018 #1
    A common use of the Fourier transform in physics is to transform between momentum-space and position-space. But what physical variables am I allowed to transform between? For instance can I use the Fourier transform to go from momentum space to frequency space or whatever?
     
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  3. Jul 5, 2018 #2

    Andy Resnick

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    Usually, one transforms between a time-varying signal and a temporal frequency, or a spatially-varying signal and 'wavenumber' (equivalently, spatial frequency or angle).

    Does that help?
     
  4. Jul 5, 2018 #3
    But why? What pair of variables are allowed and why?
     
  5. Jul 5, 2018 #4

    Nugatory

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    A Fourier transform is a way of writing a given function as a sum of sinusoids, so I can Fourier transform just about any function that meets some minimal standards for well-behavedness. The interesting question is whether that's useful: do the sinusoids correspond to any physically interesting function? For example, Fourier transforming a sound signal tells me what frequencies have been superimposed to produce that signal... but Fourier transforming the elevation above sea level along a path is unlikely to tell me anything interesting (unless the topography happens to include some unusually evenly spaced and symmetrical hills, which would show up as a spike in the transform).
     
    Last edited: Jul 5, 2018
  6. Jul 5, 2018 #5

    Andy Resnick

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    I'm not sure what you are getting at. For one thing, the product of whatever conjugate variables you choose (call them 'K' and 'L') must be dimensionless.
     
  7. Jul 6, 2018 #6
    What about the dimensions? RHS should have the same dimension as the LHS?
     
  8. Jul 6, 2018 #7

    Andy Resnick

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    In order for the functions sin(KL), cos(KL), exp(iKL), etc. to be evaluated, 'KL' must be a pure number.
     
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