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 x(n) is a discrete-time signal or sequence, then its bilateral or two-sided Z-transform is defined as −

$$\mathrm{Z[x(n)]\:=\:X(z)\:=\:\sum_{n=-\infty}^{\infty}\:x(n)z^{-n}}$$

Where, z is a complex variable.

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

$$\mathrm{z[x(n)]\:=\:X(z)\:=\:\sum_{n=0}^{\infty}\:x(n)z^{-n}}$$

Z-Transform of Decaying Exponential Sequence

The decaying causal complex exponential function is defined as −

$$\mathrm{x(n)\:=\:e^{−j\omega n}u(n)\:=\:\begin{cases}e^{-j\omega n} & \text{for }\:n \:\geq\: 0\\\\0 & \text{for }\:n \:\lt\: 0\end{cases}}$$

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

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

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

$$\mathrm{\Rightarrow\:X(z)\:=\:1\:+\:(e^{-j\omega}z^{-1})\:+\:(e^{-j\omega}z^{-1})^2\:+\:(e^{-j\omega}z^{-1})^3\:+\:\dotso}$$

$$\mathrm{\Rightarrow\:X(z)\:=\:[1\:-\:(e^{-j\omega}z^{-1})]^{-1}}$$

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

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,

$$\mathrm{e^{-j\omega n}u(n)\:\overset{ZT}\longleftrightarrow\:\frac{z}{(z\:-\:e^{-j\omega})};\:\:\text{ROC }\:\rightarrow\:|z|\:\gt\:1}$$

Z-Transform of Growing Exponential Sequence

The growing causal complex exponential function is defined as −

$$\mathrm{x(n)\:=\:e^{j\omega n}\:u(n)\:=\:\begin{cases}e^{j\omega n} & \text{for }\:n\: \geq\: 0\\\\0 & \text{for }\:n \:\lt\: 0\end{cases}}$$

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

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

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

$$\mathrm{\Rightarrow\: X(z) \:=\: 1 \:+\: (e^{j \omega} z^{-1}) \:+\: (e^{j \omega} \:z^{-1})^2 \:+\: (e^{j \omega} z^{-1})^3 \:+\: \cdots}$$

$$\mathrm{\Rightarrow\: X(z) \:=\: [1 \:-\: (e^{j \omega} z^{-1})]^{-1}}$$

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

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,

$$\mathrm{e^{j\omega n}u(n)\:\overset{ZT}\longleftrightarrow\:\frac{z}{(z\:-\:e^{j\omega})};\:\:\text{ROC }\:\rightarrow\:|z|\:\gt\:|1|}$$