Lecture 8: K-Map to POS reductions K-maps in higher dimensions CSE - - PowerPoint PPT Presentation

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

• n-setF and does not intersect with off-set R .

Prime Implicant: An implicant that is not a proper subset of any

• ther 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

• ther 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 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 1 Is M(0,2) a prime implicate?

• A. Yes
• B. No

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 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 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 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: 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 15

ab cd 00 01 11 10 00 01 11 10 K-map

Min product of sums: Ex3

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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 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

18 0 2 6 4 1 3 7 5

X 0 1 0 1 0 0 X

ab c 00 01 11 10 1

(V) (25pts) (Karnaugh Map) Use Karnaugh map to simplify function f (a, b, c) = Σ m(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.

19 0 2 6 4 1 3 7 5

X 0 1 0 1 0 0 X

ab c 00 01 11 10 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 21

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 22

bc de ab=(0,0) ab=(0,1) ab=(1,0) ab=(1,1)

Reading

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[Harris] Chapter 2, 2.7