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# lec10-handout-6up - 6.003 Signals and Systems Lecture 10...

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6.003: Signals and Systems Lecture 10 October 15, 2009 1 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 0 y [ n 1] ˙ y ( t ) = x ( t ) + s 0 y ( t ) Block diagrams: illustrate signal ﬂow paths. + R z 0 X Y + A s 0 X Y Operator representations: analyze systems as polynomials. Y X = 1 1 z 0 R Y X = A 1 s 0 A Transforms: representing diﬀ. equations with algebraic equations. H ( z ) = z z z 0 H ( s ) = 1 s s 0 Convolution Representing a system by a single signal. Responses to arbitrary signals Although we have focused on responses to simple signals ( δ [ n ] ( 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 step-by-step analysis. Check Yourself What is y [3] ? + + R R X Y x [ n ] n y [ n ] 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 .

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6.003: Signals and Systems Lecture 10 October 15, 2009 2 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 Reponses to parts are easy to compute if system is time-invariant . Time-Invariance
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lec10-handout-6up - 6.003 Signals and Systems Lecture 10...

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