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D-ConvProperties

# h t 2 d m j roberts 21807 xt h1

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Unformatted text preview: tion of x t with h t ( ) ( ) h (t ) = y t =x t x ( ) h (t ) d . () Taking the derivative of y t with respect to time, () yt= x ( ) h (t ) d ( ) h (t ) =x t and, invoking the commutativity of convolution, () ( ) h (t ) . y t =x t D.1.5 Area Property () () () Let y t be the convolution of x t with h t ( ) ( ) h (t ) = y t =x t x ( ) h (t ) d () The area under y t is () y t dt = x ( ) h (t ) d dt or, exchanging the order of integration, () y t dt = x ( )d Area of x ( ht ) dt Area of h proving that the area of y is the product of the areas of x and h. D-3 . )d M. J. Roberts - 2/18/07 D.1.6 Scaling Property ( ) ( ) h (t ) and z (t ) = x ( at ) h ( at ) , a > 0 . Let y t = x t ( )(( () )) d Then zt = =a Making the change of variable, xa hat d = d / a , for a > 0 we get () zt = 1 a x ( ) h ( at )d . . Since ( ) ( ) h (t ) = y t =x t () ( x...
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