chap2(4_in_1)

chap2(4_in_1) - ELEC151 Digital Circuits and Systems Ho-Chi...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ELEC151 Digital Circuits and Systems Ho-Chi Huang, Lecture Notes, No. 2-1 Lecture Note #2 Boolean Algebra • Basic definitions and axioms 2-1, 2-2 • Laws and theorems of Boolean algebra 2-3, 2-4 – Design process and Boolean algebra • Two-level canonical forms 2-5 – Incompletely specified functions • More logic functions 2-6, 2-7 – NOT, AND, OR, NAND, NOR, XOR, XNOR, and more – Positive and negative logic • Integrated circuits 2-8 – CMOS, TTL, … – OrCAD (in laboratory) • Reading Assignments: – Chapter 2 ELEC151 Digital Circuits and Systems Ho-Chi Huang, Lecture Notes, No. 2-2 Boolean Algebra/Logic Circuits • Boolean Algebra – In 1854, a mathematician, George Boole , developed an algebraic system to handle only two variables, TRUE and FALSE • Digital Circuits – In 1938, Claude Shannon (father of information theory) showed (in his Master’s thesis!) how to map Boolean Algebra to digital circuits: • Structure of Boolean Algebra – A set of elements B – Two binary operators, AND (•) and OR (+) – One unary operator, NOT (’) • Primitive Gates – AND, OR and NOT ELEC151 Digital Circuits and Systems Ho-Chi Huang, Lecture Notes, No. 2-3 Boolean Algebra • Algebraic Structure – A set of elements B – Two binary operators, AND (•) and OR (+) – One unary operator, NOT (’) • Axioms – 1. B contains at least two elements, a , b , such that a ≠ b – 2. Closure : » (i) a + b in B (ii) a • b in B – 3. Commutative Laws : » (i) a + b = b + a (ii) a • b = b • a – 4. Identities : 0, 1 in B » (i) a + 0 = a (ii) a • 1 = a – 5. Distributive Laws : » (i) a + ( b • c ) = ( a + b ) • ( a + c ) (ii) a • ( b + c ) = a • b + a • c – 6. Complement : » ( i ) a + a' = 1 (ii) a • a' = 0 ELEC151 Digital Circuits and Systems Ho-Chi Huang, Lecture Notes, No. 2-4 Boolean Algebra and Design • Theorem: – Any Boolean function that can be expressed as a truth table can be written as an expression in Boolean Algebra using ', +, • • Design Sequence: – 1. Truth Table (input and output relations) » Derive truth table from project specifications – 2. Boolean Expressions » Derived from the truth table – 3.Logic Minimization (optimized for cost, speed, and power) » Use Laws of Boolean Algebra to reduce the complexity of Boolean expressions while maintaining the same function – 4.Digital Circuit » Map the simplified Boolean expressions to digital circuits – 5.Waveforms » CAD tool simulation and experiments ELEC151 Digital Circuits and Systems Ho-Chi Huang, Lecture Notes, No. 2-5 Laws of Boolean Algebra • Duality: – A dual of a Boolean expression is derived by interchanging OR and AND operations, and 0s and 1s (literals are left unchanged) – Any law that is true for an expression is also true for its dual....
View Full Document

This note was uploaded on 09/16/2010 for the course ELEC 151 taught by Professor Cy during the Spring '10 term at HKUST.

Page1 / 9

chap2(4_in_1) - ELEC151 Digital Circuits and Systems Ho-Chi...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online