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Signal and SystemsProf. H. Sameti
Chapter 9: Laplace Transform
Motivation and Definition of the (Bilateral) Laplace Transform Examples of Laplace Transforms and Their Regions of Convergence
(ROCs) Properties of ROCs Inverse Laplace Transforms Laplace Transform Properties The System Function of an LTI System Geometric Evaluation of Laplace Transforms and Frequency Responses
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Book Chapter#: Section#
2
Motivation for the Laplace Transform
CT Fourier transform enables us to do a lot of things, e.g.• Analyze frequency response of LTI systems • Sampling• Modulation
Why do we need yet another transform? One view of Laplace Transform is as an extension of the
Fourier transform to allow analysis of broader class of signals and systems
In particular, Fourier transform cannot handle large (and important) classes of signals and unstable systems, i.e. when
Computer Engineering Department, Signal and Systems
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Book Chapter#: Section#
3
Motivation for the Laplace Transform (continued)
In many applications, we do need to deal with unstable systems, e.g.• Stabilizing an inverted pendulum• Stabilizing an airplane or space shuttle• Instability is desired in some applications, e.g. oscillators and lasers
How do we analyze such signals/systems? Recall from Lecture #5, eigenfunction property of LTI systems:
is an eigenfunction of any LTI system can be complex in general
Computer Engineering Department, Signal and Systems
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Book Chapter#: Section#
4
The (Bilateral) Laplace Transform
s = + σ j is a complex variable – Now we explore the full ωrange of
Basic ideas:
1. A critical issue in dealing with Laplace transform is convergence:—X(s) generally exists only for some values of s, located in what is called the region of convergence(ROC): so that
2. If = is in the ROC (i.e. = 0), then σ
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absolute integrability needed
absolute integrability condition
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Book Chapter#: Section#
5
Example #1: (a – an arbitrary real or complex number)
This converges only if Re(s+a) > 0, i.e. Re(s) > -Re(a)
Computer Engineering Department, Signal and Systems
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Book Chapter#: Section#
6
Example #2:
This converges only if Re(s+a) < 0, i.e. Re(s) < -Re(a) Same as (s), but different ROC Key Point (and key difference from FT): Need both X(s)
and ROC to uniquely determine x(t). No such an issue for FT.
Computer Engineering Department, Signal and Systems
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Book Chapter#: Section#
7
Graphical Visualization of the ROC Example1: Example2:
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Book Chapter#: Section#
8
Rational Transforms Many (but by no means all) Laplace transforms of interest to us are
rational functions of s (e.g., Examples #1 and #2; in general, impulse responses of LTI systems described by LCCDEs), where
X(s) = N(s)/D(s), N(s),D(s) – polynomials in s
Roots of N(s)= zeros of X(s)
Roots of D(s)= poles of X(s)
Any x(t) consisting of a linear combination of complex exponentials for t > 0 and for t < 0 (e.g., as in Example #1 and #2) has a rational Laplace transform.
Computer Engineering Department, Signal and Systems
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Book Chapter#: Section#
9
Example #3
Computer Engineering Department, Signal and Systems
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Book Chapter#: Section#
10
Laplace Transforms and ROCs Some signals do not have Laplace Transforms (have no ROC) for all t since for all
for all t for all X(s) is defined only in ROC; we don’t allow impulses in LTs
Computer Engineering Department, Signal and Systems
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Book Chapter#: Section#
11
Properties of the ROCThe ROC can take on only a small number of
different forms1. 1) The ROC consists of a collection of lines
parallel to the j -axis in the ω s-plane (i.e. the ROC only depends on ).Why? σdepends only on
2. If X(s) is rational, then the ROC does not contain any poles. Why?
Poles are places where D(s) = 0 ⇒ X(s) = N(s)/D(s) = ∞ Not convergent.
Computer Engineering Department, Signal and Systems
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Book Chapter#: Section#
12
More Properties If x(t) is of finite duration and is absolutely integrable, then the ROC is
the entire s-plane.
