chapter4 pp101-111
TRANSCRIPT
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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( .+.
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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
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4m5 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 91;99;+-11-+ ==++=++ yynunnnynyny n