(New page: <math>X(w) = F{x[n]} = \sum_{n=-\infty}^\infty x[n]e^{-jwn}</math> <math>X(z)|_{z=e^{jw}} = X(e^{jw})</math> Can compute Z-Transform as a DTFT write <math>X(z)=X(re^{jw})</math> then <m...) |
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| + | [[Category:discrete time Fourier transform]] | ||
| + | [[Category:z-transform]] | ||
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| + | =Relationship between DTFT and z-transform= | ||
<math>X(w) = F{x[n]} = \sum_{n=-\infty}^\infty x[n]e^{-jwn}</math> | <math>X(w) = F{x[n]} = \sum_{n=-\infty}^\infty x[n]e^{-jwn}</math> | ||
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<math> = F{x[n]r^{-n}}</math> | <math> = F{x[n]r^{-n}}</math> | ||
| + | ---- | ||
Revision as of 07:18, 14 November 2011
Relationship between DTFT and z-transform
$ X(w) = F{x[n]} = \sum_{n=-\infty}^\infty x[n]e^{-jwn} $
$ X(z)|_{z=e^{jw}} = X(e^{jw}) $
Can compute Z-Transform as a DTFT write $ X(z)=X(re^{jw}) $
then $ X(z)= \sum_{-\infty}^\infty x[n]z^{-n} $
$ X(z)= \sum_{-\infty}^\infty x[n](re^{jw})^{-n} $
$ X(z)= \sum_{-\infty}^\infty x[n]r^{-n}e^{-jwn} $
$ = F{x[n]r^{-n}} $
