Digital Signal Processing, Fall 2006 - Aalborg Universitetkom.aau.dk/~zt/courses/DSP/MM3/Digital...

1

Digital Signal Processing, III, Zheng-Hua Tan, 20061

Digital Signal Processing, Fall 2006

Zheng-Hua Tan

Department of Electronic Systems Aalborg University, Denmark

[email protected]

Lecture 3: The z-transform


Course at a glance

Discrete-time signals and systems

Fourier-domain representation

DFT/FFT

Systemstructures

Filter structures Filter design

Filter

z-transform

MM1

MM2

MM9,MM10MM3

MM6

MM4

MM7 MM8

Sampling andreconstruction MM5

Systemanalysis

System

2


Part I: z-transform

z-transformProperties of the ROC Inverse z-transformProperties of z-transform


Limitation of Fourier transform

Fourier transform

Condition for the convergence of the infinite sum

If x[n] is absolutely summable, its Fourier transform exists (sufficient condition). Example

)(21][

][)(

∫

∑

−

−∞

−∞=

=

=

π

πωω

ωω

ωπ

deeXnx

enxeX

njj

nj

n

j

∞<≤≤= ∑∑∑∞

−∞=

−∞

−∞=

−∞

−∞= n

nj

n

nj

n

j nxenxenxeX |][| |||][| |][| |)(| ωωω

- :1||

)2(1

1 )( :1

11 )( :1|| ][][

>

++−

==

−=<=

∑∞

−∞=−

−

a

ke

eXa

aeeXanuanx

kj

j

jjn

πωπδωω

ωω

3


z-transform

Fourier transform

z-transform

The complex variable z in polar form

)()( 1 jωeXzX,r|z| ===

jωrez =

nj

n

j enxeX ][)( ωω −∞

−∞=∑=

)(][ ][)( zXnxznxzXZ

n

n↔= −

∞

−∞=∑

njnnjj ernxrenxreXzX ωωω −−∞

∞−

−∞

∞−∑∑ === )][()(][)()(


z-plane

z-transform is a function of a complex variable using the complex z-plane

Z-transform on unit circle<-> Fourier transform

Linear frequency axis in Fourier transform

Unit circle in z-transform(periodicity in freq. of Fourier transform)

4


Region of convergence – ROC

Fourier transform does not converge for all sequences

z-transform does not converge for all sequences or for all values of z.

ROC – for any given seq., the set of values of z for which the z-transform converges

nj

n

j enxeX ωω −∞

−∞=∑= ][)(

∞<∑∞

−∞=

− || |][| n

nznx∑∞

−∞=

− ∞<n

nrnx |][|

nj

n

nn

n

j ernxznxreX ωω −∞

−∞=

−−∞

−∞=∑∑ === )][( ][)(X(z)

ROC is ring!


ROC

Outer boundary is a circle (may extend to infinity)Inner boundary is a circle (may extend to include the origin)If ROC includes unit circle, Fourier transform converges

5


Zeros and poles

The most important and useful z-transforms –rational function:

Zeros: values of z for which X(z)=0.Poles : values of z for which X(z) is infinite. Close relation between poles and ROC

zzQzPzQzPzX

in spolynomial are )( and )()()()( =


Right-sided exponential sequence

ROC

z-transform

∑

∑∞

=

−

∞

−∞=

−

=

=

=

0

1)(

][)(

][][

n

n

n

nn

n

az

znuazX

nuanx

∞<∑∞

=

−

0

1 ||n

naz

1||,1

1][ 1 >−

↔ − zz

nuZ

||||,1

1)()( 10

1 azazz

azazzX

n

n >−

=−

== −

∞

=

−∑

6


Left-sided exponential sequence

ROC

z-transform

∑∑

∑∞

=

−−

−∞=

−

∞

−∞=

−

−=−=

−−−=

−−−=

0

11

)(1

]1[)(

]1[][

n

n

n

nn

n

nn

n

zaza

znuazX

nuanx

∞<∑∞

=

−

0

1 ||n

nza

||||,1

1)( 1 azazz

azzX <

−=

−= −


Sum of two exponential sequence

ROC)

31)(

21(

)121(2

311

1

211

1

)31()

21()(

][)31(][)

21(][

11

0

1

0

1

+−

−=

++

−=

−+=

−+=

−−

∞

=

−∞

=

− ∑∑

zz

zz

zz

zzzX

nununx

n

n

n

n

nn

21||

31|| and

21||

1|)31(| and 1|

21| 11

>

>>

<−< −−

z

zz

zz

7


Sum of two exponential sequence

Another way to calculate:

