Z-Transform

Table of Contents

Introduction to the Z-Transform

The Z-Transform is a fundamental tool in digital signal processing that extends the concept of the Discrete-Time Fourier Transform (DTFT) to handle a broader class of signals. Where the DTFT exists only on the unit circle in the complex plane, the Z-Transform operates over a region of the complex $z$-plane.

Definition

For a discrete-time signal $x[n]$, the Z-Transform is defined as:

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

where $z$ is a complex variable. This can be viewed as the DTFT of the signal $x[n]r^{-n}$ evaluated at $z = re^{j\omega}$:

$X(z) = X(re^{j\omega}) = \text{DTFT}\{x[n]r^{-n}\}$

Geometric Interpretation

The Z-Transform extends the frequency domain analysis by:

• Unit Circle: The DTFT corresponds to $|z| = 1$ (unit circle)
• Complex Plane: The Z-Transform exists over regions in the complex $z$-plane
• Frequency Mapping: We essentially wrap the linear frequency axis around the unit circle

Region of Convergence (ROC)

The Region of Convergence is the set of all complex values $z$ for which $X(z)$ converges:

$\text{ROC} = \{z \in \mathbb{C} : \sum_{n=-\infty}^{\infty} |x[n]||z|^{-n} < \infty\}$

Properties of ROC

1. Ring or Disk: ROC is always a ring or disk in the $z$-plane
2. No Poles: ROC cannot contain any poles of $X(z)$
3. DTFT Existence: DTFT exists if and only if ROC includes the unit circle

Examples of ROC

Right-Sided Exponential Signal

For $x[n] = a^n u[n]$ (where $u[n]$ is the unit step):

$X(z) = \sum_{n=0}^{\infty} a^n z^{-n} = \sum_{n=0}^{\infty} \left(\frac{a}{z}\right)^n = \frac{1}{1 - az^{-1}} = \frac{z}{z-a}$

ROC: $|z| > |a|$ (exterior of circle with radius $|a|$)

Left-Sided Exponential Signal

For $x[n] = -a^n u[-n-1]$:

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

ROC: $|z| < |a|$ (interior of circle with radius $|a|$)

Two-Sided Signal

For a combination of right-sided and left-sided exponentials: $x[n] = b^n u[n] + c^n u[-n-1]$

If $|c| < |b|$, then ROC: $|c| < |z| < |b|$ (annular region)

Rational Z-Transforms

Most practical Z-transforms are rational functions:

$X(z) = \frac{P(z)}{Q(z)} = \frac{\sum_{k=0}^M b_k z^{-k}}{\sum_{k=0}^N a_k z^{-k}}$

where $P(z)$ and $Q(z)$ are polynomials in $z$.

Poles and Zeros

• Zeros: Values of $z$ where $X(z) = 0$ (roots of numerator)
• Poles: Values of $z$ where $X(z) = \infty$ (roots of denominator)

For a rational transform with $M$ zeros and $N$ poles:

• If $M < N$: $(N-M)$ zeros at $z = 0$
• If $M > N$: $(M-N)$ poles at $z = 0$

Pole-Zero Plot Example

Consider: $X(z) = \frac{2z^2 - (b+a)z}{(z-a)(z-b)}$

• Zeros: $z = 0, z = \frac{b+a}{2}$
• Poles: $z = a, z = b$
• ROC: Depends on signal causality and stability requirements

Inverse Z-Transform

Partial Fraction Expansion

For $M \geq N$ with distinct poles:

$X(z) = \sum_{r=0}^{M-N} B_r z^{-r} + \sum_{k=1}^{N} \frac{A_k}{1 - d_k z^{-1}}$

where:

• $B_r$ coefficients obtained by long division
• $A_k$ are residues: $A_k = (1 - d_k z^{-1})X(z)\Big|_{z=d_k}$
• $d_k$ are the pole locations

Multiple Poles

For a pole of order $s$ at $z = d_i$:

$X(z) = \sum_{m=1}^{s} \frac{C_m}{(1 - d_i z^{-1})^m} + \text{other terms}$

Methods for Inverse Transform

1. Partial Fraction Expansion
2. Power Series Expansion (long division)
3. Residue Method (contour integration)
4. Table Lookup with properties

Z-Transform Properties

Linearity

$ax_1[n] + bx_2[n] \leftrightarrow aX_1(z) + bX_2(z)$ ROC: At least $R_1 \cap R_2$

Time Shifting

$x[n-k] \leftrightarrow z^{-k}X(z)$ ROC: Same as $X(z)$ (except possibly $z = 0$ or $z = \infty$)

Scaling in Z-Domain

$a^n x[n] \leftrightarrow X(z/a)$ ROC: $|a|R$ where $R$ is the ROC of $X(z)$

Time Reversal

$x[-n] \leftrightarrow X(z^{-1})$ ROC: $1/R$ where $R$ is the ROC of $X(z)$

Convolution

$x_1[n] * x_2[n] \leftrightarrow X_1(z)X_2(z)$ ROC: At least $R_1 \cap R_2$

Initial Value Theorem

If $x[n]$ is causal: $x[0] = \lim_{z \to \infty} X(z)$

Final Value Theorem

If $x[n]$ is causal and $(z-1)X(z)$ has no poles on or outside unit circle: $\lim_{n \to \infty} x[n] = \lim_{z \to 1} (z-1)X(z)$

System Analysis with Z-Transform

System Function

For a linear time-invariant system with impulse response $h[n]$: $H(z) = \sum_{n=-\infty}^{\infty} h[n] z^{-n}$

Input-Output Relationship: $Y(z) = H(z)X(z)$

Difference Equations

Linear constant-coefficient difference equation: $\sum_{k=0}^{N} a_k y[n-k] = \sum_{k=0}^{M} b_k x[n-k]$

Taking Z-transform: $\sum_{k=0}^{N} a_k z^{-k} Y(z) = \sum_{k=0}^{M} b_k z^{-k} X(z)$

System Function: $H(z) = \frac{Y(z)}{X(z)} = \frac{\sum_{k=0}^{M} b_k z^{-k}}{\sum_{k=0}^{N} a_k z^{-k}}$

Stability and Causality

• Causal System: ROC is exterior of outermost pole
• Stable System: ROC includes unit circle
• Causal and Stable: All poles inside unit circle

Example: Second-Order System

$y[n] - 1.5y[n-1] + 0.5y[n-2] = x[n]$

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

Poles: $z = 1, z = 0.5$

For stability and causality: ROC must be $|z| > 1$

Connection to Frequency Response

When the ROC includes the unit circle: $H(e^{j\omega}) = H(z)\Big|_{z=e^{j\omega}}$

This gives the frequency response of the system, relating input and output in the frequency domain.