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o e a e o o a n rans er unc on o
open loop systemsTo be able to solve difference equations
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For
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Sam lin and Reconstruction
Input)(teo samper
outputOf zero
orderhold
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sG)(te )( kTe
The Output signal of an ideal sampler is the signal whose Laplace transform is
0
( ) ( ) n s
n
E s e nT e
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The star transform is so named because it is frequently
represented by an asterisk or "star" in the notation
The star transform, is a discrete-time variation of
.
e apace rans orm a represen s anideal sampler with period of time T.
Similar to Z-Transformwith a simple change of variables,
but the star transform ex licitl identifies each sam le interms of the sampling period (T), while the Z transformonly refers to each sample by integer index value.
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Relationshi between X* s and X z
X s X* s
( ) ( ) stX s x t e dt
The function called the star transform is defined as:*( )X s
0k
s x e
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X s X* s
Star transform is related to Laplace Transformby taking
1)(sX (3-10) from Textbook
andA endix 3
the residues of the Laplace transform of a function
1)(Residues)(
1 TXofzX
)(1111
)( 2
2
11
2,1
TT
TT
TT
eez
zX
)()(*
2
21
TTsTeee
sX
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eeee (c) Khoukhi 2010
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polesall
kzzEofresidueskx
k
1)()(
1
polessimplefor
azaz
mtymultipliciwithpolesfor
zzazzzores ues
az
km
m
m
az
k zzEaz
dz
d
m
zzEofresidues
11
11 )()(
)!1(
1)(
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X s X* s
-The change of variable Tsz e
)2)(1()( sssX
)()(
2
2
TT
TTeez
zX
TsezsXzX
)(*)(
)(*
2TTsTeee
sX
CISE318_Topic 4 10))((
ss
eeee
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Example
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Example
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G(s)
X(s) X*(s) Y(s)p
*****
)(*)()( sXsGsY
)()()( zXzGzY
The pulse transfer function is the transfer function between thesampled input and the output at the sampling instants
G(s)
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X(s) X*(s) Y(s)A(s)
1 2
zXzGGzY
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GGGG
Tsez
sGzGsGsGsG )(*)()()()( 11123
TsTsezez
sGzGsGzGG
)(*)()(*)( 22312
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X(s) X*(s) Y(s)A(s)
1 2
12 zzzz
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Y(s)A(s)B(s)X(s)
2 31
)()()()()( 123 zXzGzGzGzY
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Y(s)A(s)B(s)X(s)
2 31
zXzGzGGzY
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X(s) Y(s)A(s)
1 2
zXGzGzY We can not write a transfer function for
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Y(s)A(s)B(s)X(s)
2 31
zXGzGGzY
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ore xamp es
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More Examples
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Check the course webCT for the Homework
Assignment
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