21||

311

1

211

1][)31(][)

21(

31|| ,

311

1][)31(

21|| ,

211

1][)21(

][)31(][)

21(][

11

1

1

>+

+−

↔−+

>+

↔−

>−

↔

−+=

−−

−

−

zzz

nunu

zz

nu

zz

nu

nununx

Znn

Zn

Zn

nn


Two-sided exponential sequence

21|| ,

31|| ,

211

1

311

1)(

21|| ,

211

1]1[)21(

31|| ,

311

1][)31(

11

1

1

<>−

++

=

<−

↔−−−

>+

↔−

−−

−

−

zzzz

zX

zz

nu

zz

nu

Zn

Zn

]1[)21(][)

31(][ −−−−= nununx nn

)31)(

21(

)121(2

+−

−=

zz

zz

8


Finite-length sequence

⎩⎨⎧ −≤≤

= otherwise. ,0

10 ,][

Nnanx

n

azaz

zazaz

zazazX

NN

N

N

nN

n

nN

n

n

−−

=−

−=

==

−−

−

−−

=

−−

=∑∑

11

1

11

0

1

0

11

)(1

)()(

0 and ||

|| ROC1

0

1

≠∞<

∞<∑−

=

−

za

azN

n

n

1,...,1 ,at pole

1,...,1,0 ,

)/2(

)/2(

−==

=−==

Nkaezaz

Nkaez

Nkjk

Nkjk

π

π


Some common z-transform pairs

9


Part II: Properties of the ROC



Properties of the ROC

10


Properties of the ROC

The algebraic expressionor pole-zero pattern doesnot completely specifythe z-transform of a sequence the ROC must be specified!Stability, causality and the ROC


Part III: Inverse z-transform


11


Inverse z-transform

Needed for system analysis: 1) z-transform, 2) manipulation, 3) inverse z-transform.Approaches:

Inspection methodPartial fraction expansionPower series expansion


Inspection method

By inspection, e.g.

Make use of

21|| ),

211

1()(1

>−

=−

zz

zX

?21|| if <z

][)21(][ nunx n=∴

|||| ),1

1()( 1 azaz

nuaZ

n >−

↔ −

]1[)21(][ course, of −−−= nunx n

12


Partial fraction expansion

For rational function, get the format of a sum of simpler terms, and then use the inspection method.


Second-order z-transform

)211)(

411(

1)(11 −− −−

=zz

zX

][)41(][)

21(2][ nununx nn −=

)211(

2

)411(

1)(

2|)()211(

1|)()411(

)211()

411(

)(

11

2/11

2

4/11

1

1

2

1

1

−−

=−

=−

−−

−+

−

−=

=−=

−=−=

−+

−=

zzzX

zXzA

zXzA

z

A

z

AzX

z

z

∑

∑

=

−

=

−

= N

k

kk

M

k

kk

za

zbzX

0

0)(

∑=

−−=

N

k k

k

zdAzX

111

)(

∑

∑

=

−

=

−

−

−= N

kk

M

kk

zd

zc

abzX

1

1

1

1

0

0

)1(

)1()(

kdzkk zdzXA =−−= |)1)(( 1

21|| ,

81

431

1)(21

>+−

=−−

zzz

zX

13


What about M>=N?

)1)(211(

)1()(11

21

−−

−

−−

+=

zz

zzX

][8][)21(9][2][ nununnx n +−= δ

)1(8

)211(

92)(

8|)()1(

9|)()211(

division. longby Found 2

)1()211(

)(

11

11

2

2/11

1

0

12

1

10

−−

=−

=−

−−

−+

−−=

=−=

−=−=

=

−+

−+=

zzzX

zXzA

zXzA

B

zA

z

ABzX

z

z

1|| ,

21

231

21)(21

21

>+−

++=

−−

−−

zzz

zzzX

15 23

2121

23

21

1

12

1212

−−

+−

+++−

−

−−

−−−−

zzz

zzzz


Power series expansion

By long division

][][ nuanx n=

|||| ,1

1)( 1 azaz

zX >−

= −

...11

1 2211 +++=

−−−

− zaazaz

...

