5.2: Common Z-Transforms
- Page ID
- 103986
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)| Signal, x[n] | Z-transform, X(z) | Region of Convergence (ROC) |
|---|---|---|
| \(\delta[n]\) | \(1\) | all \(z\) |
| \(\delta[n-m]\) | \(z^{-m}\) | all \(z \neq 0\) |
| \(u[n]\) | \(\Large \frac{1}{1-z^{-1}}\) | \(\lvert z \rvert > 1\) |
| \(a^{n}u[n]\) | \(\Large \frac{1}{1-az^{-1}}\) | \(\lvert z \rvert > \lvert a \rvert\) |
| \(na^{n}u[n]\) | \(\Large \frac{az^{-1}}{(1-az^{-1})^2}\) | \(\lvert z \rvert > \lvert a \rvert\) |
| \(-a^{n} u[-n-1]\) | \(\Large \frac{1}{1-az^{-1}}\) | \(\lvert z \rvert < \lvert a \rvert\) |
| \(-na^{n} u[-n-1]\) | \(\Large \frac{az^{-1}}{(1-az^{-1})^2}\) | \(\lvert z \rvert < \lvert a \rvert\) |
| \(\cos (\omega_0 n) u[n]\) | \(\Large \frac{1 - z^{-1} cos(\omega_0)}{1-2z^{-1}cos (\omega_0) + z^{-2}}\) | \(\lvert z \rvert > 1\) |
| \(\sin (\omega_0 n)u[n]\) | \(\Large \frac{z^{-1} sin(\omega_0)}{1-2z^{-1}cos(\omega_0) + z^{-2}}\) | \(\lvert z \rvert > 1\) |
| \(a^{n}\cos (\omega_0 n) u[n]\) | \(\Large \frac{1 - az^{-1} cos(\omega_0)}{1-2az^{-1}cos (\omega_0) + a^{2}z^{-2}}\) | \(\lvert z \rvert > \lvert a \rvert \) |
| \(a^{n}\sin (\omega_0 n)u[n]\) | \(\Large \frac{az^{-1} sin(\omega_0)}{1-2az^{-1}cos(\omega_0) + a^{2}z^{-2}}\) | \(\lvert z \rvert > \lvert a \rvert \) |