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4. The z Transform

AFTER COMPLETION OF THIS UNIT

YOU SHOULD BE ABLE TO:

1. Know the mathemat!a" #o$m%"a& o# the m'o$tant (&!$ete tme &)na"& an( how to &*et!h them.

+. Know the m'o$tant &e,%en!e o'e$aton& an( the$ mathemat!a" notaton.

-. Know the mathemat!a" (e#nton o# the z t$ano$m.. E/a"%ate the z t$ano$m o# e0'onenta" &e,%en!e&.. Know the m'o$tant '$o'e$te& o# the zt$ano$m.2. E/a"%ate the z t$ano$m o# (&!$ete tme &e,%en!e& an( e&ta3"&h $e)on& o# !on/e$)en!e 4ROC&5.

6. Know the mathemat!a" $e"aton&h' 3etween the n/e$&e z t$ano$m an( the !om'"e0 n/e$&on nte)$a".

7. Ca"!%"ate !om'"e0 n/e$&on nte)$a"& %&n) the $e&(%e theo$em.8. Ca"!%"ate n/e$&e z t$ano$m& wth the !om'"e0 n/e$&on nte)$a".19. Ca"!%"ate n/e$&e z t$ano$m& %&n) 'a$ta" #$a!ton&.11. Ca"!%"ate n/e$&e z t$ano$m& %&n) "on) (/&on.1+. So"/e (##e$en!e e,%aton& wth the a( o# the z t$ano$m.1-. So"/e the '$o3"em& n E0e$!&e& .1 an( .+.

191

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4.0. Introduction

A #%n!ton &%!h a& ( )tfy= that & (e#ne( #o$ a"" $ea" /a"%e& o# t !an 3e !on&(e$e( a& a!ontn%o%&tme &)na". I# th& &)na" & &am'"e( at $e)%"a$ tme nte$/a"& 9ntt= whe$en & an nte)e$ 4'o&t/e o$ ne)at/e5; then the !ontn%o%&tme &)na" ( )tf 3e!ome& a

(&!$etetme &)na" ( )9ntf . The &am'"n) o# ( )tf to

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Figure 4-1 E0am'"e& o# !ontn%o%&tme &)na"&

Figure 4-2 B"o!* (a)$am o# a !ontn%o%&tme &

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Definition:A (&!$etetme &e$o &%!h that ( ) ( )nxPnx =+ #o$ a"" n . The &ma""e&t nte)e$ P #o$ wh!h the!on(ton & &ate( & !a""e( the 'e$o( o# the (&!$etetme &)na" ( )nx .

E0am'"e& o# (&!$etetme &)na"& a$e &hown n F)%$e . Note that ( )nw & 'e$o(!

wth 'e$o( . an( that ( )nw an( ( )nz ta*e on on"< a #nte n%m3e$ o# (##e$ent /a"%e&;wh"e ( )nx an( ( )ny ta*e on a !o%nta3"e n#nte n%m3e$ o# /a"%e&.

Figure 4-3A )ene$a" (&!$etetme &

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Sn%&o(a" &e,%en!e( ) nAnx

9&n = #o$ a"" n

The %nt &am'"e &e,%en!e & &ometme& $e#e$$e( to a& a (&!$etetme m'%"&e an( ha& atn) '$o'e$t< &m"a$ to that o# the !ontn%o%&tme m'%"&e #%n!ton. It (##e$&;

howe/e$; 3< 3en) we"" (e#ne( #o$ a"" /a"%e& o# t& n(e'en(ent /a$a3"e. The sifting'$o'e$t< #o$ the %nt &am'"e &e,%en!e & )/en 3e$o& 3< an a& at 9=z .

Figure 4-+Ann%"a$ $e)on o# !on/e$)en!e n the z '"ane

)(ample 4.2: Fn( the z t$ano$m n!"%(n) $e)on o# !on/e$)en!e o#( ) ( )1= nubnx n

*olution

The zt$ano$m o# ( )nx & #o%n( #$om the 3a&! (e#nton

( ) ( )[ ] ( ) ( )

=

=

===n n

nnnnzbznubnubZzX

1

C11

I# we "et mn = ; then ( )zX 3e!ome&

Be$o '"ot #o$ th& ne)at/e tme &e,%en!e a$e &hown

n F)%$e 8. Note that the 'o&t/e tme e0'onenta" &e,%en!e ha& a t$ano$m wth

$e)on o# !on/e$)en!e o%t&(e a !$!"e n the z '"ane; wh"e the ne)at/e tmee0'onenta" &e,%en!e ha& a t$ano$m wth $e)on o# !on/e$)en!e n&(e a !$!"e.

