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5.2: Common Z-Transforms

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    103986
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    Table \(\PageIndex{1}\): z-Transform Pairs
    Signal, x[n] Z-transform, X(z) Region of Convergence (ROC)
    \(\delta[n]\) \(1\) all \(z\)
    \(\delta[n-m]\) \(z^{-m}\) all \(z \neq 0\)
    \(u[n]\) \(\Large \frac{1}{1-z^{-1}}\) \(\lvert z \rvert > 1\)
    \(a^{n}u[n]\) \(\Large \frac{1}{1-az^{-1}}\) \(\lvert z \rvert > \lvert a \rvert\)
    \(na^{n}u[n]\) \(\Large \frac{az^{-1}}{(1-az^{-1})^2}\) \(\lvert z \rvert > \lvert a \rvert\)
    \(-a^{n} u[-n-1]\) \(\Large \frac{1}{1-az^{-1}}\) \(\lvert z \rvert < \lvert a \rvert\)
    \(-na^{n} u[-n-1]\) \(\Large \frac{az^{-1}}{(1-az^{-1})^2}\) \(\lvert z \rvert < \lvert a \rvert\)
    \(\cos (\omega_0 n) u[n]\) \(\Large \frac{1 - z^{-1} cos(\omega_0)}{1-2z^{-1}cos (\omega_0) + z^{-2}}\) \(\lvert z \rvert > 1\)
    \(\sin (\omega_0 n)u[n]\) \(\Large \frac{z^{-1} sin(\omega_0)}{1-2z^{-1}cos(\omega_0) + z^{-2}}\) \(\lvert z \rvert > 1\)
    \(a^{n}\cos (\omega_0 n) u[n]\) \(\Large \frac{1 - az^{-1} cos(\omega_0)}{1-2az^{-1}cos (\omega_0) + a^{2}z^{-2}}\) \(\lvert z \rvert > \lvert a \rvert \)
    \(a^{n}\sin (\omega_0 n)u[n]\) \(\Large \frac{az^{-1} sin(\omega_0)}{1-2az^{-1}cos(\omega_0) + a^{2}z^{-2}}\) \(\lvert z \rvert > \lvert a \rvert \)

    This page titled 5.2: Common Z-Transforms is shared under a CC BY license and was authored, remixed, and/or curated by Richard Baraniuk et al..

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