Computer Engineering Department, Signal and Systems
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Book Chapter#: Section#
13
ROC Properties that Depend on Which Side You Are On - I
If x(t) is right-sided (i.e. if it is zero before some time), and if Re(s) = is in the ROC, then all values of s for which Re(s) > are also in the ROC.
Computer Engineering Department, Signal and Systems
ROC is a right half plane (RHP)
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Book Chapter#: Section#
14
ROC Properties that Depend on Which Side You Are On -II
If x(t) is left-sided (i.e. if it is zero after some time), and if Re(s) = is in the ROC, then all values of s for which Re(s) < are also in the ROC.
Computer Engineering Department, Signal and Systems
ROC is a left half plane (LHP)
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Book Chapter#: Section#
15
Still More ROC Properties If x(t) is two-sided and if the line Re(s) = is in the ROC,
then the ROC consists of a strip in the s-plane
Computer Engineering Department, Signal and Systems
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Book Chapter#: Section#
16
Example:
Computer Engineering Department, Signal and Systems
Intuition?
Okay: multiply by constant () and will be integrable
Looks bad: no will dampen both sides
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Book Chapter#: Section#
17
Example (continued):
Overlap if , with ROC:
What if b < 0? No overlap No Laplace Transform⇒ ⇒
Computer Engineering Department, Signal and Systems
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Book Chapter#: Section#
18
Properties, Properties If X(s) is rational, then its ROC is bounded by poles or extends to
infinity. In addition, no poles of X(s) are contained in the ROC. Suppose X(s) is rational, then
a) If x(t) is right-sided, the ROC is to the right of the rightmost pole.b) If x(t) is left-sided, the ROC is to the left of the leftmost pole.
If ROC of X(s) includes the j -axis, then FT of x(t) exists.ω
Computer Engineering Department, Signal and Systems
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Book Chapter#: Section#
19
Example: Three possible ROCs
x(t) is right-sided ROC: III No x(t) is left-sided ROC: I Nox(t) extends for all time ROC: II Yes
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Fourier Transform exists?
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20
The z-Transform
• Last time:
• Unit circle (r = 1) in the ROC ⇒DTFT exists
• Rational transforms correspond to signals that are linear combinations of DT exponentials
Computer Engineering Department, Signal and Systems
-depends only on r = |z|, just like the ROC in s-plane only depends on Re(s)
nxZznxzXnxn
n
)(
n
nj rnxrez ||at which ROC
( )jX e
Book Chapter10: Section 1
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21
Some Intuition on the Relation between ZT and LT
Computer Engineering Department, Signal and Systems
The (Bilateral) z-Transform
Can think of z-transform as DTversion of Laplace transform with
( ) ( ) ( ) { ( )}stx t X s x t e dt L x t
TenTx nsT
n nxT
)()(lim
0
n
nsT
TenxT )(lim
0
}{)( nxzznxzXnxn
n
sTez
Let t=nT
Book Chapter10: Section 1
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22
More intuition on ZT-LT, s-plane - z-plane relationship
LHP in s-plane, Re(s) < 0 |⇒ z| = | esT| < 1, inside the |z| = 1 circle.
Special case, Re(s) = -∞ ⇔|z| = 0. RHP in s-plane, Re(s) > 0 |⇒ z| = | esT| > 1, outside the |z| = 1 circle.
Special case, Re(s) = +∞ ⇔|z| = ∞. A vertical line in s-plane, Re(s) = constant⇔| esT| = constant, a circle in z-plane.
Computer Engineering Department, Signal and Systems
zesT )( plane-sin axis jsj plan-zin circleunit a1 Tjez
Book Chapter10: Section 1
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23Computer Engineering Department, Signal and SystemsComputer Engineering Department, Signal and Systems 23
Inverse Laplace Transform
})({
,)()(
t
st
etxF
ROCjsdtetxsX
Fix σ ROC and apply the inverse Fourier ∈
dejXetx tjt )(2
1)(
dejXtx tj )()(2
1)(
But s = σ + jω (σ fixed) ds =⇒ jdω
j
j
stdsesXj
tx )(2
1)(
Book Chapter 9 : Section 2
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24Computer Engineering Department, Signal and Systems
Inverse Laplace Transforms Via Partial FractionExpansion and Properties
Example:
3
5 ,
3
2
21)2)(1(
3)(
BA
s
B
s
A
ss
ssx
Three possible ROC’s — corresponding to three different signals
Recall sided-left )(}{ ,1
tueaseas
at
1, { } ( ) right-sidedate s a e u t
s a
Book Chapter 9 : Section 2
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25Computer Engineering Department, Signal and Systems
ROC I: — Left-sided signal.
