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Lecture 10Digital Logic Design

-- Basics

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Logic Design

Fromnow,wellstudyhowamodernprocessorisbuiltstartingwithbasiclogic

elementsasbuildingblocks.Whystudylogicdesign?Understandwhatprocessorscandofastandwhat

theycantdofast(avoidslowthingsifyouwantyour

codetorunfast!)Backgroundformoredetailedhardwarecourses

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Logic Gates

Basicbuildingblocksarelogicgates.Inthebeginning,didadhocdesigns,andthen

sawpatternsrepeated,gavenamesCanbuildgateswithtransistorsandresistors

ThenfoundtheoreticalbasisfordesignCanrepresentandreasonaboutgateswith

truthtablesandBooleanalgebraAssumeknowtruthtablesandBoolean

algebrafromamathorcircuitscourse.

SectionB.2inthetextbookhasareview

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Gates

A transistor inverteror A NOT gate

A NAND gate A NOR gate

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Gates and Boolean Algebra

Fivebasicgates

Truth Table

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Digit Display

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Building a Circuit

UseAnd/Orgatestoimplementthisexpression

Howmanygatesdoweuse?Canweuseless

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Boolean Algebra 101How to simplify Boolean expression?

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Boolean Algebra (1)

Usingjust0and1s(afterGeorgeBoole)TruthTableABooleanfunctionwithnvariableshasonly2^npossibleinputcombinationsAtablewith2^nrows

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Boolean Algebra (2)

Algebrarepresentations

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Boolean Algebra (3)

Anotherexamplewith3inputs

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Boolean Algebra (4)

AllBooleanfunctionscanbeimplementedbylogicgates(AND,OR,NOT)andviceversa.

BooleanoperationsAND:XY,X^Y,X&YOR:X+Y,XvY,X|YNOT:X,X,/X,\X

Order:NOT,AND,OR

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Basics

P1:X=0orX=1P2:0.0=0P3:1+1=1P4:0+0=0

P5:1.1=1P6:1.0=0.1=0P7:1+0=0+1=1

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Boolean Laws

T1:CommutativeLawA+B=B+A

AB=BA

T2:AssociateLaw(A+B)+C=A+(B+C)(AB)C=A(BC)

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Boolean Laws

T3:DistributiveLawA(B+C)=AB+AC

A+(BC)=(A+B)(A+C)

T4:IdentityLawA+A=AAA=A

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Laws

T6:RedundancyLawA+AB=A

A(A+B)=A

T70+A=A0A=0

T81+A=11A=A

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Other LawsSomeidentitiesofBooleanalgebra.

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Example

Idempotent theorem +

commutative lawdistributive lawcomplementarity lawidentity law

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Question 1SimplifytheBooleanexpression

(A+B+C)(D+E)'+(A+B+C)(D+E)andchoosethe

bestanswer.

A+B+CD+EA'B'C'D'E'Noneoftheabove

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Question 2GiventhefunctionF(X,Y,Z)=XZ+Z(X'+XY),the

equivalentmostsimpliiedBoolean

representationforFis:Z+YZZ+XYZXZX+YZNoneoftheabove

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Question 3AnequivalentrepresentationfortheBoolean

expressionA'+1is

AA'10

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Question 4SimpliicationoftheBooleanexpression

AB+ABC+ABCD+ABCDE+ABCDEF

yieldswhichofthefollowingresults?

ABCDEFABAB+CD+EFA+B+C+D+E+FA+B(C+D(E+F))

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Question 5GiventhatF=A'B'+C'+D'+E',whichofthe

followingrepresenttheonlycorrect

expressionforF'

?F'=A+B+C+D+EF'=ABCDEF'=AB(C+D+E)F'=AB+C'+D'+E'F'=(A+B)CDE

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Deriving Complement of a Boolean

Expression DeMorgans LAW: The complement of a Boolean

expression is formed by replacing all literals

by their compliments; ANDs become ORs,ORs become ANDs

YXYXYXYX +==+

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YXYXYXYX +==+ Using DeMorgans Law

?=+ ZYX

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Question 5GiventhatF=A'B'+C'+D'+E',whichofthe

followingrepresenttheonlycorrect

expressionforF'?F'=A+B+C+D+EF'=ABCDEF'=AB(C+D+E)F'=AB+C'+D'+E'F'=(A+B)CDE

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Question 6GiventhefunctionF(A,B,X,Y)=AB+X'Y,the

mostsimpliiedBooleanrepresentationforF'

is

(AB)'+(X'Y)A'B'+XY'(A'+B')(X+Y')(AB+X'Y)'(AB)'(X'Y)'

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Question 7GiventhefunctionF(X,Y,Z)=XZ+Z(X'+XY),the

equivalent,mostsimpliiedBoolean

representationforFis

Z+YZZ+XYZXZZnoneoftheabove

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Question 8SimpliicationoftheBooleanexpressionAB+A

(BC)'yieldswhichoffollowingresults?

ABCBAB(BC)'