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ECE 3110: Introduction to Digital Systems

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?.

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ECE 3110: Introduction to Digital Systems

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  1. ECE 3110: 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 • 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, multiple-outputs, 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. • Multiple-outputs • Term sharing can reduce costs

  14. 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.

  15. 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

  16. 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

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