(New page: Back to ECE438 course page ==Linearity == <math>\mathfrak{Z}\big\{ax[n]+by[n]\big\}=a\mathfrak{Z}\big\{x[n]\big\}+b\mathfrak{Z}\big\{y[n]\big\}\;\;\;\;\;\for...) |
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<math>\mathfrak{Z}\big\{x[n-n_0]\big\}=\sum_{n=-\infty}^{\infty}x[n-n_0]z^{-n}\;\;\;let\;\;\;k = n-n_0</math> | <math>\mathfrak{Z}\big\{x[n-n_0]\big\}=\sum_{n=-\infty}^{\infty}x[n-n_0]z^{-n}\;\;\;let\;\;\;k = n-n_0</math> | ||
| − | = <math>\sum_{k=-\infty}^{\infty}x[k]z^{-k-n_0} = z^ | + | = <math>\sum_{k=-\infty}^{\infty}x[k]z^{-k-n_0} = z^{-n_0}\sum_{k=-\infty}^{\infty}x[k]z^{-k}</math> |
| − | = <math>z^{n_0}X(Z)</math> | + | = <math>z^{-n_0}X(Z)</math> |
Latest revision as of 14:22, 21 September 2009
Linearity
$ \mathfrak{Z}\big\{ax[n]+by[n]\big\}=a\mathfrak{Z}\big\{x[n]\big\}+b\mathfrak{Z}\big\{y[n]\big\}\;\;\;\;\;\forall a,b \in \mathbb{C} \;\;\;, \forall signals \; x[n],y[n] $
if
$ \mathfrak{Z}\big\{x[n]\big\} = X(Z)\;with\;ROC = R_1 $ $ \mathfrak{Z}\big\{y[n]\big\} = Y(Z)\;with\;ROC = R_2 $
then
$ ax[n]+bx[n]\longrightarrow aX(Z)+bY(Z)\;\;\;with\;ROC = R_1 \cap R_2 $
Time Shifting
$ \mathfrak{Z}\big\{x[n-n_0]\big\} = z^{-n_0}\mathfrak{Z}\big\{x[n]\big\}\;same\;ROC\;as\;x[n] $
Proof:
let $ n_o $ be a finite integer
$ \mathfrak{Z}\big\{x[n-n_0]\big\}=\sum_{n=-\infty}^{\infty}x[n-n_0]z^{-n}\;\;\;let\;\;\;k = n-n_0 $
= $ \sum_{k=-\infty}^{\infty}x[k]z^{-k-n_0} = z^{-n_0}\sum_{k=-\infty}^{\infty}x[k]z^{-k} $
= $ z^{-n_0}X(Z) $
