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EECS 20 Lecture 6 (January 29, 2001) Tom Henzinger

Transducive Systems. EECS 20 Lecture 6 (January 29, 2001) Tom Henzinger. Continuous-Time Signals. sound position. Discrete-Time Signals. sampledSound movie. ContinuousTime = { x  Reals | x  0 } DiscreteTime = { 0, 1, 2, 3, 4, … }.

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EECS 20 Lecture 6 (January 29, 2001) Tom Henzinger

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  1. Transducive Systems EECS 20 Lecture 6 (January 29, 2001) Tom Henzinger

  2. Continuous-Time Signals sound position Discrete-Time Signals sampledSound movie

  3. ContinuousTime = { x  Reals | x  0 } DiscreteTime = { 0, 1, 2, 3, 4, … } continuousTimeSignal : ContinuousTime  Values discreteTimeSignal : DiscreteTime  Values continuousTimeSignal = evolution of values discreteTimeSignal = stream of values

  4. More Discrete-Time Signals text : DiscreteTime  Chars file : DiscreteTime  Bins vendingMachine : DiscreteTime  Events Chars = { a, b, c, … } Bins = { 0, 1 } Events = { dropQuarter, requestCoke, deliverCoke, … }

  5. Transducive or Combinational System Input Value Output Value transduciveSystem : Values  Values

  6. Transducive or Combinational System  true false  : Bools  Bools

  7. Transducive or Combinational System f frame frame f : Frames  Frames

  8. Reactive or Sequential System Input Signal Output Signal reactiveSystem : [ Time  Values ]  [ Time  Values ]

  9. Reactive or Sequential System F movie movie F : [ Time  Frames ]  [ Time  Frames ]

  10. Reactive or Sequential System F sound text F: [ ContinuousTime  Pressure ]  [ DiscreteTime  Chars ]

  11. Reactive or Sequential System G text sound G: [ DiscreteTime  Chars ]  [ ContinuousTime  Pressure ]

  12. Identity ? F G sound sound text

  13. Identity ? G F text text sound

  14. Reactive Systems • Memory-free systems • Delays • Causality • Finite-memory systems • Infinite-memory systems

  15. Every transducive system is a reactive system f : Values  Values F : [ Time  Values ]  [ Time  Values ] such that  x  [ Time  Values ],  y  Time, ( F (x) ) (y) = f ( x (y) )

  16. Every transducive system is a reactive system f : Values  Values F : [ Time  Values ]  Time  Values such that  x  [ Time  Values ],  y  Time, F (x,y) = f ( x (y) )

  17. Every transducive system is a continuous-time reactive system f : Values  Values F : [ ContTime  Values ]  [ ContTime  Values ] such that  x  [ ContTime  Values ],  y  ContTime, ( F (x) ) (y) = f ( x (y) )

  18. normalize : Reals  Reals such that  x  Reals, normalize (x) = x - 50 Normalize : [ Reals+ Reals ]  [ Reals+  Reals ] such that  x  [ Reals+ Reals ] ,  y  Reals+, ( Normalize (x) ) (y) = normalize ( x (y) ) = x (y) - 50

  19. Normalize (sin) : Reals+ Reals such that  y  Reals+ , Normalize (sin) (y) = sin (y) - 50 . Normalize (id) : Reals+ Reals such that  y  Reals+ , Normalize (id) (y) = y - 50 . Normalize (53) : Reals+ Reals such that  y  Reals+ , Normalize (53) (y) = 3 .

  20. Normalize (sin) : Reals+ Reals such that  y  Reals+ , Normalize (sin) (y) = sin (y) - 50 . Normalize (id) : Reals+ Reals such that  y  Reals+ , Normalize (id) (y) = y - 50 . Normalize (53) : Reals+ Reals such that Normalize (53) = 3 .

  21. trunc : Reals  Reals 256 if x > 256 x if -256  x  256 -256 if x < -256 { such that  x  Reals, trunc (x) = Trunc : [ Reals+ Reals ]  [ Reals+  Reals ] such that  x  [ Reals+ Reals ] ,  y  Reals+, ( Trunc (x) ) (y) = trunc ( x (y) )

  22. Trunc (sin) : Reals+ Reals such that  y  Reals+ , Trunc (sin) (y) = sin (y) . Trunc (id) : Reals+ Reals such that  y  Reals+ , Trunc (id) (y) = y if y < 256 256 if y  256 { Trunc (500) : Reals+ Reals such that Trunc (500) = 256 .

  23. Every transducive system is a discrete-time reactive system f : Values  Values F : [ DiscTime  Values ]  [ DiscTime  Values ] such that  x  [ DiscTime  Values ],  y  DiscTime, ( F (x) ) (y) = f ( x (y) )

  24. quantize : Reals  ComputerInts such that  x  Reals, quantize (x) = trunc ( x ) Quantize : [ Nats0 Reals ]  [ Nats0  ComputerInts ] such that  x  [ Nats0 Reals ] ,  y  Nats0 , ( Quantize (x) ) (y) = quantize ( x (y) ) = trunc ( x(y) )

  25. Quantize (squareroot) : Nats0 ComputerInts such that Quantize (squareroot) = { (0,0), (1,1), (2,1), (3,1), (4,2), (5,2), … , (1000000,256), … } . Quantize (id) : Nats0 ComputerInts such that Quantize (id) = { (0,0), (1,1), (2,2), (3,3), … , (256,256), (257,256), … } . Quantize (pi) : Nats0 ComputerInts such that Quantize (pi) = 3 .

  26. negate : Bools  Bools such that  x  Bools, negate (x) =  x Negate : [ Nats0 Bools ]  [ Nats0  Bools ] such that  x  [ Nats0 Bools ] ,  y  Nats0 , ( Negate (x) ) (y) = negate ( x (y) ) =  x(y)

  27. alt : Nats0 Bools such that  y  Nats0 , alt (y) = true if y odd false if y even { Negate (alt) : Nats0 Bools such that Negate (alt) = { (0,true), (1, false), (2, true), (3, false) , (4, true), … } . Negate (true) : Nats0 Bools such that Negate (true) = false .

  28. A reactive system F : [ Time  Values ]  [ Time  Values ] is memory-free iff there exists a transducive system f : Values  Values such that  x  [ Time  Values ] ,  y  Time, ( F (x) ) (y) = f ( x (y) ) .

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