ttuee
tuBetuAetx
tt
tt
as es Diverg)(3
5
3
2
)()()(
2
2
ROC II: — Two-sided signal, has Fourier Transform.
ttuetue
tuBetuAetx
tt
tt
as erges Div)(3
5)(
3
2
)()()(
2
2
ROC III: — Right-sided signal.
tu(t)ee
tuBetuAetx
tt
tt
as erges Div3
5
3
2
)()()(
2
2
Book Chapter 9 : Section 2
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Book Chapter 9 : Section 2
26
Many parallel properties of the CTFT, but for Laplace transforms we need to determine implications for the ROC
For example:
Computer Engineering Department, Signal and Systems
Properties of Laplace Transforms
Linearity
)()()()( 2121 sbXsaXtbxtax
ROC at least the intersection of ROCs of X1(s) and X2(s)
ROC can be bigger (due to pole-zero cancellation)
0)(0)()( Then
and )()( E.g.
21
21
sXtbxtax
batxtx
⇒ ROC entire s-
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27Computer Engineering Department, Signal and Systems
Time Shift
)( as ROCsame ),()( sXsXeTtx sT
? 2}{ ,2
:Example3
ses
e s
Ttttueses
e tsT
|)( 2}{ ,2
2
3 T
)3( 2}{ ,2
)3(23
tueses
e ts
Book Chapter 9 : Section 2
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28Computer Engineering Department, Signal and Systems
Time-Domain Differentiation
j
j
j
j
stst dsessXjdt
tdxdsesX
jtx )(
2
1)( ,)(
2
1)(
ROC could be bigger than the ROC of X(s), if there is pole-zero cancellation. E.g.,
)( of ROC thecontaining ROC with),()(
sXssXdt
tdx
plane-s entire ROC1
1)( )(
0}{ ,1
)( )(
sst
dt
tdx
ses
tutx
s-Domain Differentiation
X(s)ds
sdXttx as ROCsame with,
)()(
(Derivation is
similar to )d
sdt
aseasasds
dtute at
}{ ,)(
11)( E.g.
2
Book Chapter 9 : Section 2
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29Computer Engineering Department, Signal and Systems
Convolution Property
x(t) y(t)=h(t)* x(t)h(t)
For
Then )()()(
)()(),()(),()(
sXsHsY
sHthsYtysXtx
• ROC of Y(s) = H(s)X(s): at least the overlap of the ROCs of H(s) & X(s)
• ROC could be empty if there is no overlap between the two ROCs
E.g.
x(t)=etu(t),and h(t)=-e-tu(-t)
• ROC could be larger than the overlap of the two.
