# 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 )}}$1All 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|}}$