L77 - Fall 2003 Math 308/501502 7 The Laplace Transform 7.7...

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Fall 2003 Math 308/501–502 7 The Laplace Transform 7.7 Convolution Mon, 20/Oct c ± 2003, Art Belmonte Summary Definition The convolution of two piecewise continuous functions f and g is ( f * g )( t ) = Z t 0 f ( t - v) g (v) d v Properties of the convolution For piecewise continuous functions f , g ,and h ,wehave 1. f * g = g * f (The convolution is commutative .) 2. f * ( g + h ) = f * g + f * h (It is distributive .) 3. ( f * g ) * h = f * ( g * h ) (It is associative .) 4. f * 0 = 0 (A function convolved with zero is zero.) The Convolution Theorem If f and g are piecewise continuous functions on [0 , ) of exponential order with Laplace transforms L { f } = F ( s ) and L { g } = G ( s ) then L { f * g } = F ( s ) G ( s ). Initial value problems The transfer function H ( s ) of a linear system L [ y ] = g ( t ) with all initial conditions zero is H ( s ) = Y ( s )/ G ( s ) ; i.e., the Laplace transform of the output function y ( t ) divided by that of the input function g ( t ) .If L [ y ] has constant coefficients, then H ( s ) is the reciprocal of p ( s ) , the characteristic polynomial of the corresponding homogeneous equation; i.e., H ( s ) = 1 / p ( s ) . The impulse response function is the inverse Laplace transform of the transfer function; i.e., h ( t ) = L - 1 { H ( s ) } . Physically, it
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L77 - Fall 2003 Math 308/501502 7 The Laplace Transform 7.7...

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