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The Z-Transform

March 16, 2019 by 3200 Creative

The z-transform is an important signal-processing tool for analyzing the interaction between signals and systems. A significant advantage of the z-transform over the discrete-time Fourier transform is that the z-transform exists for many signals that do not have a discrete-time Fourier transform. Thus, it is a more general analysis tool. In this series of 13 lessons you will learn how to work with the z-transform and use it to characterize signal processing systems. You will learn how the poles and zeros of a system tell us whether the system can be both stable and causal, and whether it has a stable and causal inverse system. You will also learn how the pole and zero locations of a system give us insight into the nature of its frequency and impulse response. The insights gained with the z-transform are particularly useful for designing frequency-selective filters.

Course Content

Lessons Status
1

Introduction to the z-Transform

2

The Region of Convergence for the z-Transform

3

Poles and Zeros of the z-Transform

4

Properties of the Region of Convergence

5

Inversion of the z-Transform via Power Series Expansion

6

Inversion of the z-Transform: Partial Fraction Expansion

7

Properties of the z-Transform

8

z-Transform Analysis of LTI Systems

9

Stability and Causality of LTI Systems Described by Difference Equations

10

Inverse Systems for LTI Systems Described by Difference Equations

11

Minimum-Phase and All-Pass Systems

12

Frequency Response Magnitude and Poles and Zeros

13

Impulse Response and Poles and Zeros

Filed Under: Uncategorized

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Course Lessons

  • Introduction to the [latex]z[/latex]-Transform

  • The Region of Convergence for the [latex]z[/latex]-Transform

  • Poles and Zeros of the [latex]z[/latex]-Transform

  • Properties of the Region of Convergence

  • Inversion of the [latex]z[/latex]-Transform via Power Series Expansion

  • Inversion of the [latex]z[/latex]-Transform: Partial Fraction Expansion

  • Properties of the [latex]z[/latex]-Transform

  • [latex]z[/latex]-Transform Analysis of LTI Systems

  • Stability and Causality of LTI Systems Described by Difference Equations

  • Inverse Systems for LTI Systems Described by Difference Equations

  • Minimum-Phase and All-Pass Systems

  • Frequency Response Magnitude and Poles and Zeros

  • Impulse Response and Poles and Zeros

Courses

  • Foundations

  • Time Domain LTI Systems

  • Fourier Series and Transforms

  • Sampling and Reconstruction

  • The DFT and Applications

  • The Z-Transform

  • Intro to Filter Design

  • IIR Filter Design

  • FIR Filter Design

  • Random Signal Characterization

  • Basis Representations of Signals

  • Estimation of Power Spectra and Coherence

  • Introduction to Signal Estimation and Detection Theory

  • MMSE Filtering and Least-Squares Problems

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