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Math Help - Finite Fourier Transform

  1. #1
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    Finite Fourier Transform

    Hi,
    As you know Fourier Tansform bounds are from -\infty to +\infty and for sine and cosine transforms is from 0 to \infty. How can apply Fourier Transform to a finite domain? I mean by some kind of transform function!
    If the transformations are different for different Fourier transforms, Please give me all.
    Thanks
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by Ehsan View Post
    Hi,
    As you know Fourier Tansform bounds are from -\infty to +\infty and for sine and cosine transforms is from 0 to \infty. How can apply Fourier Transform to a finite domain? I mean by some kind of transform function!
    If the transformations are different for different Fourier transforms, Please give me all.
    Thanks
    If by finite domein you mean some interval then you have Fourier series,

    The Discrete Fourier Transform is what you have if the domain is discrete and finite, think of it as the FT on the vertices of a regular n-gon, where n is the number of points.

    RonL
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by Ehsan View Post
    I have seen this formula in a PhD thesis and the author calls is "generalized finite Fourier Transform" but I could not find it anywhere else. Can you tell me where does it come from?

     <br />
w(n)=F_g\{v(z)\}=\int_{0}^{h}v(z)sin((n-\frac{1}{2})\frac{\pi z}{h})dz<br />
    1. Are you sure that you have not lost a factor of \sqrt{2} somewhere?

    2. For any complete ortho-normal basis \{\phi_i; \ i=1,2, ..\} for an appropriate class of functions on (a,b) (with respect to some weight w if we want) we have:

     <br />
f(x)=\sum_{k=0}^{\infty} a_n \phi_n(x)<br />

    where:

     <br />
a_n=\int_a^b f(x) \phi_n(x) w(x) ~dx<br />

    Now the basis functions in you post are certainly orthogonal, and with w(x)=2 are orthonormal on (0,1), so we need only show that they are a complete basis (which I don't have time to do at present, I may come back to this).

    RonL
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  4. #4
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    Quote Originally Posted by CaptainBlack View Post
    1. Are you sure that you have not lost a factor of \sqrt{2} somewhere?

    2. For any complete ortho-normal basis \{\phi_i; \ i=1,2, ..\} for an appropriate class of functions on (a,b) (with respect to some weight w if we want) we have:

     <br />
f(x)=\sum_{k=0}^{\infty} a_n \phi_n(x)<br />

    where:

     <br />
a_n=\int_a^b f(x) \phi_n(x) w(x) ~dx<br />

    Now the basis functions in you post are certainly orthogonal, and with w(x)=2 are orthonormal on (0,1), so we need only show that they are a complete basis (which I don't have time to do at present, I may come back to this).

    RonL
    The equation is exactly as it is in the PhD thesis and he has given the inversion formula as:

    v(z)=\frac{2}{h}\sum_{n=1}^{\infty} w(n)sin((n-\frac{1}{2})\frac{\pi z}{h})
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  5. #5
    Grand Panjandrum
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    Quote Originally Posted by Ehsan View Post
    The equation is exactly as it is in the PhD thesis and he has given the inversion formula as:

    v(z)=\frac{2}{h}\sum_{n=1}^{\infty} w(n)sin((n-\frac{1}{2})\frac{\pi z}{h})
    Ther 2 in front of the integral is the weight (or the factor of \sqrt{2}, these are the analogue of the odd factors which appear in the fourier transform and inversion formula's definition in random locations), and this is exactly what I wrote (or rather this is a special case of the generalized Fourier series). No great mystery here, second or third year undergraduate real analysis in my day.

    RonL
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