midterm solution i (1)

8/14/2019 Midterm Solution I (1)

1/6

Midterm Solution 1 Hour, 50 minutes

Do any 10 of the 12 problems. Make sure it is clear which one you donot want graded (or I will grade the first 10 problems). All problems areworth 0 points.

1. !he following fun"tion is alread# in minimum sum of produ"ts form. Showthe e$uation anda blo"% diagram for an implementation using onl# &input'A' gates. 'o gate ma# be used as a '!. All *ariables are a*ailable bothun"omplemented and "omplemented. +or full "redit, find a solution using 1gates- partial "redit for more gates or one or two &input gates. /onus of upto 5 points for less.

+ A / 2 3 / 2 4 3 / 3 / 2 4 3 2 4 3 2 4

/ (A 2 3 2 4) 3 (2 4 3 2 4) 3 / ( 3 2 4)

3 2 4 3 2 46 / 3 (A 2 3 2 4)6 3 / ( 3 2 4) 3 (2 3 4) (2 3 4)6 / 3 (A 2 3 2 4)6 3 / ( 3 2 4)where the underlined terms use the same gate.

!his uses 1 gates.

+

A

2

2

4

/

2

4

/


2/6

. Show an algebrai" e7pression for the following "ir"uit. 8edu"e it to aminimum sum of produ"ts e7pression ( terms, literals.)

a

b

"

d

"

g

w

7

#

9

w (a b) a 3 b # " w " (a b)7 " 3 d 9 w 3 7 a b 3 " d

g (# 3 9) (" w 3 w 3 7) (" 3 w 3 " d) (" 3 w) (" 3 a b) " (a 3 b) a " 3 b "

. 2on*ert the following to a sum of minterms. 8emo*e an# dupli"ates fromthe list.

f (a 3 b 3 ") (a 3 " 3 d) (b 3 ") (b 3 d) (a 3 " 3 b d) (b d 3 b ") a b d 3 a b " 3 b " d 3 b " d a b " d 3 a b " d 3 a b " d 3 a b " d 3 a b " d 3 a b

" d 3 a b " d 3 a b " d a b " d 3 a b " d 3 a b " d 3 a b " d 3 a b " d 3 a b

" d

. +or the following fun"tion, find !H4 minimum sum of produ"ts solution.

00 01 11 10

00

01

11

10

a b

" d

1

1

1

1

1

1 1

1

f a b d 3 a b d 3 a b " 3 b " d5. +or the following fun"tion, list all of the prime impli"ants and find /!Hminimum sum of produ"t e7pressions. (e7tra maps shown for #our"on*enien"e)


3/6

h (w, 7, #, 9) : m (, , 5, ;, 7 # w 7 9 w # 9 7 # 9 w 7 9

h # 9 3 w # 3 7 # 9 3 w 7 9

h # 9 3 7 # 3 w # 9 3 w 7 9

;. +or the fun"tion, g, shown, show the e$uations and a blo"% diagram for aminimum "ost two&le*el 'A' implementation a minimum two&le*el '8implementation (spare map shown). All *ariables are a*ailable bothun"omplemented and "omplemented.

00 01 11 10

00

01

11

10

w 7

#9

1 1

1 1

1

1

1

1

1

00 01 11 10

00

01

11

10

w 7

#9

1 1 1

1

1

1 1

g g

g 7 9 3 w 7 3 w # g w 7 3 w 9 3 7 # 9g (w 3 7) (w 3 9) (7 3 # 3 9)


4/6

7

9

w

7

w

#

w

7

w

9

7

#

9

gg


5/6

=. +or the following 5&*ariable fun"tion, +, find /!H minimum sum of produ"tsolutions. (!wo "opies of the map are shown.)

00 01 11 10

00

01

11

10

00 01 11 10

00

01

11

10

A0/ 2

4

/ 2

4

1

1

1 1 1 1

1

1

1

1 1

1 1

1 1

+ A 2 3 A / 3 2 4 3 EA / 2 or / 2 F

10. +or the following 5&*ariable fun"tion, +, find the minimum sum of produ"tsolution. (!wo "opies of the map are shown.) (; terms, 0 literals)

00 01 11 10

00

01

11

10

00 01 11 10

00

01

11

10

A0/ 2

4

/ 2

4

1

1 1

1 1

1 1

1 1

1

1

1

1

1 1 1

1

1

1

+ A / 2 4 3 A 2 3 A 2 4 3 / 2 3 / 2 3 / 4


6/6

11. +or the two fun"tions shown, find a minimum two&le*el solution (map ande$uations) "orresponding to 5 A' gates and 8 gates with a total of 1=inputs. All inputs are a*ailable both un"omplemented and "omplemented.(Show onl# the maps and e$uations.)

00 01 11 10

00

01

11

10

00 01 11 10

00

01

11

10

w 7

# 9

w 7# 9

f g

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

f w 7 3 w 7 # 3 7 9g 7 # 9 3 w 7 # 3 7 9 3 w 9

1. +or the three fun"tions shown, find a minimum two&le*el solution (mapand e$uations) "orresponding to < A' gates and 8 gates with a total of 0inputs. All inputs are a*ailable both un"omplemented and "omplemented.(Show the maps and e$uations.)

00 01 11 10

00

01

11

10

00 01 11 10

00

01

11

10

00 01 11 10

00

01

11

10

a b

" d

a b

" d a b" d

f g h

1 1

1 1

1 1 1

1 1

1 1

1 1

1 1

1

1 1 1

1 1

1

f b " d 3 b " d 3 a' d 3 a " dg b " d 3 b " d 3 b " d

h b " d 3 a b d 3 a' d

midterm solution i (1)

Documents