This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 1–7 Recitations 1–8 Homeworks 1–4 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 oﬃce 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 −R−R 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 −R−R 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 + τ + 1 τ X − Impulse Response h ( t ) = 1 τ e − t/τ u ( t ) Differential Equation System Function τr ˙ 1 ( t ) = r ( t ) − r 1 ( t ) H ( s ) = Y ( s ) = 1 X ( s ) 1 + τs heck Yourself What is the “step response” of the leaky tank system?...
View
Full
Document
 Fall '11
 Freeman

Click to edit the document details