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MIT6_003S10_lec08_handout

MIT6_003S10_lec08_handout - 6.003 Signals and Systems...

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X 6.003: Signals and Systems 6.003: Signals and Systems Convolution March 2, 2010 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. Lecture 8 March 2, 2010 Mid-term Examination #1 Tomorrow, Wednesday, March 3, 7:30-9:30pm. 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. Convolution Representing a system by a single signal. + R Y X z 0 + A Y s 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 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 step-by-step analysis. 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 1
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6.003: Signals and Systems Lecture 8 March 2, 2010 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 Reponses to parts are easy to compute if system is time-invariant . 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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