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ECE 301 – Digital Electronics

Learn about minterms, maxterms, expansions, and logic circuits in digital electronics through examples and illustrations from ECE 301 lectures.

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ECE 301 – Digital Electronics

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  1. Minterm and Maxterm Expansions and Incompletely Specified Functions (Lecture #6) ECE 301 – Digital Electronics The slides included herein were taken from the materials accompanying Fundamentals of Logic Design, 6th Edition, by Roth and Kinney, and were used with permission from Cengage Learning.

  2. ECE 301 - Digital Electronics Minterms and Maxterms

  3. ECE 301 - Digital Electronics Minterm • In general, a minterm of n variables is a product (ANDing) of n literals in which each variable appears exactly once in either true or complemented form, but not both. • A literal is a variable or its complement. • For a given row in the truth table, the corresponding minterm is formed by • Including the true form a variable if its value is 1. • Including the complemented form of a variable if its value is 0.

  4. ECE 301 - Digital Electronics Minterms

  5. ECE 301 - Digital Electronics Minterm Expansion • When a function f is written as a sum (ORing) of minterms, it is referred to as a minterm expansion or a standard sum of products. • aka. “canonical sum of products” • aka. “disjunctive normal form” • If f = 1 for row i of the truth table, then mi must be present in the minterm expansion. • The minterm expansion for a function f is unique. • However, it is not necessarily the lowest cost.

  6. ECE 301 - Digital Electronics Minterm Expansion • The minterm expansion for a general function of 3 variables can be written as follows: 3 variables Denotes the logical sum operation ai = 0 or 1. This can be extended to n variables

  7. ECE 301 - Digital Electronics Determine the minterm expansion for the function defined by the following truth table: Minterm Expansion: Example #1

  8. ECE 301 - Digital Electronics Determine the minterm expansion for each of the following Boolean expressions: F1(A,B,C) = A.B.C' + A.B'.C + A'.B'.C + A.B.C F2(A,B,C) = A.C' + A.B + B'.C Minterm Expansion: Example #2

  9. ECE 301 - Digital Electronics Maxterm • In general, a maxterm of n variables is a sum (ORing) of n literals in which each variable appears exactly once in either true or complemented form, but not both. • A literal is a variable or its complement. • For a given row in the truth table, the corresponding maxterm is formed by • Including the true form a variable if its value is 0. • Including the complemented form of a variable if its value is 1.

  10. ECE 301 - Digital Electronics Maxterms

  11. ECE 301 - Digital Electronics Maxterm Expansion • When a function f is written as a product (ANDing) of maxterms, it is referred to as a maxterm expansion or a standard product of sums. • aka. “canonical product of sums” • aka. “conjunctive normal form” • If f = 0 for row i of the truth table, then Mi must be present in the maxterm expansion. • The maxterm expansion for a function f is unique. • However, it is not necessarily the lowest cost.

  12. ECE 301 - Digital Electronics Maxterm Expansion • The maxterm expansion for a general function of 3 variables can be written as follows: 3 variables Denotes the logical product operation ai = 0 or 1. This can be extended to n variables

  13. ECE 301 - Digital Electronics Determine the maxterm expansion for the function defined by the following truth table: Maxterm Expansion: Example #1

  14. ECE 301 - Digital Electronics Determine the maxterm expansion for each of the following Boolean expressions: F1(A,B,C) = (A+B+C').(A+B'+C).(A'+B'+C).(A+B+C) F2(A,B,C) = (A+C').(A+B).(B'+C) Maxterm Expansion: Example #2

  15. ECE 301 - Digital Electronics What is the relationship between the minterm expansion and maxterm expansion for the same function? Minterm and Maxterm Expansions

  16. ECE 301 - Digital Electronics What is the relationship between the minterm expansion for a function and that for the complement of the function? What about the maxterm expansion? Minterm and Maxterm Expansions

  17. ECE 301 - Digital Electronics Minterm and Maxterm Expansions

  18. ECE 301 - Digital Electronics Logic Circuits • A function f can be represented by either a minterm expansion or a maxterm expansion. • Both forms of the function can be realized using logic gates that implement the basic logic operations. • Minterm Expansion (Standard SOP) • Consists of the sum (OR) of product (AND) terms. • Realized using an AND-OR circuit. • Maxterm Expansion (Standard POS) • Consists of the product (AND) of sum (OR) terms. • Realized using an OR-AND circuit.

  19. ECE 301 - Digital Electronics Logic Circuits: Example For the function defined by the following truth table, 1. Determine the minterm expansion 2. Draw the circuit diagram

  20. ECE 301 - Digital Electronics Logic Circuits: Example For the same function, 1. Determine the maxterm expansion 2. Draw the circuit diagram Which logic circuit is “cheaper”?

  21. ECE 301 - Digital Electronics Incompletely Specified Functions

  22. ECE 301 - Digital Electronics Incompletely Specified Functions • A function f is completely specified when its output is defined (i.e. either 0 or 1) for all combinations of its inputs. • However, if the output of a function f is not defined for all combinations of its inputs, then it is said to be incompletely specified. • Those combinations of the inputs for which the output of function f is not defined are referred to as “don't care” outputs.

  23. ECE 301 - Digital Electronics Incompletely Specified Functions • The truth table representing an incompletely specified function includes an “x” (or a “d”) in each row corresponding to an input combination for which the output is not defined. “don't care” for ABC = 001 “don't care” for ABC = 011 “don't care” for ABC = 110

  24. ECE 301 - Digital Electronics Incompletely Specified Functions The minterm expansion is: • A “don't care” can be either a 0 or 1. • Select a value for each “don't care” that will help simplify the function. F(A,B,C) = Sm(2,4,7) + Sd(1,3,6) “don't care” minterms The maxterm expansion is: F(A,B,C) = P M(0,5) . P D(1,3,6) “don't care” maxterms

  25. ECE 301 - Digital Electronics Incompletely Specified Functions Assume X1 = 0, X2 = 0, X3 = 0: F(A,B,C) = A'BC' + AB'C' + ABC Assume X1 = 1, X2 = 1, X3 = 1: F(A,B,C) = B + AC' + A'C Assume X1 = 0, X2 = 1, X3 = 1: F(A,B,C) = B + AC'

  26. ECE 301 - Digital Electronics Questions?

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