1. SYEN 3330 Digital Systems Jung H. Kim Chapter 2-5 1 SYEN 3330 Digital Systems Chapter 2 – Part 5

2. SYEN 3330 Digital Systems Chapter 2-5 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 teach other.

3. SYEN 3330 Digital Systems Chapter 2-5 3 Three-Variable Maps Topological warps of 3-variable K-maps that show all adjacencies:  Venn Diagram Cylinder YZ X 1 3 7 65 4 2 0

4. SYEN 3330 Digital Systems Chapter 2-5 4 Three-Variable Maps Example Shapes of Rectangles: 0 1 3 2 5 6 4 7 X Y Z XX YY ZZ ZZ

5. SYEN 3330 Digital Systems Chapter 2-5 5 Three Variable Maps F(x,y,z) =  x y+ z

6. SYEN 3330 Digital Systems Chapter 2-5 6 Three-Variable Map Simplification F(X,Y,Z) =  (0,1,2,4,6,7)

7. SYEN 3330 Digital Systems Chapter 2-5 7 Four Variable Maps x y z w w'x'y'z'w'x'y'zw'x'yz' w'xy'z'w'xy'zw'xyz'w'xyz wx'y'z'wx'y'zwx'yz'wx'yz wxy'z'wxy'zwxyz'wxyz x y z w m0m0 m1m1 m2m2 m3m3 m4m4 m5m5 m6m6 m7m7 m8m8 m9m9 m 10 m 11 m 12 m 13 m 14 m 15

8. SYEN 3330 Digital Systems Chapter 2-5 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")

9. SYEN 3330 Digital Systems Chapter 2-5 9 89 1011 12 13 1415 Four-Variable Maps Example Shapes of Rectangles: 0 1 3 2 5 6 4 7 X Y Z XX YY ZZ ZZ XX W WW

10. SYEN 3330 Digital Systems Chapter 2-5 10 Four-Variable Map Simplification F(W,X,Y,Z) =  (0, 2,4,5,6,7,8,10,13,19)

11. SYEN 3330 Digital Systems Chapter 2-5 11 Four-Variable Map Simplification F(W,X,Y,Z) =  (3,4,5,7,13,14,15,17)

12. SYEN 3330 Digital Systems Chapter 2-5 12 Systematic Simplification A Prime Implicant is a product term obtained by combining the maximum possible number of adjacent squares in the map. A is 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.

13. SYEN 3330 Digital Systems Chapter 2-5 13 Example of Prime Implicants

14. SYEN 3330 Digital Systems Chapter 2-5 14 Prime Implicant Practice F(A,B,C,D) =  (0,2,3,8,9,10,11,12,13,14,15)

15. SYEN 3330 Digital Systems Chapter 2-5 15 Systematic Approach (No Don’t Cares) 1.Select all Essential PI’s 2.Find and delete all Less Than PI’s 3.Repeat 1) and 2) until all minterms are covered If Cycles Occur: 4.Arbitrarily select a PI and generate a cover. 5.Delete the selected PI and generate a new cover 6.Select the cover with fewer literals 7.If a new cycle appears, repeat steps 4), 5), and 6) and compare all solutions for the best.

16. SYEN 3330 Digital Systems Chapter 2-5 16 Other PI Selection

17. SYEN 3330 Digital Systems Chapter 2-5 17 Example 2 from Supplement 1

18. SYEN 3330 Digital Systems Chapter 2-5 18 Example 2 (Continued)

19. SYEN 3330 Digital Systems Chapter 2-5 19 Another Example G(A,B,C,D) =  (0,2,3,4,7,12,13,14,15)

20. SYEN 3330 Digital Systems Chapter 2-5 20 Five Variable or More K-Maps

21. SYEN 3330 Digital Systems Chapter 2-5 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.

22. SYEN 3330 Digital Systems Chapter 2-5 22 Example: BCD “5 or More”

23. SYEN 3330 Digital Systems Chapter 2-5 23 Product of Sums Example F(A,B,C,D) =  (3,9,11,12,13,14,15) +  d (1,4,6) Use and take complement of result: