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Gate-Level Minimization: Mastering the Map Method for Boolean Function Simplification

Chapter 3 delves into the Map Method for simplifying Boolean functions using two, three, four, and five-variable maps. It highlights how to identify adjacent squares, the rules for combining them effectively, and how these combinations can reduce complex functions into minimal forms. Through examples and problems, the chapter teaches how to express functions in their simplest sum-of-products form. Special conditions such as "don't care" scenarios are also explored, along with practical implementations using NAND and NOR gates.

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Gate-Level Minimization: Mastering the Map Method for Boolean Function Simplification

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  1. Gate-Level Minimization Chapter 3

  2. The Map Method Two-variable map

  3. Two-variable map

  4. Three-variable map Adjacent Adjacent Adjacent when minterms differ by one variable

  5. Graphical view ofadjacency Taking the difference between adjacent squares gives:

  6. Example 3.1 Simplify the Boolean function

  7. Example 3.2 Simplify the Boolean Function

  8. Three variable map 00 01 11 10 0 1

  9. Three-variable map • The number of adjacent squares that may be combined must always represent a number that is a power of two: • One square represents one minterm which in this case gives a term with three literals • Two adjacent squares represent a term with two literals • Four adjacent squares represent a term with one literal • Eight adjacent squares encompass the entire map and produce a function equal to 1

  10. Three-variable map 00 01 11 10 0 1

  11. Three-variable map 00 01 11 10 0 1

  12. Three-variable map 00 01 11 10 0 1

  13. Example 3.3

  14. Example 3.4 Let the Boolean Function (a) Express this function as a sum of minterms

  15. Example 3.4 (b) Find the minimal sum-of-products expression 00 01 11 10 0 1 Find adjacent squares

  16. Problem 3.3 (a) Simplify the following Boolean function, using three-variable maps 00 01 11 10 0 1 Find adjacent squares

  17. Problem 3.3 (a) Another solution (without a three-variable map) Factorize the expression Voila!

  18. Four-Variable Map

  19. Four-variable map • The number of adjacent squares that may be combined must always represent a number that is a power of two: • One square represents one minterm which in this case gives a term with four literals • Two adjacent squares represent a term with three literals • Four adjacent squares represent a term with two literals • Eight adjacent squares represent a term with one literal • Sixteen adjacent squares encompass the entire map and produce a function equal to 1

  20. Adjacency in a four-variable map

  21. Adjacency in a four-variable map

  22. Example 3.5 Simplify the Boolean Function 00 01 11 10 00 1 1 1 01 1 1 1 11 1 1 1 1 1 10

  23. Example 3.6 Simplify the Boolean Function 00 01 11 10 00 1 1 1 01 1 11 10 1 1 1

  24. Prime implicants • All the minterms are covered when combining the squares • The number of terms in the expression is minimized • There are no redundant terms (minterms already covered by other terms) When choosing adjacent squares in a map, make sure that:

  25. Example 3.5 (revisited) Simplify the Boolean Function 00 01 11 10 00 1 1 1 01 1 1 1 11 1 1 1 1 1 10

  26. Example 3.6 (revisited) Simplify the Boolean Function 00 01 11 10 00 1 1 1 01 1 11 10 1 1 1

  27. Prime implicants Simplify the function 00 01 11 10 00 1 1 1 01 1 1 11 1 1 1 1 1 10 1

  28. Prime implicants Simplify the function 00 01 11 10 00 1 1 1 01 1 1 11 1 1 1 1 1 10 1

  29. Prime implicants Simplify the function 00 01 11 10 00 1 1 1 01 1 1 11 1 1 1 1 1 10 1

  30. Prime implicants Simplify the function 00 01 11 10 00 1 1 1 01 1 1 11 1 1 1 1 1 10 1

  31. Five-Variable Map

  32. Five-Variable Map How can adjacency be visualized in a five-variable map? 00 01 11 10 00 01 11 10

  33. Five-Variable Map

  34. Five-Variable Map Simplify the Boolean function

  35. Product-Of-Sums Simplification Take the squares with zeros and obtain the simplified complemented function Complement the above expression and use DeMorgan’s

  36. Product-Of-Sums Simplification Sum-of-products Product-of-sums

  37. Don’t-Care Conditions • Used for incompletely specified functions, e.g. BCD code in which six combinations are not used (1010, 1011, 1100, 1101, 1110, and 1111) • Those unspecified minterms are neither 1’s nor 0’s • Unspecified terms are referred to as “don’t care” and are marked as X • In choosing adjacent squares, don’t care squares can be chosen either as 1’s or 0’s to give the simplest expression

  38. Don’t-Care Conditions Simplify the Boolean function which has the don’t care conditions

  39. NAND and NOR Implementation

  40. NAND and NOR Implementation

  41. Two-level implementation

  42. NAND and NOR Implementation Implement the following Boolean function with NAND gates

  43. Multilevel NAND circuits

  44. Multilevel NAND circuits

  45. NOR Implementation

  46. NOR Implementation

  47. NOR Implementation

  48. NOR Implementation

  49. Exclusive OR Function Exclusive-OR or XOR performs the following logical operation Exclusive-NOR or equivalence performs the following logical operation Identities of the XOR operation XOR is commutative and associative

  50. Exclusive-OR implementations

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