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# Class Note for ECE 274 with Professor Lysecky at UA

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This 15 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Arizona taught by a professor in Fall. Since its upload, it has received 14 views.

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Date Created: 02/06/15

ECE 274 Digital Logic Lezluve a LEELUYE a a mom 4 5 Lu mom 4 a Mulllderoulpul vall Multilevel Swim MaliSixufwllilevalcmuil Cubalkevvexenlzllon ECE 274 Digital Logic Mdllderoulvul Gmull Sumctiituitsiequiicmultiplc uutputs mom diplzv emoie can we ticatc a less Expensive tiituit bv shallrl availaupina 921257 ECE 274 Digital Logic Mullith am We 7 WWW will mmLmD mamm a wwz Wx vz wwz Wm sz WM wwz waz b wwzi wwz Wx vz o Wx vzo wwz o WW1 m 0 WW ECE 274 Digital Logic Muln lerou throJiis Example 2 We can also use Krmaps and share same optimized variables Find low oostimplemermauon ofeadi micum GroLp like variables x2 xi x i ii xi xi x2 x2 xi M W xiga r2 m oxig 4 xzm xW x ECE 274 Digital Logic Mul pleroulput rmiis Example 3 Let in mode example same procedure Find low oostimpiemermon ofeadw fmcuom GroLpliKe variables x j a xix 2 a xix xi 3 xi x2 4 n i i I r xixmxzxm m xixaoxzxm i xixzxi xixzxixx quotminim immune 5 ECE 274 Digital Logic Mul pleroulput rcms Example 3 Alternanve Stramgv Using same Krmaps let in someming different Tryto nd overlapping lmpllcal ils fomd if both maps Then w m nd minmum cover for eadw xi x a A xi xi l l n x ix n i i gtltgt4 i n xz ifo x1 xmm i n n n i Wm xi am xzxa we cauld 5 mm mm 7 but twauld mm impimmmi enemiy mo ECE 274 Digital Logic Muln lerou mormis ernanve SUaEgVCamparism Different technique yields cost improvement other tedmmcpes exist sumrofrprod icis w productrofrsum implememz on CAD Tools typicallyperform mam mos ofopumizaums to determine best implememznon i 1 r3 xi x xi x 1 xi ms M m cm a z 7 ECE 274 Digital Logic Mulnrlevel Svr hesis inuodm on Many of the CerulB We have built are tworlevels First level comprised ofAND gems followed try a secmd level OR gem I Womctrofrsums form i 2 i u z x 39 ligt7 i3 7 43 i 2 37 if W m m m m l a gay 37 u my a ECE 274 Digital Logic Faninhoblem Depending on me underlying technology ramm can pose a problem Famm is line number ofinpuB toa logc gem x 1mm 3 f 2 i mquot 3 537 i i mm a Ms W m w a l a 2 37 um um 9 ECE 274 Digital Logic Mulnrlevel Funcums as Agduum to me Fawn Problem Soluuon Mululevellogic expression Logic quann m a form mm 5 1 more than two levels oflogic How do We synthesize multilevel CircuiB7 253wa Funcumal Deoomposmon n ECE 274 Digital Logic Factoring We can take advantage of dlSU lbuDVe property 12a x y oz w oxz How much facmrirvg should We performv 2n x o v z w m ASSLITVE we can have a maximum nin cf 2 V numubvewm v codeobvcode i v v we at am am Vde We bvde V m we at n ECE 274 Digital Logic Facmring Example 2 implement F x1x239y3x439x5x5 gtlt1gtlt2gtlt339gtlt439gt639gtlt6 Assume We can orlyl39iave a fawn of4 i v xi mgmoximmgm xwmxnggong r gtlt1gtQgtGx4gt6gt O V xmwmgmwa xingxm m murmur m ECE 274 Digital Logic Facmnn Exam e3 Desigi a ormitwim me following spea caums