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Unformatted text preview: Properties of CT Convolution and Convolution Systems, Examples Outline: Properties of convolution commutativity, associativity, distributivity. Properties of continuoustime (CT) convolution systems linearity and time invariance, composition, derivative property. Examples: applying convolution properties to simplify convolution computation. Reading : Sections 97 and 98. EE 224, #14 1 Properties of Convolution The convolution of any two functions x and h is ( x ? h )( t ) = Z +  x ( ) h ( t ) d. If we make the substitution 1 = t , then = t 1 , d = d 1 , and ( x ? h )( t ) = Z + x ( t 1 ) h ( 1 ) ( d 1 ) = Z +  x ( t 1 ) h ( 1 ) d 1 = ( h ? x )( t ) implying that CT convolution is commutative . Note: If we have two signals to convolve, we can choose which of the two we hold constant, and which we flip and drag. Typically, one way will be easier than the other. EE 224, #14 2 Back to Example from handout #13: If we convolve three functions f, g , and h , then ( x ? ( g ? h ))( t ) = (( x ? g ) ? h )( t ) implying that convolution is associative . EE 224, #14 3 Associativity of convolution can be shown as follows: ( x ? ( g ? h ))( t ) = Z +  x ( 1 ) [ ( g ? h )( t 1 ) ] d 1 = Z +  x ( 1 ) h Z +  g ( 2 ) h ( t 1 2 ) d 2 i d 1 3 = 1 + 2 , d 3 = d 2 = Z +  x ( 1 ) h Z +  g ( 3 1 ) h ( t 3 ) d 3 i d 1 = Z +  h Z +  x ( 1 ) g ( 3 1 ) d 1 i  {z } ( x?g )( 3 ) h ( t 3 ) d 3 = (( x ? g ) ? h )( t ) . Combining the commutative and associate properties, x ? g ? h = x ? h ? g = = h ? g ? x. We can perform the convolutions in any order. EE 224, #14 4 Convolution is also distributive : x ? ( g + h ) = x ? g + x ? h easily shown by writing out the convolution integral, ( x ? ( g + h ))( t ) = Z +  x ( ) [ g ( t ) + h ( t )] d = Z +  x ( ) g ( t ) d + Z +  x ( ) h ( t ) d = ( x ? g )( t ) + ( x ? h )( t ) . Note: Together, the commutative, associative, and distributive properties mean that there is an algebra of signals , where addition is like arithmetic or ordinary algebra, and multiplication is replaced by convolution. EE 224, #14 5 Properties of Convolution Systems (Sec. 98) Convolution integral properties have important consequences for convolution systems: Convolution systems are linear : for all signals x 1 and x 2 and arbitrary finite constants , , h ? ( x 1 + x 2 ) = ( h ? x 1 ) + ( h ? x 2 ) . Composition of convolution systems corresponds to convolution of impulse responses. The cascade connection of two convolution systems y = ( x ? f ) ? g is the same as a single convolution system with impulse response h = f ? g ....
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 Spring '09

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