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Lecture 3:

The -Transform

Reza Mohammadkhani, Digital Signal Processing, 2014University of Kurdistan eng.uok.ac.ir/mohammadkhani

The -Transform2

� Counterpart of the Laplace transform (for CT signals)

� Generalization of the Fourier transform

� The z-transform definition:

� � = � � � ���

��� The DTFT definition:

� �� = � � � ��� �

��

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� Two-sided or

Bilateral z-transform:

� � = � � � ���

��

� One-sided or

Unilateral z-transform:

� � = � � � ���

��

� The �-Transform is a function of the complex �

Convergence of the -Transform4

� Convergence of the DTFT

� � �

��< ∞

� Convergence of the �-tansform

� � � ���

��< ∞

� Some � � may not have a DTFT, but can have a �-

transform for some values of � = ��� � Example: �[�] = ���[�] for |�| > 1 does not have a DTFT

but it has z-transform

Region of Convergence (ROC)5

� The set of values of � for which the z-transform

converges

� Each value of � represents a circle of radius �� The region of convergence is made of circles

Example 16

� Right-sided exponential sequences

� �[�] � ������

� is real or complex?

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Pole-zero plot and ROC of � � = ��� � with (a) decaying amplitude (0 < a < 1),

(b) fixed amplitude (unit step sequence), and (c) growing amplitude (a > 1).

Example 28

� Left-sided exponential sequences

It converges if

Example 39

� Two-sided exponential sequences

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Right-sided exp sequence Left-sided exp sequence Two-sided exp sequence

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Properties of ROC12

� The ROC cannot include any poles.

� The ROC is a connected (a single contiguous) region.

� For finite duration sequences the ROC is the entire z-

plane, with the possible exception of � � 0 or � � ∞.

� For infinite duration sequences the ROC can have one of

the following shapes:

Stability, Causality, and the ROC13

� Consider a system with impulse response h[n] and

the z-transform H(z) with the pole-zero plot shown

below. Three possible ROC:

� |�| > 2 or |�| < ½ or ½ < |�| < 2� If system is stable, ROC must

include unit-circle: ½ < |�| < 2� If system is causal must be right-

sided: |�| > 2

The Inverse -Transform14

The Inverse -Transform15

� Formal inverse z-transform is based on a Cauchy

integral

� Less formal ways sufficient most of the time

� Inspection method

� Partial fraction expansion

� Power series expansion

Inspection method16

� Make use of known z-transform pairs such as

� Example: The inverse z-transform of

[ ] az az1

1nua

1

Zn >−

→← −

( ) [ ] [ ]nu2

1nx

2

1z

z2

11

1zX

n

1

=→>−

=−

Partial Fraction Expansion17

� Assume that a given z-transform can be expressed

as the ratio of polynomials in �� :

� Apply partial fractional expansion

( )∑

∑

=

−

=

−

=N

0k

k

k

M

0k

k

k

za

zb

zX

( ) ( )∑∑∑=

−≠=

−

−

=

−

−+

−+=

s

mm

i

mN

ikk k

kNM

r

rr

zd

C

zd

AzBzX

11

,11

0 11

It exists only if M>N

represents all first order poles

represents an order ! pole

Partial Fraction Expansion (2)18

� Coefficients are given as

( ) ( )∑∑∑=

−≠=

−

−

=

−

−+

−+=

s

mm

i

mN

ikk k

kNM

r

rr

zd

C

zd

AzBzX

11

,11

0 11

( ) ( )kdzkk zXzdA

=−−= 11

( ) ( ) ( ) ( )[ ]1

11!

1

−=

−−

−

−

−−−

=idw

sims

ms

msi

m wXwddw

d

dmsC

Example19

( )2

1z :ROC

z2

11z

4

11

1zX

11

>

−

−=

−−

( )

−+

−=

−− 1

2

1

1

z2

11

A

z4

11

AzX

( ) 1

4

1

2

11

1zXz

4

11A

1

4

1z

1

1−=

−

=

−=−

=

−

( ) 2

2

1

4

11

1zXz

2

11A

1

2

1z

1

2=

−

=

−=−

=

−

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� ROC extends to infinity

� Indicates right sided sequence

( )2

1z

z2

11

2

z4

11

1zX

11

>

−+

−

−=−−

Example 221

� Long division to obtain Bo

( ) ( )( )

1z

z1z2

11

z1

z2

1z

2

31

zz21zX

11

21

21

21

>−

−

+=+−

++=−−

−

−−

−−

1z5

2z3z

2

1z2z1z2

3z

2

1

1

12

1212

−

+−

+++−

−

−−

−−−−

( )( )11

1

z1z2

11

z512zX

−−

−

−

−

+−+=

( )1

2

1

1

z1

A

z2

11

A2zX −

− −+

−+=

( ) 9zXz2

11A

2

1z

1

1−=

−==

− ( ) ( ) 8zXz1A1z

1

2=−=

=

−

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� ROC extends to infinity

� Indicates right-sides sequence

( ) 1z z1

8

z2

11

92zX

11

>−

+−

−= −−

[ ] [ ] [ ] [ ]n8u-nu2

19n2nx

n

−δ=

Power Series Expansion23

� The z-transform is power series and can be

expanded as:

� Z-transforms of this form can generally be inversed

easily, especially for finite-length series

� Example

( ) [ ]∑∞

−∞=

−=n

nz nxzX

( ) [ ] [ ] [ ] [ ] [ ] LL ++++−+−+= −− 2112z 2xz 1x 0xz 1xz 2xzX

( ) ( )( )12

1112

z2

11z

2

1z

z1z1z2

11z zX

−

−−−

+−−=

−+

−=

[ ] [ ] [ ] [ ] [ ]1n2

1n1n

2

12nnx −δ+δ−+δ−+δ=

[ ]

=

==−

−=−−=

=

2n0

1n2

1

0n1

1n2

12n1

nx

� Linear

� Time and frequency shifting

The z-Transform Properties24

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References26

� A. V. Oppenheim and R. W. Schafer, Discrete-Time

Signal Processing, 3rd Edition, Prentice Hall, 2009.

� D. Manolakis and V. Ingle, Applied Digital Signal

Processing, Cambridge University Press, 2011.

� Güner Arslan, EE351M Digital Signal Processing,

Lecture notes, Dept. of Electrical and Computer

Engineering, The University of Texas at Austin, 2007.

Available at:www.ece.utexas.edu/~arslan/351m.html