Fuuv mums x1 x2 x3 x4 mum must be 1 Wm Mleleuneuflhemwlxlzr deixemzllulzmbmh zndxbzveeqmllul n 2nd xZ m em in n 2nd eilhev x3 mm mm in All mle 2525 n is H 0mm 2 is m all 2525 except n 2nd xZ m em in n a when bah x3 2nd x6 m equal ion 11 mox2x3mowx2 x3om 12 m1 xaw 2 x139x239gtlt339gtlt43939 m XSN o m mm m WWW DeMufgan s Law 2 x1x2x3 x4 13 ECE 274 Digital Logic Facmnng Example 3 X3X4 x1 x2 x3 x4 f2 x1 x2x3 x4 11 xam o MonMXSo x6 quot539 fzf gi i f We em mm xi We en New m 1 I 3 D 392 14 ECE 274 Digital Logic chuonai Deoomposmon Soluuon Mululevellogic expression Logic ecuauon inaform We more than two levels oflogic How do We synthesize multilevel M quotn quot1 CerulB7 I Facmring I Funcumal Deoomposmon n vwlace Waled ltgi Equannn Wm Wu m me new amass ans mm ave mm crnbired m de ne mamad uvmit ECE 274 Digital Logic Funcumal Deoom osmoij By Egtlta mple Let s look at an example F xl XZX gtlt1gtlt2 gtlt3 gtlt1gtlt2gtlt4 gtlt1 lt2 gtlt4 was was mm mzm m2 mzm m2 wsz Hv I at 4 2 La gm X2 W2 X1x239 g XI39XZ P 1 We39ve leamad hm m ma But mmmun vaviatlz 3 Otsave g39 a 7 m2 my 4 We can verwvita m2 aquannn F 9x3 9 usirg g ECE 274 Digital Logic Funcumal Deoomposi non By Egtlta mple Did Weimprove7 original auaion malarial demmvusmun F mm was mm mzw m2 W F 7 9x3 gm cog ns 21 mg 512 E max an in A max anin 2 delay 2gaedelay delay Agaedelay 17 ECE 274 Digital Logic Funcumal Deoomposmon Example 2 Let W anomer example Fumnon spea caum provided as a Krmap For a good decomposium We need to look for good sutrfmcuors Platters Bf 139s Whene 17 Hmmuwa cuvevlhe NEWng mwsv Fl 2a n o ab39 0 2a n o ab da b o ab R e o c axon o ab39 ECE 274 Digital Logic Funcumal Deoom osmoij Example 2 Example 2 cont39 What abuui these 157 r c d 2 a b eh we saw Mman 1 mewqw tboab mm Shmninexamplelskpl F 19 ECE 274 Digital Logic Funcumal Deoomposmon Example 2 Example 2 cont39 Tu cuvev all 1 s numbing n and r r FHF Fkgk39g39 r 2da39bab 2 c39d an Assumev we k mu ECE 274 Digital Logic Funcumal Deoomposmon Example 2 How did We do7 Min mum sumrofrprocucm form F mm WM mm aw X3lX4X5 X3X4lgtlt5 mzxams39 mmws Decomposed eqmnon F kg k39D39 g 2 c a k at ab a new 2 Qatarddsv East 75 22 min 3 ddav macemiav 21 ECE 274 Digital Logic XORXNOR LO C Gates Other basic iogic gates ii i addmon to AND OR NOT NAND NOR XOR gate 7 eXdLsive OR srmiar m OR gate extwtuumuts u Mm mm mm are 1 Tqu Mama arr and number at mm is hue XNOR gate 7 obtained m compiemermrvg XOR gate Invasim bmbie On nuruuttu vwvzmn e Empiemmt cutth sigmi max is the Dwusite cumut ivtm xoR 115 hue Mama evm number er hue mums 2D wm EDP mm mm XN nR 22 ECE 274 Digital Logic gtltOR Gate impiementauon Lets buiid gtltOR gate using tworievei mm ANDOR gates We can aiso impiement using oniy NAND XDR gates b F Invert can be impiemented by tying both 3 j j inpuBtogeiher 0 o land 1 1 o n i i n r Flte39b7eb39 F s39 b a EV F 2 b 39 2 b39 t 23 ECE 274 Digital Logic improved gtltOR Gate impiernentauon Using Functionai Decomposmon Can We use funcuonai decomposmon to nd a better irrpiementation7 aw l a b3939 a b D 2 b39 239 b D a b a 33939 39 b b b39 a b39 s3939 b a39 b3939 a a w u 13339 lt2 w We fumd a tcmrnurr term a a b F a 9 b 9 ECE 274 Digital