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# lec08-handout-6up - 6.003 Signals and Systems Lecture 8...

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6.003: Signals and Systems Lecture 8 October 6, 2009 1 6.003: Signals and Systems Operator Representations for Continuous-Time Systems October 6, 2009 Mid-term Examination #1 Tomorrow, October 7, 7:30-9:30pm, Walker Memorial. No recitations tomorrow. Coverage: DT Signals and Systems Lectures 1–5 Homeworks 1–4 Homework 4 includes practice problems for mid-term 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 1-hour exam; two hours to complete. Analyzing CT Systems We have used differential equations to represent CT systems. r 0 ( t ) r 1 ( t ) h 1 ( t ) τ ˙ r 1 ( t ) = r 0 ( 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. + 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 ) = t −∞ x ( τ ) 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 ) + t −∞ x ( τ ) y ( t ) = w ( t ) + t −∞ w ( τ ) y ( t ) = x ( t ) + t −∞ x ( τ ) + t −∞ x ( τ ) + t −∞ τ 2 −∞ 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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6.003: Signals and Systems Lecture 8 October 6, 2009 2 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 Continuous-Time Feedback What is the response of a CT system with feedback?
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lec08-handout-6up - 6.003 Signals and Systems Lecture 8...

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