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ECE 124 – Assignment #4: Quine McCluskey Assignment #5: Multilevel Gate Networks
Presented by: Mahmoud Elmohr Supervised by: Prof. Cathy Gebotys
05/29/18
Assign#4 - Problem #1,2 (a) § Find a minimum sum of products solution using Quine McCluskey:
f(a,b,c,d) = m(1,5,7,9,12,14,15)
PAGE 2 ECE 124 – Assignment #3
Group(0) Group(1) Group(2)
Group(3) Group(4)
Assign#4 - Problem #1,2 (a) § Find a minimum sum of products solution using Quine McCluskey:
f(a,b,c,d) = m(1,5,7,9,12,14,15)
PAGE 3 ECE 124 – Assignment #3
Group(0) Group(1)
1 0001
Group(2)
5 9 12
0101 1001 1100
Group(3)
7 14
0111 1110
Group(4)
15 1111
Assign#4 - Problem #1,2 (a) § Find a minimum sum of products solution using Quine McCluskey:
f(a,b,c,d) = m(1,5,7,9,12,14,15)
PAGE 4 ECE 124 – Assignment #3
Group(0) Group(1)
1 0001 0-01 (1,5) -001 (1,9)
Group(2)
5 9 12
0101 1001 1100
Group(3)
7 14
0111 1110
Group(4)
15 1111
Assign#4 - Problem #1,2 (a) § Find a minimum sum of products solution using Quine McCluskey:
f(a,b,c,d) = m(1,5,7,9,12,14,15)
PAGE 5 ECE 124 – Assignment #3
Group(0) Group(1)
1 0001 0-01 (1,5) -001 (1,9)
Group(2)
5 9 12
0101 1001 1100
01-1 (5,7) 11-0 (12,14)
Group(3)
7 14
0111 1110
Group(4)
15 1111
Assign#4 - Problem #1,2 (a) § Find a minimum sum of products solution using Quine McCluskey:
f(a,b,c,d) = m(1,5,7,9,12,14,15)
PAGE 6 ECE 124 – Assignment #3
Group(0) Group(1)
1 0001 0-01 (1,5) -001 (1,9)
Group(2)
5 9 12
0101 1001 1100
01-1 (5,7) 11-0 (12,14)
Group(3)
7 14
0111 1110
-111 (7,15) 111- (14,15)
Group(4)
15 1111
Assign#4 - Problem #1,2 (a) § Find a minimum sum of products solution using Quine McCluskey:
f(a,b,c,d) = m(1,5,7,9,12,14,15)
PAGE 7 ECE 124 – Assignment #3
Group(0) Group(1)
1 0001 0-01 (1,5) > a’c’d -001 (1,9) > b’c’d
Group(2)
5 9 12
0101 1001 1100
01-1 (5,7) > a’bd 11-0 (12,14) > abd’
Group(3)
7 14
0111 1110
-111 (7,15) > bcd 111- (14,15) > abc
Group(4)
15 1111
Assign#4 - Problem #1,2 (a) § Find a minimum sum of products solution using Quine McCluskey:
f(a,b,c,d) = m(1,5,7,9,12,14,15)
PAGE 8 ECE 124 – Assignment #3
1 5 7 9 12 14 15
a’c’d
b’c’d
a’bd
abd’
bcd
abc
Assign#4 - Problem #1,2 (a) § Find a minimum sum of products solution using Quine McCluskey:
f(a,b,c,d) = m(1,5,7,9,12,14,15)
PAGE 9 ECE 124 – Assignment #3
1 5 7 9 12 14 15
a’c’d x x
b’c’d x x
a’bd x x
abd’ x x
bcd x x
abc x x
Assign#4 - Problem #1,2 (a) § Find a minimum sum of products solution using Quine McCluskey:
f(a,b,c,d) = m(1,5,7,9,12,14,15)
PAGE 10 ECE 124 – Assignment #3
1 5 7 9 12 14 15
a’c’d x x
b’c’d x x
a’bd x x
abd’ x x
bcd x x
abc x x
F= b’c’d
Assign#4 - Problem #1,2 (a) § Find a minimum sum of products solution using Quine McCluskey:
f(a,b,c,d) = m(1,5,7,9,12,14,15)
PAGE 11 ECE 124 – Assignment #3
1 5 7 9 12 14 15
a’c’d x x
b’c’d x x
a’bd x x
abd’ x x
bcd x x
abc x x
F= b’c’d + abd’
Assign#4 - Problem #1,2 (a) § Find a minimum sum of products solution using Quine McCluskey:
f(a,b,c,d) = m(1,5,7,9,12,14,15)
PAGE 12 ECE 124 – Assignment #3
1 5 7 9 12 14 15
a’c’d x x
b’c’d x x
a’bd x x
abd’ x x
bcd x x
abc x x
F= b’c’d + abd’ + a’c’d + bcd
Assign#4 - Problem #3 (c) § Find a minimum sum of products solution using Quine McCluskey:
f(a,b,c,d) = m(0,1,5,6,8,9,11,13) + d(7,10,12)
PAGE 13 ECE 124 – Assignment #3
Group(0) 0 0000
Group(1)
1 8
0001 1000
Group(2)
5 6 9 10 12
0101 0110 1001 1010 1100
Group(3)
7 11 13
0111 1011 1101
Assign#4 - Problem #3 (c) § Find a minimum sum of products solution using Quine McCluskey:
f(a,b,c,d) = m(0,1,5,6,8,9,11,13) + d(7,10,12)
PAGE 14 ECE 124 – Assignment #3
Group(0) 0 0000 000- (0,1) -000 (0,8)
Group(1)
1 8
0001 1000
Group(2)
5 6 9 10 12
0101 0110 1001 1010 1100
Group(3)
7 11 13
0111 1011 1101
Assign#4 - Problem #3 (c) § Find a minimum sum of products solution using Quine McCluskey:
f(a,b,c,d) = m(0,1,5,6,8,9,11,13) + d(7,10,12)
PAGE 15 ECE 124 – Assignment #3
Group(0) 0 0000 000- (0,1) -000 (0,8)
Group(1)
1 8
0001 1000
-001 (1,9) 0-01 (1,5) 100- (8,9) 10-0 (8,10) 1-00 (8,12)
Group(2)
5 6 9 10 12
0101 0110 1001 1010 1100
Group(3)
7 11 13
0111 1011 1101
Assign#4 - Problem #3 (c) § Find a minimum sum of products solution using Quine McCluskey:
f(a,b,c,d) = m(0,1,5,6,8,9,11,13) + d(7,10,12)
PAGE 16 ECE 124 – Assignment #3
Group(0) 0 0000 000- (0,1) -000 (0,8)
Group(1)
1 8
0001 1000
-001 (1,9) 0-01 (1,5) 100- (8,9) 10-0 (8,10) 1-00 (8,12)
Group(2)
5 6 9 10 12
0101 0110 1001 1010 1100
10-1 (9,11) 101- (10,11) 1-01 (9,13) 110- (12,13) 011- (6,7) 01-1 (5,7) -101 (5,13)
Group(3)
7 11 13
0111 1011 1101
Assign#4 - Problem #3 (c) § Find a minimum sum of products solution using Quine McCluskey:
f(a,b,c,d) = m(0,1,5,6,8,9,11,13) + d(7,10,12)
PAGE 17 ECE 124 – Assignment #3
Group(0) 0 0000 000- (0,1) -000 (0,8)
-00- (0,1;8,9) -00- (0,8;1,9)
Group(1)
1 8
0001 1000
-001 (1,9) 0-01 (1,5) 100- (8,9) 10-0 (8,10) 1-00 (8,12)
Group(2)
5 6 9 10 12
0101 0110 1001 1010 1100
10-1 (9,11) 101- (10,11) 1-01 (9,13) 110- (12,13) 011- (6,7) 01-1 (5,7) -101 (5,13)
Group(3)
7 11 13
0111 1011 1101
Assign#4 - Problem #3 (c) § Find a minimum sum of products solution using Quine McCluskey:
f(a,b,c,d) = m(0,1,5,6,8,9,11,13) + d(7,10,12)
PAGE 18 ECE 124 – Assignment #3
Group(0) 0 0000 000- (0,1) -000 (0,8)
-00- (0,1;8,9) -00- (0,8;1,9)
Group(1)
1 8
0001 1000
-001 (1,9) 0-01 (1,5) 100- (8,9) 10-0 (8,10) 1-00 (8,12)
--01 (1,9;5,13) --01 (1,5; 9,13) 1-0- (8,12;9,13) 10-- (8,9;10,11) 1-0- (8,9;12,13) 10– (8,10;9,11)
Group(2)
5 6 9 10 12
0101 0110 1001 1010 1100
10-1 (9,11) 101- (10,11) 1-01 (9,13) 110- (12,13) 011- (6,7) 01-1 (5,7) -101 (5,13)
Group(3)
7 11 13
0111 1011 1101
Assign#4 - Problem #3 (c) § Find a minimum sum of products solution using Quine McCluskey:
f(a,b,c,d) = m(0,1,5,6,8,9,11,13) + d(7,10,12)
PAGE 19 ECE 124 – Assignment #3
Group(0) 0 0000 000- (0,1) -000 (0,8)
-00- (0,1;8,9) > b’c’ -00- (0,8;1,9)
Group(1)
1 8
0001 1000
-001 (1,9) 0-01 (1,5) 100- (8,9) 10-0 (8,10) 1-00 (8,12)