)()(*)( tthtx
Book Chapter 9 : Section 2
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30Computer Engineering Department, Signal and Systems
The System Function of an LTI System
x(t) y(t)h(t)
function system the)()( sHth
The system function characterizes the system⇓
System properties correspond to properties of H(s) and its ROC
A first example:
dtth )( stable is System
axis theincludes
of ROC
jω
H(s)
Book Chapter 9 : Section 2
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31Computer Engineering Department, Signal and Systems
Geometric Evaluation of Rational Laplace Transforms
Example #1: A first-order zeroassX )(1
Book Chapter 9 : Section 2
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32Computer Engineering Department, Signal and Systems
Example #2: A first-order pole
)(
11)(
12 sXas
sX
)( )(
)|)(|log log(or )(
1 )(
12
121
2
sXsX
sX(s)||XsX
sX
Still reason with vector, but remember to "invert" for poles
Example #3: A higher-order rational Laplace transform
)(
)( )(
1
1
jPj
iRi
s
sMsX
jPj
iRi
s
sMsX
1
1 )(
R
i
p
jji ssMsX
1 1
)()( )(
Book Chapter 9 : Section 2
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33Computer Engineering Department, Signal and Systems
First-Order System
Graphical evaluation of H(jω)
1
}{,/1
/1
1
1)(
se
sssH
)(1
)( / tueth t
)(1)( / tuets t
/1
11
/1
/1 )(
jjjH
Book Chapter 9 : Section 2
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34Computer Engineering Department, Signal and Systems
Bode Plot of the First-Order System
1/ 2/
1/ 4/
0 0
)(tan )(
1/ /1
1/ 2/1
0 1
)/1(
/1 )(
1/j
1/ )(
1
22
jH
jH
jH
Book Chapter 9 : Section 2
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35Computer Engineering Department, Signal and Systems
Second-Order System
e(pole)e{s} ROC2
)(22
2
nn
n
sssH
10 complex poles— Underdamped
1
1
double pole at s = −ωn
— Critically damped
2 poles on negative real axis— Overdamped
Book Chapter 9 : Section 2
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36Computer Engineering Department, Signal and Systems
Demo Pole-zero diagrams, frequency response, and step response of first-order and second-order CT causal systems
Book Chapter 9 : Section 2
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37Computer Engineering Department, Signal and Systems
Bode Plot of a Second-Order System
Top is flat whenζ= 1/√2 = 0.707⇒a LPF for
ω < ωn
Book Chapter 9 : Section 2
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38Computer Engineering Department, Signal and Systems
Unit-Impulse and Unit-Step Response of a Second- Order System
No oscillations when ≥ 1ζ
⇒ Critically (=) and over (>) damped.
Book Chapter 9 : Section 2
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39Computer Engineering Department, Signal and Systems
First-Order All-Pass System
1. Two vectors have the same lengths
2.
)0( }{ ,)(
aaseas
assH
a
a
jH
0~
2/
0
2
)(
)(
2
22
21
Book Chapter 9 : Section 2
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CT System Function Properties
x(t) y(t)h(t)
)()()( sXsHsY H(s) = “system function”
1) System is stable ROC of H(s) includes jω axis
2) Causality => h(t) right-sided signal => ROC of H(s) is a right-half plane
dtth )(
Question:If the ROC of H(s) is a right-half plane, is the system causal?
E.x. sided-right )(1}{ ,1
)( thees
esH
eT
Tttt
Ttt
eT
tues
Ls
eLth
|)(
1
1
1 )( 11
0at 0)( )( tTtue Tt Non-causal40
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Properties of CT Rational System Functions
a) However, if H(s) is rational, then
The system is causal ⇔ The ROC of H(s) is to the right of the rightmost pole
b) If H(s) is rational and is the system function of a causal system, then
The system is stable ⇔ jω-axis is in ROC ⇔ all poles are in
41
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Checking if All Poles Are In the Left-Half Plane
)(
)()(
sD
sNsH
Poles are the root of D(s)=sn+an-1sn-1+…+a1s+a0
Method #1: Calculate all the roots and see!Method #2: Routh-Hurwitz – Without having to solve for roots.
210
01201
22
3
01012
00
and
0,0,0order-Third
0,0order-Second
0order-First LHP thein are roots
all that so ConditionPolynomial
aaa
aaaasasas
aaasas
aas
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Initial- and Final-Value Theorems
If x(t) = 0 for t < 0 and there are no impulses or higher order discontinuities at the origin, then
)(lim )0( ssXxs
)(lim )(0
ssXxs
Initial value
Final value
If x(t) = 0 for t < 0 and x(t) has a finite limit as t → ∞, then
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Applications of the Initial- and Final-Value Theorem
For
n-order of polynomial N(s), d – order of polynomial D(s)
• Initial value:
• Final value:
1
1 0 finite
1 0
)(lim )0(
nd
nd
nd
ssXxs
?)0( 1
1)( E.g.
x
ssX
)(
)()(
sD
sNsX
0at poles No
)(lim0)(lim)( If00
s
sXssXxss
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LTI Systems Described by LCCDEs
N
k
M
kk
k
kk
k
k dt
txdb
dt
tyda
0 0
)()(
N
k
M
k
kk
kk
kk
k
sXsbsYsa
sdt
ds
dt
d
0 0
)()(
,property ationdifferenti of use Repeated :
)( where
)()()(
Rational
0
0
N
k
kk
M
k
kk
sa
sbsH
sXsHsY
roots of numerator ⇒ zeros
roots of denominator ⇒ poles
ROC =? Depends on: 1) Locations of all poles. 2) Boundary conditions, i.e. right-, left-, two-sided signals.
