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16.362 Signal and System I • Analysis and characterization of the LTI system using the Laplace transform
0)( thCausal
)()()( sXsHsY
ROC associate with a causal system is a right-half plane
for t<0 Right-side
Converse
)(thCausal ROC: right half plane
)(thCausal ROC: right half plane
Unless rational
16.362 Signal and System I Converse
)(thCausal ROC: right half plane
Unless rational
Example
1)(
s
esH
s
ROC: s>-1
)1()( )1( tueth t Not causal
1
')'(
)1()1(
)1()(
''
)1()1(
)1(
s
e
dttueee
tdtueee
dttueesH
s
stts
tsts
stt
16.362 Signal and System I • Analysis and characterization of the LTI system using the Laplace transform
0)( thCausal
)()()( sXsHsY
ROC associate with a causal system is a right-half plane
for t<0 Right-side
then converse
)(thCausal ROC: right half plane
)(thCausal ROC: right half plane
If rational
16.362 Signal and System I Example
1
1
1
1)(
sssH ROC: -1<s<1
teth )( Not causal
ROC is not to right of the rightmost pole.
16.362 Signal and System I Stability
Bounded output for EVERY bounded input .System stable
dtth )(If and only if
')'(
')'()'(
')'()'()(
dtthB
dtttxth
dtttxthty
dtethsH st
)()(
If and only if the ROC of H(s) contains entire j axis, i.e. Re(s) = 0.
dteth
dtethjH
tj
tj
)(
)()0(
If the system stable, then ROC contains the j axis.
16.362 Signal and System I Stability
Bounded output for EVERY bounded input .System stable
dtth )(If and only if
If and only if the ROC of H(s) contains entire j axis, i.e. Re(s) = 0.
dteth tj)( for any
dtth )(
Bdteth tj
)(
2'* ')'()( Bdtethdteth tjtj
2)'(* ')'()( Bdtdtethth ttj
''')'()( )')('(2)')('(* deBdtdtdethth ttjttj
)'()'(')'()( 2* ttBttdtdtthth
22
)( Bdtth
16.362 Signal and System I Example
)2)(1(
1)(
ss
ssH
)(3
1)(
3
2)( 2 tuetueth tt
-1 2
)2(
3/1
)1(
3/2)(
sssH
)(3
1)(
3
2)( 2 tuetueth tt
-1 2
)(3
1)(
3
2)( 2 tuetueth tt
-1 2
16.362 Signal and System I
A causal LTI system with rational H(s) is stable if and only if all poles of H(s) lie in the left-half of the s-plane, i.e. all poles have negative real parts
If: all poles of H(s) lie in the left-half of the s-plane
ROC associate with a causal system is a right-half plane
ROC includes Re(s) = 0 Stable
Only if:
all poles have negative real parts
Stable ROC includes Re(s) = 0 + causal
No poles in the right-half s-plane
16.362 Signal and System I Example
Not stable
)()( 2 tueth t)2(
1)(
s
sH
-1 2
16.362 Signal and System I
0
122 nnc
)()( 21 tueeMth tctc 12 2
nM
21
2
)(cscs
sH n
121 nnc
c1 c2
Not stable
16.362 Signal and System I • LTI Systems characterized by linear constant-coefficient differential equations
)()(3)(
txtydt
tdy )()(3)( sXsYssY
3
1)(
s
sH
Casual
3
1)(
s
sH
ROC to the right most pole
ROC Re(s) >-3
)()( 3 tueth t
16.362 Signal and System I • LTI Systems characterized by linear constant-coefficient differential equations
M
kk
k
k
N
kk
k
k dt
txdb
dt
tyda
00
)()(
N
k
kk
M
k
kk
sa
sbsH
0
0)(
Always rational
00
M
k
kksb
00
N
k
kksa
Zeros:
Poles:
Initial rest condition Causal ROC: to the rightmost pole
Causal and stable: ROC: to the rightmost pole & include the Re(s) = 0.
16.362 Signal and System I Example
Causal and stable: ROC: to the rightmost pole & include the Re(s) = 0.
1
1)(
2
RCsLCssH
LCL
R
L
Rs
14
2
1
2
2
Re(s)<0
)(tx )(ty
R L
C
dt
tdyCti
)()(
)()()()(
2
2
txtydt
tydLC
dt
tdyRC
16.362 Signal and System I Example
)()( 3 tuetx t
)()( 2 tueety tt
Causal and stable: ROC: to the rightmost pole & include the Re(s) = 0.
)2)(1(
1)(
sssY
)3(
1)(
s
sX
)2)(1(
)3()(
ss
ssH
Re(s)<0
)2(
1
)1(
1)(
sssY
16.362 Signal and System I Example: an LTI system
1. causal
)4)(2(
)()(
ss
spsH
tetx 01)(
2. H(s) is rational and has two poles, at s = -2, and s = 4
3. If x(t) =1, then y(t) = 0.
4. The value of the impulse response at t = 0+ is 4.
tesHty 0)()( 0)4)(2(
)(
ss
sp
4)4)(2(
)(lim)(lim2
0
ss
KsssHth
st
)()( ssqsp
)4)(2(
4)(
ss
ssH
16.362 Signal and System I Example: an LTI system
1. Causal & stable
teth 3)(
2. H(s) is rational and has one pole, at s = -2, and doesn’t have a zero at the origin.
3. The locations of other poles and zeros are unknown.
)(tth
)(tth
)(sH
)(sHds
d
converge 0)(
dtth 0|)()( 0
0
st sHdteth
ROC contain -3
is the impulse response of a causal and stable system
)(tth Same ROC
Stable ROC contains j axis
Stable
16.362 Signal and System I Example: an LTI system
1. Causal & stable
2. H(s) is rational and has one pole, at s = -2, and doesn’t have a zero at the origin.
3. The locations of other poles and zeros are unknown.
)()( sHsH
)(th
)(thdt
d
Has pole at s = 2 Not stable
ROC contains whole s-plane
No sufficient info.
contains at least one pole in its Laplace transform
has finite duration
2)(lim
sH
s
)()2(
2)(
sps
assH
16.362 Signal and System I Butterworth filters
N
cjj
jB 2
2
1
1)(
Restrict the impulse response of the Butterworth filter is real
)()()( *2 jBjBjB
)()(* jBjB
jssBjB |)()( **
N
cjs
sBsB 2
1
1)()(
Roots: cNp js 2/1)1( cps
22
12
N
ksp
16.362 Signal and System I Butterworth filters
N
cjs
sBsB 2
1
1)()(
Roots: cNp js 2/1)1( cps
22
12
N
ksp
Poles appear in pairs
Restrict B(s) causal and stable Poles are in the left-half plane
16.362 Signal and System I Butterworth filters
N
cjs
sBsB 2
1
1)()(
Roots: cNp js 2/1)1( cps
22
12
N
ksp
N=1
c
Restrict B(s) causal and stable Poles are in the left-half plane
x x
cc
ssB
)(
N=2
c
x x
xx
4
5
4
3
2
)(j
c
j
c
c
eses
sB
16.362 Signal and System I N
cjs
sBsB 2
1
1)()(
Roots: cNp js 2/1)1( cps
22
12
N
ksp
N=3
c
x
x
xx
6
8
6
6
6
4
3
)(j
c
j
c
j
c
c
eseses
sB
x
xN=4
8
11
8
9
8
7
8
5
4
)(j
c
j
c
j
c
j
c
c
eseseses
sB
c
x
x
x
x
x
x
x
x
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