--01 (1,9;5,13) > c’d --01 (1,5; 9,13) 1-0- (8,12;9,13) > ac’ 10-- (8,9;10,11) > ab’ 1-0- (8,9;12,13) 10– (8,10;9,11)
Group(2)
5 6 9 10 12
0101 0110 1001 1010 1100
10-1 (9,11) 101- (10,11) 1-01 (9,13) 110- (12,13) 011- (6,7) > a’bc 01-1 (5,7) > a’bd -101 (5,13)
Group(3)
7 11 13
0111 1011 1101
Assign#4 - Problem #3 (c) § Find a minimum sum of products solution using Quine McCluskey:
f(a,b,c,d) = m(0,1,5,6,8,9,11,13) + d(7,10,12)
PAGE 20 ECE 124 – Assignment #3
0 1 5 6 8 9 11 13
b’c’
c’d
ac’
ab’
a’bc
a’bd
Assign#4 - Problem #3 (c) § Find a minimum sum of products solution using Quine McCluskey:
f(a,b,c,d) = m(0,1,5,6,8,9,11,13) + d(7,10,12)
PAGE 21 ECE 124 – Assignment #3
0 1 5 6 8 9 11 13
b’c’ x x x x
c’d x x x x
ac’ x x x
ab’ x x x
a’bc x
a’bd x
Assign#4 - Problem #3 (c) § Find a minimum sum of products solution using Quine McCluskey:
f(a,b,c,d) = m(0,1,5,6,8,9,11,13) + d(7,10,12)
PAGE 22 ECE 124 – Assignment #3
0 1 5 6 8 9 11 13
b’c’ x x x x
c’d x x x x
ac’ x x x
ab’ x x x
a’bc x
a’bd x
F= b’c’
Assign#4 - Problem #3 (c) § Find a minimum sum of products solution using Quine McCluskey:
f(a,b,c,d) = m(0,1,5,6,8,9,11,13) + d(7,10,12)
PAGE 23 ECE 124 – Assignment #3
0 1 5 6 8 9 11 13
b’c’ x x x x
c’d x x x x
ac’ x x x
ab’ x x x
a’bc x
a’bd x
F= b’c’ + a’bc
Assign#4 - Problem #3 (c) § Find a minimum sum of products solution using Quine McCluskey:
f(a,b,c,d) = m(0,1,5,6,8,9,11,13) + d(7,10,12)
PAGE 24 ECE 124 – Assignment #3
0 1 5 6 8 9 11 13
b’c’ x x x x
c’d x x x x
ac’ x x x
ab’ x x x
a’bc x
a’bd x
F= b’c’ + a’bc + ab’
Assign#4 - Problem #3 (c) § Find a minimum sum of products solution using Quine McCluskey:
f(a,b,c,d) = m(0,1,5,6,8,9,11,13) + d(7,10,12)
PAGE 25 ECE 124 – Assignment #3
0 1 5 6 8 9 11 13
b’c’ x x x x
c’d x x x x
ac’ x x x
ab’ x x x
a’bc x
a’bd x
F= b’c’ + a’bc + ab’ + c’d
Assign#5 - Problem #1 § using and, or gates find the minimum network to realize:
f(a,b,c,d)=M1 M2 M5 M9 M10 M14 using: (a) two level logic (b) using three level logic.
PAGE 26 ECE 124 – Assignment #3
ab
Assign#5 - Problem #1 § using and, or gates find the minimum network to realize:
f(a,b,c,d)=M1 M2 M5 M9 M10 M14 using: (a) two level logic (b) using three level logic.
§ SOP: f = c’d’ + cd + abc’ + a’bc (5 gates, 14 inputs)
PAGE 27 ECE 124 – Assignment #3
00 01 11 10
00 1 1 1 1
01 0 0 1 0
11 1 1 1 1
10 0 1 0 0
ab
cd
0
1
4
5
3 7
6
12
13
15
14
8
9
11
10 2
Assign#5 - Problem #1 § using and, or gates find the minimum network to realize:
f(a,b,c,d)=M1 M2 M5 M9 M10 M14 using: (a) two level logic (b) using three level logic.
§ SOP: f = c’d’ + cd + abc’ + a’bc (5 gates, 14 inputs)
§ POS: f = (b+c’+d)(a+c+d’)(a’+c’+d)(b+c+d’) (5 gates, 16 inputs)
PAGE 28 ECE 124 – Assignment #3
00 01 11 10
00 1 1 1 1
01 0 0 1 0
11 1 1 1 1
10 0 1 0 0
ab
cd
0
1
4
5
3 7
6
12
13
15
14
8
9
11
10 2
Assign#5 - Problem #1 § using and, or gates find the minimum network to realize:
f(a,b,c,d)=M1 M2 M5 M9 M10 M14 using: (a) two level logic (b) using three level logic.
