Introduction z -Transform and DTFT Region of Convergence (RoC)

The z -Transform
Dr. Muhammad Sarwar Ehsan sarwar.ehsan@ee.uol.edu.pk
Department of Electrical Engineering, The University of Lahore

April 9, 2013

Dr. M. Sarwar Ehsan

The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

Outline

Introduction

Dr. M. Sarwar Ehsan

The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

Outline

Introduction z -Transform and DTFT

Dr. M. Sarwar Ehsan

The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

Outline

Introduction z -Transform and DTFT

Region of Convergence (RoC)

Dr. M. Sarwar Ehsan

The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

Introduction

z -transform of a sequence x[n] is dened as:

X (z ) =
n=

x[n]z n

Dr. M. Sarwar Ehsan

The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

Introduction

z -transform of a sequence x[n] is dened as:

X (z ) =
n=

x[n]z n

z is complex i.e. z = rej

Dr. M. Sarwar Ehsan

The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

Introduction

z -transform of a sequence x[n] is dened as:

X (z ) =
n=

x[n]z n

z is complex i.e. z = rej Powerful tool for analyzing & designing DT systems

Dr. M. Sarwar Ehsan

The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

Introduction

z -transform of a sequence x[n] is dened as:

X (z ) =
n=

x[n]z n

z is complex i.e. z = rej Powerful tool for analyzing & designing DT systems Generalization of the DTFT

Dr. M. Sarwar Ehsan

The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

z -Transform and DTFT

X (z ) =
n=

x[n]z n

Dr. M. Sarwar Ehsan

The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

z -Transform and DTFT

X (z ) =
n=

x[n]z n x[n](rej )n

=
n=

Dr. M. Sarwar Ehsan

The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

z -Transform and DTFT

X (z ) =
n=

x[n]z n x[n](rej )n

=
n=

X (rej ) =
n=

x[n]rn ejn

Dr. M. Sarwar Ehsan

The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

z -Transform and DTFT

X (z ) =
n=

x[n]z n x[n](rej )n

=
n=

X (rej ) =
n=

x[n]rn ejn forr = 1

Dr. M. Sarwar Ehsan

The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

z -Transform and DTFT

X (z ) =
n=

x[n]z n x[n](rej )n

=
n=

X (rej ) =
n=

x[n]rn ejn forr = 1

X (ej ) =
n=

x[n](ejn )

Dr. M. Sarwar Ehsan

The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

Region of Convergence (RoC)

Critical question: Does summation X (z ) = converge (to ne value)?
n n= x[n]z

Dr. M. Sarwar Ehsan

The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

Region of Convergence (RoC)

Critical question: n Does summation X (z ) = n= x[n]z converge (to ne value)? In general, depends on the value of z

Dr. M. Sarwar Ehsan

The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

Region of Convergence (RoC)

Critical question: n Does summation X (z ) = n= x[n]z converge (to ne value)? In general, depends on the value of z Region of Convergence: Portion of complex z-plane for which a particular X (z ) will converge.

Dr. M. Sarwar Ehsan

The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

ROC Example
Example
Let x[n] = n u[n]

X (z ) =
n=0

n z n =

1 1 z 1

Dr. M. Sarwar Ehsan

The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

ROC Example
Example
Let x[n] = n u[n]

X (z ) =
n=0

n z n =

1 1 z 1

converges for |z 1 | i.e. RoC is |z | > ||

Dr. M. Sarwar Ehsan

The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

ROC Example
Example
Let x[n] = n u[n]

X (z ) =
n=0

n z n =

1 1 z 1

converges for |z 1 | i.e. RoC is |z | > || || < 1 (e.g. 0.8)- nite energy sequence

Dr. M. Sarwar Ehsan

The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

ROC Example
Example
Let x[n] = n u[n]

X (z ) =
n=0

n z n =

1 1 z 1

converges for |z 1 | i.e. RoC is |z | > || || < 1 (e.g. 0.8)- nite energy sequence || > 1 (e.g. 1.2) - Divergent sequence, innite energy, DTFT does not exist but still has ZT when |z | > 1.2 (in RoC)
Dr. M. Sarwar Ehsan The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

About ROCs
ROCs always dened in terms of |z | circular regions on z-plane (inside circles/outside circles/rings)

Dr. M. Sarwar Ehsan

The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

About ROCs
ROCs always dened in terms of |z | circular regions on z-plane (inside circles/outside circles/rings) If ROC includes unit circle (|z | = 1), x[n] has a DTFT

