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Unit 5: Karnaugh Maps
EEC180A
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5.1 Minimum Forms of Switching Functions
( ) ( )∑= 7,6,5,2,1,0,, mcbaF
abcabccabbcacbacbaF +++++= '''''''''
Find a minimum sum-of products expression for:
F = a’b’ b’c bc’ ab+ + +
abcabccabbcacbacbaF +++++= '''''''''
None of these terms can be eliminated
However, if we combine in a different way.
F = a’b’ bc’ ac+ +
Note: Use XY’ + XY = X
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5.2 Karnaugh Maps
We can represent a 1 and 2-input truth table as 1-D and 2-D cube
XY00011011
F
X F01
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XY + XY’ = XX’Y + X’Y’ = X’
X’Y+XY = Y
X’Y’+XY’ = Y’
5.2 Karnaugh Maps
Allows for easy application of XY + XY’ = X
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5.2 Two-Variable Karnaugh Maps
A
B
AB
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5.2 Two-Variable Karnaugh Maps
Two Variable Karnaugh Map Example:
• Minterms in adjacent squares on the map can be combined sincethey differ in only one variable (i.e. XY’ + XY = X)
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5.2 Three-Variable Karnaugh Maps
F
We can represent a 3-input truth table as a 3-D cube
X Y Z
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5.2 Three-Variable Karnaugh Maps
Location of Minterms on a Three Variable Karnaugh Map
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Truth Table and resulting Karnaugh Map for Three-Variable Function
5.2 Three-Variable Karnaugh Maps
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5.2 Three-Variable Karnaugh Maps
Location of Minterms on a Three Variable Karnaugh Map
( ) ( ) ( )∏∑ == 7,6,4,2,05,3,1,, mcbaF
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5.2 Three-Variable Karnaugh Maps
''' acbabcF ++=Karnaugh Map for
cb'
'a
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5.2 Three-Variable Karnaugh MapsKarnaugh Maps for Product Terms
abc
a
bc
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5.2 Three-Variable Karnaugh Maps
Simplification of a Three-Variable Function
( )∑= 5,3,1mF
cbcaF '' +=
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5.2 Three-Variable Karnaugh Maps
Simplification of F’ ( )∑= 5,3,1mF
( )∑= 7,6,4,2,0' mF
abcF += ''F’
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5.2 Three-Variable Karnaugh Maps
Karnaugh Maps which illustrate the Consensus Theorem
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5.2 Three-Variable Karnaugh Maps
Function with Two Minimum Forms
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5.3 Four-Variable Karnaugh Maps
Adjacent squares should differ by only one variable
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5.3 Four-Variable Karnaugh Maps
Location of Minterms on a Four-Variable Karnaugh Map
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5.3 Four-Variable Karnaugh MapsSample 4-variable Karnaugh Map
'' dbaacdF ++=
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5.3 Four-Variable Karnaugh MapsSimplification of Four-Variable Functions
( )∑= 13,12,10,5,4,3,1mF
''''' cdabdbabcF ++=
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5.3 Four-Variable Karnaugh MapsSimplification of Incompletely Specified Function
( ) ( )∑∑ += 13,12,69,7,5,3,1 dmF
dcdaF '' +=
c’da’d
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5.3 Four-Variable Karnaugh MapsFinding Minimum Product of Sums from Karnaugh Maps
yxzywwyzzxF '''''' +++=
F’F
1
0
1
1
1
0
0
0
0
0
1
0
1
0
1
1
wxyz 00 01 11 10
00
01
11
10
0
1
0
0
0
1
1
1
1
1
0
1
0
1
0
0
wxyz 00 01 11 10
00
01
11
10
y’z
wxz’
w’xy
xywwxzzyF '''' ++=Using DeMorgan’s
( )( )( )''''' yxwzxwzyF +++++=
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5.4 Determination of Minimum Expressions
Implicant – any single 1 or any group of 1’s which can be combined together on a map of the function F
1
1 1
1
1
1
1
wxyz 00 01 11 10
00
01
11
10
wy’z’
wy’z
wxy’ wx’y’
wy’
w’yz’
w’x’y
List of Implicants – wxy’, wx’y’, wy’z’, wy’z, wy’, w’x’y, w’yz’and all single 1’s
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Prime Implicant – an implicant which can not be combinedwith another term to eliminate a variable.
1
1 1
1
1
1
1
wxyz 00 01 11 10
00
01
11
10
1
1 1
1
1
1
1
wxyz 00 01 11 10
00
01
11
10
wy’
w’yz’
w’x’y
List of Prime Implicants: w’x’y, w’yz’, wy’
5.4 Determination of Minimum Expressions
Implicants Prime Implicants
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Find the prime implicants:
1
1
1
1
1
1
1
1
1
1
abcd 00 01 11 10
00
01
11
10
5.4 Determination of Minimum Expressions
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Find the prime implicants:
1
1
1
1
1
1
1
1
1
1
abcd 00 01 11 10
00
01
11
10
1
1
1
1
1
1
1
1
1
1
abcd 00 01 11 10
00
01
10
11
5.4 Determination of Minimum Expressions
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1
1
1
1
1
1
1
1
1
1
abcd 00 01 11 10
00
01
11
10
1
1
1
1
1
1
1
1
1
1
abcd 00 01 11 10
00
01
10
11
5.4 Determination of Minimum Expressions
Minimum Solution might not utilize all prime implicants
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1 1
Essential Prime Implicant –A prime implicant that contains a minterm that is coveredby only one prime implicant
5.4 Determination of Minimum Expressions
1
1
1
1
1
1
1 1
abcd 00 01 11 10
00
01
10
11
- minterm covered by morethan 1 prime implicant
- minterm covered byonly 1 prime implicant
List of Essential Prime Implicants: bc’, ac
bc’
ac
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1 1
To find minimum expression:
5.4 Determination of Minimum Expressions
1
1
1
1
1
1
1 1
abcd 00 01 11 10
00
01
10
11
find simplestexpressionfor remaining1’s
a’b’d
F = a’b’d + bc’ + ac
bc’
ac
• Find all Prime Implicants• Determine Essential Prime Implicants• Find Simplest Expression for remaining uncovered 1’s
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Find Minimum Sum-of-Products Expression
5.4 Determination of Minimum Expressions
1
1
1
1
1
1 1 1
ABCD 00 01 11 10
00
01
11
10
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5.4 Determination of Minimum Expressions
1
1
1
1
1
1 1 1
ABCD 00 01 11 10
00
01
11
10
1
1
1
1
1
1 1 1
ABCD 00 01 11 10
00
01
11
10
First find all Prime Implicants
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5.4 Determination of Minimum Expressions
1
1
1
1
1
1 1 1
ABCD 00 01 11 10
00
01
11
10
Next find all essential Prime Implicants
1
1
1
1
1
1 1 1
ABCD 00 01 11 10
00
01
11
10
- mintern covered byonly 1 prime implicant
ACD
A’C’
A’B’D’
List of Essential Prime Implicants: A’C’, ACD, A’B’D’
Minimum Solution: F = A’C’ + ACD + A’B’D’ + A’BDor
BCD
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