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PPT – CSE20 Lecture 15 Karnaugh Maps PowerPoint presentation | free to download - id: 741f16-NGYzY

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CSE20 Lecture 15 Karnaugh Maps

- Professor CK Cheng
- CSE Dept.
- UC San Diego

Example

Given F Sm (3, 5), D Sm (0, 4)

b

0 2 6

4

- 0 0 -

1 3 7

5

c

0 1 0 1

a

Primes Sm (3), Sm (4, 5) Essential Primes

Sm (3), Sm (4, 5) Min exp f(a,b,c) abc

ab

Boolean Expression K-Map

Variable xi and its compliment xi

Two half planes Rxi, and Rxi

?

Product term P (Pxi e.g. bc)

?

Intersect of Rxi for all i in P e.g. Rb

intersect Rc

Each minterm

?

One element cell

Two minterms are adjacent iff they differ by one

and only one variable, eg abcd, abcd

The two cells are neighbors

?

Each minterm has n adjacent minterms

Each cell has n neighbors

?

Procedure Input Two sets of F R D

- Draw K-map.
- Expand all terms in F to their largest sizes

(prime implicants). - Choose the essential prime implicants.
- Try all combinations to find the minimal sum of

products. (This is the most difficult step)

Example

Given F Sm (0, 1, 2, 8, 14) D

Sm (9, 10) 1. Draw K-map

b

0 4

12 8

1 0 0 1

1 5

13 9

1 0 0 -

d

3 7 15

11

0 0 0 0

c

2 6

14 10

1 0 1 -

a

2. Prime Implicants Largest rectangles that

intersect On Set but not Off Set that

correspond to product terms. Sm (0, 1, 8, 9), Sm

(0, 2, 8, 10), Sm (10, 14) 3. Essential Primes

Prime implicants covering elements in F that

are not covered by any other primes. Sm (0, 1,

8, 9), Sm (0, 2, 8, 10), Sm (10, 14) 4. Min exp

Sm (0, 1, 8, 9) Sm (0, 2, 8, 10) Sm (10,

14) f(a,b,c,d) bc bd acd

Another example

Given F Sm (0, 3, 4, 14, 15) D

Sm (1, 11, 13) 1. Draw K-map

b

0 4

12 8

1 1 0 0

1 5

13 9

- 0 - 0

d

3 7

15 11

1 0 1 -

c

2 6

14 10

0 0 1 0

a

2. Prime Implicants Largest rectangles that

intersect On Set but not Off Set that

correspond to product terms. E.g. Sm (0, 4), Sm

(0, 1), Sm (1, 3), Sm (3, 11), Sm (14, 15), Sm

(11, 15), Sm (13, 15) 3. Essential Primes Prime

implicants covering elements in F that are

not covered by any other primes. E.g. Sm (0, 4),

Sm (14, 15) 4. Min exp Sm (0, 4), Sm (14, 15),

( Sm (3, 11) or Sm (1,3) ) f(a,b,c,d) acd

abc bcd (or abd)

Five variable K-map

c

c

0 4 12 8

16 20 28 24

1 5 13 9

17 21 29 25

e

e

3 7 15 11

19 23 31 27

d

d

2 6 14 10

18 22 30 26

b

b

a

Neighbors of m5 are minterms 1, 4, 7, 13, and

21 Neighbors of m10 are minterms 2, 8, 11, 14,

and 26

Six variable K-map

d

d

0 4 12 8

16 20 28 24

1 5 13 9

17 21 29 25

f

f

3 7 15 11

19 23 31 27

e

e

2 6 14 10

18 22 30 26

c

c

d

d

48 52 60 56

32 36 44 40

49 53 61 57

33 37 45 41

a

f

f

51 55 63 59

35 39 47 43

e

e

50 54 62 58

34 38 46 42

c

c

b

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 covered by 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 covered by 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.

Min product of sums

Given F Sm (3, 5), D Sm (0, 4)

b

0 2 6

4

- 0 0 -

1 3 7

5

c

0 1 0 1

a

Prime Implicates PM (0,1), PM (0,2,4,6), PM

(6,7) Essential Primes Implicates PM (0,1), PM

(0,2,4,6), PM (6,7) Min exp f(a,b,c)

(ab)(c )(ab)

Corresponding Circuit

a

b

f(a,b,c,d)

a

b

c

Quiz

- Given F Sm (0, 6), D Sm (2, 7),
- Fill the Karnaugh map.
- Identify all prime implicates
- Identify all essential primes.
- Find a minimal expression in product of sums

format.

Another min product of sums example

Given R Sm (3, 11, 12, 13, 14) D

Sm (4, 8, 10) K-map

b

0 4

12 8

1 - 0 -

1 5

13 9

1 1 0 1

d

3 7

15 11

0 1 1 0

c

2 6

14 10

1 1 0 -

a

Prime Implicates PM (3,11), PM (12,13),

PM(10,11), PM (4,12), PM

(8,10,12,14) Essential Primes PM

(8,10,12,14), PM (3,11),

PM(12,13) Exercise Derive f(a,b,c,d) in minimal

product of sums expression.

Summary

- Karnaugh Maps Two dimensional truth table

which mimics an n-variable cube with imaginary

adjacency. - Theme Relation between Boolean algebra and

Karnaugh maps. - Key words Primes, Essential Primes
- Goal Minimal expression in the format of
- sum-of-products or product-of-sums.