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More Simplification

More Simplification. Lecture Notes 7. 1. Introduction. In this note, the method of using Karnaugh Maps is going to be introduced.

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More Simplification

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  1. More Simplification Lecture Notes 7

  2. 1. Introduction • In this note, the method of using Karnaugh Maps is going to be introduced. • A Karnaugh map (also known as ‘K-map’) provides a systematic method for simplifying Boolean expressions and, if properly used, will produce the simplest SOP or POS expression possible, know as the minimum expression.

  3. 1. Introduction (cont’d) • Remember that the effectiveness of algebraic simplification depends on your familiarity with all the laws, rules, and theorems of Boolean algebra and on your ability to apply them. • On the other hand, the K-map provides a “cookbook” method for simplification.

  4. 2. The Karnaugh Map • A K-map is similar to a truth table because it presents all of the possible values of input variables and the resulting output for each value. • Instead of being organized into columns and rows like a truth table, the K-map is an array of cells in which each cell represents a binary value of the input variables.

  5. i. The 3-variable K-map • The 3-variable K-map is an array of eight cells. • In this case, we will use the variables A, B, and C. • Notice the sequence: A B C

  6. ii. The 4-variable K-map • The 4-variable K-map is an array of sixteen cells. • In this case, we will use the variables A, B, C, and D. • Notice the sequence: A B C D

  7. iii. Cell Adjacency • The cells in a K-map are arranged so that there is only a single-variable change between adjacent cells. • Adjacency is defined by a single-variable change.

  8. 3. Karnaugh Map SOP Minimization • The K-map is used for simplifying Boolean expressions to their minimum form. • A minimized SOP expression contains the fewest possible terms with fewest possible variables per term.

  9. Mapping a Standard SOP Expression • Step 1: Determine the binary value of each product term in the standard SOP expression. • Step 2: As each product term is evaluated, place a 1 on the K-map in the cell having the same value as the product term.

  10. Mapping a Standard SOP Expression (cont’d) • Examples: • Map the following standard SOP expression on a K-map: ABCD + ABCD + ABCD + ABCD + ABCD

  11. ii. Mapping a Nonstandard SOP Expression • If an expression is not in standard form, the it must be converted to standard form. • Example: A + AB + ABC 000 100 110 001 101 010 011

  12. iii. K-map Simplification of SOP Expressions • The process that results in an expression containing the fewest possible terms with fewest possible variables is called minimization. • The minimum SOP expression is obtained by grouping the 1s and determining the minimum SOP expression from the map.

  13. iii. K-map Simplification of SOP Expressions (cont’d) • Grouping the 1s: You can group 1s on the K-map according to the following rules by enclosing those adjacent cells containing 1s. The goal is to maximize the size of the groups and to minimize the number of groups. • A group must contain a power of two number of cells. • Each cell in a group must be adjacent to one or more cells in that same group, but all cells in the group do not have to be adjacent to each other. • Always include the largest possible number of 1s in a group in accordance with rule 1. • Each 1 on the map must be include in at least one group. The 1s already in a group can be included in another group as long as the overlapping groups include noncommon 1s.

  14. iii. K-map Simplification of SOP Expressions (cont’d) • Example: Group the 1s in each of the next K-maps:

  15. iii. K-map Simplification of SOP Expressions (cont’d) • Determining the minimum SOP expression from the map: • Group the cells that have 1s. Each group of cells containing 1s creates one product term composed of all variables that occur in only one form (either uncomplemented or complemented) with the group. Variables that occur both uncomplemented and complemented within the group are eliminated. These are called contradictory variables.

  16. iii. K-map Simplification of SOP Expressions (cont’d) • (Determining the minimum SOP expression from the map): • Determine the minimum product term for each group. • For a 3-variable map: • A 1-cell group yields a 3-variable product term. • A 2-cell group yields a 2-variable product term. • A 4-cell group yields a 1-variable product term. • An 8-cell group yields a value of 1 for the expression. • For a 4-variable map: • A 1-cell group yields a 4-variable product term. • A 2-cell group yields a 3-variable product term. • A 4-cell group yields a 2-variable product term. • An 8-cell group yields a 1-variable product term. • A 16-cell group yields a value of 1 for the expression.

  17. iii. K-map Simplification of SOP Expressions (cont’d) • (Determining the minimum SOP expression from the map): • When all the minimum product terms are derived from the K-map, they are summed to form the minimum SOP expression.

  18. iii. K-map Simplification of SOP Expressions (cont’d) • Example: Determine the product terms for each of the K-maps in the next figure and write the resulting minimum SOP expression.

  19. iii. K-map Simplification of SOP Expressions (cont’d) • Example: Use K-map to minimize the following SOP expression:

  20. iv. Mapping directly from a Truth Table

  21. v. Don’t Care Conditions • Sometimes, a situation arises in which some input variable combinations are not allowed. • For example, recall that in the BCD code, there are six invalid combinations: 1010, 1011, 1100, 1101, 1110, and 1111. • Since theses unallowed states will never occur in a an application involving the BCD code, they can be treated as “don’t care” terms with respect to their effect on the output. • That is, for these “don’t care” terms either a 1 or a 0 may be assigned to the output; it really does not matter since they will never occur.

  22. v. Don’t Care Conditions (cont’d) • The “don’t care” terms can be used to advantage on the K-map. • In the next example, the figure shows that for each “don’t care” term, an X is placed in the cell. When grouping the 1s, the Xs can be treated as 1s to make a larger grouping or as 0s if they cannot be used to advantage.

  23. v. Don’t Care Conditions (cont’d) • Remember: the larger a group, the simpler the resulting term will be.

  24. vi. Another form to use the K-maps

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