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**EECS 20. Midterm No. 1
March 7, 2002.
**

Please use these sheets for your answer and
your work. Use the backs if necessary. **Write
clearly and put a box around your an-
swer, and show your work.
**

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

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Classify whether these compositions are well-formed. Explain your answer.

(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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**Use this page for over¤ow
**

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