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# Class Note for ECE 474A with Professor Lysecky at UA 2

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Date Created: 02/06/15

ECE AMA57A CumputerAided Lugi Design Lecture 1 Review of Combinational Logic What is Combinational Logic Cumbmatiunai civcuit Adigitaicivmn Museumpmsdependsnieivnntnegresentcnmbmatmn at me Now mputs39vaues Digitai civcuit 3 Emmi t i DJ aimquot Quilum v n guicmn Representations of Boolean Functions Samefunctinncan be 573m in raresmted in amenways Hm Enuiish BuuieanEquatiun 323vavu I mm Tabie Rymhxxmhxuiile 0mm VennDiaqvam BMW m n Can Dnvert from any reresmtmn in any Ether W W 39 mm mm Converting English to 3 Logic Expression Cunyerting Engiiei Pmblem tn aeeieen LDglE A i iie spiinidei 5Y5l2m shuuld spisy Water ir hl h heat is sensed and me SYSLEM is set m enabled identify and label yaiiabies 5 iepiexenig wig hut i urier new emblerf semen VEDannglevquot Wiite BEIEIlEal39i Eduatiun emiessind funttiunalilY dESUled VFhANDe Converting English to 3 Logic Expression Seat semeimnd LivMSVStem Cimdimeseiiptien Tum wsinind iidm un ir diiyei in diiyei39s seat key inserted seat belt nut fastened Identify aid liel ya39iables s l seatbeltfastened k4 kEYinsevted DEVSEIH ll i seat wsinind iidmen W p AND NOTsAND y vyiice aeeieai Equatinn eypieesmg functlnnallty described neisen ll i seat and seat belt nut fastened and key inseined Representations of Boolean Functions Same mnetien an be represented in drreent Ways Enulish addiem Enuzllnn mm Table Clltull yenn Diadisiii Boolean Algebra eiiemime Hstaiical yeispeewe anniean algebia EDDe s intent imam merBUUS human thought Swinhes For telephone i aans 5mm smother electronic uses algetwa to design oimtchrbased nigiiai design sin muded Hoolezn iqeim lo mum mus Mumu rum um lo my mu degq Boolean Algebra and its Relation to Digital Circuits I Bunlean Algebra m I Vaiiables iEDiesentU Di 1 urilY D Oueiatuis iemm u m l unlv am UDEiatuis s Am r 1AM bveium x mlv Menbulh 2 me bl okr Mew i Mimmmimet m b Nov 7 W me mm L 2 u i zen n i I NlEE VEEUJYES r destiibe iituits usiriu math FWWW Famzllylunxfuvm an a lmplementannlem waamrsusing at i transstcrs all time implementatiuns lDElE Estes Boolean Algebra AWN aeeieen meme based an set Dfrules awed from a small number of assumiens These assumptiuns ave tailed Mums Buulean aluebia invnlves elements that Like an me ur twu values 7 n m 1 Mn at an 1 431MEhtfienx 4t lfxlthenx Boolean Algebra Tneurems We can derive single variable theorems ruies Assume X l a variable MW Hang75 Idawry IdaUpoan Compemer Immen sue wer Boolean Algebra Duality Principle ofduality Given a logic expression is dueis obtained by replacing aii operators With w operators arid aii 039 With 139 The dual of any true statement is also true Axioms arid theorem listed ll l pairs to iiiusirate duality 1a 0 0 53 X 0 0 1b Sb gtlt1 1 2a 6a gtlt1 gtlt 2b 6b gtlt07 3b o11o1 4a lfgtlt 0d1erigtlt 4b irx1menx39o sue wer Boolean Algebra Pmpemes 27 and Srvariable identities 7 Properties X y aridzare variables mmmuzewe m x Y z x Y z Answerv2 11b H0 xyz X W Z x w x z minimrive w Y H z IhEDnEEthky 13a ADMprion 13b 145 x yx y Combinwg 14b Xy xy x pmuryan39s IbewEm 15b x x yx y sue wer Boolean Algebra Perfect lnductlun Example How can we prove these properties Truth table Algebraic Manipulation Show DeMorgan s Theorem works using a truth mole X v X t v T Ll77 tabes produce same ESLf 7 W ms are vaaEn 23 m Boolean Algebra Algebraic Manlpulatlun Example Prove x1 x3 x1 x3 manipulation x1 x3 x1quotx3 using algebraic LHS X1 X3 X1 X3 UJEDsmbunEDIDDEIW 1257 x w z x y 1 ms x1 X3 X1 x1 gt6 x3 memobumpme agen