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Math Help - Pulse response and Phase function (image processing-Wavelets)?

  1. #1
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    Pulse response and Phase function (image processing-Wavelets)?

    Hello,

    How can I calculate the Fourier transform of impulse response ^h(w) and phase function psi(w) of the given impulse response h= (kh) is defined by: h0= 1/2, h1= 1, h2= 1/2, otherwise= 0 ?


    Thanks for your help.
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  2. #2
    MHF Contributor chisigma's Avatar
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    The [discrete] 3-points Fourier Tranform of h(n) is given by...

    \displaystyle H(k)= \sum_{n=0}^{2} h(n)\ e^{- j 2 \pi \frac{k n}{3} , k=0,1,2 (1)

    Kind regards

    \chi \sigma
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  3. #3
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    Thanks for the answer, how can I now calculate the pulse response h^(w) and psi (w) on this formula? I need really a complete solution.
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  4. #4
    MHF Contributor chisigma's Avatar
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    The Z-Transform os the finite 3-points sequence h(n) by definition is given by...

    \displaystyle H(z)= \sum_{n=0}^{2} h(n)\ z^{-n} (1)

    The complex transfer function [FIR filter...] is easily obtained from (1) setting z=e^{j \omega}. A more complex alternative is the derivation of the H(z) fron the H(k) I have defined in my previous post using the inverse discrete Fourier transform...

    \displaystyle H(z)=  \frac{1}{3}\ \sum_{k=0}^{2} H(k)\ \frac{1-z^{-3}}{1- e^{-j 2 \pi \frac{k}{3}}\ z^{-1}} (2)

    ... and again You can find the complex transfer function setting in (2) z=e^{j \omega}...

    Kind regards

    \chi \sigma
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