Inverse Laplace Transform-Theory of Signals And Systems-Lecture Slides, Slides for Signals and Systems. Birla Institute of Technology and Science

Signals and Systems

Description: This lecture is part of lecture series for Signals and Systems course. Dr. Aishwarya Vyasa delivered this lecture at Birla Institute of Technology and Science. It includes: Inverse, Laplace, Transform, Shift, Property, Rational, Exponential, partial, Fraction, Expansion
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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!sT X(s) (Similar to the FT property
x(t!T)" # $ e!j
%
TX(j
%
)
X(s)=e!sT N(s) / D(s)
( )
X1(s)"x1(t)
1 2 4 4 3 4 4
Then b
x(t)=x1(t)t!t"T=x1(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) =
!2
3
e!tu(t)+5
3
e2tu(!t)
"
#
$ %
&
' (0 as t(±)
Ae!tu(t)+Be 2tu(t)
=2
3
e!t!5
3
e2t"
#
$ %
&
' u(!t) Diverges as t( ! )
=!2
3
e!t+5
3
e2t"
#
$ %
&
' 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 X1(s) and X2(s)
ROC can be bigger (due to pole-zero cancellation)
E.g. x1(t) = x2(t) and a = -b
Then a x1(t) + b x2(t) = 0 X(s) = 0
ROC entire s-plane
ax1(t)+bx2(t)! " # aX1(s)+bX2(s)
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