1. ECE 3110: Introduction to Digital Systems Chapter #4 Review

2. 2 Switching Algebra Variables, expressions, equations Axioms (A1-A5 pairs) Theorems  Single variable  2- or 3- variable  N-variables Prime, complement, logic multiplication/addition, precedence

3. 3 How to prove a theorem? Perfect induction (1,2,3-variable) Method used in Exercise 4.29

4. 4 Duality Swap 0 & 1, AND & OR  Result: Theorems still true Principle of Duality  Any theorem or identity in switching algebra remains true if 0 and 1 are swapped and and + are swapped throughout. Fully parenthesized before taking its duality

5. 5 DeMorgan Symbol Equivalence

6. 6 Likewise for OR

7. 7 Representations for a combinational logic function Truth table Algebraic sum of minterms (canonical sum) Minterm list Algebraic product of maxterms (canonical product) Maxterm list

8. 8 Combinational-circuit analysis  Obtain a formal representation of a given circuit  Truth table: axioms, exhaustive  Logic expression: algebraic approach  Simulation: LogiSim

9. 9 Combinational circuit synthesis Description--->combinational logic circuit. Description:  Word description of a problem using English-language connectives Write corresponding logic expression/truth table Manipulate the expression if necessary. Build a circuit from the expression.

10. 10 Minimization u Logic Function minimization : Simplifying the logic function to reduce the number and size of gates. u Minimization methods: 1- Algebraic simplification: Using theorems T9,T9’, T10,T10’ 2- Karnaugh map (SOP, POS, multiple-outputs, Don’t Cares) 3- Computer tools, e.g. Logisim

11. 11 Simplifying SOP:  Draw K-map  Find prime implicants (circle largest rectangular sets of 1s: …16,8,4,2,1)  Find distinguished 1-cell  Determine essential prime implicants if available  Select all essential prime implicants and the minimal set of the remaining prime implicants that cover the remaining 1’s.

12. 12 Simplifying POS Products-Of-Sums (POS) minimization  Duality: circle 0s on the K-map  (Other way) F=(F’)’ Draw a K-map for F’ Simplifying SOP for F’ Get POS for F using DeMorgan theorems repeatedly=(F’)’

13. 13 Other minimization issues Don’t care conditions  d  Since the output function for those minterms (maxterms) is not specified, those minterms (maxterms) could be combined with the adjacent 1 cells(0-cells) to get a more simplified sum-of-products (product-of-sums) expression.  d cells are only combined when we have to.

14. 14 Chapter Summary Boolean Algebra is used to represent, manipulate and simplify logic functions. Truth table represents the logic function by listing the output for each possible combination of the inputs. Combinational circuit analysis: - The logic function is obtained from the logic circuit. - The truth table is obtained from the logic circuit by evaluating the logic function for each combination of the input variables. - The Canonical sum ( sum-of-products ) is the sum of all minterms in the truth table. - The Canonical product ( product-of-sums ) is the product of all maxterms terms in the truth table. - Boolean algebra theorems are used to simplify the canonical forms and obtain a simplified representation of the logic function

15. 15 Chapter summary Combinational circuit synthesis: - The logic circuit is obtained from the logic function. - There are four equivalent canonical implementations of a logic function: - AND-OR & NAND-NAND - OR-AND & NOR-NOR - Karnuagh map is used to simplify the canonical forms: 1- The canonical sum expression is simplified by combining the 1’s to obtain the minimal sum. 2- The canonical product is simplified by combining the 0’s to obtain the minimal product.