Chapter 4: Z-Transform

Introduction

The Z-transform is the discrete-time counterpart of the Laplace transform, providing a powerful tool for analyzing discrete-time systems.

Definition

Definition: Bilateral Z-Transform

The bilateral Z-transform of $x[n]$ is:

$$X(z)=\mathcal{Z}\{x[n]\}=\sum _{n=-\mathrm{\infty }}^{+\mathrm{\infty }}x[n]z^{-n}$$

where $z$ is a complex variable.

Definition: Unilateral Z-Transform

For causal signals:

$$X(z)=\sum _{n=0}^{+\mathrm{\infty }}x[n]z^{-n}$$

Region of Convergence (ROC)

Definition: ROC

The ROC is the set of values of $z$ for which the Z-transform sum converges. The ROC is always an annular region in the $z$-plane.

Proposition: ROC Properties

1. The ROC does not contain any poles
2. The ROC is a connected annular region
3. For right-sided sequences: ROC extends outward from the outermost pole
4. For left-sided sequences: ROC extends inward from the innermost pole
5. For two-sided sequences: ROC is an annular ring
6. For causal and stable systems: ROC includes the unit circle $|z|=1$

Relationship with DTFT

Theorem: DTFT as Z-Transform on Unit Circle

If the ROC includes the unit circle, then:

$$X(e^{j\omega })=X(z){|}_{z=e^{j\omega }}$$

The DTFT is the Z-transform evaluated on the unit circle.

Common Z-Transform Pairs

Signal $x[n]$	Z-Transform $X(z)$	ROC
$\delta [n]$	$1$	All $z$
$u[n]$	$\frac{1}{1-z^{-1}}=\frac{z}{z-1}$	$|z|>1$
$a^{n}u[n]$	$\frac{1}{1-a{z}^{-1}}=\frac{z}{z-a}$	$|z|>|a|$
$-a^{n}u[-n-1]$	$\frac{1}{1-a{z}^{-1}}$	$|z|<|a|$
$n{a}^{n}u[n]$	$\frac{a{z}^{-1}}{(1-a{z}^{-1}{)}^2}$	$|z|>|a|$
$\cos ({\omega }_{0}n)u[n]$	$\frac{1-z^{-1}\cos {\omega }_0}{1-2z^{-1}\cos {\omega }_0+z^{-2}}$	$|z|>1$
$a^n\cos ({\omega }_{0}n)u[n]$	$\frac{1-a{z}^{-1}\cos {\omega }_0}{1-2a{z}^{-1}\cos {\omega }_0+a^2{z}^{-2}}$	$|z|>|a|$

Properties

Theorem: Z-Transform Properties

Let $x[n]\leftrightarrow X(z)$ with ROC $R_x$.

Linearity:

$$ax[n]+by[n]\leftrightarrow aX(z)+bY(z)$$

Time Shift:

$$x[n-n_0]\leftrightarrow z^{-n_0}X(z)$$

Scaling in z-Domain:

$$a^{n}x[n]\leftrightarrow X(z/a)$$

with ROC $|a|R_x$

Time Reversal:

$$x[-n]\leftrightarrow X(1/z)$$

with ROC $1/R_x$

Differentiation in z-Domain:

$$nx[n]\leftrightarrow -z\frac{dX(z)}{dz}$$

Convolution:

$$x[n]\ast h[n]\leftrightarrow X(z)H(z)$$

Initial Value Theorem (causal):

$$x[0]=\underset{z\to \mathrm{\infty }}{lim}X(z)$$

Final Value Theorem: If $(z-1)X(z)$ has poles inside unit circle:

$$\underset{n\to \mathrm{\infty }}{lim}x[n]=\underset{z\to 1}{lim}(z-1)X(z)$$

Transfer Function

Definition: System Function

The transfer function of a discrete LTI system is the Z-transform of its impulse response:

$$H(z)=\mathcal{Z}\{h[n]\}=\frac{Y(z)}{X(z)}$$

For a system described by:

$$\sum _{k=0}^N{a}_{k}y[n-k]=\sum _{k=0}^M{b}_{k}x[n-k]$$

The transfer function is:

$$H(z)=\frac{\sum _{k=0}^M{b}_k{z}^{-k}}{\sum _{k=0}^N{a}_k{z}^{-k}}$$

Poles and Zeros

Definition: Poles and Zeros

For $H(z)=\frac{N(z)}{D(z)}$:

• Zeros: values of $z$ where $H(z)=0$
• Poles: values of $z$ where $H(z)=\mathrm{\infty }$

Theorem: Stability Criterion

A causal discrete LTI system is BIBO stable if and only if all poles of $H(z)$ are inside the unit circle:

$$|p_i|<1\text{for all poles }{p}_i$$

Inverse Z-Transform

Definition: Partial Fraction Expansion

For a rational $X(z)$:

$$X(z)=\sum _i\frac{A_{i}z}{z-p_i}$$

Then for a causal signal (ROC: $|z|>|p_i|$ for all $i$):

$$x[n]=\sum _i{A}_i{p}_i^{n}u[n]$$

Example: Second-Order System

Consider:

$$H(z)=\frac{1}{1-1.5z^{-1}+0.5z^{-2}}=\frac{z^2}{z^2-1.5z+0.5}$$

Poles at $z=1$ and $z=0.5$. Since there's a pole at $z=1$ (on the unit circle), the system is marginally stable.