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Electrical Engineering 20N - Spring 2000 - Midterm 1

Electrical Engineering 20N - Spring 2000 - Midterm 1 - EECS...

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Unformatted text preview: EECS 20. Midterm No. 1 Solution, February 23, 2000. 1. 40 points. Consider the state machine below {1} / 1 {1} / 0 b a c d {1} / 0 {1} / 1 where Inputs = 1; absent f g and Outputs = 0; 1; absent f g a Is this machine deterministic or nondeterministic? Answer: Deterministic. b Give the update table. Answer: The update function is given by: state  next state, output 1 absent a b,1 a,absent  b c,0 b,absent  c d,1 c,absent  d a,0 d,absent  c Find a deterministic state machine that is bisimilar to this one and has only two states Give it as a state transition diagram by completing the diagram below: Answer: {1} / 1 e f {1} / 0 1 d Give the bisimulation relation. Answer: The bisimulation relation is S = a; e; b; f ; c; e; d; f  ; or equivalently, S = e; a; e; b; f; c; f; d ; f 0 g f g 2. 30 points. Let X = a; b; c represent a set of circles in the following picture: f g 1 0 b 1 0 a c 0 1 Consider the following relations, all subsets of X X :  F0 F1 F0and1 F0or1 = = = = x1 ; x2  x1 ; x2  x1 ; x2  x1 ; x2  f j f j f j f j there is an arc going from x1 to x2 with a 0 there is an arc going from x1 to x2 with a 1 there are two arcs going from x1 to x2 , one with a 0 and one with a 1 there is an arc going from x1 to x2 with a 0 or one with a 1 g g g g a Give the elements of the four relations. Answer: F0 = a; b; b; c; c; a F1 = a; a; b; b; c; c F0and1 = F0or1 = a; b; b; c; c; a; a; a; b; b; c; c b Which of the four relations are the graph of a function of the form f : X List all that are such a graph. Answer: F0 and F1 . f g f g ; f g 2 ! X? c Are the following assertions true or false? F0and1 = F0 F1 F0or1 = F0 F1 Answer: Both are true. 3. 20 points Consider all state machines with Inputs = 1; 2; absent States = a; b; c; d : f f g and Outputs = 1; 2; absent f g g Assume all these state machines stutter, as usual, when presented with the stuttering input, absent . a Give a state machine B that simulates all of these state machines. You will lose points if your machine is more complicated than it needs to be. b Give the simulation relation. Answer {1,2}/1 e {1,2, absent}/ absent {1,2}/2 The simulation relation is S = fa; e; b; e; c; e; d; eg: 4. 30 points Consider the functions g: Y ! Reals and f : Nats ! Y: where Y is a set. a Draw a block diagram for g f , with one block for each of g and f , and label the inputs and output of the blocks with the domain and range of g and f . Nats Y f 3 g Reals b Suppose Y is given by Y = 1; ; 100 Reals Thus, the function f takes a natural number and returns a sequence of length 100, while the function g takes a sequence of length 100 and returns a real number. Suppose further that g is given by: for all y Y , f gy =  g ! X yi = y1 + y2 + 2 100 i=1  + y100; and f by: for all x Nats and z 1; ; 100 , f xz  = cos2z=x: Thus, x gives the period of a cosine waveform, and f gives 100 samples of that waveform. Give a one-line Matlab expression that evaluates g f x for any x Nats . Assume the value of x is already in a Matlab variable called x. Answer: 2 2 f 2 sumcos2*pi* 1:100 x c Find g f 1. Answer: 100 4  g ...
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