(→Relationship between Z-Transform and F.T.) |
(→Relationship between Z-Transform and F.T.) |
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<math>X(z) = \sum_{n = -\infty}^\infty x[n]z^{-n} = \sum_{n = -\infty}^\infty x[n](re^{j\omega})^{-n} = \sum_{n = -\infty}^\infty x[n]r^{-n}e^{-j\omega n}</math> | <math>X(z) = \sum_{n = -\infty}^\infty x[n]z^{-n} = \sum_{n = -\infty}^\infty x[n](re^{j\omega})^{-n} = \sum_{n = -\infty}^\infty x[n]r^{-n}e^{-j\omega n}</math> | ||
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Where <math>\sum_{n = -\infty}^\infty x[n]r^{-n}e^{-j\omega n}</math> is the F.T! | Where <math>\sum_{n = -\infty}^\infty x[n]r^{-n}e^{-j\omega n}</math> is the F.T! | ||
Revision as of 14:09, 30 November 2008
Z Transform
Discrete analog of Laplace Transform
$ X(z) = \sum_{n = -\infty}^\infty x[n]z^{-n} $
Where z is a complex variable.
Relationship between Z-Transform and F.T.
$ X(\omega) = X(e^{j\omega}) $
$ X(z)=X(re^{j\omega}) $ Then $ X(z) = F(x[n]r^{-n}) $ $ X(z) = \sum_{n = -\infty}^\infty x[n]z^{-n} = \sum_{n = -\infty}^\infty x[n](re^{j\omega})^{-n} = \sum_{n = -\infty}^\infty x[n]r^{-n}e^{-j\omega n} $ Where $ \sum_{n = -\infty}^\infty x[n]r^{-n}e^{-j\omega n} $ is the F.T!
