L78 - Spring 2006 Math 308-505 7 The Laplace Transform 7.8...

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

View Full Document Right Arrow Icon
Spring 2006 Math 308-505 7 The Laplace Transform 7.8 Impulses and The Delta Function Wed, 08/Mar c ± 2006, Art Belmonte Summary Construction of the delta function Let p 0. Recall from §7 . 6 the translated Heaviside function u p ( t ) = u ( t - p ) = ± 0 , t < p ; 1 t p . For p 0, define δ ± p ( t ) = 1 ± ( u p ( t ) - u p + ± ( t ) ) or ² 1 ± , for p t < p + ± ; 0f o r t < p or t p + ±. The Dirac delta function centered at p is defined as the limit δ p ( t ) = lim ± 0 δ ± p ( t ). We denote δ 0 by δ .NOTETHAT δ p ( t ) = δ( t - p ) . The delta “function” is an example of a generalized function or distribution . Colloquially speaking, t ) = ± 0 , t 6= 0 , , t = 0 . Physically, it models a force that concentrates a great energy over a short duration, such as a hammer hitting a nail or a bat hitting a baseball. The delta function is known by its properties, which we now discuss. Properties of the delta function In the following p is a nonnegative constant, φ a function that is continuous near p ,and f a piecewise continuous function. Moreover, we define δ * f = lim ± 0 ( δ ± 0 * f ) .Then ³ -∞ δ p ( t )φ( t ) dt = ³ -∞ t - p t ) = φ( p ) L ´ δ p ( t ) µ = L { t - p ) } = e - ps L { δ 0 ( t ) } = L { t ) } = 1 d u ( t - a ) = t - a ) f * δ = δ * f = f (The first item is called the sifting property . The last item says that δ is the identity for the convolution product.) Impulse response functions [revisited] For constants a , b , c ,the (unit) impulse response function is the solution h ( t )
Background image of page 1

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

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

This note was uploaded on 03/29/2008 for the course MATH 308 taught by Professor Comech during the Spring '08 term at Texas A&M.

Page1 / 3

L78 - Spring 2006 Math 308-505 7 The Laplace Transform 7.8...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online