Final Exam ReviewFinal Exam Review

Wade FifeWade Fife

ECEn/CS 224ECEn/CS 224

August 13, 2007August 13, 2007

2

Loose EndsLoose Ends

• Check your grades onlineCheck your grades online

• Weighting of grades and breakdown found Weighting of grades and breakdown found on syllabuson syllabus– A curve will be applied if neededA curve will be applied if needed

• Labs 9-12 should be graded by end of Labs 9-12 should be graded by end of weekweek

• Estimate your missing scores and you Estimate your missing scores and you should be able to calculate your gradeshould be able to calculate your grade

3

Exam SummaryExam Summary

• Do NOT write on exam!Do NOT write on exam!– Bring scratch paperBring scratch paper– Throw it away before leaving the testing centerThrow it away before leaving the testing center

• 50 Questions50 Questions– 1-17: True/False, 1 point each1-17: True/False, 1 point each– 18-25: Multiple choice, 1 point each18-25: Multiple choice, 1 point each– 26-50: Multiple choice, 3 points each26-50: Multiple choice, 3 points each

• 4 hour time limit4 hour time limit• Some reference material providedSome reference material provided• Wednesday and Thursday in the Testing CenterWednesday and Thursday in the Testing Center

– 8:00 am to 8:00 pm (tests collected at 9:00 pm)8:00 am to 8:00 pm (tests collected at 9:00 pm)

4

Study TipsStudy Tips

• Review topics onlineReview topics online• Previous semesters’ review slides onlinePrevious semesters’ review slides online• Midterm exam study questionsMidterm exam study questions• Homework solutions onlineHomework solutions online• Come see me to go over past tests, ask Come see me to go over past tests, ask

questionsquestions– Wednesday, 8:00 am to 5:00 pmWednesday, 8:00 am to 5:00 pm– Other times by appointment (email me first)Other times by appointment (email me first)– Room 435 CBRoom 435 CB

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New for the Final ExamNew for the Final Exam

• Equivalent gatesEquivalent gates– An application of DeMorgan’s laws, truth An application of DeMorgan’s laws, truth

tablestables

• Excitation tablesExcitation tables• Flip flops with control inputsFlip flops with control inputs• FET operations and gates built from FETsFET operations and gates built from FETs• Need to have FF behavior memorized (D, Need to have FF behavior memorized (D,

T, JK)T, JK)• Possibly others…Possibly others…

6

Quick ReviewQuick Review

• FPGAsFPGAs• ROMsROMs• Mealy vs. MooreMealy vs. Moore• State encoding impact on circuitsState encoding impact on circuits• LC-3LC-3• VerilogVerilog• Bit order in questions and answersBit order in questions and answers

– QQ33QQ22QQ11QQ00 is different from Q is different from Q00QQ11QQ22QQ33

– Pay attention to the notation used!Pay attention to the notation used!

7

K-mapsK-maps• Give the minimum SOP expression for the function Give the minimum SOP expression for the function

F(A,B,C,D,E) = F(A,B,C,D,E) = m(2,5,7,12,13,14,15,16,17,18,22,23,24,25,26,28,29,30,31)m(2,5,7,12,13,14,15,16,17,18,22,23,24,25,26,28,29,30,31)

DEDE

BCBC0000 0101 1111 1010

0000

0101

1111

1010

A = 0A = 0

DEDE

BCBC0000 0101 1111 1010

0000

0101

1111

1010

A = 1A = 1

8

K-mapsK-maps• Give the minimum SOP expression for the function Give the minimum SOP expression for the function

F(A,B,C,D,E) = F(A,B,C,D,E) = m(2,5,7,12,13,14,15,16,17,18,22,23,24,25,26,28,29,30,31)m(2,5,7,12,13,14,15,16,17,18,22,23,24,25,26,28,29,30,31)

DEDE

BCBC0000 0101 1111 1010

0000 11

0101 11 11

1111 11 11 11 11

1010

A = 0A = 0

DEDE

BCBC0000 0101 1111 1010

0000 11 11 11

0101 11 11

1111 11 11 11 11

1010 11 11 11

A = 1A = 1

9

K-mapsK-maps• Give the minimum SOP expression for the functionGive the minimum SOP expression for the function