Figure 4-,Re)on o# !on/e$)en!e #o$ the z t$ano$m o# ( ) ( )nuanx n=

)(ample 4.3: Fn( the z t$ano$m an( $e)on o# !on/e$)en!e o# ( )ny ; whe$e ( )ny &the &%m o# the 'o&t/e an( ne)at/e tme &e,%en!e& )/en n E0am'"e& .1 an(.+:

( ) ( ) ( )1= nubnuany nn

*olution

F$om the (e#nton t #o""ow& that

( ) ( )[ ] ( ) ( )

( ) ( )[ ]

bz

z

az

z

zbza

znubnua

nubnuaZnyZz

n

nn

n

nn

n

nnn

nn

+

=

=

=

==

=

=

=

1

9

1

1

The #$&t &%mmatonaz

z

ha& $e)on o# !on/e$)en!e az > ; wh"e the &e!on(

&%mmatonbz

z

ha& $e)on o# !on/e$)en!e bz < . Th%& the t$ano$m ( )z & the$

&%m wth $e)on o# !on/e$)en!e e,%a" to the nte$&e!ton o# the $e)on& o# !on/e$)en!e:

11+

( )zRe

( )zIm

ROC

a

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( )bz

z

az

zz

+

= wth ROC { } { }bzaz

Figure 4- Re)on o# !on/e$)en!e #o$ the z t$ano$m o#( ) ( )1= nubnx n

I# ab < the a3o/e nte$&e!ton & the em't< &et; .e. the t$ano$m (oe& not !on/e$)e;

wh"e # ab > the t$ano$m !on/e$)e& n the ann%"a$ $e)on &hown n F)%$e 19.

Important properties of the z transform

In the '$e!e(n) e0am'"e& the zt$ano$m o# 'o&t/e an( ne)at/e tme e0'onenta"&e,%en!e& we$e o3tane( 3< %&n) the #%n(amenta" (e#nton o# the z t$ano$m. The$e)on& o# !on/e$)en!e we$e o3tane( nat%$a""< n the e/a"%aton o# the &%mmaton&. In

man< !a&e& we w"" 3e a3"e to o3tan the zt$ano$m o# &e,%en!e&; n!"%(n) $e)on& o#!on/e$)en!e; wtho%t %&n) the 3a&! (e#nton. The #o""own) '$o'e$te& )/e %&

a((tona" "e/e$a)e n o3tann) z o# /a$o%& o'e$aton& on &e,%en!e& who&e t$ano$m&a$e *nown.F%n(amenta" o'e$aton& n!"%(e( a$e the a((ton o# &e,%en!e&; t$an&"aton o# &e,%en!e&;

/a$o%& m%"t'"!aton o# &e,%en!e&; an( !on/o"%ton o# &e,%en!e&. The&e '$o'e$te& a$e

&tate( wtho%t '$oo#.

11-

b ( )zRe

( )zIm

ROC

( )zIm

( )zIm

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Figure 4-10Re)on o# !on/e$)en!e #o$ the zt$ano$m o# ( ) ( ) ( )1= nubnuanx nn

ropert/ 1: 'inearit/

I# ( )[ ] ( )z!nfZ 11 = wth ROC +

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then ( ) ( )[ ] ( ) ( )zzXnynxZ = wth ROC yx RRz 4.1+5

ropert/ $: onolution zdomain

I# ( )[ ] ( )zXnxZ = wth ROC +

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( )nun - ( ) ( )+ 11 ++ zzzz 1>z

