Lecture Signal processing: The z – Transform - Nguyễn Công Phương

Lecture Signal processing: The z – Transform include all of the following: The z – transform, the inverse z – transform, properties of the z – transform, system function of LTI systems, LTI systems characterized by linear constant – coefficient difference equations, connections between pole – zero locations and time – domain behavior, the one – sided z – transform.

Nguyễn Công Phương

SIGNAL PROCESSING

The z – Transform

I Introduction

II Discrete – Time Signals and Systems

III The z – Transform

IV Fourier Representation of Signals

V Transform Analysis of LTI Systems

VI Sampling of Continuous – Time Signals

VII.The Discrete Fourier Transform

VIII.Structures for Discrete – Time Systems

IX Design of FIR Filters

X Design of IIR Filters

XI Random Signal Processing

The z – Transform

1 The z – Transform

2 The Inverse z – Transform

3 Properties of the z – Transform

4 System Function of LTI Systems

5 LTI Systems Characterized by Linear

Constant – Coefficient Difference Equations

6 Connections between Pole – Zero Locations

and Time – Domain Behavior

7 The One – Sided z – Transform

The z – Transform (2)

the set of values of z for which

1 1

( )

( )

M z

1

az− < → > z a ROC: |z| > a

Re

ROC

a

The z – Transform (9)

Ex 6

0 0

, [ ]

,

n n

1 2 20

1

1 2

(cos ) ( )

The z – Transform (13)

• ROC

entire z – plane, sometimes except for z = 0 and

z = ∞

• The z – transform

– X(z) is not defined outside the ROC

The z – Transform (14)

The z – Transform

1 The z – Transform

2 The Inverse z – Transform

3 Properties of the z – Transform

4 System Function of LTI Systems

5 LTI Systems Characterized by Linear

Constant – Coefficient Difference Equations

6 Connections between Pole – Zero Locations

and Time – Domain Behavior

7 The One – Sided z – Transform

The Inverse z – Transform (1)

1

1 2

N N

The Inverse z – Transform (2)

The Inverse z – Transform (3)

Ex 1

1

1 1

n

u n z

u n z

n

z

u n z

The Inverse z – Transform (4)

The z – Transform

1 The z – Transform

2 The Inverse z – Transform

3 Properties of the z – Transform

4 System Function of LTI Systems

5 LTI Systems Characterized by Linear

Constant – Coefficient Difference Equations

6 Connections between Pole – Zero Locations

and Time – Domain Behavior

7 The One – Sided z – Transform

Properties of the z – Transform

The z – Transform

1 The z – Transform

2 The Inverse z – Transform

3 Properties of the z – Transform

4 System Function of LTI Systems

5 LTI Systems Characterized by Linear

Constant – Coefficient Difference Equations

6 Connections between Pole – Zero Locations

and Time – Domain Behavior

System Function of LTI Systems

System Function of LTI Systems

1 1

1

1 1

System Function of LTI Systems

(3)

exterior of a circle, extending to infinity, is a

necessary condition for a discrete – time LTI

system to be causal, but not a sufficient one

the system function H(z) includes the unit circle

|z| = 1

and stable if and only if all poles of H(z) are

inside the unit circle and its ROC is on the

exterior fo a circle, extending to infinity

System Function of LTI Systems

The z – Transform

1 The z – Transform

2 The Inverse z – Transform

3 Properties of the z – Transform

4 System Function of LTI Systems

5 LTI Systems Characterized by Linear

Constant – Coefficient Difference Equations

6 Connections between Pole – Zero Locations and

Time – Domain Behavior

7 The One – Sided z – Transform

LTI Systems Characterized by

LTI Systems Characterized by

M

k k k N

k k k

( ) ( )

LTI Systems Characterized by

Consider a system function

Find its corresponding difference equation?

1

2

1 2

( ) [ ] ( ) [ ]

LTI Systems Characterized by

LCCDE (4)1

k k k

A causal LTI with a rational system function is stable if and only if all poles

of H(z) are inside the unit circle in the z – plane The zeros can be anywhere.

LTI Systems Characterized by

The z – Transform

1 The z – Transform

2 The Inverse z – Transform

3 Properties of the z – Transform

4 System Function of LTI Systems

5 LTI Systems Characterized by Linear

Constant – Coefficient Difference Equations

6 Connections between Pole – Zero

Locations and Time – Domain Behavior

7 The One – Sided z – Transform

Connections between Pole – Zero Locations and Time – Domain Behavior (1)

th th

1

1

first-order system order FIR

1

order system

1 1

k

N

M N k

Connections between Pole – Zero Locations and Time – Domain Behavior (2)

1

…

…

Growing exponential 0

Connections between Pole – Zero Locations and Time – Domain Behavior (3)

Connections between Pole – Zero Locations and Time – Domain Behavior (4)

The z – Transform

1 The z – Transform

2 The Inverse z – Transform

3 Properties of the z – Transform

4 System Function of LTI Systems

5 LTI Systems Characterized by Linear

Constant – Coefficient Difference Equations

6 Connections between Pole – Zero Locations

and Time – Domain Behavior

7 The One – Sided z – Transform

The One – Sided z – Transform