§ f = c’d’ + cd + abc’ + a’bc
§ f = c’d’ + cd + b(ac’ + a’c)
§ X+YZ = (X+Y)(X+Z)
§ f = (c’d’+c)(c’d’+d) + b(ac’+a’)(ac’+c)
§ XY’+Y = X+Y
§ f = (d’+c)(c’+d) + b(c’+a’)(a+c)
(7 gates, 15 inputs)
PAGE 29 ECE 124 – Assignment #3
ab
cd
§ f = (b+c’+d)(a+c+d’)(a’+c’+d)(b+c+d’)
§ X+YZ = (X+Y)(X+Z)
§ f = (c+d’+ab)(c’+d+a’b)
(5 gates, 12 inputs)
Assign#5 - Problem #2 [a] § using and or gates find the network which minimizes the number of gate
inputs (there is no limit on the number of gates which can be cascaded): acd' + b'cd' + b'e + ac'd + ae
§ f = acd' + b'cd' + b'e + ac'd + ae (18 inputs)
§ f = cd’(a+b’) + e(a+b’) + ac’d
§ f = (a+b’)(cd’+e) + ac'd
§ X+YZ = (X+Y)(X+Z)
§ f = (ac’d+ a+b’)(ac’d+ cd’+e)
§ X+XY = X
§ f = (a+b’)(ac’d+ cd’+e) (12 inputs)
PAGE 30 ECE 124 – Assignment #3
ab
Assign#5 - Problem #2 [b] § using and or gates find the network which minimizes the number of gate
inputs (there is no limit on the number of gates which can be cascaded): abdf + abef + abg + cdf + cef + cg
§ f = abdf + abef + abg + cdf + cef + cg (25 inputs)
§ f = ab(df + ef + g) + c(df + ef + g)
§ f = ( ab+c )(df + ef + g)
§ f = ( ab+c ) ( f (d + e) + g) (12 inputs)
PAGE 31 ECE 124 – Assignment #3
ab
Assign#5 - Problem #3 [a] § find a minimum and or two level network to realize the function:
a'c + bc'd + ac'd
PAGE 32 ECE 124 – Assignment #3
ab
Assign#5 - Problem #3 [a] § find a minimum and or two level network to realize the function:
a'c + bc'd + ac’d
§ SOP: f = a’c + bc’d + ac’d (4 gates, 11 inputs)
PAGE 33 ECE 124 – Assignment #3
00 01 11 10
00 0 0 0 0
01 0 1 1 1
11 1 1 0 0
10 1 1 0 0
ab
cd
0
1
4
5
3 7
6
12
13
15
14
8
9
11
10 2
Assign#5 - Problem #3 [a] § find a minimum and or two level network to realize the function:
a'c + bc'd + ac’d
§ SOP: f = a’c + bc’d + ac’d (4 gates, 11 inputs)
§ POS: f = (a+b+c) (c+d) (a’+c’) (4 gates, 10 inputs)
PAGE 34 ECE 124 – Assignment #3
00 01 11 10
00 0 0 0 0
01 0 1 1 1
11 1 1 0 0
10 1 1 0 0
ab
cd
0
1
4
5
3 7
6
12
13
15
14
8
9
11
10 2
Assign#5 - Problem #3 [b] § find a minimum and or two level network to realize the function:
(b' + c)( a + b' + d)( a + b + c' + d )
PAGE 35 ECE 124 – Assignment #3
ab
Assign#5 - Problem #3 [b] § find a minimum and or two level network to realize the function:
(b' + c)( a + b' + d)( a + b + c' + d )
§ SOP: f = cd + ac + b’c’ (4 gates, 9 inputs)
PAGE 36 ECE 124 – Assignment #3
00 01 11 10
00 1 0 0 1
01 1 0 0 1
11 1 1 1 1
10 0 0 1 1
ab
cd
0
1
4
5
3 7
6
12
13
15
14
8
9
11
10 2
Assign#5 - Problem #3 [b] § find a minimum and or two level network to realize the function:
(b' + c)( a + b' + d)( a + b + c' + d )
§ SOP: f = cd + ac + b’c’ (4 gates, 9 inputs)
§ POS: f = (a+c’+d) (b’+c) (3 gates, 7 inputs)
PAGE 37 ECE 124 – Assignment #3
00 01 11 10
00 1 0 0 1
01 1 0 0 1
11 1 1 1 1
10 0 0 1 1
ab
cd
0
1
4
5
3 7
6
12
13
15
14
8
9
11
10 2