UnitB2_PartialFraction - EE3210 Signal and Systems Unit B2...

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Unformatted text preview: EE3210 Signal and Systems Unit B2 Partial Fraction Expansion EE3210 Signal and Systems: Unit B2 2 Rational Function ♦ A function F(x) is rational if it is a ratio of two polynomials in x: ♦ F(x) is proper if m < n. Otherwise, it is improper. ) ( ) ( ) ( 1 1 1 1 1 1 x Q x P a x a x a x b x b x b x b x F n n n m m m m = + + + + + + = − − − − L L EE3210 Signal and Systems: Unit B2 3 Improper Functions ♦ Improper functions can be written as a sum of polynomial and a proper function, e.g., (which can be obtained by long division.) ♦ Therefore, it is sufficient to consider proper functions only. 3 4 1 1 2 3 4 2 11 9 2 ) ( 2 2 2 3 + + − + + = + + + + + = x x x x x x x x x x F EE3210 Signal and Systems: Unit B2 4 Case 1: Distinct Factor of Q(x) ♦ Consider the case where Q(x) has distinct factors: ♦ F(x) can be expressed as the sum of partial fractions: ) ( ) )( ( ) ( , ) ( 2 1 1 1 1 1 1 1 n n n n m m m m x x x x P n m a x a x a x b x b x b x b x F λ λ λ − − − = <...
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UnitB2_PartialFraction - EE3210 Signal and Systems Unit B2...

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