MIT6_003S10_lec08

# MIT6_003S10_lec08 - 6.003 Signals and Systems Convolution...

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6.003: Signals and Systems Convolution March 2, 2010

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Mid-term Examination #1 Tomorrow, Wednesday, March 3, No recitations tomorrow. Coverage: Representations of CT and DT Systems Lectures 1–7 Recitations 1–8 Homeworks 1–4 Homework 4 will not collected or graded. Solutions are posted. Closed book: 1 page of notes ( 8 1 2 × 11 inches; front and back). Designed as 1-hour exam; two hours to complete. 7:30-9:30pm.
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. X + Y X R A Y + s 0 z 0 Operator representations: analyze systems as polynomials. Y 1 Y A = = X 1 z 0 R X 1 s 0 A Transforms: representing diﬀ. equations with algebraic equations. z 1 H ( z H ( s z z 0 s s 0

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Convolution Representing a system by a single signal.
Responses to arbitrary signals Although we have focused on responses to simple signals ( δ [ n ] ( t ) ) are generally interested in responses to more complicated signals. How do compute responses to a more complicated input signals? No problem for diﬀerence equations / block diagrams. use step-by-step analysis.

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Check Yourself Example: Find y [3] + + R R X Y when the input is x [ n ] n 1. 1 2. 2 3. 3 4. 4 5. 5 0. none of the above
Responses to arbitrary signals Example. + + R R 0 0 0 0 x [ n ] y [ n ] n n

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Responses to arbitrary signals Example. + + R R 1 0 0 1 x [ n ] y [ n ] n n
Responses to arbitrary signals Example. + + R R 1 1 0 2 x [ n ] n y [ n ] n

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Responses to arbitrary signals Example. + + R R 1 1 1 3 x [ n ] n y [ n ] n
Responses to arbitrary signals Example. + + R R 0 1 1 2 x [ n ] n y [ n ] n

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Responses to arbitrary signals Example. + + R R 0 0 1 1 x [ n ] n y [ n ] n
Responses to arbitrary signals Example. + + R R 0 0 0 0 x [ n ] n y [ n ] n

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Check Yourself What is y [3] ? 2 + + R R 0 0 0 0 x [ n ] n y [ n ] n
Alternative: Superposition Break input into additive parts and sum the responses to the parts. x [ n ] n y [ n ] n = n + + + + = n 10123 4 5 n n n n 4 5

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Superposition Break input into additive parts and sum the responses to the parts. x [ n ] n y [ n ] n = n + + + + = n 10123 4 5 n n n n 4 5 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 y 1 [ n ] x 1 [ n ] and system x 2 [ n ] y 2 [ n ] the system is linear if αx 1 [ n ]+ βx 2 [ n ] αy 1 [ n βy 2 [ n ] system is true for all α and β .

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Superposition Break input into additive parts and sum the responses to the parts. x [ n ] n y [ n ] n = n + + + + = n 10123 4 5 n n n n 4 5 Superposition works if the system is linear .
Superposition Break input into additive parts and sum the responses to the parts. x [ n ] n y [ n ] n = n + + + + = n 10123 4 5 n n n n 4 5 Reponses to parts are easy to compute if system is time-invariant .

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Time-Invariance A system is time-invariant if delaying the input to the system simply delays the output by the same amount of time.
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MIT6_003S10_lec08 - 6.003 Signals and Systems Convolution...

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