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) ✓ Stable (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) ✓ Stable (includes unit circle) Two-sided Signal Two-sided (nonzero for all n) x[n] n 0 |z|=1 Re Im Pole₁ Pole₂ ROC |pole₁| < |z| < |pole₂| (annular) ✓ Stable (includes unit circle) Stability Condition BIBO stable ⟺ ROC (region where $\Sigma x[n] z^{-n}$ 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)

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