. E0'onenta" ( )nuan ( )azz az >

( )1 nubn ( )bzz bz

( )nuan n+ ( ) ( )- azazaz + az >

( )nuan n- ( ) ( )++ C azaazzaz ++ az >

( )nuan n 1 ( ) + azz az >

( )( )nua

nn n +

D+

1 ( )- azz az >

( )( )( )nua

nnn n -

D-

+1 ( ) azz az >

( ) ( )( )a

k

knnn kn 1

D

+1 + ( )k

azz az >

. Sn%&o(a" ( )nun9&n

1!o&+

&n

9

+

9

+

zz

z 1>z

( )nun9!o&

1!o&+

!o&

9

+

9

+

+

zz

zz 1>z

( ) ( )nun +9&n ( )[ ]

1!o&+

&n&n

9

+

9

++

zz

zz 1>z

( ) ( )nun +9!o& ( )[ ]

1!o&+

!o&!o&

9

+

9

+

zz

zz 1>z

. H

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( ) ( ) ( )=

=#

i

n

i nuanx1

Ta*n) the z t$ano$m o# ( )nx )/e&

( )[ ] ( ) =

==#

i iaz

zzXnxZ

1

The $e)on o# !on/e$)en!e R & the nte$&e!ton o# the $e)on& o# !on/e$)en!e #o$ ea!he0'onenta" &e,%en!e:

iRR = ; whe$e { }ii azzR >= :

The$e#o$e

{ }iazzR o#"a$)e&t: >= .Sn!e the $e)on o# !on/e$)en!e #o$ a t$an&"ate( e0'onenta" $eman& the &ame a& that #o$the o$)na" e0'onenta" &e,%en!e; a"" $)hthan( &e,%en!e& that a$e &%m& o# t$an&"ate(

e0'onenta"& ha/e $e)on& o# !on/e$)en!e &m"a$ to that e0'$e&&e( a3o/e.

Sm"a$"< we !an &how that a"" "e#than( &e,%en!e& e0'$e&&3"e a& a &%m o# t$an&"ate(!om'"e0 e0'onenta"& ha/e a $e)on o# !on/e$)en!e L )/en 3z we ha/e a 'o&t/e tme &e,%en!e. Th%&

( ) ( ) ( )[ ]

( )( ) ( )( )

( )

( ) ( )( )

( ) ( )

( )

( ) ( ){ } ( )( ) ( ){ } ( )nu

nu

nu

nu

nuz

z

z

z

zz

zz

zz

z

zzXdzzzXj

nx

nn

nn

n

n

nn

z

n

z

n

nn

c

nn

==

=

+

=

+

=

=

=

=

=

+

+

++

=

+

=

+

+

C1+C1+

C1+C1+C1

C1

C1+C1

+C1C1

C1

C1+C1

+C1

+C1C1

C1+C1Re&

C1+C1Re&

Re&+

1

1

1

11

C1

1

+C1

1

11

+

11

435 Fo$ ROC .C1

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( ) ( ) ( ){ } ( )1C1+C1+ = nunx nn4!5 Fo$ ROC .C1>z we ha/e a 'o&t/e tme &e,%en!e. Th%&

( ) ( ) ( )[ ]

( )

( )

( )( )

( )

[ ] ( )

( ) ( )nun

nuzdz

d

nuz

zz

dz

d

z

z

zz

z

zzXdzzzXj

nx

n

z

n

z

n

n

n

c

nn

+=

=

=

=

=

=

=

=

+

=

+

+

11

11

1Re&

1Re&

Re&+

1

1

1

1

+

1+

+

1

1

+

+

11

artial fraction e(pansions

=hen ( )zX & e0'$e&&e( a& a $atona" #%n!ton o# z ; that &; a $ato o# two 'o"

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In the 'a$ta" #$a!ton e0'an&on metho( we (ete$mne the 'a$ta" #$a!ton& o#( )

z

zX; &az ; we ha/e a 'o&t/e tme &e,%en!e; .e. 3oth 'o"e& 1=z an(-=z !ont$3%te to a 'o&t/e tme &e,%en!e.