Dr. M. Sarwar Ehsan

The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

ROC Another Example

Example
Let x[n] = n u[n 1] an anti-causal (Left-sided) sequence
1

X (z ) =
n=

()n z n =

1 1 z 1

Dr. M. Sarwar Ehsan

The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

ROC Another Example

Example
Let x[n] = n u[n 1] an anti-causal (Left-sided) sequence
1

X (z ) =
n=

()n z n =

1 1 z 1

Dr. M. Sarwar Ehsan

The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

ROC Another Example

Example
Let x[n] = n u[n 1] an anti-causal (Left-sided) sequence
1

X (z ) =
n=

()n z n = =

1 1 z 1

n z n
m=1

Dr. M. Sarwar Ehsan

The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

ROC Another Example

Example
Let x[n] = n u[n 1] an anti-causal (Left-sided) sequence
1

X (z ) =
n=

()n z n = =

1 1 z 1

n z n
m=1

1 z 1 1 z

1 1 z 1

Same ZT as n u[n], dierent sequence?

Dr. M. Sarwar Ehsan

The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

RoC is Necessary!
To completely dene a ZT, you must specify the ROC

Dr. M. Sarwar Ehsan

The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

RoC is Necessary!
To completely dene a ZT, you must specify the ROC
n u[n]
1 1z 1

RoC is |z | > ||

Dr. M. Sarwar Ehsan

The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

RoC is Necessary!
To completely dene a ZT, you must specify the ROC
n u[n] n u[n]
1 1z 1 1 1z 1

RoC is |z | > || RoC is |z | < ||

Dr. M. Sarwar Ehsan

The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

RoC is Necessary!
To completely dene a ZT, you must specify the ROC
n u[n] n u[n]
1 1z 1 1 1z 1

RoC is |z | > || RoC is |z | < ||

A single X (z ) can describe several sequences with dierent RoCs

Dr. M. Sarwar Ehsan

The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

RoC is Necessary!
To completely dene a ZT, you must specify the ROC
n u[n] n u[n]
1 1z 1 1 1z 1

RoC is |z | > || RoC is |z | < ||

A single X (z ) can describe several sequences with dierent RoCs

Dr. M. Sarwar Ehsan

The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

RoC is Necessary!
To completely dene a ZT, you must specify the ROC
n u[n] n u[n]
1 1z 1 1 1z 1

RoC is |z | > || RoC is |z | < ||

A single X (z ) can describe several sequences with dierent RoCs

Dr. M. Sarwar Ehsan

The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

RoCs and Sidedness

Two sequences have X (z ) = 1 1 z 1

Dr. M. Sarwar Ehsan

The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

ROC intersections
Consider X (z ) = with |1 | < 1 and |2 | > 1 1 1 + 1 1 1 z 1 2 z 1

Dr. M. Sarwar Ehsan

The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

ROC intersections
Consider X (z ) = 1 1 + 1 1 1 z 1 2 z 1

with |1 | < 1 and |2 | > 1 Two possible sequences for 1 term ...

Dr. M. Sarwar Ehsan

The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

ROC intersections
Consider X (z ) = 1 1 + 1 1 1 z 1 2 z 1

with |1 | < 1 and |2 | > 1 Two possible sequences for 1 term ...

Similarly for 2 ...

Dr. M. Sarwar Ehsan

The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

ROC intersections
Consider X (z ) = 1 1 + 1 1 1 z 1 2 z 1

with |1 | < 1 and |2 | > 1 Two possible sequences for 1 term ...

Similarly for 2 ...

4 possible x[n] sequences and RoCs

Dr. M. Sarwar Ehsan The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

ROC intersections
Consider X (z ) = 1 1 + 1 1 1 z 1 2 z 1

with |1 | < 1 and |2 | > 1 Two possible sequences for 1 term ...

Similarly for 2 ...

4 possible x[n] sequences and RoCs

Dr. M. Sarwar Ehsan The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

ROC intersections
Consider X (z ) = 1 1 + 1 1 1 z 1 2 z 1

with |1 | < 1 and |2 | > 1 Two possible sequences for 1 term ...

Similarly for 2 ...

