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Unformatted text preview: .003: Signals and Systems elations between CT and DT: Insights from Operators and Transforms February 25, 2010 ednesday, March 3, o recitations on the day of the exam. overage: Representations of CT and DT Systems Lectures 17 Recitations 18 Homeworks 14 omework 4 will not collected or graded. Solutions will be posted. losed book: 8 1 1 page of notes ( 2 11 inches; front and back). esigned as 1hour exam; two hours to complete. eview sessions during open oce hours. idterm Examination #1 N C H D R C 7:309:30pm. x ( t ) x ( t ) X A X oncept Map: ContinuousTime Systems Relations among CT representations. Block Diagram System Functional + Y Y 2 A 2 = X 2 + 3 A + A 2 + 1 1 2 X Impulse Response h ( t ) = 2( e t/ 2 e t ) u ( t ) Differential Equation System Function 2 y ( t ) + 3 y ( t ) + y ( t ) = 2 x ( t ) Y ( s ) 2 X ( s ) = 2 s 2 + 3 s + 1 index shift Delay R oncept Map: DiscreteTime Systems Relations among DT representations. Block Diagram System Functional Y Y 1 = Delay X 1 RR 2 + Delay + X UnitSample Response h [ n ] : 1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 , 34 , 55 ,... Difference Equation System Function z y [ n ] = x [ n ] + y [ n 1] + y [ n 2] H ( z ) = Y ( z ) = 2 X ( z ) 1 z z 2 x ( t ) x ( t ) X A X index shift Delay R oncept Map Relations between CT and DT representations. Block Diagram System Functional Y Y 1 = X 1 RR 2 + Delay + Delay X System Function y [ n ] + y [ n 1] + y [ n 2] H ( z ) = Y ( z ) z 2 System Functional Impulse Response + + Y X = 2 A 2 2 + 3 A + A 2 e t/ 2 e t ) u ( t ) Difference Equation UnitSample Response n ] = x [ h [ n ] : 1 , 1 , 2 C T D T C T D T X ( z ) = 1 z z 2 Block Diagram X Y 1 1 2 h ( t ) = 2( , 3 , 5 , 8 , 13 , 21 , 34 , 55 ,... Differential Equation System Function Y ( s ) 2 2 y ( t ) + 3 y ( t ) + y ( t ) = 2 x ( t ) X ( s ) = 2 s 2 + 3 s + 1 x ( t ) x ( t ) X A X FirstOrder CT System Example: leaky tank. r ( t ) r 1 ( t ) h 1 ( t ) Block Diagram System Functional Y A Y = X A + + 1 X Impulse Response h ( t ) = 1 e t/ u ( t ) Differential Equation System Function r 1 ( t ) = r ( t ) r 1 ( t ) H ( s ) = Y ( s ) = 1 X ( s ) 1 + s heck Yourself What is the step response of the leaky tank system?...
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 Fall '11
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