{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

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

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

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 1 , for p t < p + ; 0 for 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 . NOTE THAT δ p ( t ) = δ( t - p ) . The delta “function” is an example of a generalized function or distribution . Colloquially speaking, δ( 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 -∞ δ p ( t )φ( t ) dt = -∞ δ( t - p )φ( t ) dt = φ( p ) L δ p ( t ) = L { δ( t - p ) } = e - ps L { δ 0 ( t ) } = L { δ( t ) } = 1 d dt u ( t - a ) = δ( 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 ) to the system represented by the symbolic initial

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 ]}