DEDE

BCBC0000 0101 1111 1010

0000 11

0101 11 11

1111 11 11 11 11

1010

A = 0A = 0

DEDE

BCBC0000 0101 1111 1010

0000 11 11 11

0101 11 11

1111 11 11 11 11

1010 11 11 11

A = 1A = 1

10

K-mapsK-maps• Give the minimum SOP expression for the functionGive the minimum SOP expression for the function

DEDE

BCBC0000 0101 1111 1010

0000 11

0101 11 11

1111 11 11 11 11

1010

A = 0A = 0

DEDE

BCBC0000 0101 1111 1010

0000 11 11 11

0101 11 11

1111 11 11 11 11

1010 11 11 11

A = 1A = 1

F = BC + A’CE + B’C’DE’ + AC’D’ + …

Essential Prime Implicants

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K-mapsK-maps• Give the minimum SOP expression for the functionGive the minimum SOP expression for the function

DEDE

BCBC0000 0101 1111 1010

0000 11

0101 11 11

1111 11 11 11 11

1010

A = 0A = 0

DEDE

BCBC0000 0101 1111 1010

0000 11 11 11

0101 11 11

1111 11 11 11 11

1010 11 11 11

A = 1A = 1

F = BC + A’CE + B’C’DE’ + AC’D’ + …

Essential Prime Implicants

12

K-mapsK-maps• Give the minimum SOP expression for the functionGive the minimum SOP expression for the function

DEDE

BCBC0000 0101 1111 1010

0000 11

0101 11 11

1111 11 11 11 11

1010

A = 0A = 0

DEDE

BCBC0000 0101 1111 1010

0000 11 11 11

0101 11 11

1111 11 11 11 11

1010 11 11 11

A = 1A = 1

F = BC + A’CE + B’C’DE’ + AC’D’ + ACD + …

Essential Prime Implicants

13

K-mapsK-maps• Give the minimum SOP expression for the functionGive the minimum SOP expression for the function

DEDE

BCBC0000 0101 1111 1010

0000 11

0101 11 11

1111 11 11 11 11

1010

A = 0A = 0

DEDE

BCBC0000 0101 1111 1010

0000 11 11 11

0101 11 11

1111 11 11 11 11

1010 11 11 11

A = 1A = 1

F = BC + A’CE + B’C’DE’ + AC’D’ + ACD + ADE’

Essential Prime Implicants

14

K-mapsK-maps• Give the minimum SOP expression for the functionGive the minimum SOP expression for the function

DEDE

BCBC0000 0101 1111 1010

0000 11

0101 11 11

1111 11 11 11 11

1010

A = 0A = 0

DEDE

BCBC0000 0101 1111 1010

0000 11 11 11

0101 11 11

1111 11 11 11 11

1010 11 11 11

A = 1A = 1

F = BC + A’CE + B’C’DE’ + AC’D’ + ACD + ADE’ … + ACD + AC’E’

Essential Prime Implicants

15

K-mapsK-maps• Give the minimum SOP expression for the functionGive the minimum SOP expression for the function

DEDE

BCBC0000 0101 1111 1010

0000 11

0101 11 11

1111 11 11 11 11

1010

A = 0A = 0

DEDE

BCBC0000 0101 1111 1010

0000 11 11 11

0101 11 11

1111 11 11 11 11

1010 11 11 11

A = 1A = 1

F = BC + A’CE + B’C’DE’ + AC’D’ + ACD + ADE’ … + ACD + AC’E’ … + ACD + ABE’Essential Prime Implicants

16

K-mapsK-maps• Give the minimum SOP expression for the functionGive the minimum SOP expression for the function

DEDE

BCBC0000 0101 1111 1010

0000 11

0101 11 11

1111 11 11 11 11

1010

A = 0A = 0

DEDE

BCBC0000 0101 1111 1010

0000 11 11 11

0101 11 11

1111 11 11 11 11

1010 11 11 11

A = 1A = 1

F = BC + A’CE + B’C’DE’ + AC’D’ + ACD + ADE’ … + ACD + AC’E’ … + ACD + ABE’ … + CDE + …

Essential Prime Implicants

17

K-mapsK-maps• Give the minimum SOP expression for the functionGive the minimum SOP expression for the function

DEDE

BCBC0000 0101 1111 1010

0000 11

0101 11 11

1111 11 11 11 11

1010

A = 0A = 0

DEDE

BCBC0000 0101 1111 1010

0000 11 11 11

0101 11 11

1111 11 11 11 11

1010 11 11 11

A = 1A = 1

Essential Prime Implicants

F = BC + A’CE + B’C’DE’ + AC’D’ + ACD + ADE’ … + ACD + AC’E’ … + ACD + ABE’ … + CDE + ADE’

