ECE
ECE634S09_L2_PartialFraction

ECE634S09_L2_PartialFraction - ECE634 Signals and Systems...

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ECE634 Signals and Systems II, Spring 2009 - Lecture 2, January 23 Daniel S. Brogan 1 B.5 Partial Fraction Expansion: B.5-1 Method of Clearing Fractions ( ) ( )( ) ( ) ( ) 3 2 1 2 2 4 16 14 1 2 3 1 2 3 2 s s s A B C D T s s s s s s s s + + = = + + + + + + + + + + Clear fractions by multiplying by denominator… ( ) ( ) ( )( )( ) ( )( ) ( )( ) 2 2 3 2 4 16 14 2 3 1 2 3 1 3 1 2 s s s A s s B s s s C s s D s s + + = + + + + + + + + + + + + ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 2 2 2 2 2 4 16 14 4 4 3 3 2 3 4 3 1 4 4 s s s A s s s B s s s C s s D s s s + + = + + + + + + + + + + + + + + ( ) ( ) ( ) ( ) 3 2 3 2 3 2 2 3 2 4 16 14 7 16 12 6 11 6 4 3 5 8 4 s s s A s s s B s s s C s s D s s s + + = + + + + + + + + + + + + + + ( ) ( ) ( ) ( ) 3 2 3 2 4 16 14 7 6 5 16 11 4 8 12 6 3 4 s s s A B D s A B C D s A B C D s A B C D + + = + + + + + + + + + + + + + + 4 (1) 7 6 5 16 (2) 16 11 4 8 14 (3) 12 6 3 4 0 (4) A B D A B C D A B C D A B C D + + = + + + = + + + = + + + = Multiply Eq. (2) by 4 and subtract Eq. (3) from it, and multiply Eq. (2) by 3 and subtract Eq. (4) from it to remove C . Then subtract equation (6) from Eq. (5) to obtain Eq. (7). Finally, subtract Eq. (1) from Eq. (7) and divide by 2 to find A . 4 (1) 12 13 12 50 (5) 9 12 11 48 (6) A B D A B D A B D + + = + + = + + = 4 (1) 12 13 12 50 (5) 3 2 (7) A B D A B D A B D + + = + + = + + = 2 2 A = − 1 A = − Substitute in A and combine Eq. (1) and Eq. (5) to find B and D .
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  • Spring '09
  • messner

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