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Common Z-Transform Pairs
Z-Transform
Z-transform is a mathematical tool which is used to convert the difference equations in time domain into the algebraic equations in the frequency domain.
Mathematically, if $\mathrm{\mathit{x\left ( n \right )}}$ is a discrete-time sequence, then its Z-transform is defined as −
$$\mathrm{\mathit{X\left ( z \right )\mathrm{\, =\,}\sum_{n\mathrm{\, =\,}-\infty }^{\infty }x\left ( n \right )z^{-n}\; \; \; \cdot \cdot \cdot \left ( \mathrm{1} \right )}}$$
Where, z is a complex variable. The z-transform defined in eq. (1) is called bilateral or two-sided z-transform.
The unilateral or one-sided z-transform is defined as −
$$\mathrm{\mathit{X\left ( z \right )\mathrm{\, =\,}\sum_{n\mathrm{\, =\,}\mathrm{0}}^{\infty }x\left ( n \right )z^{-n}\; \; \; \cdot \cdot \cdot \left ( \mathrm{2} \right )}}$$
Common Z-Transform Pairs
The following table gives a number of unilateral and bilateral z-transforms along with their region of convergence (ROC) −
| Discrete-Time Sequence, $\mathrm{\mathit{x\left ( n \right )}}$ | Z-Transform, $\mathrm{\mathit{X\left ( z \right )}}$ | ROC |
|---|---|---|
| $\mathrm{\mathit{\delta \left ( n \right )}}$ | 1 | All 𝑧 |
| $\mathrm{\mathit{u \left ( n \right )}}$ | $\mathrm{\mathit{\frac{z}{\left ( z-\mathrm{1} \right )}\mathrm{\, =\,}\frac{\mathrm{1}}{\left ( \mathrm{1}-z^{-\mathrm{1}} \right )} }}$ | $\mathrm{\mathit{\left|z \right|>\mathrm{1}}}$ |
| $\mathrm{\mathit{u \left ( -n \right )}}$ | $\mathrm{\mathit{\frac{\mathrm{1}}{\mathrm{1}-z}}}$ | $\mathrm{\mathit{\left|z \right|<\mathrm{1}}}$ |
| $\mathrm{\mathit{u \left ( -n-\mathrm{1} \right )}}$ | $\mathrm{\mathit{-\frac{z}{\left ( z-\mathrm{1} \right )}}}$ | $\mathrm{\mathit{\left|z \right|<\mathrm{1}}}$ |
| $\mathrm{\mathit{u \left ( -n-\mathrm{2} \right )}}$ | $\mathrm{\mathit{-\frac{z^{\mathrm{2}}}{\left ( z-\mathrm{1} \right )}}}$ | $\mathrm{\mathit{\left|z \right|<\mathrm{1}}}$ |
| $\mathrm{\mathit{u \left ( -n-k \right )}}$ | $\mathrm{\mathit{-\frac{z^{k}}{\left ( z-\mathrm{1} \right )}}}$ | $\mathrm{\mathit{\left|z \right|<\mathrm{1}}}$ |
| $\mathrm{\mathit{\delta \left ( n-k \right )}}$ | $\mathrm{\mathit{z^{-k}}}$ | If 𝑘 > 0, all 𝑧
except at 𝑧 = 0 If 𝑘 < 0, all 𝑧 except at 𝑧 = ∞ |
| $\mathrm{\mathit{\frac{\mathrm{1}}{n};\; \; n> \mathrm{0}}}$ | $\mathrm{-ln\mathit{\left ( \mathrm{1}-z^{-\mathrm{1}} \right )}}$ | $\mathrm{\mathit{\left|z \right|>\mathrm{1}}}$ |
| $\mathrm{\mathit{a^{\left| n\right|};\; \; }for \: all\: \mathit{n}}$ | $\mathrm{\mathit{\frac{\left ( \mathrm{1}-a^{\mathrm{2}} \right )}{\left [ \left ( \mathrm{1}-az \right )\left ( \mathrm{1}-az^{-\mathrm{1}} \right ) \right ]}}}$ | $\mathrm{\mathit{\left|a \right|<\left|z \right|<\left|\frac{\mathrm{1}}{a} \right|}}$ |
| $\mathrm{\mathit{a^{n}u\left ( n \right )}}$ | $\mathrm{\mathit{\frac{z}{z-a}}}$ | $\mathrm{\mathit{\left|z \right|>\left|a \right|}}$ |
| $\mathrm{\mathit{-a^{n}u\left ( -n \right )}}$ | $\mathrm{\mathit{\frac{a}{\left ( z-a \right )}}}$ | $\mathrm{\mathit{\left|z \right|<\left|a \right|}}$ |
| $\mathrm{\mathit{-a^{n}u\left ( -n-\mathrm{1} \right )}}$ | $\mathrm{\mathit{\frac{z}{\left ( z-a \right )}}}$ | $\mathrm{\mathit{\left|z \right|<\left|a \right|}}$ |
