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