CS 140 Lecture 5

1
CS 140 Lecture 5

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

2
Part I. Combinational Logic

• Specification
• Implementation
• K-map
• Sum of products
• Product of sums

3
Implicant A product term tat covers at least an
element in F and has no intersect with R . Prime
Implicants Largest rectangles that intersect
on-set but not off-set that correspond to
product terms. Essential Primes Prime
implicants covering elements in F that are
not covered by any other primes.
4
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
5
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 minterm 5 are 1, 4, 13, 7, and
21 Neighbors of minterm 10 are 2, 8, 11, 14, and
26
6
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
7
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)
8
Corresponding Circuit
a
b
f(a,b,c,d)
a
b
c
9
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
10
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)