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Fourier transform me

  1. Dec 12, 2009 #1
    fourier transform plz help me

    if the fourier transform of f(x,y) is f(u,v) what is the fourier transform of the following:

    f(x+6,y)
    f(x,-y)
    f(2x+6,y)

    plz solve it and help me
     
  2. jcsd
  3. Dec 12, 2009 #2
    Re: fourier transform plz help me

    Just try substituting these into the definition

    [tex]F[f(x,y)](u,v)=\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}f(x,y)e^{-i(ux+vy)}dxdy[/tex]

    f.e. lets do the first one:

    f(x+6,y):

    [tex]F[f(x+6,y)]=\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}f(x+6,y)e^{-i(ux+vy)}dxdy[/tex]

    I'm gonna make the next change of variables:
    [tex]x->\tilde{x}-6[/tex]

    The boundaries of the integral remains unchanged, and so does the differential, but we get:

    [tex]\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}f(\tilde{x},y)e^{-i(u\tilde{x}-6u+vy)}d\tilde{x}dy[/tex]

    Notice that [tex]e^{i6u}[/tex] is constant wrt to the integration, so we pull it out and what we have left of the integral is simply the fourier transform of f(x,y) [because x* is just a dummy variable]:

    [tex]e^{i6u}\int^{\infty}_{-\infty}\int^{\infty}_{-\infty}f(\tilde{x},y)e^{-i(u\tilde{x}+vy)}d\tilde{x}dy=e^{i6u}f(u,v)[/tex]

    This technique of variable change is the standard technique to observe how shift & scale of the time-domain (the original function) affects the frequency-domain (its fourier transform).
     
  4. Dec 12, 2009 #3
    Re: fourier transform plz help me

    i can't understand, plz explain more
    thanks for your attention
     
  5. Dec 30, 2009 #4
    Re: fourier transform plz help me

    You obviously don't understand simple substitution in integrals, maybe you should brush up on that first.
     
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