LHSgtlt1 gtlt1 gtlt3 gtlt1 gtlt1 X3 gtG x33 USEL DWEWBXDIDIEWSS39X X39 0 LHS o X3 X1 x1 gtG o meldmvwmmo X LHSgtlt3 gtlt1 gtlt1 gtG USEWWISPVEDWEWIRF39X V V X andij LHSgtlt1 x3x139 x3 V V tHsmermes Myoeran mum sue new Boolean Algebra Terminology Variable Rayon a bc o abc o ab o c RepreserlB a value 0 or 1 Literal Appearance ora yanaole ll l true or complemented rorm mm at Product term noduct orlmerals mnuumums and am any 2 Mnterm prooLlct term wrose llmrals lrldude eyery yanaole ofihe runcnon exactly once ll l mmgm a Dr W true or oomplemenmd rorm X W Su mrofrproducs sop lt Eduanon wrltuerr as OR or prod lct terms a only Above equanorl lS ll l sumrofrprod lcm rorm r a bc are o Cl rot let a by on a by and 2 sum mnrnuuus ab abc ab 2 sm new Canonical Form Sum of Minterms Sumroferlinterms Ham an abc ab c Equanon Writmn as OR of minerms only 39 Standard for 0f WWWquot Know 35 sum of mmeims a bc o abc o ab o abc momcalm a b c o ab c For gven fmcuo Orlyme verle ll l standard form exists 0 Determine it Fa baba is same mnetiun as Fa ba b a bab by Dnvemng first Equatiun tn anunical term secund already in anunical term F apa already sum or pmducts p a bb Expanding term p an a b SAME 77 same three terms as Dtnerequatiun am new Boolean Algebra NDtaIlEIri and Precedence Different notations for AND operation A AND symbol can also be omitted aba b a baANDb ab Different notations for NOT operation v OR abavbaORb Different notations for NOT operation overbar gtlt139 gtlt1Ngtlt1gtlt1 Representations of Boolean Functions Same function can be represenmd in different ways Engirsn Boolean Equau on Truth Table Circuit Venn Diagram am new Truth Table Representation of Boolean Functions Shows all possible combinations 3 o r a o a a t t t t t t mus u 1 u u u 1 Define outputvaiue for each 1 3 3 3 1 i Input combination D 1 D D Truth moles quickiy expand 3 1 g g Rows 2 i iis number or E 1 1 1 inpuB 1 u u u 1 u t 1 1 t 1 t QUsetIud1 tableto define g g Fquot 1 3 g 339 function Fabcd1atisl when E E 1 n 1 1 E 1 abcissargreamr in binary u 1 u i 1 1 1 t t 1 1 i 1 1 1 1 1 t t i 1 t 1 i 1 1 t i 1 1 1 swim Converting among Representations Common conversions O Convento equation Truth tabie to equation 3 E D D EasijustORead inputmrm u u 1 D Ramwa matsnouidoutpuu u 1 u a 13m Equation to truth tabie j39 g g E Easvssjustevaiuam eqtauonror 1 E 1 1 am eadnirpttcomonaumoow 1 1 n 1 anti oeaungintermediatecdumns 1 1 1 1 ab neips a Cunventutrutntabie Fa b a o inputs output a so so r u u 1 u 1 u 1 u 1 1 1 u u u n 1 1 u u n sue wer Standard Representation Truth Table r np np c n I How can we determine if two functions are the same r np p 1 n p Use aigebraic methods V E W t 5WD But it We raiied does that r n 1 n p prove not equai7 No wnaz it we stopped hele7 r nc n pc 39 Solution Convert to truth tables a DetermineirFaba is same function as Fa39b39a39bab by converting 39 OW ONE mm tab e each to truth table rst representation of a given function r we r so not ah Standard representation for given funcuon10niv one version ii i standard form exisB gait 1 r o r 1 u 1 1 1 1 u u u 1 1 1 sue wer Representations of Boolean Functions Same funcu39on can be represean in dfferent ways English Boolean Equauon iruih Table Circuit Venn Diagram sue wer Boolean Equation to a Circuit of Logic Gates Circuit representation may represent actual physical implemenmu on which is end goal Graphical melhod easily comprehended by humans A Ra b we am sun wer Representations of Boolean Functions Same function can be represenmd in different ways English Boolean Equau on Truih Table Circuit Venn Diagram cm sun wer Venn Diagrams To verify theorems properties and other