##### Document information

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**How Do We Perform Inverse Laplace Transform?
**

• In 6.003, we will only deal with Laplace transform that are:
1) Rational, *i.e.* *X*(*s*) = *N*(*s*)/D(*s*) ;

And/or:
2) exponential, *i.e.* *X*(*s*) = *e-sT* .

• For case 2), use shift property

• For case 1), use PFE.

• For case 1) & 2), *i.e.
*

*x*(*t *! *T *)" # $ *e*! *sTX*(*s*) (Similar to the *FT* property

*x*(*t *! *T *)" # $ *e
*! *j*%*T
*

*X*( *j*%)

*X*(*s*) = *e
*!*sT
*

*N*(*s*) / *D*(*s*)( )
*X
*1
(*s*)"*x
*

1
(*t *)

1 2 4 3 4 4

*Then *b

*x*(*t*) = *x
*1
( *t*)

*t*!*t*"*T *= *x*1 (*t *" *T *)

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**Inverse Laplace Transforms Via Partial Fraction
Expansion and Properties
**

**Example:
**

Three possible ROC’s — corresponding to three *different* signals

Recall

*X*(*s*) =
*s *+ 3

*s *+1( ) *s *! 2( )
=

*A
*

*s *+1
+

*B
*

*s *! 2

*A *= !
2

3
, *B *=

5

3

1

*s *+ *a
*, Re(*s*) < !*a *" # $ ! *e
*

!*at
u*(!*t*) left - sided

*and
*1

*s *+ *a
*, Re(*s*) > !*a *" # $ *e
*

!*at
u*(*t*) right - sided

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**Example (cont.)
**ROC I: — Left-sided signal.

*x*(*t*) =

ROC II: — Two-sided signal, has Fourier Transform.

*x*(*t*) =

ROC III: — Right-sided signal.

*x*(*t*) =

! *Ae
*!*t
u*(!*t*) ! *Be
*

2*t
u*(!*t*)

! 2

3
*e
*! *t
u*(*t*) +

5

3
*e
*

2*t
u*(!*t*)"

# $
%
& '
( 0 as *t*( ± )

*Ae
*! *t
u*(*t*) + *Be
*

2*t
u*(*t*)

= 2

3
*e
*! *t *!

5

3
*e
*

2*t*"
# $

%
& '
*u*(!*t*) Diverges as *t*( !)

= ! 2

3
*e
*!*t
*

+ 5

3
*e
*

2*t*"
# $

%
& '
*u*(*t*) Diverges as *t*( + )

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**Properties of Laplace Transforms
**

• Many parallel properties of the CTFT, but for Laplace transforms we need to determine implications for the ROC

• For example:

Linearity

ROC at least the intersection of ROC’s of *X*1(*s*) and *X*2(*s*)

ROC can be bigger (due to pole-zero cancellation)

*E.g. x*1(*t*) = *x*2(*t*) and *a* = -*b
*Then *a x*1(*t*) + *b x*2(*t*) = 0 → *X*(*s*) = 0

⇒ ROC entire *s*-plane

*ax
*1
( *t*) + *bx
*

2
(*t*) ! " # *aX
*

1
(*s*) + *bX
*

2
(*s*)

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**Time Shift
**

*x*(*t *! *T *) " # $ *e
*!*sT
*

*X*(*s*) , same ROC as *X*(*s*)

*e
*! *sT
*

*s *+ 2
, Re(*s*) > !2 " # $ *e
*

!2 *t
u*(*t*)

*t*# *t*!*T
*

! *T *= " 3

*e
*3*s
*

*s *+ 2
, Re(*s*) > "2 # $ % *e*"2(*t*+3)*u*(*t *+ 3)

*E*.*g*.,

*e*3*s
*

*s *+ 2
, Re(*s*) > !2 " # $ ?

Causal or not?

8

**Time-Domain Differentiation
**

*x*(*t*) =
1

2!*j
X*(*s*)*e
*

*st
ds
*

" # *j*$

"+ *j*$

% ,
*dx*(*t*)

*dt
*=
1

2!*j
sX*(*s*)*e
*

*st
ds
*

" # *j*$

" + *j*$

%

ROC could be bigger than the ROC of *X*(*s*), if there is pole-zero
cancellation. * E.g.
*

*x*(*t*) = *u*(*t*) ! " #
1

*s
*, Re(*s*) > 0

** s**-

**Domain Differentiation**

!*tx*(*t *) " # $
*dX*(*s*)

*ds
*, with same ROC as *X*(*s*) (Derivation is

similar to
*d
*

*dt
*

! *s*)

!

*dx*(*t*)
*dt
*

" # $ *sX*(*s*), with ROC containing the ROC of *X*(*s*)

*dx*(*t*)

*dt
*= !(*t *) " # $ 1 = *s *%

1

*s
*ROC = entire *s *& plane

*E*.*g*. *te
*!*at
u*( *t*) " # $ !

*d
*

*ds
*

1

*s *+ *a
*

% & '

( ) *

= 1

*s *+ *a*( )2
, Re(*s*) > !*a
*

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**Convolution Property
**

*x*(*t*) *h*(*t*) *y*(*t*) = *h*( *t*)! *x*(*t*)

For *x*(*t*)! " # *X*(*s*), *y*(*t*)! " # *Y*(*s*), *h*(*t*)! " # *H*(*s*)

• This is the main reason why Laplace transform is useful
in solving problems involving LTI systems, just like the
Fourier transform. *However*, because Laplace transform
and its inverse transform are not *symmetric*, the reverse
relation

*x*(*t*)•*h*(*t*) *X*(*s*)∗*H*(*s*)

is *not* true, different from Fourier transform. Thus,
L-transform is *not* useful in modulation and sampling.

