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# l14 - Properties of CT Convolution and Convolution Systems...

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Properties of CT Convolution and Convolution Systems, Examples Outline: Properties of convolution - commutativity, - associativity, - distributivity. Properties of continuous-time (CT) convolution systems - linearity and time invariance, - composition, - derivative property. Examples: applying convolution properties to simplify convolution computation. Reading : Sections 9-7 and 9-8. EE 224, #14 1

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Properties of Convolution The convolution of any two functions x and h is ( x h )( t ) = + -∞ x ( τ ) h ( t - τ ) dτ. If we make the substitution τ 1 = t - τ , then τ = t - τ 1 , = - 1 , and ( x h )( t ) = -∞ + x ( t - τ 1 ) h ( τ 1 ) ( - 1 ) = + -∞ x ( t - τ 1 ) h ( τ 1 ) 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

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Associativity of convolution can be shown as follows: ( x ( g h ))( t ) = + -∞ x ( τ 1 ) [ ( g h )( t - τ 1 ) ] 1 = + -∞ x ( τ 1 ) + -∞ g ( τ 2 ) h ( t - τ 1 - τ 2 ) 2 1 τ 3 = τ 1 + τ 2 , dτ 3 = 2 = + -∞ x ( τ 1 ) + -∞ g ( τ 3 - τ 1 ) h ( t - τ 3 ) 3 1 = + -∞ + -∞ x ( τ 1 ) g ( τ 3 - τ 1 ) 1 ( x g )( τ 3 ) h ( t - τ 3 ) 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 ) = + -∞ x ( τ ) [ g ( t - τ ) + h ( t - τ )] = + -∞ x ( τ ) g ( t - τ ) + + -∞ x ( τ ) h ( t - τ ) = ( 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

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Properties of Convolution Systems (Sec. 9-8) 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 α, β , h ( α x 1 + β x 2 ) = α ( h x 1 ) + β ( 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 . EE 224, #14 6
Since convolution is commutative, the convolution systems are also commutative, i.e. these two cascade connections have the same response: Many operations are linear time-invariant (LTI), e.g. - integration, - differentiation, and - time shift and can therefore be written as convolutions, implying that they all commute. Convolution systems are time-invariant : If we shift the input signal x ( t ) by t 0 , i.e. apply the input x t 0 ( t ) = x ( t - t 0 ) to the convolution system H , the output is w ( t ) = y ( t - t 0 ) EE 224, #14 7

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where w ( t ) = H{ x t 0 ( t ) } and y ( t ) = H{ x ( t ) } . In words: convolution systems commute with time shift.
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l14 - Properties of CT Convolution and Convolution Systems...

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