1 Chapter 5 Karnaugh Maps Mei Yang ECG Logic Design 1

2 Minimum Forms of Switching Functions When a function is realized using AND and OR gates, the cost is directly related the number of gates and gate inputs used. Minimum SOP: A SOP form which (a) has a minimum number of terms and (b) for those expressions which have the same minimum number of terms, has a minimum number of literals. Minimum POS: A POS form which (a) has a minimum number of terms and (b) for those expressions which have the same minimum number of terms, has a minimum number of literals.

3 Minimizing from Minterm form

4 Why Kmap? Problems with the simplification method using Boolean algebra – The procedures are difficult to apply in systematic way. – It is difficult to tell when you have arrived at a minimum solution. Kmap is a systematic method, which is especially useful for simplifying functions with three or four variables.

5 Plotting Kmaps From truth tables From minterm/maxterm expansions From algebraic expressions Examples on pp

6 2-Variable Karnaugh Map

7 3-Variable Karnaugh Map Boolean adjacency -difference in one variable Grouping adjacent 1’s

8 Grouping Principle Groupings can contain only 1s; no 0s. Groups can be formed only at right angles; diagonal groups are not allowed. The number of 1s in a group must be a power of 2 – even if it contains a single 1. The groups must be made as large as possible. Groups can overlap and wrap around the sides of the Kmap. Use the fewest number of groups possible.

9 Grouping of adjacent 1 ’ s 3-Variable Karnaugh Map Multiple grouping

10 3-Variable Karnaugh Map Corresponding minterms Adjacent minterms

11 3-Variable Karnaugh Map

12 4-Variable Karnaugh Map

13 4-Variable Karnaugh Map

14 Boolean Adjacency

15 Grouping Example

16 Grouping Example

17 More than One Way of Grouping

18 A Systematic Way for Minimization IMPLICANT Any single one or any group of ‘1’ s is called an IMPLICANT of F. PRIME IMPLICANT. A group (covering) that cannot be combined with some other 1’s or coverings to eliminate a variable is called PRIME IMPLICANT.

19 Minimization with Don’t Cares Threat X’s as 1’s if you will get a larger grouping. Otherwise, treat them as 0’s.

20 Obtaining Minimum POS Equation Grouping 0’s means obtaining a minimum SOP for F’.

21 Grouping Zeros

22 Minimization Rules The minimum SOP expression consists of some (but not necessarily all) of the prime implicants of a function. If a SOP expression contains a term which is NOT a prime implicant, then it CANNOT be minimum.

23 Minimization Rules How to use prime implicants for obtaining a minimum SOP equation? How to use prime implicants for obtaining a minimum SOP equation?

24 F(A,B,C,D)= BC’ +A’B’D Essential Prime Implicants BC’ and A’B’D are called the Essential Prime Implicants, because they cover ‘1’s that cannot be covered by any other coverings. A’CD is not an essential, and it is not included, because all 1’s are already covered, and there is no reason to add an extra term. Essential Prime Implicants

25 More than one solution

26 Two solutions

27 Minimum SOP To obtain a minimum SOP equation Include all essential prime implicants to the equation. Check if all ones are covered by the essential prime implicants. If there are remaining ‘1’s, include non- essential prime implicants. There can be more than one minimum SOP equation equally valid.

28 Minimization with Don’t Cares

29 5-Variable Karnaugh Map

30 5-Variable Karnaugh Map

31 Example 1: F = ∑m(1,2,3,5,6,8,9,10,11,13,24, 25, 26, 27,29)

32 Example 2: F = ∑m (0,1,2,4,5,8,9,10,17,18,19,25,26) F = A'B'D'+C’D'E+C'DE’+ {AB’C’E or AB’C’D}+{A’BC’D’ or A’BC’E’} Essential prime implicants are shown in sold line groups and other prime implicants are shown in dotted line groups.