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Thread: fourrier transform proof

  1. #1
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    fourrier transform proof

    is there a proof 4 fourrier's theory;that any signal is composed of infinite number of sinusoids?

    another question : what is the reason that made fourrier think of signals in that way? i mean , of course he faced some problem or had a certain idea before he put his theory
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by allah's_slave View Post
    is there a proof 4 fourrier's theory;that any signal is composed of infinite number of sinusoids?

    For a continuous absolutely integrable function $\displaystyle f(x)$ with absolutly integrable Fourier transform $\displaystyle [\mathcal{F}f](\xi)=F(\xi)$ the result follows from:

    $\displaystyle f(0)=\int_{-\infty}^{\infty} F(\xi)~d\xi \ \ \ \ \ \ \ ...(1)$

    by the translation theorem.

    To prove $\displaystyle (1)$ you need a sequence of well behaved functions $\displaystyle u_n(x)$ with known Fourier transforms $\displaystyle U_n(\xi)$, which as $\displaystyle n \to \infty$ approximates the behaviour of a $\displaystyle \delta$ functional.

    A suitable sequence can be constructed from Gaussians with decreasing spread parameters (who's FTs form a sequence of Gaussians with increasing spread parameters).

    Then to prove $\displaystyle (1)$ you consider:

    $\displaystyle \lim_{n \to \infty} \int f(x)u_n(x) e^{i 2 \pi \xi x}dx$$\displaystyle \ = \lim_{n \to \infty }(F*U_n)(\xi)$

    By construction the limit on the left hand side is $\displaystyle f(0)$, and on the right hand side you can exchange the limit and integral and so find it goes to right hand side of $\displaystyle (1)$.

    (Of course this is much more direct it you run amok with generalised functions or distributions, when you use $\displaystyle \delta(x)$ instead of $\displaystyle \{ u_n(x)\}$, and $\displaystyle 1$ instead of $\displaystyle \{U_n(\xi)\}$)

    (And also this is modulo the odd factor depending on how the FT has been defined)

    RonL
    Last edited by CaptainBlack; Apr 17th 2008 at 01:18 PM. Reason: did not understand my earlier argument
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