CS 140 Lecture 3

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CS 140 Lecture 3

• Professor CK Cheng
• Tuesday 4/09/02

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

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Example w/ 4 bits
4
Corresponding 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
f (a, b, c, d) bc bd acd
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Boolean Expression K-Map
Each Variable xi and its compliment xi
Two half planes Rxi, Rxi divided by xi
?
Each product term P (PXi e.g.
bc)
Intersection of Rxi for all i e P. (A
rectangle e.g. Rb Rc)
?
U
Each minterm
1-cell
?
Two minterms are adjacent if they differ by one
variable, eg
abcd is adjacent to abcd
The two cells are neighbors
?
Each minterm has n adjacent neighbors
Each cell has n neighbors
?
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Another example 3 bits
f(a, b, c, d) a bc
Id a b c f (a,b,c,d) 0 0
0 0 1 1 0 0 1
1 2 0 1 0 1 3 0
1 1 0 4 1 0 0
0 5 1 0 1 0 6 1
1 0 0 7 1 1 1
0
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Corresponding K-map
b
0 2 6
4
1 1 0 0
1 3 7
5
c
1 1 1 0
a
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One more 4 bit example f(a,b,c,d) a bc
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Corresponding K-map
b
0 4
12 8
1 1 0 0
1 5
13 9
1 1 0 0
d
3 7
15 11
1 1 1 0
c
2 6
14 10
1 1 1 0
a
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Given a K-map, derive a minimal Boolean
expression in sum of products form (or product of
sums). Obj minimal of terms, minimal of
literals. Hints of terms gt of rectangles
of literals gt inverse of the size
of rectangles (if the size of the rectangle
is larger, then we can reduce literals) We
want to find the minimum number of rectangles in
their largest sizes to cover the On Set.
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Procedure Input Two sets of F R D

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