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

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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
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