ECE 3110: Introduction to Digital Systems Chapter #4 Review

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

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

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

DeMorgan Symbol Equivalence

Likewise for OR

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

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

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.

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, multiple-outputs, Don’t Cares) 3- CAD tools, HDLs

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.

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’)’

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. Multiple-outputs Term sharing can reduce costs

Timing hazards A properly designed two-level SOP (AND-OR) circuit has no static-0 hazards. It may have static-1 hazards. A properly designed two-level POS (OR-AND) circuit has no static-1 hazards. It may have static-0 hazards. Dynamic hazards do not occur in a properly designed two-level AND-OR or OR-AND circuit. It may occur in multilevel circuits. A brute-force method of obtaining a hazard-free realization is to use the complete sum or complete product. Hazard analysis and elimination are typically needed in the design of asynchronous sequential circuits.

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

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. The minimal sum and the minimal product implementations could produce hazards