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MIT6_003S10_lec04

MIT6_003S10_lec04 - 6.003 Signals and Systems...

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6.003: Signals and Systems Continuous-Time Systems February 11, 2010
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Previously: DT Systems Verbal descriptions: preserve the rationale. “Next year, your account will contain p times your balance from this year plus the money that you added this year.” Difference equations: mathematically compact. y [ n + 1] = x [ n ] + py [ n ] Block diagrams: illustrate signal flow paths. + Delay p x [ n ] y [ n ] Operator representations: analyze systems as polynomials. (1 p R ) Y = R X
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Analyzing CT Systems Verbal descriptions: preserve the rationale. “Your account will grow in proportion to the current interest rate plus the rate at which you deposit.” Differential equations: mathematically compact. dy ( t ) = x ( t ) + py ( t ) dt Block diagrams: illustrate signal flow paths. + t −∞ ( · ) dt p x ( t ) y ( t ) Operator representations: analyze systems as polynomials. (1 p A ) Y = A X
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Differential Equations Differential equations are mathematically precise and compact. r 0 ( t ) r 1 ( t ) h 1 ( t ) dr 1 ( t ) = r 0 ( t ) r 1 ( t ) dt τ Solution methodologies: general methods (separation of variables; integrating factors) homogeneous and particular solutions inspection Today: new methods based on block diagrams and operators , which provide new ways to think about systems’ behaviors.
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Block Diagrams Block diagrams illustrate signal flow paths. DT: adders, scalers, and delays represent systems described by linear difference equations with constant coefficents. + Delay p x [ n ] y [ n ] CT: adders, scalers, and integrators represent systems described by a linear differential equations with constant coefficients. + t −∞ ( · ) dt p x ( t ) y ( t )
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Operator Representation CT Block diagrams are concisely represented with the A operator . Applying A to a CT signal generates a new signal that is equal to the integral of the first signal at all points in time. Y = A X is equivalent to t y ( t ) = x ( τ ) −∞ for all time t .
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Evaluating Operator Expressions As with R , A expressions can be manipulated as polynomials. + + A A X Y W t w ( t ) = x ( t ) + x ( τ ) −∞ t y ( t ) = w ( t ) + w ( τ ) −∞ t t t τ 2 y ( t ) = x ( t ) + x ( τ ) + x ( τ ) + x ( τ 1 ) 1 2 −∞ −∞ −∞ −∞ W = (1 + A ) X Y = (1 + A ) W = (1 + A )(1 + A ) X = (1 + 2 A + A 2 ) X
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Evaluating Operator Expressions Expressions in A can be manipulated using rules for polynomials. Commutativity: A (1 − A ) X = (1 − A ) A X Distributivity: A (1 − A ) X = ( A − A 2 ) X Associativity: (1 − A ) A (2 − A ) X = (1 − A ) A (2 − A ) X
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Check Yourself A p + X Y A p + X Y A p + X Y ˙ y ( t ) = ˙ x ( t ) + p ¨ y ( t ) ˙ y ( t ) = x ( t ) + py ( t ) ˙ y ( t ) = px ( t ) + py ( t ) Which best illustrates the left-right correspondences? 1. 2. 3. 4. 5. none
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Check Yourself A p + X Y A p + X Y A p + X Y ˙ y ( t ) = ˙ x ( t ) + p ¨ y ( t ) ˙ y ( t ) = x ( t ) + py ( t ) ˙ y ( t ) = px ( t ) + py ( t ) Which best illustrates the left-right correspondences? 4 1. 2. 3. 4. 5. none
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Elementary Building-Block Signals Elementary DT signal: δ [ n ] .
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