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System Function Algebra
Example: A basic feedback system consisting of causal blocks
)()()()()()( 2 sYsHsXsZsXsE
)()()()()()()( 211 sYsHsXsHsEsHsY
)()(1
)(
)(
)()(
21
1
sHsH
sH
sX
sYsH
More on this later in feedback
ROC: Determined by the roots of 1+H1(s)H2(s), instead of H1(s)
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Block Diagram for Causal LTI Systems with Rational System Functions
Example: )()()( sXsHsY
23
642)(
2
2
ss
sssH )642(
23
1 22
ss
ss
— Can be viewed as cascade of two systems.
)(23
1)( Define
2sX
sssW
:
)(2)(
3)()(
or
restat initially ),()(2)(
3)(
2
2
2
2
twdt
tdwtx
dt
twd
txtwdt
tdw
dt
twd
)(6)(
4)(
2)(
)()642()(
Similarly
2
2
2
twdt
tdw
dt
twdty
sWsssY
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Example (continued)
Instead of
)(64223
1)(
)(
22
tyssss
tx
sH
We can construct H(s) using:
)(6)(
4)(
2 )(
)(2)(
3)( )(
2
2
2
2
twdt
tdw
dt
twdty
twdt
tdwtx
dt
twd
Notation: 1/s—an integrator48
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Note also that
Cascade 1
)1(2
2
3
1
3
2
12 )(
s
s
s
s
s
s
s
)(ssH
connection parallel 1
8
2
62
ss
PFE
Lesson to be learned: There are many different ways to construct a system that performs a certain function.
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The Unilateral Laplace Transform(The preferred tool to analyze causal CT systems described by LCCDEs with initial conditions)
Note:
1) If x(t) = 0 for t < 0,
2) Unilateral LT of x(t) = Bilateral LT of x(t)u(t-)
3) For example, if h(t) is the impulse response of a causal LTI
system then,
4) Convolution property: If x1(t) = x2(t) = 0 for t < 0,
Same as Bilateral Laplace transform
0
)()()( txdtetxs st ULX
)()( ssX X
)()( ssH H
)()()()( 2121 sxtxtx XXUL
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Differentiation Property for Unilateral Laplace Transform
)0()()(
)()(
xssdt
tdx
stx
X
X
Initial condition!
Derivation:
Note:
0
)()(dte
dt
tdx
dt
tdx stUL
0
)(
0|)()( st
s
st etxdtetxs
X
)0()( xssX
integration by parts∫f. dg=fg-∫ g. df
)0(')0()(
)0('
)(
))0()(()()(
2
2
2
xsxss
x
dt
tdx
xsssdt
tdx
dt
d
dt
txd
X
UL
X
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Use of ULTs to Solve Differentiation Equations with Initial Conditions
Example:
Take ULT:
)()(,)0(',)0(
)()(2)(
3)(
2
2
tutxyy
txtydt
tdy
dt
tyd
ssssss
dt
dy
dt
yd
)(2))((3)(
2
2
2 YYY
ULUL
ZSR
sss
ZIR
ssss
ss
)2)(1()2)(1()2)(1(
)3()(
Y
ZIR — Response for zero input x(t)=0
ZSR — Response for zero state,β=γ=0, initially at rest
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Example (continued)
• Response for LTI system initially at rest ( = = 0 )β γ
• Response to initial conditions alone ( = 0). For αexample:
)()2)(1(
1
)(
)()( sH
sss
ss
Y
H
)0,1( 0)0(' ,1)0( ),input no(0)( yytx
0 ,2)(
2
1
1
2
)2)(1(
3)(
2
teety
ssss
ss
tt
Y
53