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
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.
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 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 .
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 ² ± τ 2 ³ y ( t x ( t x ( τ ) + x ( τ ) + x ( τ 1 ) 1 2 −∞ −∞ −∞ −∞ W =(1+ A ) X Y A ) W A )(1 + A ) X =(1+2 A + A 2 ) X

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± ² ± ² Evaluating Operator Expressions Expressions in A can be manipulated using rules for polynomials. Commutativity: A (1 −A ) X =(1 ) A X Distributivity: A (1 ) X =( A−A 2 ) X Associativity: (1 ) A (2 ) X ) A (2 ) X
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
± Elementary Building-Block Signals Elementary DT signal: δ [ n ] .

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## This note was uploaded on 12/14/2011 for the course EE 6.003 taught by Professor Freeman during the Fall '11 term at MIT.

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MIT6_003S10_lec04 - 6.003: Signals and Systems...

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