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ECE 301 - Digital Electronics. 2. Karnaugh Maps. Graphical representation of a truth tableCan be used to minimize logic functionsUses Logic Adjacency A.B A.B' = A (Boolean Algebra Law)Does not produce unique resultsNot directly transferable to computer algorithmsWhy not just use Boolean
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1. ECE 301 Digital Electronics Karnaugh Maps
and
Minimization of Boolean Expressions
(Lecture #5)
2. ECE 301 - Digital Electronics 2 Karnaugh Maps Graphical representation of a truth table
Can be used to minimize logic functions
Uses Logic Adjacency
A.B + A.B' = A (Boolean Algebra Law)
Does not produce unique results
Not directly transferable to computer algorithms
Why not just use Boolean Algebra?
Boolean algebra is useful for general proofs
Difficult to use for minimization
3. ECE 301 - Digital Electronics 3 Two-variable K-Map
4. ECE 301 - Digital Electronics 4 Three-variable K-Map
5. ECE 301 - Digital Electronics 5 Four-variable K-Map
6. ECE 301 - Digital Electronics 6 Four-variable K-Map
7. ECE 301 - Digital Electronics 7
8. ECE 301 - Digital Electronics 8 Minimization using K-maps Enter minterms (for SOP) into K-map
Identify adjacent cells
Minterms differ in only one bit
Use adjacency to minimize logic function
Gray code used for enumeration
Specifies the location of each minterm in K-map
Horizontal and Vertical adjacency
Both are logically adjacent
K-map wraps
Left and right columns are logically adjacent
Top and bottom rows are logically adjacent
9. ECE 301 - Digital Electronics 9 Minimization using K-Maps Group adjacent cells to reduce
the number of literals in a term
the number of terms in a Boolean expression
Adjacencies are of size 1, 2, 4, 8,
Number of adjacent cells is a power of 2
Adjacent cells form a rectangle
The larger the grouping of adjacent cells the greater the minimization of the logical function.
10. ECE 301 - Digital Electronics 10 Minimization using K-Maps
11. ECE 301 - Digital Electronics 11 Two-variable K-Map Example:
Using a Karnaugh Map, minimize the logic function described by the following Truth table.
12. ECE 301 - Digital Electronics 12 Two-variable K-Map
13. ECE 301 - Digital Electronics 13 Three-variable K-Map Example:
Using a Karnaugh Map, minimize the logic function described by the following Truth table.
14. ECE 301 - Digital Electronics 14 Three-variable K-Map
15. ECE 301 - Digital Electronics 15 Exercise:
Given the following Truth table,
1. Derive the Boolean Expression
2. Use a K-Map to determine the minimized Boolean Expression Minimization using K-Maps
16. ECE 301 - Digital Electronics 16 Minimization using K-Maps
17. ECE 301 - Digital Electronics 17 Four-variable K-Map Example:
Using a Karnaugh Map, minimize the logic function described by the following Truth table.
18. ECE 301 - Digital Electronics 18 Four-variable K-Map
19. ECE 301 - Digital Electronics 19 Four-variable K-Map Example:
Using a Karnaugh Map, minimize the logic function described by the following Truth table.
20. ECE 301 - Digital Electronics 20 Four-variable K-Map
21. ECE 301 - Digital Electronics 21 Exercise:
Given the following Truth table,
1. Derive the Boolean Expression
2. Use a K-Map to determine the minimized Boolean Expression Minimization using K-Maps
22. ECE 301 - Digital Electronics 22 Minimization using K-Maps
23. ECE 301 - Digital Electronics 23
24. ECE 301 - Digital Electronics 24 Minimization using K-maps Enter Maxterms (for POS) into K-map
Identify adjacent cells
Maxterms differ in only one bit
Use adjacency to minimize logic function
Gray code used for enumeration
Specifies the location of each Maxterm in K-map
Horizontal and Vertical adjacency
Both are logically adjacent
K-map wraps
Left and right columns are logically adjacent
Top and bottom rows are logically adjacent
25. ECE 301 - Digital Electronics 25 Three-variable K-Map Example:
Using a Karnaugh Map, minimize the logic function described by the following Truth table.
26. ECE 301 - Digital Electronics 26 Three-variable K-Map
27. ECE 301 - Digital Electronics 27 Four-variable K-Map Example:
Using a Karnaugh Map, minimize the logic function described by the following Truth table.
28. ECE 301 - Digital Electronics 28 Four-variable K-Map
