Z-Transform of Exponential Functions

The Z-transform (ZT) is a mathematical tool which is used to convert the difference equations in time domain into the algebraic equations in z-domain.

Mathematically, if $\mathrm{\mathit{x\left ( n \right )}}$ is a discrete-time signal or sequence, then its bilateral or two-sided Z-transform is defined as −

$$\mathrm{\mathit{Z\left [ x\left ( n \right ) \right ]\mathrm{\, =\,}X\left ( z \right )\mathrm{\, =\,}\sum_{n\mathrm{\, =\,}-\infty }^{\infty }x\left ( n \right )z^{-n}}}$$

Where, z is a complex variable.

Also, the unilateral or one-sided z-transform is defined as −

$$\mathrm{\mathit{Z\left [ x\left ( n \right ) \right ]\mathrm{\, =\,}X\left ( z \right )\mathrm{\, =\,}\sum_{n\mathrm{\, =\,}\mathrm{0}}^{\infty }x\left ( n \right )z^{-n}}}$$

Z-Transform of Decaying Exponential Sequence

The decaying causal complex exponential function is defined as −

$$\mathrm{\mathit{x\left ( n \right )\mathrm{\, =\,}e^{-j\,\omega n}u\left ( n \right )\mathrm{\, =\,}\begin{Bmatrix} e^{-j\,\omega n}& \mathrm{for\: \mathit{n}\geq 0} \ \mathrm{0} & \mathrm{for\: \mathit{n}< 0}\ \end{Bmatrix} }}$$

Therefore, the Z-transform of the decaying exponential function is obtained as −

$$\mathrm{\mathit{Z\left [ x\left ( n \right ) \right ]\mathrm{\, =\,}X\left ( z \right )\mathrm{\, =\,}Z\left [ e^{-j\, \omega n}\, u\left ( n \right ) \right ] }}$$

$$\mathrm{\mathit{\Rightarrow X\left ( z \right )\mathrm{\, =\,}\sum_{n\mathrm{\, =\,}\mathrm{0}}^{\infty }e^{-j\, \omega n}\,z^{-n}\mathrm{\, =\,}\sum_{n\mathrm{\, =\,}\mathrm{0}}^{\infty }\left ( e^{-j\, \omega}\,z^{-\mathrm{1}} \right )^{n}}}$$

$$\mathrm{\mathit{\Rightarrow X\left ( z \right )\mathrm{\, =\,}\mathrm{1}\mathrm{\, +\,}\left ( e^{-j\, \omega}\,z^{-\mathrm{1}} \right )\mathrm{\, +\,}\left ( e^{-j\, \omega}\,z^{-\mathrm{1}} \right )^{\mathrm{2}}\mathrm{\, +\,}\left ( e^{-j\, \omega}\,z^{-\mathrm{1}} \right )^{\mathrm{3}}\mathrm{\, +\,}\cdot \cdot \cdot }}$$

$$\mathrm{\mathit{\Rightarrow X\left ( z \right )\mathrm{\, =\,}\left [ \mathrm{1}-\left ( e^{-j\, \omega}\,z^{-\mathrm{1}} \right ) \right ]^{\mathrm{-1}} }}$$

$$\mathrm{\mathit{\therefore X\left ( z \right )\mathrm{\, =\,}Z\left [ e^{-j\, \omega n}\, u\left ( n \right ) \right ]\mathrm{\, =\,}\frac{\mathrm{1}}{\left [ \mathrm{1}-\left ( e^{-j\, \omega}\,z^{-\mathrm{1}} \right ) \right ]}\mathrm{\, =\,}\frac{z}{\left ( z-e^{-j\, \omega } \right )} }}$$

This series converges for |𝑍⁻¹| < 1. Therefore, the ROC of the Z-transform of the decaying exponential sequence is |𝑧| > 1. Thus, the Z-transform of the decaying complex exponential sequence with its ROC may be represented as,

$$\mathrm{\mathit{e^{-j\, \omega n}\, u\left ( n \right )\overset{ZT}{\leftrightarrow} \frac{z}{\left ( z-e^{-j\, \omega } \right )};\; \; }ROC\rightarrow \left|\mathit{z} \right|> 1}$$

Z-Transform of Growing Exponential Sequence

The growing causal complex exponential function is defined as −

$$\mathrm{\mathit{x\left ( n \right )\mathrm{\, =\,}e^{j\, \omega n}u\left ( n \right )\mathrm{\, =\,}\begin{Bmatrix} e^{j\, \omega n}& \mathrm{for\: \mathit{n}\geq 0} \ \mathrm{0} & \mathrm{for\: \mathit{n}< 0}\ \end{Bmatrix} }}$$

The Z-transform of the growing exponential sequence is obtained as follows −

$$\mathrm{\mathit{Z\left [ x\left ( n \right ) \right ]\mathrm{\, =\,}X\left ( z \right )\mathrm{\, =\,}Z\left [ e^{j\, \omega n}\, u\left ( n \right ) \right ] }}$$

$$\mathrm{\mathit{\Rightarrow X\left ( z \right )\mathrm{\, =\,}\sum_{n\mathrm{\, =\,}\mathrm{0}}^{\infty }e^{j\, \omega n}\,z^{-n}\mathrm{\, =\,}\sum_{n\mathrm{\, =\,}\mathrm{0}}^{\infty }\left ( e^{j\, \omega}\,z^{-\mathrm{1}} \right )^{n}}}$$

$$\mathrm{\mathit{\Rightarrow X\left ( z \right )\mathrm{\, =\,}\mathrm{1}\mathrm{\, +\,}\left ( e^{j\, \omega}\,z^{-\mathrm{1}} \right )\mathrm{\, +\,}\left ( e^{j\, \omega}\,z^{-\mathrm{1}} \right )^{\mathrm{2}}\mathrm{\, +\,}\left ( e^{j\, \omega}\,z^{-\mathrm{1}} \right )^{\mathrm{3}}\mathrm{\, +\,}\cdot \cdot \cdot }}$$

$$\mathrm{\mathit{\Rightarrow X\left ( z \right )\mathrm{\, =\,}\left [ \mathrm{1}-\left ( e^{j\, \omega}\,z^{-\mathrm{1}} \right ) \right ]^{\mathrm{-1}}}}$$

$$\mathrm{\mathit{\therefore X\left ( z \right )\mathrm{\, =\,}Z\left [ e^{j\, \omega n}\, u\left ( n \right ) \right ]\mathrm{\, =\,}\frac{\mathrm{1}}{\left [ \mathrm{1}-\left ( e^{j\, \omega}\,z^{-\mathrm{1}} \right ) \right ]}\mathrm{\, =\,}\frac{z}{\left ( z-e^{j\, \omega } \right )} }}$$

The ROC of the Z-transform of the growing causal complex exponential sequence is |𝑧| > |1|. Hence, the Z-transform of the decaying complex exponential sequence with its ROC can be represented as,

$$\mathrm{\mathit{e^{j\, \omega n}\, u\left ( n \right )\overset{ZT}{\leftrightarrow} \frac{z}{\left ( z-e^{j\, \omega } \right )};\; \; }ROC\rightarrow \left|\mathit{z} \right|> \left|1 \right|}$$