$\text{z}$-transform

The $z$- transform for a discrete-time signal $\text{x}[n]$ is given as:

$$\text{X}(\text{z}) = \sum_{n=-\infty}^ {\infty} \text{x}[n]~ \text{z}^{-n}$$

where $\text{X}(\text{z})$ is a complex valued function of complex variable $\text{z}$.

$\text{z}$ transform always has two parts:

(i) Mathematical expression $\text{X}(\text{z})$

(ii) Region of convergence (ROC) - region in $\text{z}$- plane where the sum of $\text{X}(\text{z})$ converges.

Examples:

(i) Let $\text{x}[n]$ be $$\delta[n] = \left\{\begin{matrix} 1 \quad ~~~~~~n=0 \\ 0 \quad \text{otherwise} \end{matrix}\right.$$

We know that $\text{z}$- tranform can be given as:

$\text{X}(\text{z}) = \sum_{n=-\infty}^ {\infty} \text{x}[n]~ \text{z}^{-n}$

$~~~~~~~~~ = \sum_{n=-\infty}^ {\infty} \delta[n]~ \text{z}^{-n}$

$~~~~~~~~~= 1$

ROC of $\delta[n]$ : whole $z$-plane

(ii) $\text{x}[n] = \delta[n-n_0]$, where $n_0 > 0$ and an integer

We know that $\text{z}$- tranform can be given as:

$\text{X}(\text{z}) = \sum_{n=-\infty}^ {\infty} \text{x}[n]~ \text{z}^{-n}$

$~~~~~~~~~ = \sum_{n=-\infty}^ {\infty} \delta[n-n_0]~ \text{z}^{-n}$

$~~~~~~~~~ = \text{z}^{-n_0}$

$~~~~~~~~~ = {\left ( \frac{1}{\text{z}} \right )}^{n_0}$

ROC: whole $\text{z}$-plane except $\text{z} = 0$

(iii) $\text{x}[n] = \delta[n+n_0]$, where $n_0 > 0$ and an integer

We know that $\text{z}$- tranform can be given as:

$\text{X}(\text{z}) = \sum_{n=-\infty}^ {\infty} \text{x}[n]~ \text{z}^{-n}$

$~~~~~~~~~ = \sum_{n=-\infty}^ {\infty} \delta[n+n_0]~ \text{z}^{-n}$

$~~~~~~~~~ = \text{z}^{n_0}$ using shifting property of impulse signals

ROC: whole $\text{z}$-plane except $|\text{z}|=\infty$

Poles and zeros

We are interested in $\text{z}$-transform in the form of ratio of polynomials in $\text{z}$

$$\text{X}(\text{z}) = \frac{N(\text{z})}{D(\text{z})}$$

Numerator $\text{N}(\text{z}) = 0$ provides zeros of $\text{X}(\text{z})$

$$\text{X}(\text{z}) = 0$$

Denominator $\text{D}(\text{z}) = 0$ provides poles of $\text{X}(\text{z})$

$$\text{X}(\text{z}) = \infty$$

Note

1. Poles play an important role in deciding the ROC, zeros do not. The ROC can not contain any poles.

2. Different time domain signals can have same $\text{z}$-transform expression $\text{X}(\text{z})$ but with different ROC.

Example:

$1.~~\text{x}[n] = a^n u[n]$

$~~~~~\text{X}[\text{z}] = \sum_{n=-\infty}^ {\infty} x[n]~ \text{z}^{-n}$

$~~~~~\text{X}[\text{z}] = \sum_{n=-\infty}^ {\infty} a^n u[n] ~ \text{z}^{-n}$

$~~~~~\text{X}(\text{z})~ = ~\frac{\text{z}}{\text{z}-a} ~;$

$~~~~~\text{x}[n]$ is a causal signal i.e. $\text{x}[n] = 0$ for $n<0$

$~~~~~\text{ROC:} ~|\text{z}| > |a|$

$2.~~\text{x}[n] = -a^n u[-n-1]$

$~~~~~\text{X}[\text{z}] = \sum_{n=-\infty}^ {\infty} x[n]~ \text{z}^{-n}$

$~~~~~\text{X}[\text{z}] = \sum_{n=-\infty}^ {\infty} -a^n u[-n-1] ~ \text{z}^{-n}$

$~~~~~\text{X}(\text{z})~ = ~\frac{\text{z}}{\text{z}-a} ~;$

$~~~~~\text{x}[n]$ is a anti-causal signal i.e. $\text{x}[n] = 0$ for $n>0$

$~~~~~\text{ROC:} ~|\text{z}| < |a|$