eattle Pacific University EE 1210 - Logic System Design KMaps-1 Two-Level Simplification • All Boolean expressions can be represented in two-level forms • Sum-of-products • Product-of-sums c b a c b a bc a c b a f ) , , ( Canonical S.O.P. form Reduced S.O.P. form • Canonical forms are very easy to produce • Just read them off of a truth table • But, they’re not the most efficient representation • Reduced two-level forms are more efficient f abc abc ab c abc f abc abc ab c c f abc abc ab (,,) (,,) ( ) (,,)
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Seattle Pacific University EE 1210 - Logic System DesignKMaps-1 Two-Level Simplification All Boolean expressions can be represented in two- level forms.
Seattle Pacific University EE Logic System DesignKMaps-3 Karnaugh maps 2-variable K-map A B F B A F(A,B) Space for A’B Space for AB Space for A’B’ Space for AB’
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Seattle Pacific University EE 1210 - Logic System Design KMaps-1
Two-Level Simplification
• All Boolean expressions can be represented in two-level forms• Sum-of-products• Product-of-sums cbacbabcacbaf ),,(
Canonical S.O.P. form
Reduced S.O.P. form
• Canonical forms are very easy to produce• Just read them off of a
truth table• But, they’re not the most
efficient representation• Reduced two-level forms
are more efficient
f a b c abc ab c ab cf a b c abc ab c cf a b c abc ab
( , , )( , , ) ( )( , , )
Seattle Pacific University EE 1210 - Logic System Design KMaps-2
Venn Diagrams
Consider a Venn Diagram for 2 sets, A and B
AB’
A’B
AB
A’B’
B=0 B=1
A=0
A=1
A
B A’B’
AB’ AB
A’B
Seattle Pacific University EE 1210 - Logic System Design KMaps-3
Karnaugh maps
2-variable K-map
A 0 0 1 1
B 0 1 0 1
F 0 110
0
1
1
0
B A 0 1
0
1
00
10
01
11
F(A,B)
Space for A’B
Space for AB
Space for A’B’
Space for AB’
Seattle Pacific University EE 1210 - Logic System Design KMaps-4
Karnaugh maps
K-maps can represent up to four variables easily
3-variableK-map 4-variable
K-map
Numbering Scheme: 00, 01, 11, 10Gray Code — only a single bit changes from one number to the next
A
f(A,B,C)f(A,B,C,D)AB
C
00
01
11
10
0 1
000 001
010 011
110 111
100 101
B
ABCD
00 01 11 10
00
01
11
10
0000 0001
0100 0101
1100 1101
1000 1001
0011 0010
0111 0110
1111 1110
1011 1010A
B
C
D
C
m0 m1
m2 m3
m6 m7
m4 m5
m0 m1
m4 m5
m12 m13
m8 m9
m3 m2
m7 m6
m15 m14
m11 m10
Seattle Pacific University EE 1210 - Logic System Design KMaps-5
A B C D W X Y Z0 0 0 0 1 0 0 10 0 0 1 0 0 0 00 0 1 0 0 0 0 10 0 1 1 0 0 1 00 1 0 0 0 0 1 10 1 0 1 0 1 0 00 1 1 0 0 1 0 10 1 1 1 0 1 1 01 0 0 0 0 1 1 11 0 0 1 1 0 0 01 0 1 0 X X X X1 0 1 1 X X X X1 1 0 0 X X X X1 1 0 1 X X X X1 1 1 0 X X X X1 1 1 1 X X X X
N1 N2
Seattle Pacific University EE 1210 - Logic System Design KMaps-17