logic expressions we used Truth tables perfect inducuon Cmslarlt 1 Algebraic manipulauol l Graphical method also exisE r Venn Diagram Universe represented by a square Boolean algebra has Orly owo values mm D Unverse B lo1 Elernens of a set are enclosed by a contour Square clrde ellipse X39 Area within contour shaded area denotes where Mam line value of line expressol l is equal to 1 Area ouslde contour not shaded denotes where line value of line expressol l is equal to o n W Venn Diagrams Twu Variables Represent two variables r x and y en pen 7 Xen y1 Draw two overlapping orcles Area the circles overlap represent Mr V the case where X y 1 H v1 DD x39Y xy xy v n n l l sue new Venn Diagrams Three Variables Represent three variables 7 x y and z Draw mree overlapping circles Fr n Hm xen H mm m Fa vl rn m m m v r Farmer immech on My m man 07 matinrssHvon wm sue new Venn Diagrams Equivalence or tugic Equations Use Venn Diagrams to verify equivalence of two expressions Dismbuuorl property X y Z X y X Z Ll S OD x xy X39htz X39r x39z i n W Combinational Logic Design Process step Description step i capturetne create a trutn taple ur eduatiunsi wnlcnwerls most runctiun natural forthe glvenpmblem tu descripe tne desired bEHaVlDr drtne cdmpinatidnal ldgic step 2 conventu This step is urily necessary irydu capmred tnerunctiun eduatiuns uslrlg a trutn taple instead or eduatiuns create an equation rur eacn output by Oerlg all tne mirterms rur tnatuutput Simplirytne eduatiuns it desired step a implement as a Fureacn uutput create a circuitcurrespundingtutne gatepased circuit uutput s eduatiun snaring gates amurig multiple outputs is on uptlurlally sue new Combinational Logic Design Process Example anee is Detectur I Problem Detect three consecutive 15 in Srbltlrlput aocoergn I 00011101 gt 1 10101011 0 6 11110000 gt 1 step 1 Capture theruncuon e Truth table mo pig peezss rows Equanon create terms for eacn possible case or tnree consecutve 1s 39 V abt lId Ededzf2 tm Step 2 Convert to equauorl already dorie Step 3 implement as a gatesbased circuit sue new 10 Combinational Logic Design Process Examwe NumbevaMs cam Rremem Output Tn emery an MD emeue yz the number Of Is an mreempue muem mem uuueuu Step 1 Cyture themnctmn Tmmememeeuemm mm mm A mmmm Step 2 Cmvertm Equatmn v a hah ahc ahc z a h a h ah c ahc a Step 3 Imp ement as a git a basadmrcmt E E E E More Gales NANO Oppeswe DfAND NOT AND NOR Oppeswe Of OR NOT OR XOR 8in 1 nput s 1 fur ernputXOR Fm move mvuts 77 add numhev of 15 mOR Oppnsne quOR NOT XOR Completeness of NAND Any Bun ean mnmen an be Tmp ementad usTng Just NANO geces OnN need AND OR and NOT NoT HHDutNAND mzrmvutNAND Wm mvuts Ted taqethev AND NAND faHawed hv NoT oR NAND Dveceded thOYs LTkeWTse fur NOR 11 NAND and NOR Logic Networks DEMurgarl s Tneurem as Lngl Circuits Any function can be implemented in two telesalesmt ate We sumsofsproducls form De new I We can transform into a network usl39lg only NAND gate x2 x2 x2 x3 h x3 h x3 xt xt xt x5 x5 x5 geneul um39oi Vumd nm zmnedlm belwun mo and Nelwak an be lumfumied ox gzle neuan z W gale tnte s nelwakuf two gale rim uml v equmtent s xquotx stempew Chapter Summary Combinational circuiE Circuttwnose outqu are funcuol39l ofpresel ltlnpuB Boolean logic gates AND OR NOT Enables use ofBoolearl algebra to design ClrculB Represenmu39ons ofBoolean functions Engltsn Boolean EquauonsTrud1 tables Circuit Venn Diagram Canonical Form Combinational design process Translate from equauon or table to Circuit mrougn wellrde ned steps More gates NAND NOR XOR XNOR also useful I Complemness of NAND and NOR sue new 12

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