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Lecture 5: K-Maps in higher dimensions, K-map to product of sum minimization CSE 140: Components and Design Techniques for Digital Systems Spring 2014 CK Cheng, Diba Mirza Dept. of Computer Science and Engineering University of California,

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Lecture 5: K-Maps in higher dimensions, K-map to product of sum minimization

CSE 140: Components and Design Techniques for Digital Systems Spring 2014

CK Cheng, 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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SLIDE 4

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

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SLIDE 5

Minimum Sum of Product

Given F (a,b,c) = Σm (3, 5), D = Σm (0, 4)

0 2 6 4 1 3 7 5 5

ab c 00 01 11 10 1 Prime Implicants: Essential Prime Implicants: Min SOP exp: f(a,b,c)=

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SLIDE 6

Minimum Sum of Product

0 2 6 4 1 3 7 5

X 0 0 X 0 1 0 1

Prime Implicant: Σm (3), Σm (4, 5) Essential Prime Implicant: Σm (3), Σm (4, 5) Min SOP exp: f(a,b,c) = a’bc + ab’

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ab c 00 01 11 10 1 Given F (a,b,c) = Σm (3, 5), D = Σm (0, 4)

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SLIDE 7

Minimum Product of Sum

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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

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SLIDE 8

Minimum Product of Sum

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Given F (a,b,c) = Σm (3, 5), D = Σm (0, 4) F (a,b,c) = Σm (1, 2, 6,7)+ Σd (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

0 2 6 4 1 3 7 5

ab c 00 01 11 10 1

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SLIDE 9

Minimum Product of Sum: Boolean Algebra Rationale

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F (a,b,c) = Σm (1, 2, 6,7)+ Σd (0, 4)

0 2 6 4 1 3 7 5

X 1 1 X 1 0 1 0

ab c 00 01 11 10 1

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SLIDE 10

Minimum Product of Sum

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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

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SLIDE 11

Minimum Product of Sum

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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 can be minimized to the following max term:

  • A. a+b
  • B. (a+b)’
  • C. a’+b’
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SLIDE 12

Minimum Product of Sum

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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) =

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SLIDE 13

Minimum Product of Sum

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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’)

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Corresponding Circuit

a b a’ b’ c f(a,b,c,d)

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Another min product of sums example

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

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SLIDE 16

Another min product of sums example

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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)

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SLIDE 17

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

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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

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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

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SLIDE 20

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

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Reading

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[Harris] Chapter 3, 3.1