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Unformatted text preview: 6.003: Signals and Systems Operator Representations for ContinuousTime Systems October 6, 2009 Midterm Examination #1 Tomorrow, October 7, 7:309:30pm, Walker Memorial. No recitations tomorrow. Coverage: DT Signals and Systems Lectures 1–5 Homeworks 1–4 Homework 4 includes practice problems for midterm 1. It will not collected or graded. Solutions are posted. Closed book: 1 page of notes ( 8 1 2 × 11 inches; front and back). Designed as 1hour exam; two hours to complete. Analyzing CT Systems We have used differential equations to represent CT systems. r ( t ) r 1 ( t ) h 1 ( t ) τ ˙ r 1 ( t ) = r ( t ) − r 1 ( t ) Methods to solve differential equations: • homogeneous and particular equations • singularity matching • Laplace transform Today: new methods based on block diagrams and operators . 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. + Ú t −∞ ( · ) dt p x ( t ) y ( t ) Operator Representation Block diagrams are concisely represented with operators . We will define the A operator for functional analysis of CT systems. 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 y ( t ) = Ú t −∞ x ( τ ) dτ for all time t . Evaluating Operator Expressions As with R , A expressions can be manipulated as polynomials. + + A A X Y W w ( t ) = x ( t ) + Ú t −∞ x ( τ ) dτ y ( t ) = w ( t ) + Ú t −∞ w ( τ ) dτ y ( t ) = x ( t ) + Ú t −∞ x ( τ ) dτ + Ú t −∞ x ( τ ) dτ + Ú t −∞ Ú τ 2 −∞ x ( τ 1 ) dτ 1 dτ 2 W = (1 + A ) X Y = (1 + A ) W = (1 + A )(1 + A ) X = (1 + 2 A + A 2 ) X 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 ContinuousTime Feedback What is the response of a CT system with feedback? + Ú t −∞ ( · ) dt p x ( t ) y ( t ) Check Yourself + Ú t −∞ ( · ) dt p x ( t ) y ( t ) Find the corresponding differential equation. 1. ˙ y ( t ) = x ( t ) + py ( t ) 2. y ( t ) = ˙ x ( t ) + p ˙ y ( t ) 3. y ( t ) = ˙ x ( t ) + py ( t ) 4. ˙ y ( t ) = x ( t ) + p ˙ y ( t ) 5. none of the above Check Yourself Find the corresponding differential equation. + Ú t −∞ ( · ) dt p x ( t ) y ( t ) ˙ y ( t ) The input to the integrator is ˙ y ( t ) ....
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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

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