4 possible x[n] sequences and RoCs

Dr. M. Sarwar Ehsan The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

ROC intersections: Case 1

X (z ) = 1 1 + 1 1 1 z 1 2 z 1

Dr. M. Sarwar Ehsan

The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

ROC intersections: Case 1

X (z ) = Right-sided sequences 1 1 + 1 1 1 z 1 2 z 1

Dr. M. Sarwar Ehsan

The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

ROC intersections: Case 1

X (z ) = Right-sided sequences
n x[n] = n 1 u[n] + 2 u[n]

1 1 + 1 1 1 z 1 2 z 1

Dr. M. Sarwar Ehsan

The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

ROC intersections: Case 1

X (z ) = Right-sided sequences
n x[n] = n 1 u[n] + 2 u[n]

1 1 + 1 1 1 z 1 2 z 1

Dr. M. Sarwar Ehsan

The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

ROC intersections: Case 1

X (z ) = Right-sided sequences
n x[n] = n 1 u[n] + 2 u[n]

1 1 + 1 1 1 z 1 2 z 1

Dr. M. Sarwar Ehsan

The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

ROC intersections: Case 2

X (z ) = 1 1 + 1 1 1 z 1 2 z 1

Dr. M. Sarwar Ehsan

The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

ROC intersections: Case 2

X (z ) = Left-sided sequences 1 1 + 1 1 1 z 1 2 z 1

Dr. M. Sarwar Ehsan

The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

ROC intersections: Case 2

X (z ) = Left-sided sequences
n x[n] = n 1 u[n 1] 2 u[n 1]

1 1 + 1 1 1 z 1 2 z 1

Dr. M. Sarwar Ehsan

The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

ROC intersections: Case 2

X (z ) = Left-sided sequences
n x[n] = n 1 u[n 1] 2 u[n 1]

1 1 + 1 1 1 z 1 2 z 1

Dr. M. Sarwar Ehsan

The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

ROC intersections: Case 2

X (z ) = Left-sided sequences
n x[n] = n 1 u[n 1] 2 u[n 1]

1 1 + 1 1 1 z 1 2 z 1

Dr. M. Sarwar Ehsan

The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

ROC intersections: Case 3

X (z ) = 1 1 + 1 1 1 z 1 2 z 1

Dr. M. Sarwar Ehsan

The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

ROC intersections: Case 3

X (z ) = Two-sided sequence 1 1 + 1 1 1 z 1 2 z 1

Dr. M. Sarwar Ehsan

The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

ROC intersections: Case 3

X (z ) = Two-sided sequence
n x[n] = n 1 u[n] 2 u[n 1]

1 1 + 1 1 1 z 1 2 z 1

Dr. M. Sarwar Ehsan

The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

ROC intersections: Case 3

X (z ) = Two-sided sequence
n x[n] = n 1 u[n] 2 u[n 1]

1 1 + 1 1 1 z 1 2 z 1

Dr. M. Sarwar Ehsan

The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

ROC intersections: Case 3

X (z ) = Two-sided sequence
n x[n] = n 1 u[n] 2 u[n 1]

1 1 + 1 1 1 z 1 2 z 1

Dr. M. Sarwar Ehsan

The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

ROC intersections: Case 4

X (z ) = 1 1 + 1 1 1 z 1 2 z 1

Dr. M. Sarwar Ehsan

The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

ROC intersections: Case 4

1 1 + 1 1 1 z 1 2 z 1 Two-sided sequences with no overlap X (z ) =

Dr. M. Sarwar Ehsan

The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

ROC intersections: Case 4

1 1 + 1 1 1 z 1 2 z 1 Two-sided sequences with no overlap X (z ) =
n x[n] = n 1 u[n 1] + 2 u[n]

Dr. M. Sarwar Ehsan

The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

ROC intersections: Case 4

1 1 + 1 1 1 z 1 2 z 1 Two-sided sequences with no overlap X (z ) =
n x[n] = n 1 u[n 1] + 2 u[n]

Dr. M. Sarwar Ehsan

The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

ROC intersections: Case 4

1 1 + 1 1 1 z 1 2 z 1 Two-sided sequences with no overlap X (z ) =
n x[n] = n 1 u[n 1] + 2 u[n]

Dr. M. Sarwar Ehsan

The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

ROC intersections

Two-sided exponential sequences x[n] = n < n < = n u[n] + n u[n 1]

Dr. M. Sarwar Ehsan

The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

ROC intersections

Two-sided exponential sequences x[n] = n < n < = n u[n] + n u[n 1] No overlap in RoCs

Dr. M. Sarwar Ehsan

The z -Transform

Introduction z -Transform and DTFT Region of Convergence (RoC)

ROC intersections

Two-sided exponential sequences x[n] = n < n < = n u[n] + n u[n 1] No overlap in RoCs ZT does not exist

Dr. M. Sarwar Ehsan

The z -Transform