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==Relationship between Z-Transform and F.T.== | ==Relationship between Z-Transform and F.T.== | ||
| − | *<math>X(\omega) = X(e^{j\omega}</math> | + | *<math>X(\omega) = X(e^{j\omega})</math> |
*<math>X(z)=X(re^{j\omega})</math> | *<math>X(z)=X(re^{j\omega})</math> | ||
Then <math>X(z) = F(x[n]r^-n)</math> | Then <math>X(z) = F(x[n]r^-n)</math> | ||
| − | <math>X(z) = \sum_{n = -\infty}^\infty x[n]z^{-n} = \sum_{n = -\infty}^\infty x[n](re^{j\omega})^ | + | <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> |
Revision as of 15:07, 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} $