| $\mathrm{\mathit{nu\left ( n \right )}}$ | $\mathrm{\mathit{\frac{z}{\left ( z-\mathrm{1} \right )^{\mathrm{2}}}}}$ | $\mathrm{\mathit{\left|z \right|>\mathrm{1}}}$ |
| $\mathrm{\mathit{n\, a^{n}u\left ( n \right )}}$ | $\mathrm{\mathit{\frac{az}{\left ( z-a \right )^{\mathrm{2}}}}}$ | $\mathrm{\mathit{\left|z \right|>\left|a \right|}}$ |
| $\mathrm{\mathit{-n\,u\left ( -n-\mathrm{1} \right )}}$ | $\mathrm{\mathit{\frac{z}{\left ( z-\mathrm{1} \right )^{\mathrm{2}}}}}$ | $\mathrm{\mathit{\left|z \right|<\mathrm{1}}}$ |
| $\mathrm{\mathit{-n\,a^{n}u\left ( -n-\mathrm{1} \right )}}$ | $\mathrm{\mathit{\frac{az}{\left ( z-a \right )^{\mathrm{2}}}}}$ | $\mathrm{\mathit{\left|z \right|<\left|a \right|}}$ |
| $\mathrm{\mathit{e^{-j\, \omega n}u\left ( n \right )}}$ | $\mathrm{\mathit{\frac{z}{\left ( z-e^{-j\omega } \right )}}}$ | $\mathrm{\mathit{\left|z \right|>\mathrm{1}}}$ |
| $\mathrm{cos\mathit{\, \omega n\: u\left ( n \right )}}$ | $\mathrm{\mathit{\frac{z\left ( z-\mathrm{cos}\, \omega \right )}{z^{\mathrm{2}}-\mathrm{2}z\, \mathrm{cos}\, \omega \mathrm{\, +\,}\mathrm{1} }}}$ | $\mathrm{\mathit{\left|z \right|>\mathrm{1}}}$ |
| $\mathrm{sin\mathit{\, \omega n\: u\left ( n \right )}}$ | $\mathrm{\mathit{\frac{z\: \mathrm{sin}\omega }{z^{\mathrm{2}}-\mathrm{2}z\, \mathrm{cos}\, \omega \mathrm{\, +\,}\mathrm{1} }}}$ | $\mathrm{\mathit{\left|z \right|>\mathrm{1}}}$ |
| $\mathrm{\mathit{a^{n}\, \mathrm{cos}\: \omega n\: u\left ( n\right )}}$ | $\mathrm{\mathit{\frac{z\left ( z-a\, \mathrm{cos}\, \omega \right )}{z^{\mathrm{2}}-\mathrm{2}az\, \mathrm{cos}\, \omega \mathrm{\, +\,}a^{\mathrm{2}}}}}$ | $\mathrm{\mathit{\left|z \right|>\left|a \right|}}$ |
| $\mathrm{\mathit{a^{n}\, \mathrm{sin}\: \omega n\: u\left ( n\right )}}$ | $\mathrm{\mathit{\frac{az\: \mathrm{sin}\, \omega }{z^{\mathrm{2}}-\mathrm{2}az\, \mathrm{cos}\, \omega \mathrm{\, +\,}a^{\mathrm{2}} }}}$ | $\mathrm{\mathit{\left|z \right|>\left|a \right|}}$ |
| $\mathrm{\mathit{\left ( n\mathrm{\, +\,}\mathrm{1} \right )a^{n}u\left ( n \right )}}$ | $\mathrm{\mathit{\frac{z^{\mathrm{2}}}{\left ( z-a \right )^{\mathrm{2}}}}}$ | $\mathrm{\mathit{\left|z \right|>\left|a \right|}}$ |
| $\mathrm{\mathit{\frac{\left ( n\mathrm{\, +\,}\mathrm{1} \right )\left ( n\mathrm{\, +\,}\mathrm{2} \right )}{\mathrm{2!}}a^{n}u\left ( n \right )}}$ | $\mathrm{\mathit{\frac{z^{\mathrm{3}}}{\left ( z-a \right )^{\mathrm{3}}}}}$ | $\mathrm{\mathit{\left|z \right|>\left|a \right|}}$ |
| $\mathrm{\mathit{\frac{n\left ( n-\mathrm{1} \right )}{\mathrm{2!}}a^{\left ( n-\mathrm{2} \right )}u\left ( n \right )}}$ | $\mathrm{\mathit{\frac{z}{\left ( z-a \right )^{\mathrm{3}}}}}$ | $\mathrm{\mathit{\left|z \right|>\left|a \right|}}$ |
| $\mathrm{\mathit{\frac{n\left ( n-\mathrm{1} \right )...\left [ n-\left ( k-\mathrm{2} \right ) \right ]}{\left ( k-\mathrm{1} \right )\mathrm{!}}a^{\left ( n-k\mathrm{\, +\,}\mathrm{1} \right )}u\left ( n \right )}}$ | $\mathrm{\mathit{\frac{z}{\left ( z-a \right )^{k}}}}$ | $\mathrm{\mathit{\left|z \right|>\left|a \right|}}$ |
Updated on: 11-Jan-2022
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