MIT6_003S10_lec04_handout

MIT6_003S10_lec04_handout - 6.003: Signals and Systems...

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± 6.003: Signals and Systems Lecture 4 February 11, 2010 6.003: Signals and Systems Continuous-Time Systems 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 February 11, 2010 Analyzing CT Systems Differential Equations 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. y ( t ) Operator representations: analyze systems as polynomials. (1 p A ) Y = A X Differential equations are mathematically precise and compact. r 0 ( t ) h 1 ( t ) r 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. + ± t −∞ ( · ) dt p x ( t ) 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 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 . 1
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± ± ² ³ ² ³ 6.003: Signals and Systems Lecture 4 February 11, 2010 Evaluating Operator Expressions Evaluating Operator Expressions As with R , A expressions can be manipulated as polynomials.
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This note was uploaded on 11/07/2011 for the course ELECTRICA 6.003 taught by Professor Staff during the Summer '10 term at MIT.

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MIT6_003S10_lec04_handout - 6.003: Signals and Systems...

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