# Machines - Signals and Systems - Exam, Exams for Signals and Systems Theory. Biju Patnaik University of Technology, Rourkela

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Main points of this exam paper are: Machines, Output Alphabet, State Transition Diagram, State Space, Input, Outputs, Well-Formed
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EECS 20. Midterm No. 1 March 7, 2002.

Print your name and lab TA’s name below

Name:

Lab TA:

Problem 1:

Problem 2:

Problem 3:

Problem 4:

Total:

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1. 15 points. Consider state machines A, B and C, described below by their state space, input alphabet, output alphabet, and state transition diagram.

Let

Set1 = {T, F, absent}

Set2 = {1, 0, absent}

State machine A:

Inputs = Set2, Outputs = Set1, States = {a, b}

State machine B:

Inputs = Set1, Outputs = Set2, States = {1, 2}

State machine C:

Inputs = Set2, Outputs = Set1, States = {x, y}

x y

{1}/T

{0}/F

{0}/T

{1}/F

C

a b

{0}/F

{0}/T

{1}/F

{1}/T

A

1 2

{T}/1

{F}/0

{T}/0

{F}/1

B

2

(a)Well-formed or not well-formed

B C

(b)Well-formed or not well-formed

A B

(c)Well-formed or not well-formed

A C

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2. 20 points. Consider the following state machine A:

Inputs={0,1,reset,absent}

Outputs={0,1,absent}

States = {a,b,c,d}

initialState = a

update function is given by the following table of values of (nextState, output symbol) under speci£ed input symbol

current state 0 1 reset absent a (b,0) (c,0) (a,0) (a,absent) b (a,0) (c,0) (a,0) (b,absent) c (b,0) (d,0) (a,0) (c,absent) d (d,0) (a,1) (a,0) (d,absent)

(a) Draw the state transition diagram for state machine A.

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(b) Construct a simpler state machine which is bisimilar to A. Give Inputs, Outputs, States, initialState, and a state transition diagram for this state machine.

(c) What is the bisimulation relation between the state machines in part (a) and (b)?

(d) Provide the state transition diagram of a one-state non-deterministic state machine which simulates A.

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3. 20 points. Consider a general £nite state machine A with a certain set of Inputs, Outputs, States, initialState and state transition diagram. Now construct a new FSM B with the same set of Inputs, Outputs, States and initialState, but with the arrows in the state transition diagram reversed; i.e., the direction of every arc in B is the reverse of the corresponding arc in A. The £gure below illustrates this for one speci£c arc of A and the corresponding arc of B .

s1 s2

{guard}/output

A

s1 s2

{guard}/output

B

(a) Does the arc-reversed machine B always make sense? That is, can we always de£ne it via a £nite state machine model? Explain.

(b) Is B always deterministic if A is? Prove or give a counterexample.

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4. 10 points. Consider a discrete-time system with one input port, one output port and one state. The current input, current output, and current state are denoted by x(n), y(n), and s(n), respectively. The equations describing this system are:

s(n + 1) = as(n) + bx(n)

y(n) = cs(n) + dx(n)

Furthermore, the output is multiplied by a feedback gain factor k and fed back as input.

x(n) = ky(n).

s(n+1)=as(n)+bx(n) y(n)=cs(n)+dx(n)

Gain factor k

x(n) y(n)

Derive conditions on the values of a, b, c, d, k under which the system is well-formed.

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