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

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• December 3rd 2010, 05:17 PM
nasil122002
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.
• December 3rd 2010, 11:09 PM
chisigma
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

$\chi$ $\sigma$
• December 4th 2010, 01:59 AM
nasil122002
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.
• December 6th 2010, 02:14 AM
chisigma
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

$\chi$ $\sigma$