chap2 - ELEC151 Digital Circuits and Systems Lecture Note#2...

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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
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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. • Laws of Boolean Algebra
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This note was uploaded on 10/16/2011 for the course ELEC 308,315,10 taught by Professor Prof.shenghuisong during the Spring '11 term at CUHK.

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chap2 - ELEC151 Digital Circuits and Systems Lecture Note#2...

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