Laplace Transform vs. z-Transform
• The Laplace Transform and $z$-Transform (along with other integral transforms such as the Wavelet Transform) are based on the concept of Hilbert spaces (vector space with inner product).
• The $z$-Transform is the discrete-time analogue of the Laplace Transform, often used for analyzing discrete-time signals and systems.
c.f. The Fourier Transform can be viewed as a special case of the Laplace Transform, where the complex variable $s$ in the Laplace domain is restricted to the imaginary axis (i.e., $s=j\omega$, where $j$ is the imaginary unit and $\omega$ is the frequency).
| Laplace Transform | $z$-Transform | |
|---|---|---|
| Transform | $F\left( s \right) = L\left\lbrack f\left( t \right) \right\rbrack = \int_{0}^{\infty}{f(t)e}^{- \text{st}}\text{dt}$ | $X\left( z \right) = Z\left\lbrack x\left( t \right) \right\rbrack = Z\left\lbrack x\left( \text{kT} \right) \right\rbrack = \sum_{k = 0}^{\infty}{x\left( \text{kT} \right)z^{- k}}$ |
| Inverse Transform | $f\left( t \right) = \frac{1}{2\text{πj}}\ \int_{c - j\infty}^{c + j\infty}{F\left( s \right)e^{\text{st}}}\text{ds},\ \ t > 0$ where $c$ is the abscissa of convergence for $F\left( s \right)$ |
$x\left( \text{kT} \right) = Z^{- 1}\left\lbrack X\left( z \right) \right\rbrack = \frac{1}{2\text{πj}}\ \oint_{C}^{}{X\left( z \right)z^{k - 1}}\text{dz}$ where $C$ is a circle with its center at the origin of the $z$ plane such that all poles of $X\left( z \right)z^{k - 1}$ are inside it |
| Graphical meaning |  |
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| $\sin \omega t$ | $\mathcal{L}[\sin \omega t]=\frac{\omega}{s^2 + \omega^2}$ |
$\mathcal{Z}[\sin \omega k T]=\frac{z \sin \omega T}{z^2 - 2 z \cos \omega T + 1} = \frac{z(e^{j\omega T} - e^{-j\omega T})/{2j}}{z^2-z(e^{j\omega T} - e^{-j\omega T})+1}$ |
| $\cos \omega t$ | $\mathcal{L}[\cos \omega t]=\frac{s}{s^2+\omega^2}$ |
$\mathcal{Z}[\cos \omega k T]=\frac{z(z- \cos \omega T)}{z^2 - 2 z \cos \omega T + 1} = \frac{z[z-(e^{j\omega T} + e^{-j\omega T})/{2}]}{z^2-z(e^{j\omega T} + e^{-j\omega T})+1}$ |
| $\sinh \omega t$ | $\mathcal{L}[\sinh \omega t]=\frac{\omega}{s^2 - \omega^2}$ |
$\mathcal{Z}[\sinh \omega k T]=\frac{z \sinh \omega T}{z^2 - 2 z \cosh \omega T + 1} = \frac{z(e^{\omega T} - e^{-\omega T})/{2}}{z^2-z(e^{\omega T} + e^{-\omega T})+1}$ |
| $\cosh \omega t$ | $\mathcal{L}[\cosh \omega t]=\frac{s}{s^2-\omega^2}$ |
$\mathcal{Z}[\cosh \omega k T]=\frac{z(z- \cosh \omega T)}{z^2 - 2 z \cosh \omega T + 1} = \frac{z[z-(e^{\omega T} + e^{-\omega T})/{2}]}{z^2-z(e^{\omega T} + e^{-\omega T})+1}$ |