( )( ) ( )-1

+

+

=zz

zz

z

zX

The (enomnato$ an( n%me$ato$ o#( )

z

zXa$e o# e,%a" (e)$ee. =e (/(e #$&t:

( ) ( )-1

11

-

11

-

+++

+

+=

++=

++

zz

z

zz

z

zz

z

The$e#o$e

( )

( ) ( )-1

11

+=zz

z

z

zX

Now (ete$mne the 'a$ta" #$a!ton& o# the &e!on( te$m on the $)ht. Let

( ) ( ) -1-1

1

+

=

z

$

z

A

zz

z

Th%&

( )

( )

+

11

1-

1-

1

1

+

-

-1

11

-

1

-

1

==

=

===

=

=

z

z

z

z$

z

zA

Th%&

( )

( )-+

11

1+

--

+11

1

+-1

+

=

+

+=

z

z

z

zzzX

zzz

zX

The$e#o$e

( ) ( ){ } { }

( ) ( ) ( ) ( )nunun

zzZ

zzZzZzXZnx

n-

+

11

+

-1

-+11

1+- 1111

++=

+

==

435 Sn!e the ROC & -1

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( )-C1

+

+=

zzX

#o$ the #o""own) $e)on& o# !on/e$)en!e:45 -C1z

435 @/en( )

( )++

-

+

+=

z

zzX wth ROC +>z .

45 U&e the (/&on metho( to e/a"%ate ( )nx at 9=n an( 1+=n .45 Che!* 17>2

-

-+

+

+

+

z

zz

z

zz

z

z

z

+

+

++

The$e#o$e( )

( )( )

( )

+

171

29

9#o$9

==

=

>=

x

x

x

nnx

45 45 Fo$ -C1

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The$e#o$e( )

( )

( )

( )

8+-

-++

+1

9#o$9

==

==

x

x

x

nnx

435

45 Sn!e ROC +>z m'"e& a $)hthan( &e,%en!e; we (/(e to o3tan ne)at/e 'owe$&

o# z . Th%&

et!.2

+1

1

+

-+

11

+1

+1

1

1

+

+1

++

++

+++++

zz

zz

z

zz

zzz

zz

=e ha/e( )

( ) 11

99

==

x

x

45 The nta" /a"%e '$o'e$t< &tate& that

( ) ( )

9991

99+1

-1"m

+

-"m

"m9

+

+

+

++ +=

+++

=

+++

=

=

zz

zzzz

z

zXx

z

z

z

A((tona" #o$m%"a& that ma< 3e %&e#%" n the e/a"%aton o# the n/e$&e z ; a$e:

1.

( ) ( )

( ) ( )

>

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+.

>

e ntowho"e #a!to$&; o$ #a!to$>e& nto !om'"e0 #a!to$&.

*olution of difference e"uations #/ ztransforms

The zt$ano$m !an 3e %&e( to &o"/e "nea$ (##e$en!e e,%aton& wth nta" !on(ton&n the &ame wa< that the La'"a!e t$ano$m & %&e( to &o"/e "nea$ (##e$enta" e,%aton&.

To th& en( we note the #o$m%"a& #o$ the zt$ano$m o# the #$&t an( &e!on( (&!e$te(e$/at/e&:

( )[ ] ( ) ( )zzynyZ 111 +=( )[ ] ( ) ( ) ( )zzyzynyZ +1 1++ ++=

Note that ( )1ny an( ( )+ny a$e te$m& that we #n( n the #o$m%"a& #o$ the #$&t an(&e!on( 3a!*wa$( (##e$en!e&; $e&'e!t/e"< 4&ee Ta3"e 15. The &ame !an 3e &a( a3o%t

( )1+ny ; ( )++ny ; an( the #$&t an( &e!on( #o$wa$( (##e$en!e #o$m%"a&. I# we '%t( ) ( ) ( ) ( )

( ) ( ) ( ) ( )ny%nyny%ny

ny%nyny%ny++

++

11

==+==+

; et!.

then we !an &m%"ate a (##e$en!e e,%aton 4(&!$ete "nea$ &

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4a5 ( ) ( ) ( ) 9;1+

1+= nnxnyny

whe$e ( ) ( )nunxn

=

+

1an( ( )

11 =y

435 ( ) ( ) ( ) 9;+ =+ nnnyny

whe$e ( ) ( ) 11;9+ == yy

*olution

4a5

( ) ( ) ( )

( ) ( )

( ){ } ( ){ } ( ) ( ){ }( ) ( ) ( )[ ]

( ) ( )

( ) ( )

( )

( ) ( )

