D-ConvProperties

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Unformatted text preview: yz xy xy yz and d = d ) z (t yz xy d yz = x ( ) y ( ) z (t xy yz make the yz )d xy d = yz x i ntegration xy yz change )d of xy d yz variable ( yz ) y ( ) z (t ) d yz d yz make the change of variable . Then x ( ) y( yz xy ) z (t yz )d xy d yz = x ( ) y( yz xy ) z (t yz )d xy d yz = xy xy . Then N ext, in the right-hand = )d i ntegration xy and d = d ( ) y( yz x ( ) y( ) z (t )d d (D.1) or xy x ( ) y( ) z (t )d d (D.2) Except for the names of the variables of integration, the two integrals (D.1) and (D.2) are the same, therefore the integrals are equal and the associativity of convolution is proven. D.1.3 Distributivity Property Convolution is also distributive, () xt () xt () () h1 t + h 2 t () () h1 t + h 2 t = () =x t () () h1 t + x t () ( x t h1 t D-2 () h2 t . ) + h (t ) 2 d M. J. Roberts - 2/18/07 () xt () () h1 t + h 2 t () () )d x t h1 t () () h1 t + h 2 t xt D.1.4 () ( = + () ( x t h2 t () () h1 t + x t =x t () h2 t Differentiation Property () () () Let y t be the convolu...
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This note was uploaded on 02/11/2014 for the course EECS 320 taught by Professor Philips during the Spring '06 term at University of Michigan.

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