lecture 8: k-map to pos reductions k-maps in higher dimensions · lecture 8: k-map to pos...
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Lecture 8: K-Map to POS reductions K-maps in higher dimensions
CSE 140: Components and Design Techniques for Digital Systems
Diba Mirza
Dept. of Computer Science and Engineering University of California, San Diego
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Part I. Combinational Logic 1. Specification 2. Implementation
K-map: Sum of products Product of sums
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Implicant: A product term that has non-empty intersection with on-setF and does not intersect with off-set R . Prime Implicant: An implicant that is not a proper subset of any other implicant. Essential Prime Implicant: A prime implicant that has an element in on-set F but this element is not covered by any other prime implicants.
Implicate: A sum term that has non-empty intersection with off-set R and does not intersect with on-set F. Prime Implicate: An implicate that is not a proper subset of any other implicate. Essential Prime Implicate: A prime implicate that has an element in off-set R but this element is not covered by any other prime implicates.
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K-Map to Minimized Product of Sum
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• Sometimes easier to reduce the K-map by considering the offset
• Usually when number of zero outputs is less than number of outputs that evaluate to one OR offset is smaller than onset
ab
cd
00
01
00 01 11 10
11
10
1 1 1 1 0 1 1 1 0 1 1 1 1 1 1 1
Minimum Product of Sum: Ex1
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Given F (a,b,c) = Σm (3, 5), D = Σm (0, 4)
0 2 6 4
1 3 7 5
X 0 0 X
0 1 0 1
ab c 00 01 11 10
0
1
Minimum Product of Sum: Ex 1
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Given F (a,b,c) = Σm (3, 5), D = Σm (0, 4)
0 2 6 4
1 3 7 5
X 0 0 X
0 1 0 1
ab c 00 01 11 10
0
1
PI Q: The adjacent cells grouped in red minimize to the following sum term: A. a+b B. (a+b)’ C. a’+b’
Minimum Product of Sum: Ex1
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Given F (a,b,c) = Σm (3, 5), D = Σm (0, 4)
0 2 6 4
1 3 7 5
X 0 0 X
0 1 0 1
ab c 00 01 11 10
0
1
Prime Implicates: Essential Primes Implicates: Min exp: f(a,b,c) =
Minimum Product of Sum: Ex 1
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Given F (a,b,c) = Σm (3, 5), D = Σm (0, 4)
0 2 6 4
1 3 7 5
X 0 0 X
0 1 0 1
ab c 00 01 11 10
0
1
Prime Implicates: ΠM (0, 1), ΠM (0, 2, 4, 6), ΠM (6, 7) Essential Primes Implicates: ΠM (0, 1), ΠM (0, 2, 4, 6), ΠM(6, 7) Min exp: f(a,b,c) = (a+b)(c )(a’+b’)
Corresponding Circuit
a
b
a’
b’
c
f(a,b,c,d)
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Min exp: f(a,b,c) = (a+b)(c )(a’+b’)
0 4 12 8
1 5 13 9
3 7 15 11
2 6 14 10
1 0 0 1
1 0 0 X
0 0 0 0
1 0 1 X
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ab cd
00
01
00 01 11 10
11
10
• Reduce the following to a POS form • First find the essential prime implicates
Minimum product of sum: Ex 2
0 4 12 8
1 5 13 9
3 7 15 11
2 6 14 10
1 0 0 1
1 0 0 X
0 0 0 0
1 0 1 X
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ab cd
00
01
00 01 11 10
11
10
• Reduce the following to a POS form • First find the essential prime implicates
Minimum product of sum: Ex2
0 4 12 8
1 5 13 9
3 7 15 11
2 6 14 10
1 0 0 1
1 0 0 X
0 0 0 0
1 0 1 X
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ab cd
00
01
00 01 11 10
11
10
• Reduce the following to a POS form • First find the essential prime implicates
Minimum product of sum: Ex 2
Min product of sums: Ex3
Given R(a,b,c,d) = Σm (3, 11, 12, 13, 14) D (a,b,c,d)= Σm (4, 8, 10)
0 4 12 8
1 5 13 9
3 7 15 11
2 6 14 10
13
ab cd 00 01 11 10
00
01
11
10
K-map
Min product of sums: Ex3
14 a
d
1 X 0 X
1 1 0 1
0 1 1 0
1 1 0 X
0 4 12 8
1 5 13 9
3 7 15 11
2 6 14 10
ab 00 01 11 10 cd
00
01
11
10
Given R(a,b,c,d) = Σm (3, 11, 12, 13, 14) D (a,b,c,d)= Σm (4, 8, 10)
Prime Implicates: ΠM (3,11), ΠM (12,13), ΠM(10,11), ΠM (4,12), ΠM (8,10,12,14) PI Q: Which of the following is a non-essential prime implicate? A. ΠM(3,11) B. ΠM(12,13) C. ΠM(10,11) D. ΠM(8,10,12,14)
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a
d
1 X 0 X
1 1 0 1
0 1 1 0
1 1 0 X
0 4 12 8
1 5 13 9
3 7 15 11
2 6 14 10
ab 00 01 11 10 cd
00
01
11
10
16
0 2 6 4
1 3 7 5
X 0 1 0
1 0 0 X
ab c 00 01 11 10
0
1
-3-
(IV) (25pts) (Karnaugh Map) Use Karnaugh map to simplify functionf (a, b, c) = Σm(2, 3, 4, 7) +Σ d(0, 5). List all possible minimal sum of productsexpressions. Show the Boolean expressions. No need for the logic diagram.
(V) (25pts) (Karnaugh Map) Use Karnaugh map to simplify functionf (a, b, c) = Σm(1, 6) +Σ d(0, 5). List all possible minimal product of sums expres-sions. Show the Boolean expressions. No need for the logic diagram.
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0 2 6 4
1 3 7 5
X 0 1 0
1 0 0 X
ab c 00 01 11 10
0
1
Five variable K-map
0 4 12 8
c
d
b
e 1 5 13 9
3 7 15 11
2 6 14 10
16 20 28 24
c
d
b
e
a
17 21 29 25
19 23 31 27
18 22 30 26
Neighbors of m5 are: minterms 1, 4, 7, 13, and 21 Neighbors of m10 are: minterms 2, 8, 11, 14, and 26
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a=0 a=1
bc de
00 01 11 10 00 01 11 10
00
01 11 10
Reading a Five variable K-map
0 4 12 8
c
d
b
e 1 5 13 9
3 7 15 11
2 6 14 10
16 20 28 24
c
d
b
e
a
17 21 29 25
19 23 31 27
18 22 30 26
19
a=0 a=1
bc de
00 01 11 10 00 01 11 10
00
01 11 10
1 1 1 1 1
1 1 1 1 1
1 1
1 1 1 1 1 1
Six variable K-map d
e
c
f
d
e
c d
e
c
f
48 52 60 56
d
e
c b
49 53 61 57
51 55 63 59
50 54 62 58
a
32 36 44 40
33 37 45 41
35 39 47 43
34 38 46 42
f
f
0 4 12 8
1 5 13 9
3 7 15 11
2 6 14 10
16 20 28 24
17 21 29 25
19 23 31 27
18 22 30 26
20
bc de
ab=(0,0) ab=(0,1)
ab=(1,0) ab=(1,1)
Reading
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[Harris] Chapter 2, 2.7