CSE20 Lecture 15 Karnaugh Maps

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CSE20 Lecture 15Karnaugh Maps

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

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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
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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
?
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Procedure Input Two sets of F R D

1. Draw K-map.
2. Expand all terms in F to their largest sizes
   (prime implicants).
3. Choose the essential prime implicants.
4. Try all combinations to find the minimal sum of
   products. (This is the most difficult step)

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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
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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
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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
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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)
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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
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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
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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 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.
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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)
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Corresponding Circuit
a
b
f(a,b,c,d)
a
b
c
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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.

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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
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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.
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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.