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# Logic Design - Chapter 4: Karnaugh Maps

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- THE THREE VARIABLE KARNAUGH MAP.
- THE FOUR VARIABLE KARNAUGH MAP.
- Karnaugh Map Simplification of SOP Expressions.
- KARNAUGH MAP PRODUCT OF SUM (POS) SIMPLIFICATION 5-VARIABLE K-MAPS.
- DON'T CARE CONDITIONS.

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### Logic Design - Chapter 4: Karnaugh Maps

1. 1. CHAPTER 4 THE KARNAUGH-VEITCHMAP
2. 2. Contents       THE THREE VARIABLE KARNAUGH MAP THE FOUR VARIABLE KARNAUGH MAP Karnaugh Map Simplification of SOP Expressions KARNAUGH MAP PRODUCT OF SUM (POS) SIMPLIFICATION 5-VARIABLE K-MAPS DON'T CARE CONDITIONS 2
3. 3. THE THREE VARIABLE KARNAUGH MAP Karnaugh Maps
4. 4. THE FOUR VARIABLE KARNAUGH MAP Karnaugh Maps
5. 5. Karnaugh Map Simplification of SOP Expressions  Grouping the 1s 1. A group must contain either 1, 2, 4, 8, or 16 cells.  2. Each cell in a group must be adjacent to one or more cells in that same group.  3. Always include the largest possible number of 1s in a group. 4. Each 1 on the map must be included in at least one group.   Karnaugh Maps
6. 6. Determining the Minimum SOP Expression from the Map    1. Group the cells that have 1s. 2. Determine the minimum product terms for each group. For a 3-variable map:       ? For a 5-variable map   A 1-cell group yields a 3-variable product term A 2-cell group yields a 2-variable product term A 4-cell group yields a 1-variable term An 8-cell group yields a value of 1 for the expression For a 4-variable map   (1) (2) (3) (4) ? 3. When all the minimum product terms are derived from the Karnaugh map, they are summed to form the minimum SOP expression. Karnaugh Maps
7. 7. EXAMPLE  Simplify the Boolean expression:  F(x,y,z) = Σ (0,1,6,7) 7
8. 8. EXAMPLE:  Simplify the Boolean expression: F(x,y,z) = Σ (0,2,5,7) 8
9. 9. EXAMPLE:  Group the 1's in each of the following Karnaugh maps: 9
10. 10. EXAMPLE: 10
11. 11. KARNAUGH MAP PRODUCT OF SUM (POS) SIMPLIFICATION If we mark the empty squares by 0's and combine them into valid adjacent squares, we obtain a simplified expression of the complement of the function, i.e., of F'.  The complement of F' gives us back the function F.  Because of Demorgan's theorem, the function so obtained is automatically in the product of sums form.  11
12. 12. EXAMPLE   Simplify the following Boolean function in (a) sum of products and (b) product of sums. F(w,x,y,z) = Σ (0,1,2,3,10,11,14) 12
13. 13. EXAMPLE     Use a Karnaugh map to minimize the following POS expression. (x+y+z)(w+x+y'+z) (w'+x+y+z') (w+x'+y+z) (w'+x'+y+z) (w+x+y+z)(w'+x+y+z) (w+x+y'+z) (w'+x+y+z') (w+x'+y+z) (w'+x'+y+z) =Π(0,8,2,9,4,12) 13
14. 14. 5-VARIABLE K-MAPS (1/2)  K-maps of more than 4 variables are more difficult to use because the geometry (hypercube configurations) for combining adjacent squares becomes more involved.  For 5 variables, e.g. v, w, x, y, z, we need 25 = 32 squares.  Each square has 5 neighbours. ELC224C Karnaugh Maps
15. 15. 5-VARIABLE K-MAPS (2/2)  Organized as two 4-variable K-maps. One for v' and the other for v. v' wx v y yz 00 01 11 m0 m1 m3 0 m4 1 m5 m7 00 m12 w wx 10 00 01 11 10 m2 00 m16 m17 m19 m18 m6 0 m20 1 m21 m23 m22 m28 m29 m31 m30 m24 m25 m27 m26 x m13 m15 m14 w 11 m8 m9 m11 y yz 11 m10 10 x 10 z z Corresponding squares of each map are adjacent. Can visualise this as one 4-variable K-map being on TOP of the other 4-variable K-map. ELC224C Karnaugh Maps
16. 16. DON'T CARE CONDITIONS      Sometimes a situation arises in which some input variable combinations are not allowed. Since these unallowed states will never occur in an application involving the BCD code, they can be treated as "don't care" terms for each "don't care" term, an X is placed in the cell. When grouping the 1's, Xs can be treated as 1's to make a larger grouping or as 0s if they cannot be used to advantage. Be careful do not make a group entirely of x's. 16
17. 17. Example x 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 Inputs y y 0 0 0 0 0 1 0 1 1 0 1 0 1 1 1 1 0 0 0 0 0 1 0 1 1 0 1 0 1 1 1 1 w 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 Output Y 0 0 0 0 0 0 0 1 1 1 X X X X X X 17
18. 18. Example    Simplify the following Boolean function F, where d represents the set of do not care conditions F(w,x,y,z) = Σ (0,1,2,8,10,11) d(w,x,y,z) = Σ (4,6,12,13) 18