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Unformatted text preview: 6.003: Signals and Systems Convolution October 15, 2009 Multiple Representations of CT and DT Systems Verbal descriptions: preserve the rationale. Difference/differential equations: mathematically compact. y [ n ] = x [ n ] + z y [ n − 1] ˙ y ( t ) = x ( t ) + s y ( t ) Block diagrams: illustrate signal flow paths. + R z X Y + A s X Y Operator representations: analyze systems as polynomials. Y X = 1 1 − z R Y X = A 1 − s A Transforms: representing diff. equations with algebraic equations. H ( z ) = z z − z H ( s ) = 1 s − s Convolution Representing a system by a single signal. Responses to arbitrary signals Although we have focused on responses to simple signals ( δ [ n ] ,δ ( t ) ) we are generally interested in responses to more complicated signals. How do we compute responses to a more complicated input signals? No problem for difference equations / block diagrams. → use stepbystep analysis. Responses to arbitrary signals Example. + + R R X Y x [ n ] n y [ n ] n Check Yourself What is y [3] ? + + R R X Y x [ n ] n y [ n ] n Responses to arbitrary signals Example. + + R R x [ n ] n y [ n ] n Responses to arbitrary signals Example. + + R R 1 1 x [ n ] n y [ n ] n Responses to arbitrary signals Example. + + R R 1 1 2 x [ n ] n y [ n ] n Responses to arbitrary signals Example. + + R R 1 1 1 3 x [ n ] n y [ n ] n Responses to arbitrary signals Example. + + R R 1 1 2 x [ n ] n y [ n ] n Responses to arbitrary signals Example. + + R R 1 1 x [ n ] n y [ n ] n Responses to arbitrary signals Example. + + R R x [ n ] n y [ n ] n Check Yourself What is y [3] ? 2 + + R R x [ n ] n y [ n ] n Alternative: Superposition Break input into additive parts and sum the responses to the parts. n x [ n ] y [ n ] n = + + + + = n − 1 0 1 2 3 4 5 n n n n − 1 0 1 2 3 4 5 n Superposition Break input into additive parts and sum the responses to the parts. n x [ n ] y [ n ] n = + + + + = n − 1 0 1 2 3 4 5 n n n n − 1 0 1 2 3 4 5 n Superposition works if the system is linear . Linearity A system is linear if its response to a weighted sum of inputs is equal to the weighted sum of its responses to each of the inputs. Given system x 1 [ n ] y 1 [ n ] and system x 2 [ n ] y 2 [ n ] the system is linear if system αx 1 [ n ] + βx 2 [ n ] αy 1 [ n ] + βy 2 [ n ] is true for all α and β . Superposition Break input into additive parts and sum the responses to the parts. n x [ n ] y [ n ] n = + + + + = n − 1 0 1 2 3 4 5 n n n n − 1 0 1 2 3 4 5 n Superposition works if the system is linear . Superposition Break input into additive parts and sum the responses to the parts....
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This note was uploaded on 08/24/2011 for the course EECS 6.003 taught by Professor Dennism.freeman during the Spring '11 term at MIT.
 Spring '11
 DennisM.Freeman
 Superposition

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