{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

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

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

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• 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.

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

• 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.

Dana University of Pennsylvania ‘17, Course Hero Intern

• 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.

Jill Tulane University ‘16, Course Hero Intern