{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

Info icon This preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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 , we have 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
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern