Copyright 2001, Regents of University of CaliforniaEECS 42 Intro. electronics for CS Fall 2001 Lecture 8: 9/26/01 A.R. NeureutherVersion Date 9/25/01Lecture 9: September 26th, 2001Digital Signals and Basic Logic BlocksA) Advantages of DigitalB) Goals: Gates Ù Logical FunctionsC) Truth Tables and Logical FunctionsD) Boolean Operations and GatesReading: Schwarz and Oldham 11.1, 11.2 pp. 92-402The following slides were derived from those prepared by Professor Oldham For EE 40 in Fall 01Copyright 2001, Regents of University of CaliforniaEECS 42 Intro. electronics for CS Fall 2001 Lecture 8: 9/26/01 A.R. NeureutherVersion Date 9/25/01GOAL FOR LECTURE 9ABTXYHGiven GatesFind the Truth Table and the Boolean Logic FunctionABT H00 001 110 01100010000111101111H = (A · B) + TCopyright 2001, Regents of University of CaliforniaEECS 42 Intro. electronics for CS Fall 2001 Lecture 8: 9/26/01 A.R. NeureutherVersion Date 9/25/01Evaluation of Logical Expressions with “Truth Tables”Suppose we have a heater which we want to operate any time anytime switch T is “on” or if both switches A and B are “on” . We would say H is true if T is True or A and B are both true.OrWe could say H is 1 if either T is 1 or A and B are both 1.A “Truth Table” is a simple table listing all possible combinations of A, B, T and the resulting value of H.ABHT000etcetc?Copyright 2001, Regents of University of CaliforniaEECS 42 Intro. electronics for CS Fall 2001 Lecture 8: 9/26/01 A.R. NeureutherVersion Date 9/25/01Evaluation of Logical Expressions with “Truth Tables”Truth Table for Heater AlgorithmABT H00 001 110 011000100001111011110111H is 1 if either T is 1 or A and B are both 1.Copyright 2001, Regents of University of CaliforniaEECS 42 Intro. electronics for CS Fall 2001 Lecture 8: 9/26/01 A.R. NeureutherVersion Date 9/25/01Logical Expressions to express Truth TablesWe need a notation for logic expressions.Standard logic notation and logic gates:AND: “dot”OR : “+ sign”NOT: “bar over symbol for complement”With these basic operations we can construct any logical expression.Example: Z = AExamples: X = A · B ; Y = A · B · CExamples: W = A+B ; Z = A+B+CCopyright 2001, Regents of University of CaliforniaEECS 42 Intro. electronics for CS Fall 2001 Lecture 8: 9/26/01 A.R. NeureutherVersion Date 9/25/01Digital Heater Control Example (cont.)• Logical Statement: H = 1 if A and B are 1 or T is 1.• Remember we use “dot” to designate logical “and” and “+” to designate logical or in switching algebra. So how can we express this as a Boolean Expression?Logical Expression : To create logical values we will define a closed switch as “True”, ie Boolean 1 (and thus an open switch as 0).Heater is on (H=1) if (A and B) or T is 1• Boolean Expression: H = (A · B) + TCopyright 2001, Regents of University of CaliforniaEECS 42 Intro. electronics for CS Fall 2001 Lecture 8: 9/26/01 A.R. NeureutherVersion Date 9/25/01The Important Logical FunctionsThe most frequent (i.e. important) logical functions are implemented as electronic “building blocks” or “gates”.We already know about AND , OR and NOT What are some others:Combination of above: inverted AND = NAND,inverted OR = NORAnd one other basic function is often used: the “EXCLUSIVE OR” … which logically is “or except not and”An EXCLUSIVE OR circuit is need in adding two binary bits to decide if the result bit is a one.Copyright 2001, Regents of University of CaliforniaEECS 42 Intro. electronics for CS Fall 2001 Lecture 8: 9/26/01 A.R. NeureutherVersion Date 9/25/01Some Important Logical Functions• “AND” • “OR”• “INVERT” or “NOT”• “not AND” = NAND• “not OR” = NOR• exclusive OR = XOR1BA wh0(onl BA==⋅)(o BA KDCBAr++++BA BA i.e., differ)BA,when1(only BA⋅+⊕exceptC)BA(or BA⋅⋅⋅)0BA when1ly n(o BA==+)1 and when 0ly (o AB=BAnAorAotnCopyright 2001, Regents of University of CaliforniaEECS 42 Intro. electronics for CS Fall 2001 Lecture 8: 9/26/01 A.R. NeureutherVersion Date 9/25/01SYMBOLS FOR LOGIC GATESABC=A·BANDC = ABNANDBA⋅NORABBAC+=NOTAAORABC=A+BEXCLUSIVE ORABBAC⊕=Copyright 2001, Regents of University of CaliforniaEECS 42 Intro. electronics for CS Fall 2001 Lecture 8: 9/26/01 A.R. NeureutherVersion Date 9/25/01ABTXYHEXAMPLE: LOGIC GATES TO LOGIC FUNCTIONBAX•=TY=YXH•=YXH+=TBAH+•=TBAH+•=De Morgan’s LawA B T X Y H0 0 0 1 1 00 0 1 1 0 10 1 0 1 1 00 1 1 1 0 1There are 4 more
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