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����PRESENTATION ON �INTRODUCTION TO Z TRANSFORM

  • BRANCH-E & TC ENGG
  • SUBJECT- DIGITAL SIGNAL PROCESSING
  • CHAPTER – 3 – THE Z TRANSFORM & ITS APPLICATION TO THE ANALYSIS OF LTI SYSTEM
  • TOPIC- INTRODUCTION Z TRANSFORM
  • SEM-6TH
  • FACULTY – Er. ARADHANA DAS (Sr. LECTURER E & TC ENGG DEPARTMENT)
  • AY-2021-2022, SUMMER-2022

​

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The z-Transform

Region of Convergence

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Content

  • Introduction
  • z-Transform
  • Zeros and Poles
  • Region of Convergence
  • Important z-Transform Pairs
  • Inverse z-Transform
  • z-Transform Theorems and Properties
  • System Function

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The z-Transform

Introduction

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Why z-Transform?

  • A generalization of Fourier transform
  • Why generalize it?
    • FT does not converge on all sequence
    • Notation good for analysis
    • Bring the power of complex variable theory deal with the discrete-time signals and systems

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The z-Transform

z-Transform

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Definition

  • The z-transform of sequence x(n) is defined by
  • Let z = e−jω.

Fourier Transform

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z-Plane

Re

Im

z = e−jω

ω

Fourier Transform is to evaluate z-transform on a unit circle.

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z-Plane

Re

Im

X(z)

Re

Im

z = e−jω

ω

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Periodic Property of FT

Re

Im

X(z)

π

−π

ω

X(ejω)

Can you say why Fourier Transform is a periodic function with period 2π?

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The z-Transform

Zeros and Poles

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Definition

  • Give a sequence, the set of values of z for which the z-transform converges, i.e., |X(z)|<∞, is called the region of convergence.

ROC is centered on origin and consists of a set of rings.

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Example: Region of Convergence

Re

Im

ROC is an annual ring centered on the origin.

r

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Stable Systems

  • A stable system requires that its Fourier transform is uniformly convergent.

Re

Im

1

  • Fact: Fourier transform is to evaluate z-transform on a unit circle.
  • A stable system requires the ROC of z-transform to include the unit circle.

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Example: A right sided Sequence

1

2

3

4

5

6

7

8

9

10

-1

-2

-3

-4

-5

-6

-7

-8

n

x(n)

. . .

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Example: A right sided Sequence

For convergence of X(z), we require that

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Example: A right sided Sequence ROC for x(n)=anu(n)

a

−a

Re

Im

1

a

−a

Re

Im

1

Which one is stable?

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Example: A left sided Sequence

1

2

3

4

5

6

7

8

9

10

-1

-2

-3

-4

-5

-6

-7

-8

n

x(n)

. . .

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Example: A left sided Sequence

For convergence of X(z), we require that

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Example: A left sided Sequence ROC for x(n)=−anu(− n−1)

a

−a

Re

Im

1

a

−a

Re

Im

1

Which one is stable?

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THANK YOU