1. Chapter 4 OPTIMIZED IMPLEMENTATION OF LOGIC FUNCTIONS

2. 4.1 Karnaugh Mapping Used to minimize the number of gates –Reduce circuit cost –Reduce physical size –Reduce gate failures –Requires SOP form Karnaugh Mapping –Graphically shows output level for all possible input combinations – Moving from one cell to an adjacent cell, only one variable changes 31 2

3. Karnaugh Mapping Steps for K-map reduction: 1.Transform the Boolean equation into SOP form. 2.Fill in the appropriate cells of the K-map. 3.Encircle adjacent cells in groups of 2, 4 or 8. 1.Adjacent means a side is touching, NOT diagonal. 2.Watch for the wraparound 4.Find each term of the final SOP equation by determining which variables remain the same within circles. 33 3

4. x 2 (a) Truth table(b) Karnaugh map 0 1 01 m 0 m 2 m 3 m 1 x 1 x 2 00 01 10 11 m 0 m 1 m 3 m 2 x 1 Figure 4.2. Location of two-variable minterms. 4

5. Figure 4.3. The function of Figure 2.15. 1 5 Figure 2.15. The function f (x 1, x 2 ) =  m(0, 1, 3).

6. Figure 4.4. Location of three-variable minterms. 6 x 1 x 2 x 3 00011110 0 1 (b) Karnaugh map x 2 x 3 00 01 10 11 m 0 m 1 m 3 m 2 0 0 0 0 00 01 10 11 1 1 1 1 m 4 m 5 m 7 m 6 x 1 (a) Truth table m 0 m 1 m 3 m 2 m 6 m 7 m 4 m 5

7. Figure 4.5. Examples of three-variable Karnaugh maps. fx 1 x 3 x 2 x 3 += x 1 x 2 x 3 00 10 11 01 (a) The function of Figure 2.18 00011110 0 1 7 Figure 2.18. The function f (x 1, x 2, x 3 ) =  m(1, 4, 5, 6).

8. Figure 4.5. Examples of three-variable Karnaugh maps. x 1 x 2 x 3 11 00 11 01 fx 3 x 1 x 2 += (b) The function of Figure 4.1 00011110 0 1 8 Figure 4.1. The function f (x 1, x 2, x 3 ) =  m(0, 2, 4, 5, 6).

9. Figure 4.6. A four-variable Karnaugh map. 9

10. Figure 4.7. Examples of four-variable Karnaugh maps. f 2 x 3 x 1 x 4 += f 3 x 2 x 4 x 1 x 3 x 2 x 3 x 4 ++= f 4 x 1 x 3 x 1 x 3 ++= x 1 x 2 x 2 x 3 or 10

11. Index of 5-variable Karnaugh map 11 x 1 x 2 x 3 x 4 0 00011110 82416 2102618 6143022 4122820 00 01 11 10 x 1 x 2 x 3 x 4 1 00011110 92517 3112719 7153123 5132921 00 01 11 10 x 1 x 2 x 3 x 4 x 5: x 5 is the least significant bit.

12. Figure 4.8. A five-variable Karnaugh map. 12