Digital Logic & Design Vishal Jethva Lecture 09

1. Digital Logic & DesignVishal JethvaLecture 09

2. Recap • Commutative, Associative and Distributive Laws • Rules • Demorgan’s Theorems

3. Recap • Boolean Analysis of Logic Circuits • Simplification of Boolean Expressions • Standard form of Boolean expressions

4. Examples • Boolean Analysis of Circuit • Evaluating Boolean Expression • Representing results in a Truth Table • Simplification of Boolean Expression into SOP or POS form • Representing results in a Truth Table • Verifying two expressions through truth tables

5. Analysis of Logic Circuits Example 1

6. Evaluating Boolean Expression • The expression • Assume and • Expression • Conditions for output = 1 X=0 & Y=0 • Since X=0 when A=0 or B=1 • Since Y=0 when A=0, B=0, C=1 and D=1

7. Evaluating Boolean Expression & Truth Table • Conditions for o/p =1 • A=0, B=0, C=1 & D=1

8. Simplifying Boolean Expression • Simplifying by applying Demorgan’s theorem =

9. Truth Table of Simplified expression

10. Simplified Logic Circuit

11. Simplified Logic Circuit • Simplified expression is in SOP form • Simplified circuit

12. Second Example • Evaluating Boolean Expression • Representing results in a Truth Table • Simplification of Boolean Expression results in POS form and requires 3 variables instead of the original 4 • Representing results in a Truth Table • Verifying two expressions through truth tables

13. Analysis of Logic Circuits Example 2

14. Evaluating Boolean Expression • The expression • Assume and • Expression • Conditions for output = 1 X=0 OR Y=0 • Since X=0 when A=1,B=0 or C=1 • Since Y=0 when C=1 and D=0

15. Evaluating Boolean Expression & Truth Table • Conditions for o/p =1 • (A=1,B=0 OR C=1) OR (C=1 AND D=0)

16. Rewriting the Truth Table • Conditions for o/p =1 • (A=1,B=0 OR C=1) OR (C=1 AND D=0)

17. Simplifying Boolean Expression • Simplifying by applying Demorgan’s theorem =

18. Truth Table of Simplified expression

19. Simplified Logic Circuit

20. Simplified Logic Circuit • Simplified expression is in POS form representing a single Sum term • Simplified circuit

21. Standard SOP and POS form • Standard SOP and POS form has all the variables in all the terms • A non-standard SOP is converted into standard SOP by using the rule • A non-standard POS is converted into standard POS by using the rule

22. Standard SOP form

23. Standard POS form

24. Why Standard SOP and POS forms? • Minimal Circuit implementation by switching between Standard SOP or POS • Alternate Mapping method for simplification of expressions • PLD based function implementation

25. Minterms and Maxterms • Minterms: Product terms in Standard SOP form • Maxterms: Sum terms in Standard POS form • Binary representation of Standard SOP product terms • Binary representation of Standard POS sum terms

26. Minterms and Maxterms & Binary representations

27. SOP-POS Conversion • Minterm values present in SOP expression not present in corresponding POS expression • Maxterm values present in POS expression not present in corresponding SOP expression

28. SOP-POS Conversion • Canonical Sum • Canonical Product • =

29. Boolean Expressions and Truth Tables • Standard SOP & POS expressions converted to truth table form • Standard SOP & POS expressions determined from truth table

30. SOP-Truth Table Conversion

31. POS-Truth Table Conversion

32. Karnaugh Map • Simplification of Boolean Expressions • Doesn’t guarantee simplest form of expression • Terms are not obvious • Skills of applying rules and laws • K-map provides a systematic method • An array of cells • Used for simplifying 2, 3, 4 and 5 variable expressions

33. 3-Variable K-map • Used for simplifying 3-variable expressions • K-map has 8 cells representing the 8 minterms and 8 maxterms • K-map can be represented in row format or column format

34. 4-Variable K-map • Used for simplifying 4-variable expressions • K-map has 16 cells representing the 16 minterms and 8 maxterms • A 4-variable K-map has a square format