( )

( ) +

1

1

1

1

+1

12178+1

12178

+1

+171+1

+17

1

+

11

+1+

1

7

1+1

1

+

1

+11

+

1

+11+1

+1

1+1

=

=

+=

+=

=

=

+

=+

==

=

z

z

z

zz

z

z

zz

z

zz

z

zzz

z

zzzz

z

zzzz

z

zzzyz

nuZnyZnyZ

nu

nxnyny

n

n

P%t

( ) ( ) ++ +C1+C1+C1

12C17C8

+

=

z

$

z

A

z

z

Th%&

$zAz +

=

+

1

12

1

7

8 415

I#+

1=z n 415; then

+

1

12

7

12

1

+

1

7

8===

$$

E,%atn) !oe##!ent& o# z n 415; we )et

7

8=A

1--

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)(ercise 4.2

1. U&n) the 'a$ta" #$a!ton e0'an&on metho(; #n( the n/e$&e zt$ano$m o# ( )zX)/en 3z ; 435 .1

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6. @/en ( ) ( ) ( -C1 ++ += zzzzzX ; #n( ( )nx #o$ the #o""own) $e)on& o#!on/e$)en!e %&n) the metho( n(!ate(:

4a5 ROC ->z 3< 'a$ta" #$a!ton&

435 ROC 1

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4a5 ( ) ( )

( )( ) ++11

1

+=

zz

zzzX #o$ ROC 1>z

435 ( )( ) ( )+-C1

++

+-

+++

=zzz

zzzzX #o$ ROC +>z

4!5 ( ) 1+- 1+

1

+=

zzzzX #o$ ROC 1>z

4(5 ( )( )-C1

=

zzzX #o$ ROC -C1>z

4e5 ( ) ( ) ( )( )[ ]-+1+ += zzzzX #o$ ROC -+ 4)5 ( ) ( )

++1 = zzzX #o$ ROC 1>z

11. Let ( ) ( ) ( )nxnxnx +1 = ; whe$e ( ) ( )nunxn

=

+

11 an( ( ) ( )nunx

n

=

-

1+

4a5 Fn( ( )zX 3< %&n) the !on/o"%ton '$o'e$t< o# z t$ano$m&

435 Fn( ( )nx 3< ta*n) the n/e$&e t$ano$m o# ( )zX %&n) the 'a$ta" #$a!tone0'an&on metho(.

1+. So"/e the #o""own) (##e$en!e e,%aton& #o$ ( )ny %&n) z t$ano$m& an( the&'e!#e( nta" !on(ton&:

4a5 ( ) ( ) ( ) 9;+1-1 += nnxnyny

whe$e ( ) ( ) ( ) -11an(-

1=

= ynunx

n

435 ( ) ( ) ( ) 9;11 = nnxnyny

whe$e ( ) ( ) ( ) +1an(+

1=

= ynunx

n

4!5 ( ) ( ) ( ) 9;+1- += nnynynywhe$e ( ) ( ) -1an(++ == yy4(5 ( ) ( ) ( ) 9;1++ =+ nnnynywhe$e ( ) ( ) 91an(1+ == yy4e5 ( ) ( ) ( ) ( ) 9;++-1 =+ nnxnynyny

whe$e ( ) ( ) ( ) -1an(1+;7

1==

=

yynx

n

4#5 ( ) ( ) ( ) ( ) 19;+

1-1 =

=+ ynunyny

n

4)5 ( ) ( ) ( ) ( ) ( ) +1;19;91+ ===+++ yynynyny

4h5 ( ) ( ) ( ) ( ) ( ) ( ) -1;99;1-1+ ===++ yynunyny n

45 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 91;-9;-+1++ ==+=+++ yynunnynyny n

45 ( ) ( ) ( ) ( ) ( ) -1;9;1-6+ + ===+ yynunyny n

4*5 ( ) ( ) ( ) ( ) ( ) ( ) ( ) 91;+9;-1+- ==+=++ yynunnynyny n

4"5 ( ) ( ) ( ) ( ) ( ) ( ) 11;19;-

+1+ ==

=+ yynunynyny

n

1-7

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4m5 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 91;99;+-11-+ ==++=++ yynunnnynyny n