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K-mapsK-maps

• Essential prime implicants ARE also prime Essential prime implicants ARE also prime implicantsimplicants

• Many prime implicants may not be used in Many prime implicants may not be used in the final solutionthe final solution

19

K-mapsK-maps• Non-essential prime implicants (some unused)Non-essential prime implicants (some unused)

DEDE

BCBC0000 0101 1111 1010

0000 11

0101 11 11

1111 11 11 11 11

1010

A = 0A = 0

DEDE

BCBC0000 0101 1111 1010

0000 11 11 11

0101 11 11

1111 11 11 11 11

1010 11 11 11

A = 1A = 1

F = BC + A’CE + B’C’DE’ + AC’D’ + ACD + ADE’ … + ACD + AC’E’ … + ACD + ABE’ … + CDE + ADE’

20

K-mapsK-maps• Give the minimum Give the minimum POSPOS expression for the function expression for the function

DEDE

BCBC0000 0101 1111 1010

0000 11

0101 11 11

1111 11 11 11 11

1010

A = 0A = 0

DEDE

BCBC0000 0101 1111 1010

0000 11 11 11

0101 11 11

1111 11 11 11 11

1010 11 11 11

A = 1A = 1

21

K-mapsK-maps• Give the minimum Give the minimum POSPOS expression for the function expression for the function

DEDE

BCBC0000 0101 1111 1010

0000 00 00 00

0101 00 00

1111

1010 00 00 00 00

A = 0A = 0

DEDE

BCBC0000 0101 1111 1010

0000 00

0101 00 00

1111

1010 00

A = 1A = 1

22

K-mapsK-maps• Give the minimum Give the minimum POSPOS expression for the function expression for the function

DEDE

BCBC0000 0101 1111 1010

0000 00 00 00

0101 00 00

1111

1010 00 00 00 00

A = 0A = 0

DEDE

BCBC0000 0101 1111 1010

0000 00

0101 00 00

1111

1010 00

A = 1A = 1

F’ = A’BC’ + C’DE + A’C’D’ + A’B’CE’ + AB’CD’ (all are essential)F = (A’BC’ + C’DE + A’C’D’ + A’B’CE’ + AB’CD’)’F = (A+B’+C)(C+D’+E’)(A+C+D)(A+B+C’+E)(A’+B+C’+D), by DeMorgan’s

23

Timing DiagramsTiming Diagrams

A

B=1C=1

D

G

A

D

5ns

20ns 30ns

32ns

TypeType DelayDelay

AND2AND2 3 ns3 ns

OR2OR2 4 ns4 ns

G

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Timing DiagramsTiming Diagrams

A

B=1C=1

D

G

F

E

A

D

E

F

G

5ns

8ns

12ns

20ns

23ns

30ns

33ns

27ns 37ns

35ns

32ns

39ns

TypeType DelayDelay

AND2AND2 3 ns3 ns

OR2OR2 4 ns4 ns

25

Transistor Level SchematicsTransistor Level Schematics

What is it?What is it?

aa bb cc outout

00 00 00

00 00 11

00 11 00

00 11 11

11 00 00

11 00 11

11 11 00

11 11 11

Assume positive logic

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Transistor Level SchematicsTransistor Level Schematics

What is it?What is it?

aa bb cc outout

00 00 00 11

00 00 11 11

00 11 00 11

00 11 11 11

11 00 00 11

11 00 11 00

11 11 00 00

11 11 11 00

out = a’ + b’c’

Assume positive logic

27

A B

C D 00 01 11 10

00

01

11

10

Z (A,B,C,D) = m(3,5,10,11,12,15) + d(4,8,14)

Implementing Logic with MuxesImplementing Logic with Muxes

4-to-1MUX Z

A C

I0

I1

I2

I3

28

A B

C D 00 01 11 10

00 X 1 X

01 1

11 1 1 1

10 X 1

Z (A,B,C,D) = m(3,5,10,11,12,15) + d(4,8,14)

4-to-1MUX Z

A C

I0

I1

I2

I3

B

D’

1

B'

D

F = A’B’CD = (0)’B’(1)D = B’D

Implementing Logic with MuxesImplementing Logic with Muxes