1. SYEN Digital Systems
Chapter 2 – Part 5
SYEN 3330 Digital Systems

2. Three-Variable Maps
Reduced literal product terms for SOP standard forms correspond to rectangles on K-maps containing cell counts that are powers of 2.
Rectangles of 2 cells represent 2 adjacent minterms; of 4 cells represent 4 minterms that form a “pairwise adjacent” ring.
Rectangles can be in many different positions on the K-map since adjacencies are not confined to cells truly next to each other.
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3. Three-Variable Maps
Topological warps of 3-variable K-maps that show all adjacencies:
Venn Diagram  Cylinder X 4 Y 6 Z 5 7 3 2 1
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4. Three-Variable Maps Example Shapes of Rectangles: X Y Z X Y Z 1 3 21 3 2 4 5 7 6
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5. Three Variable Maps
F(x,y,z) = x y+ z
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6. Three-Variable Map Simplification
F(X,Y,Z) = (0,1,2,4,6,7)
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7. Four Variable Maps x y z w x y z w m0 m1 m2 m3 m4 m5 m6 m7 m8 m9 m10
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8. Four Variable Terms Four variable maps can have terms of:
Single one = 4 variables, (i.e. Minterm)
Two ones = 3 variables,
Four ones = 2 variables
Eight ones = 1 variable,
Sixteen ones = zero variables (i.e. Constant "1")
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9. Four-Variable Maps Example Shapes of Rectangles: Y Y 1 3 2 X W 4 51 3 2 X W 4 5 7 6 X 12 13 15 14 W 11 10 8 9 X Z Z Z
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10. Four-Variable Map Simplification
F(W,X,Y,Z) = (0, 2,4,5,6,7,8,10,13)
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11. Four-Variable Map Simplification
F(W,X,Y,Z) = (3,4,5,7,13,14,15)
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12. Systematic Simplification
A Prime Implicant is a product term obtained by combining the maximum possible number of adjacent squares in the map.
A prime implicant is called an Essential Prime Implicant if it is the only prime implicant that covers (includes) one or more minterms.
Prime Implicants and Essential Prime Implicants can be determined by inspection of the K-Map.
A set of prime implicants that "covers all minterms" means that, for each minterm of the function, there is at least one prime implicant in the selected set of prime implicants that includes the minterm.
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13. Example of Prime Implicants
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14. Prime Implicant Practice
F(A,B,C,D) = (0,2,3,8,9,10,11,12,13,14,15)
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15. Systematic Approach (No Don’t Cares) Select all Essential PI’s
Find and delete all Less Than PI’s
Repeat 1) and 2) until all minterms are covered
If Cycles Occur:
Arbitrarily select a PI and generate a cover.
Delete the selected PI and generate a new cover
Select the cover with fewer literals
If a new cycle appears, repeat steps 4), 5), and 6) and compare all solutions for the best.
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16. Other PI Selection
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17. Example 2 from Supplement 1
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18. Example 2 (Continued)
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19. Another Example G(A,B,C,D) =  (0,2,3,4,7,12,13,14,15)
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20. Five Variable or More K-Maps
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21. Don't Cares in K-Maps
Sometimes a function table contains entries for which it is known the input values will never occur. In these cases, the output value need not be defined. By placing a “don't care” in the function table, it may be possible to arrive at a lower cost logic circuit.
“Don't cares” are usually denoted with an "x" in the K-Map or function table.
Example of “Don't Cares” - A logic function defined on 4-bit variables encoded as BCD digits where the four-bit input variables never exceed 9, base 2. Symbols 1010, 1011, 1100, 1101, 1110, and 1111 will never occur. Thus, we DON'T CARE what the function value is for these combinations.
“Don't cares“are used in minimization procedures in such a way that they may ultimately take on either a 0 or 1 value in the result.
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22. Example: BCD “5 or More”
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23. Product of Sums Example
F(A,B,C,D) =  (3,9,11,12,13,14,15) d (1,4,6)
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