1

...1 11

22

221

1

1

221

1

−

−−

−

−

−−

−

−

−

+++−

zazaaz

azaz

zaazaz

14


Finite-length sequence

12

1112

211

21)(

)1)(1)(211()(

−

−−−

+−−=

−+−=

zzzzX

zzzzzX

]1[21][]1[

21]2[][ −+−+−+= nnnnnx δδδδ


Part IV: Properties of z-transform


15


Linearity

Linearity

21 contains ROC ),()(][][ 2121 xx

ZRRzbXzaXnbxnax ∩+↔+

zzznnunu

zzznu

zz

nu

Z

Z

Z

All ,111][]1[][

1|| ,1

]1[

1|| ,1

1][

1

1

1

1

1

=−−

↔=−−

>−

↔−

>−

↔

−

−

−

−

−

δ

2

1

ROC ),(][

ROC ),(][

22

11

x

Z

x

Z

RzXnx

RzXnx

=↔

=↔ ][)( n

nznxzX −

∞

−∞=∑=

at least


Time shifting

Example

)

411

1(]1[)41(

411

1][)41(

)

411

1(

411

)(

41|| ,

41

1)(

1

11

1

1

1

1

1

−

−−

−

−

−

−

−

−↔−

−↔

−=

−=

>−

=

zznu

znu

zz

z

zzX

zz

zX

Zn

Zn

)or 0for (except ROC ),(][

1

00

∞=↔− −

x

nZ

RzXznnx ][)( n

nznxzX −

∞

−∞=∑=

16


Multiplication by exponential sequence

Examples

azazaz

nua

zz

nu

Zn

Z

>−

=−

↔

>−

↔

−−

−

|| ,1

1)/(1

1][

1|| ,1

1][

11

1

)(][

ROC ),/(][

)(

000

00 ωωω −↔

=↔

jF

nj

x

Zn

eXnxe

|R|zzzXnxz ][)( n

nznxzX −

∞

−∞=∑=

1|| ,cos21

)cos1(1

21

121

)(

][)(21][)(

21][)cos(][

210

10

11

0

00

00

>+−

−=

−+

−=

+==

−−

−

−−−

−

zzz

zzeze

zX

nuenuenunnx

jj

njnj

ωω

ω

ωω

ωω


Differentiation of X(z)

Example

]1[)(][

]1[)(][

|||| ,1

)(][

1)(

|||| ),1log()(

1

1

1

1

1

2

1

−−

=

−−=

>+

=−↔

+−

=

>+=

−

−

−

−

−

−

−

nunaanx

nuaannx

azazaz

dzzdXznnx

azaz

dzzdX

azazzX

n

n

Z

x

ZR

dzzdXznnx =−↔ ROC ,)(][ ][)( n

nznxzX −

∞

−∞=∑=

17


Conjugation of a complex sequence

x

ZRzXnx =↔ ROC ),(][ *** ][)( n

nznxzX −

∞

−∞=∑=


Time reversal

Example

|||| ,1

1)(

][][

1−

−

<−

=

−=

azaz

zX

nuanx n

x

ZRzXnx /1ROC ),/1(][ =↔−

][)( n

nznxzX −

∞

−∞=∑=

|||| ,1

1)(

][][

1 azaz

zX

nuanx n

>−

=

=

−

18


Convolution of sequences

Example

?][][][

][)21(][

nynunh

nunx n

=

=

21 contains ROC

),()(][*][ 2121

xx

Z

RRzXzXnxnx

∩↔ ][)( n

nznxzX −

∞

−∞=∑=

))

211(21

)1(1(

211

1

21|| ,

)1)(211(

1)(

1|| ,1

1)(

21|| ,

211

1)(

11

11

1

1

−−

−−

−

−

−−

−−=

>−−

=

>−

=

>−

=

zz

zzz

zY

zz

zH

zz

zX

])[)21(][(

211

1][ 1 nununy n+−−

=


Initial-value theorem

)(lim]0[ zXxz ∞→

= ][)( n

nznxzX −

∞

−∞=∑=

x[0]

lim][

][lim)(lim

=

=

=

−

∞→

∞

−∞=

−∞

−∞=∞→∞→

∑

∑n

zn

n

nzz

znx

znxzX

19


Properties of z-transform


Summary


20


Course at a glance

Discrete-time signals and systems

Fourier-domain representation

DFT/FFT

Systemstructures

Filter structures Filter design

Filter

z-transform

MM1

MM2

MM9,MM10MM3

MM6

MM4

MM7 MM8

Sampling andreconstruction MM5

Systemanalysis

System

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