CS 140 Lecture 4 Combinational Logic: K-Map

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CS 140 Lecture 4Combinational Logic K-Map

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

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• Part I. Combinational Logic
• Implementation
• K-map

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4-Variable K-Maps An example f(a,b,c,d)
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Corresponding K-map
b
0 4
12 8
0 0 0 0
1 5
13 9
0 0 0 0
d
3 7
15 11
1 1 1 1
c
2 6 14
10
1 1 1 1
a
f (a, b, c, d) c
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Another example w/ 4 bits
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Corresponding 4-variable K-map
b
0 4
12 8
1 0 0 1
1 5
13 9
1 0 0 -
d
0 0 0 0
3 7
15 11
c
1 0 1 -
2 6
14 10
a
f (a, b, c, d) bc bd acd
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Boolean Expression K-Map
Variable xi and its compliment xi
Two half planes Rxi, and Rxi
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Product term P (Pxi e.g. bc)
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Intersect of Rxi for all i in P e.g. Rb
intersect Rc
Each minterm
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One element cell
Two minterms are adjacent iff they differ by one
and only one variable, eg abcd, abcd
The two cells are neighbors
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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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4-input K-map
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4-input K-map
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K-maps with Dont Cares
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K-maps with Dont Cares
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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)