Plotting functions not in canonical form

1. Plotting functions not in canonical form • Plot the function f(a, b, c) = a + bc ab a ab c 00 01 11 10 c 00 01 11 10 0 1 1 0 0 2 6 4 1 1 1 1 1 1 3 7 5 b The squares are numbered – derive the canonical form

2. 00 00 01 01 11 11 10 10 18 0 1 19 3 17 16 2 00 00 22 4 23 5 7 21 6 20 01 01 30 12 31 13 15 29 14 28 11 11 26 8 9 27 11 25 24 10 10 10 5-variable K-maps - alternative 0 1

3. 00 00 00 00 01 01 01 01 11 11 11 11 10 10 10 10 18 62 0 40 63 1 19 41 61 3 43 17 16 42 2 60 00 10 10 00 22 58 44 4 45 59 23 5 47 57 7 21 46 20 6 56 01 11 11 01 54 36 12 30 13 31 55 37 39 15 53 29 28 38 14 52 11 01 01 11 32 26 50 8 33 51 9 27 25 49 35 11 34 48 24 10 00 10 10 00 6-variable K-maps - alternative 00 01 10 11

4. Simplifying functions using K-maps • Why is simplification possible • Logically adjacent minterms are physically adjacent on the K-map • Adjacent minterms can be combined by eliminating the common variable • abc and ābc are adjacent • abc + ābc = bc  variable a eliminated • Done by drawing on the map a ring around the terms that can be combined

5. Simplifying functions using K-maps

6. Simplifying functions using K-maps

7. Simplifying functions using K-maps • Definition of terms • Implicant  product term that can be used to cover minterms • Prime implicant  implicant not covered by any other implicant • Essential prime implicant  a prime implicant that covers at least one minterm not covered by any other prime implicant • Cover  set of prime implicants that cover each minterm of the function • Minimizing a function  finding the minimum cover

8. Simplifying functions using K-maps • Definition of terms • Implicants:

9. Simplifying functions using K-maps • Definition of terms • Prime implicants: only B and AC • Essential prime implicants: B and AC • Cover: { B, AC }

10. Simplifying functions using K-maps • Definition of terms • Implicate  sum term that can be used to cover maxterms (0’s on the K-map) • Prime implicate  implicate not covered by any other implicate • Essential prime implicate  a prime implicate that covers at least one maxterm not covered by any other prime implicate • Cover  set of prime implicates that cover each maxterm of the function

11. Simplifying functions using K-maps • Algorithm 1: • Fast and easy, not optimal

12. Simplifying functions using K-maps • Algorithm 2: • More work than the first • Can give better results, because all prime implicants are considered • Still not optimal

13. Simplifying functions using K-maps • Algorithm 2: 1: Identify all PIs

14. Simplifying functions using K-maps • Algorithm 2: 2: Identify EPIs

15. Simplifying functions using K-maps • Algorithm 2: 3: Select cover

16. The Quine-McCluskey minimization method • Tabular • Systematic • Can handle a large number of variables • Can be used for functions with more than one output

17. The Q-M minimization method

18. The Q-M minimization method

19. The Q-M minimization method

20. The Q-M minimization method • Combine minterms from List 1 into pairs in List 2 • Take pairs from adjacent groups only, that differ in 1 bit • Combine entries from List 2 into pairs in List 3

21. The Q-M minimization method

22. The Q-M minimization method

23. The Q-M minimization method

24. The Q-M minimization method