×

Then *Y *(*s*) = *H*(*s*) ! *X*(*s*)

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**Convolution Property (cont.)
**

ROC of *Y*(*s*) = *H*(*s*)*X*(*s), *at least the overlap of the ROC’s
of *H*(*s*) & *X*(*s*)

a) ROC could be empty if there is no overlap between the two ROCs

*E.g. x*(*t*) = *etu*(*t*) *Re*(*s*) > 1 and *h*(*t*) = -*e-tu*(*-t*) *Re*(*s*) < -1

b) ROC could be larger than the overlap of the two.

*E.g. x*(*t*)∗*h*(*t*) = δ(*t*) ⇒ ROC entire *s*-plane.

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**The System Function of an LTI System
**

*h*(*t*) ! " # *H*(*s*) — the system function

The system function characterizes the system ⇓

System dynamics correspond to properties of *H*(*s*) and its ROC

A first example:

System is stable

⇒ The frequency response *H*(*j*ω), which is the Fourier transform of
*h*(*t*), is well defined.

! " # *h*(*t*) *dt
*$%

+%

& < % ' ROC of *H *(*s*)

includes the *j*( axis

*x*(*t*) *h*(*t*) *y*(*t*)

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**Geometric Evaluation of Rational Laplace
Transforms
**

**Example #1: ***X*1(*s*) = *s* - *a *A first-order zero

Graphic evaluation
of *X*1(*s*1) = *s*1 - *a
*

⇒|*X*1(*s*1)| = |*s*1 - *a|
& *∠*X*1(*s*1) = ∠(*s*1 - *a*)

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**Example #2: **A first-order pole

*X
*2
(*s*) =

1

*s *! *a
*=

1

*X
*1
(*s*)

**Example #3: **A higher-order rational Laplace transform

*and *!*X*(*s*) = !*M *+ !
*i*=1

*R
*

" *s *# $*i*( ) # !
*j*=1

*P
*

" *s *#% *j*( )

! *X*2 (*s*) =
1

*X*1(*s*)
log *X*2 (*s*) = " log*X*1(*s*)( )

*and *!*X
*2
(*s*) = "!*X
*

1
(*s*)

*X*(*s*) = *M
s *! "*i*( )*i*=1

*R
*

#
*s *!$ *j*( )*j*=1

*P
*

#

*X*(*s*) = *M
s *! "*ii*=1

*R
*

#
*s *! $ *jj*=1

*P
*

#

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**First-Order System ***H*(*s*) = 1
*s*! +1

= 1 / !

*s *+1 / !
, Re(*s*) > "

1

!

Graphical evaluation of *H*(*j*ω):
*H*( *j*! ) =

1/ "

*j*! + 1 / "

*h*(*t*) =
1

!
*e
*" *t*! *u*(*t*) *s*(*t*) = [1 " *e
*

" *t*! ] *u*(*t *)

!*H*( *j*") = #$ = # tan#1
"

1 / %

& ' (

) * + = # tan#1("% )

| *H*( *j*! ) | =
1 / "

! 2 + (1 / " )2 =

1 ! = 0

1 / 2 ! = 1 / "

~ 1 /!" ! >>1 / "

#

$ %

& %

=

0 ! = 0

"# / 4 ! =1 / $

"# / 2 ! >>1 / $

%

& '

( '

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**Bode Plot of the first-order system
**

*H*( *j*! ) =
1/ "

*j*! + 1 / "

20 dB/decade

changes by -π/2 =

0 ! = 0

"# / 4 ! =1 / $

"# / 2 ! >>1 / $

%

& '

( '

| *H*( *j*! ) | =
1 / "

! 2 + (1 / " )2

=

1 ! = 0

1/ 2 ! =1 / "

~1 / !" ! >>1 / "

#

$ %

& %

!*H*( *j*") = #$ = # tan
#1
("% )

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**Second-Order System **(causal)
*E.g.* *LCR* circuit, a mechanical system with a spring and friction

*H*(*s*) =
! *n
*

2

*s
*2
+ 2"!*ns *+!*n
*

2
ROC Re(*s*) > Re(*pole*)

0 < ! <1 " complex poles

! =1 "

— *Under*damped

double pole at *s *= #$*n
*

! >1 "

— *Critically * damped

2 poles on negative real axis

— *Over*damped

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**Demo **Pole-zero diagrams, frequency response, and step
response of first-order and second-order CT causal systems

Important point: Dynamics of a
*rational *system is completely
characterized by its pole-zero
diagram (except for a prefactor).

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**Bode Plot of a Second-order System
**

40 dB/decade Top is flat when ζ= 1/√ 2 = 0.707 ⇒ a LPF for ω < ωn

changes by -π

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**Unit-impulse and Unit-step Response of a
Second-order System
**

No oscillations when ζ ≥ 1

⇒ Critically (=) and over (>) damped.

20

**Next lecture covers
**O&W pp. 698-720

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First-order **All-Pass System
**

*H*(*s*) =
*s *! *a
*

*s *+ *a
*, Re(*s*) > !*a *(*a *> 0)

(a) Two vectors have

the same length

= (! "# 2 ) "#

2

(b) !*H*( *j*" ) = #1 $ #2

= ! " 2# 2

=

! " = 0

! / 2 " = *a
*

~ 0 " >> *a
*

#

$ %

& %

⇒ |*H*(*j*ω)| = 1

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