1. ECE 2110: Introduction to Digital Systems
Chapter #4 Review

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

4. Duality Swap 0 & 1, AND & OR Principle of Duality
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. DeMorgan Symbol Equivalence

6. Likewise for OR

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. Combinational-circuit analysis
Obtain a formal representation of a given circuit
Truth table: axioms, exhaustive
Logic expression: algebraic approach
Simulation/ test bench: HDL

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

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

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. 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. 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 The canonical product is simplified by combining the 0’s to obtain the minimal product.