EE3TP4_11b_CTConvolution_Lecture 14

EE3TP4_11b_CTConvolution_Lecture 14 - C-T convolution...

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) ( ) ( ) ( ) ( )] ( ) ( [ t v t x t v t x t v t x dt d = = C-T convolution properties Many of these are the same as for DT convolution. We only discuss the new ones here. See the next slide for the others derivative −∞ t y λ d λ = [ −∞ t x λ ] h t = x t ∗ [ −∞ t h λ ] The properties of convolution help perform analysis and design tasks that involve convolution. For example, the associative property says that (in theory) we can interchange to order of two linear systems … in practice, before we can switch the order we need to check what impact that might have on the physical interface conditions. 1. Derivative Property : 2. Integration Property Let y ( t ) = x ( t ) * h ( t ), then These overheads were originally developed by Mark Fowler at Binghamton University, State University of New York.
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Convolution Properties These are things you can exploit to make it easier to solve convolution problems 1. Commutativity You can choose which signal to “flip” x t ∗ h t = h t ∗ x t 2. Associativity Can change order sometimes one order is easier than another x t ∗ v t ∗ w t = x t ∗ v t ∗ w t 3. Distributivity may be easier to split complicated system h(t) into sum of simple ones we can split complicated input into sum of simple ones (nothing more than “linearity”) OR 4. Convolution with impulses x t ∗ δ t τ = x t τ x t ∗ h 1 t  h 2 t = x t ∗ h 1 t  x t ∗ h 2 t How to do continuous-time graphical convolution?
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Computing” CT Convolution -For D-T systems, convolution is something we do for analysis and for implementation (either via H/W or S/W). -For C-T systems, we do convolution for analysis… nature does convolution for implementation. If we are
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EE3TP4_11b_CTConvolution_Lecture 14 - C-T convolution...

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