6.6we - ) d d where = + = Z t Z f ( ) g ( - ) h ( t- ) d d...

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6.6: THE CONVOLUTION INTEGRAL KIAM HEONG KWA Let f and g be two functions on [0 , ). The convolution of these functions is the function (1) ( f * g )( t ) = Z t 0 f ( τ ) g ( t - τ ) whenever the integral is defined. It is easy to see that the convolution is commutative in the sense that (2) f * g = g * f because, upon the substitution σ = t - τ , ( f * g )( t ) = Z t 0 g ( σ ) f ( t - σ ) = ( g * f )( t ) . Some other elementary properties of the convolution are as follows: c ( f * g ) = ( cf ) * g = f * ( cg ), where c R ; f * ( g + h ) = ( f * g ) + ( f * h ) (distributive property); f * ( g * h ) = ( f * g ) * h (associative property). Date : February 28, 2011. 1
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2 KIAM HEONG KWA The first and the second properties follow readily from the basic prop- erties of integration. As for the last property, we have f * ( g * h )( t ) = Z t 0 f ( τ )( g * h )( t - τ ) = Z t 0 f ( τ ) ±Z t - τ 0 g ( ρ ) h ( t - τ - ρ ) ² = Z t 0 Z t - τ 0 f ( τ ) g ( ρ ) h ( t - τ - ρ ) dρdτ = Z t 0 Z t τ f ( τ ) g ( σ - τ ) h ( t - σ
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Unformatted text preview: ) d d where = + = Z t Z f ( ) g ( - ) h ( t- ) d d = Z t Z f ( ) g ( - )] d h ( t- ) d = Z t ( f * g )( ) h ( t- ) d = ( f * g ) * h ( t ) . Theorem 1 (The convolution theorem) . Let f and g be piecewise con-tinuous and have exponential order . If F ( s ) = L f ( s ) and G ( s ) = L g ( s ) , then (3) F ( s ) G ( s ) = L ( f * g )( s ) for s > . A proof can be found in [1, Theorem 2.39, pp. 92-93]. References [1] Joel L. Schi. The Laplace Transform Theory and Applications . Springer-Verlag New York, Inc., 1999....
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This note was uploaded on 11/11/2011 for the course MATH 255.01 taught by Professor Kwa during the Winter '11 term at Ohio State.

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6.6we - ) d d where = + = Z t Z f ( ) g ( - ) h ( t- ) d d...

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