2. EE365 Adv. Digital Circuit Design Clarkson University Lecture #2 Boolean Laws and Methods

3. Boolean algebra a.k.a. “switching algebra” –deals with boolean values -- 0, 1 Positive-logic convention –analog voltages LOW, HIGH --> 0, 1 Signal values denoted by variables (X, Y, FRED, etc.) Rissacher EE365Lect #2

4. Boolean operators Complement:X (opposite of X) AND:X  Y OR:X + Y binary operators, described functionally by truth table. Rissacher EE365Lect #2

5. More definitions Literal: a variable or its complement –X, X, FRED, CS_L Expression: literals combined by AND, OR, parentheses, complementation –X+Y –P  Q  R –A + B  C –((FRED  Z) + CS_L  A  B  C + Q5)  RESET Equation: Variable = expression –P = ((FRED  Z) + CS_L  A  B  C + Q5)  RESET Rissacher EE365Lect #2

6. Logic symbols Rissacher EE365Lect #2

7. Theorems Rissacher EE365Lect #2

8. More Theorems Rissacher EE365Lect #2

9. Duality Swap 0 & 1, AND & OR –Result: Theorems still true –Note duals in previous 2 tables (e.g. T6 and T6’) –Example: Rissacher EE365Lect #2

10. N-variable Theorems Most important: DeMorgan theorems Rissacher EE365Lect #2

11. DeMorgan Symbol Equivalence Rissacher EE365Lect #2

12. Likewise for OR Rissacher EE365Lect #2

13. DeMorgan Symbols Rissacher EE365Lect #2

14. Even more definitions Product term –WX’Y Sum-of-products expression –(WX’Y)+(XZ)+(W’X’Y’) Sum term –A+B’+C Product-of-sums expression –(A+B’+C)(D’+A’)(D+B+C) Normal term –No variable appears more than once –(WX’Y)+(AZ)+(B’C’) Minterm (n variables) Maxterm (n variables) Rissacher EE365Lect #2

15. Minterm An n-variable minterm is a normal product term with n literals There are 2 n possibilities 3-variable example: X’Y’Z or Σ X,Y,Z (1) A minterm is a product term that is 1 in exactly one row of the truth table: Rissacher EE365Lect #2 XYZF 0000 0011 0100 0110 1000 1010 1100 1110 new notation

16. Maxterm An n-variable maxterm is a normal sum term with n literals There are 2 n possibilities 3-variable example: X’+Y’+Z or Л X,Y,Z (6) A maxterm is a sum term that is 0 in exactly one row of the truth table: Rissacher EE365Lect #2 XYZF 0001 0011 0101 0111 1001 1011 1100 1111 new notation

17. Truth table vs. minterms & maxterms Rissacher EE365Lect #2

18. Combinational analysis Rissacher EE365Lect #2

19. Signal expressions Multiply out: F = ((X + Y)  Z) + (X  Y  Z) = (X  Z) + (Y  Z) + (X  Y  Z) Rissacher EE365Lect #2

20. New circuit, same function Rissacher EE365Lect #2 F = ((X + Y)  Z) + (X  Y  Z) = (X  Z) + (Y  Z) + (X  Y  Z)

21. “Add out” logic function Circuit: Rissacher EE365Lect #2

22. Shortcut: Symbol substitution Rissacher EE365Lect #2

23. Different circuit, same function Rissacher EE365Lect #2

24. Practice Rissacher EE365Lect #2 Convert the following function into a POS: F = ((X + Z) Y) + (X’ Z’ Y’)

25. Convert the following function into a POS: F = ((X + Z) Y) + (X’ Z’ Y’) F = (X + Z + X’) (X + Z + Z’) (X + Z + Y’) (Y + X’) (Y + Z’) (Y + Y’) F = 1 1 (X + Z + Y’) (Y + X’) (Y + Z’) 1 F = (X + Z + Y’) (Y + X’) (Y + Z’) Practice Rissacher EE365Lect #2

26. Next Class Rissacher EE365Lect #2 Building Combination Circuits Minimization Karnaugh Maps