MIT6_003S10_lec06

# MIT6_003S10_lec06 - 6.003: Signals and Systems Z Transform...

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6.003: Signals and Systems Z Transform February 23, 2010

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Mid-term Examination #1 Wednesday, March 3, No recitations on the day of the exam. Coverage: Representations of CT and DT Systems Lectures 1–7 Recitations 1–8 Homeworks 1–4 Homework 4 will not collected or graded. Solutions will be posted. × 11 inches; front and back). Designed as 1-hour exam; two hours to complete. Review sessions during open oﬃce hours. 1 8 Closed book: 1 page of notes ( 2 7:30-9:30pm.
Z Transform Z transform is discrete-time analog of Laplace transform.

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Z Transform Z transform is discrete-time analog of Laplace transform. Furthermore, you already know about Z transforms (we just haven’t called them Z transforms) ! Example: Fibonacci system diﬀerence equation y [ n ]= x [ n ]+ y [ n 1] + y [ n 2] operator expression Y = X + R Y + R 2 Y Y 1 system functional = X 1 −R−R 2 unit-sample response h [ n ]: 1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 , 34 , 55 , 89 ,...
Check Yourself Example: Fibonacci system diﬀerence equation y [ n ]= x [ n ]+ y [ n 1] + y [ n 2] operator expression Y = X + R Y + R 2 Y system functional Y X = 1 1 −R−R 2 unit-sample response h [ n ]: 1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 , 34 , 55 , 89 ,... What is the relation between system functional and h [ n ] ?

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Check Yourself 1 −R−R 2 1 1 −R −R 2 Y 1 system functional = X 1 2 unit-sample response h [ n ]: 1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 , 34 , 55 , 89 ,... Expand functional in a series: 1 + R +2 R 2 +3 R 3 +5 R 4 +8 R 5 + ··· R + R 2 R −R 2 −R 3 2 R 2 + R 3 2 R 2 2 R 3 2 R 4 3 R 3 +2 R 4 3 R 3 3 R 4 3 R 5 Y 1 = 1 2 =1+ R +2 R 2 +3 R 3 +5 R 4 +8 R 5 +13 R 6 + X = h [0] + h [1] R + h [2] R 2 + h [3] R 3 + h [4] R 4 + n = h [ n ] R n
Check Yourself Example: Fibonacci system diﬀerence equation y [ n ]= x [ n ]+ y [ n 1] + y [ n 2] operator expression Y = X + R Y + R 2 Y system functional Y X = 1 1 −R−R 2 unit-sample response h [ n ]: 1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 , 34 , 55 , 89 ,... What is the relation between system functional and h [ n ] ? Y X = n h [ n ] R n

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Check Yourself Example: Fibonacci system diﬀerence equation y [ n ]= x [ n ]+ y [ n 1] + y [ n 2] operator expression Y = X + R Y + R 2 Y system functional Y X = 1 1 −R−R 2 unit-sample response h [ n ]: 1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 , 34 , 55 , 89 ,... Y X = n h [ n ] R n What’s the relation between H ( z ) and h [ n ] ?
Check Yourself Series expansion of system functional: Y n = h [ n ] R X n 1 Substitute R→ : z H ( z )= h [ n ] z n n

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Check Yourself Example: Fibonacci system diﬀerence equation y [ n ]= x [ n ]+ y [ n 1] + y [ n 2] operator expression Y = X + R Y + R 2 Y system functional Y X = 1 1 −R−R 2 unit-sample response h [ n ]: 1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 , 34 , 55 , 89 ,... Y X = n h [ n ] R n What’s the relation between H ( z ) and h [ n ] ? H ( z )= n h [ n ] z n
index shift Delay →R Concept Map: Discrete-Time Systems Multiple representations of DT systems. Block Diagram System Functional Y Y 1 = Delay X 1 −R−R 2 + Delay + X Unit-Sample 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

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index shift Delay →R Concept Map: Discrete-Time Systems Relation between Unit-Sample Response and System Functional.
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## MIT6_003S10_lec06 - 6.003: Signals and Systems Z Transform...

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