Z-Transform Basic Level

Properties and Inverse Transform (Undergraduate Level)

Overview

Three Patterns of the Region of Convergence (ROC) Conceptual diagram illustrating the region of convergence and waveform examples for causal, anti-causal, and two-sided signals on the z-plane Causal Signal Causal (nonzero for n ≥ 0) x[n] n 0 |z|=1 Re Im Pole ROC |z| > |pole| (exterior region) ✓ ROC includes unit circle Anti-causal Signal Anti-causal (nonzero for n ≤ −1) x[n] n 0 |z|=1 Re Im Pole ROC |z| < |pole| (interior region) ✓ ROC includes unit circle Two-sided Signal Two-sided (nonzero on both sides) x[n] n 0 |z|=1 Re Im Pole₁ Pole₂ ROC |pole₁| < |z| < |pole₂| (annular) ✓ ROC includes unit circle Stability Condition BIBO stable ⟺ ROC (region where Σx[n] z−n converges) includes the unit circle |z|=1 Summary: Causal → ROC is exterior Anti-causal → ROC is interior Two-sided → ROC is annular = unit circle (|z| = 1) ROC = Region of Convergence × = pole (systems often have multiple poles)
Figure 1: Three patterns of the Region of Convergence (ROC) — ROC and waveform examples for causal, anti-causal, and two-sided signals. The note "ROC includes unit circle" means that an LTI system whose impulse response $h[n]$ is that signal is BIBO stable; stability is a property of the system, not of the signal itself.

In the introduction, we learned the definition of the unilateral Z-transform, basic transform formulas, and the concepts of the transfer function and poles. At the elementary level, we deepen our understanding of the theoretical foundations of the Z-transform. We study the meaning of the complex variable $z$, the concept of the Region of Convergence (ROC), the bilateral Z-transform and its relationship to causality and stability, and techniques for computing the inverse transform.

Learning Objectives

  • Understand the polar form representation of the complex variable $z$ and its relationship to the DTFT
  • Apply the linearity and time-shift properties of the Z-transform
  • Understand the Region of Convergence (ROC) and the bilateral Z-transform
  • Compute inverse Z-transforms using partial fraction decomposition
  • Understand the relationship between causality, stability, and the ROC
  • Apply the initial value theorem and final value theorem

Table of Contents

  1. Chapter 1 Properties of the Z-Transform

    Meaning of the complex variable $z$, relationship with the DTFT, z-domain differentiation, scaling, time reversal, conjugation, convolution

  2. Chapter 2 Region of Convergence (ROC)

    Convergence of series, shape of the ROC, bilateral Z-transform, causal vs. anti-causal signals

  3. Chapter 3 Convolution Theorem

    Discrete convolution theorem, LTI systems, transfer functions, series and parallel connections

  4. Chapter 4 Inverse Z-Transform

    Partial fraction decomposition, power series expansion

  5. Chapter 5 Initial Value Theorem and Final Value Theorem

    Finding initial and steady-state values without computing the inverse transform

Prerequisites

  • Content from Z-Transform Introduction (definition of the unilateral Z-transform, basic transform formulas, transfer function and poles)
  • Complex number arithmetic (polar form $z = re^{j\omega}$)
  • Partial fraction decomposition

Frequently Asked Questions

Q1. What is the Region of Convergence (ROC) of the Z-transform?

The Region of Convergence (ROC) is the set of values of $z$ for which the Z-transform $X(z) = \sum x[n] z^{-n}$ converges. For causal signals the ROC takes the form $|z| > R$ (the exterior of a circle); for anti-causal signals, $|z| < R$ (the interior); and for two-sided signals, $R_1 < |z| < R_2$ (an annular region). Even if two signals share the same $X(z)$, different ROCs correspond to different signals, which is why specifying the ROC is essential for the bilateral Z-transform.

Q2. How is the inverse Z-transform computed?

The most commonly used method in practice is partial fraction decomposition. One decomposes $X(z)/z$ into partial fractions and matches each term against known Z-transform pairs (for example $z/(z-a) \leftrightarrow a^n u[n]$) to recover the time-domain signal. Power series expansion (long division) is another available method.

Q3. What is the stability condition in the Z-transform?

For a discrete-time system to be BIBO stable (bounded-input bounded-output stable), the ROC of $H(z)$, the Z-transform of its impulse response, must include the unit circle $|z| = 1$. For causal systems this is equivalent to requiring that all poles lie inside the unit circle ($|z| < 1$).