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ELEC361: Signals And SystemsDr. Aishy Amer, Concordia University, Electrical and Computer Engineering

Review: Partial Fraction ExpansionPartial fractions are several fractions whose sum equals a given fraction

Purpose: Working with transforms requires breaking complex fractions into simpler fractions to allow use of tables of transforms

1111

)1)(1()1(5)1(6

15

16

15

16

1111

2

2

−−

=−+++−

=−

++

•

−+

+=

−−

•

ss

ssss

ss

ssss

2

Review:Partial Fraction Expansion

32)3()2(1

++

+=

+++

sB

sA

sss

( ))3()2(

2)3()3()2(

1++

+++=

+++

sssBsA

sss

32

21

)3()2(1

++

+−

=++

+ssss

s

1=+ BA 123 =+ BA

Expand into a term for each factor in the denominator.Recombine right hand side

Equate terms in s and constant terms. Solve.Each term is in a form so that inverse Laplace transforms can be applied.

3

Review:Partial Fraction Expansion: Different terms of 1st degree

To separate a fraction into partial fractions when its denominator can be divided into different terms of first degree, assume an unknown numerator for each fraction

561

11)1()111(

)1)(1()()(

)1)(1()1()1(

)1()1()1()111(

2

2

==⇒⎭⎬⎫

−=−=+

−−

=−+−++

=−+

++−=

−+

+=

−−

BAABBA

ss

ssABsBA

sssBsA

sB

sA

ss

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Review: Partial Fraction Expansion: Repeated terms of 1st degree

When the factors of the denominator are of the first degree but some are repeated, assume unknown numerators for each factor

If a term is present twice, make the fractions the corresponding term and its second powerIf a term is present three times, make the fractions the term and its second and third powers

2114

321

)()2()1()1(43)()()(

)1()1()1()1(

)1()1()1()1(43

2

22

33

2

323

2

===⇒⎪⎭

⎪⎬

⎫

=++=+

=

+++++=

++++=++=

=+

++

+++=

++

++

+=

+++

CBACBA

BAA

CBAsBAAsCsBsAsssN

sDsN

sC

ssBsA

sC

sB

sA

sss

5

Review: Partial Fraction Expansion: Different quadratic terms

When there is a quadratic term, assume a numerator of the form Bs + C

05.05.0120

0)2()()(1

)1()1()2(1)2()1()2)(1(

1

2

2

22

===⇒⎪⎭

⎪⎬

⎫

=+=++

=+

++++++=

++++++=

+++

++

=+++

CBACACBA

BACAsCBAsBA

sCsBsssAss

CBssA

sss

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Review:Partial Fraction Expansion: Repeated quadratic terms

When there is repeated quadratic term, assume two numerator of the form Bs + C and Ds+E

05.0025.025.0

1240324

0235022

0)1()1()2)(1(

)2)(1()2(1)2()2()1()2)(1(

1

2

222

22222

=−==−==

⇒

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

=++=++++

=+++=++

=+

++++++++

++++++=

+++

+++

++

+=

+++

EDCBA

ECAEDCBA

DCBACBA

BAsEsDssssC

sssBsssAss

EDsss

CBssA

sss