Logic XOR Gate lm lementauofv How did We do7 Fa b39 239 b 25 ECE 274 Digital Logic Mulnlevel NANDNOR Qrmit Synthesis Previously converted Worlevel ANDOR circuiB to NANDNOR circuiB Same approadw can be Lsed for mulnrlevel CirOJiS x x x2 x2 a a x x x x ml WNW W gm minimum m x t x 1 x2 x2 a a x x x x mm m be quotmm mm name New be quotmm mm mm a W W m m as ECE 274 Digital Logic Mulnlevel NAND 6am swmesis DeMaqarv39 thr m Convert to NANDrgate Circuit x1 x2 x3 x6 F x5 x5 ECE 274 Digital Logic MulnlevelNANDGam quesis Convert to NANDrgate Circuit mmquot rimm 221 it 323 nut its dtr x F x5 x5 x7 x u a x F x5 x5 x7 25 ECE 274 Digital Logic Mulnlevel NOR Gate Synthesis Convert same initial circuit to NORV DEMWAIH39X Theorem gate circuit iii i 13 x e a x F x5 x5 x7 x u a x F x5 x5 x7 29 ECE 274 Digital Logic Mulnlevel NOR Gate synthesis Convert same initial circuit to NORV atrium mum gate circuit 223w ECE 274 Digital Logic Anal is ofMunievei OrcuB We39ve snown now to create various mululevel circuit Given a mululevel circuit can We determine wnat tuncuon it implemens7 r a w a39 w Fx1 x2m29xax1 x2x1x27x4 31 ECE 274 Digital Logic Ariale ofmunievei OrcuB rEXample 1 Deterrnine function implemented by mululevel circuit below Corsideririternaipanm mus st psepcmemxs Ham ms mm m7 xzxzms xzxzms mm ECE 274 Digital Logic Analvns ofmunievei OrcuB rEXample 2 Determine function irrplemented by Mmgmmm rnuiuievei circuit Sllg y more dif cult because eadn gate involves an inversion mum xS39 pm xS39 mm xS39 Dl mm xS39 m m xS39 my m xS39 mm m xS39 Alternauve memod to determlrre funcuorr ECE 274 Digital Logic Anal ls ofMunlevel OrcuB rEXample 2b W uniHi mm mm x6 gm look gm 6 remove Emmi 6 NOV gale Wf r 34 ECE 274 Digital Logic Arlalws ofMunlevel OrcuB rEXample 3 Carl you determll39le ml clrcult s funcuorl7 7 x2 a F x 5 Solution in book Exercise 413 g ECE 274 Digital Logic CLblCal Represermon We have explored several dlffererlt Ways to represent loglc runcuons mm table Algebralc empresle Verll39l Dagam Kamaugw maps introduce new representau on e Cubl cal Representau on ECE 274 Digital Logic 01ml Re esenbanon Twordimensional cube nu m vevlex m lwo murdimle havixmlzl wadinle eenexpmee lo 2 meme vulin zuadimle eenexpmee la b mue Fm eemee ye memes zavewmd in m m uflm m lzble ECE 274 Digital Logic CLblCal Representanon Twordimensional cube m Le nu sz mm min tube m healing wih blue wile meee vulize rm Mizh z Exvvexx iurrlionz 2 el men eee F m m m ECE 274 Digital Logic CLblCal Representanon Twordimensional cube m M 11 1x L an in melee vuliue mlv r1va me mme a win lwovevlze when kl Me in We w me mue x deride meme 2an l mu ngle edge denule mmbn n pvwevlv m W39va K Twordimensional cube La ECE 274 Digital Logic 01ml Re esenbanon denole b m M M 1x mm 2 an in Edge n Art u may lhe mm mm can de m b lereerdi mensional cube ECE 274 Digital Logic CLblCal Representanon Ill w mu m mm m nun mu sign mm m Anus mvmivmd lo gm m 0mm um m vulex m lhvee zmvdmle 41 lereerdi mensional cube ECE 274 Digital Logic CLblCal Representanon Ill w mu m mm m nu mu m Mam m be m in We 42 ECE 274 Digital Logic 01ml Re esenbanon lereerdi mensional cube Ill w mu m mm mm m vevlim m rum mm edge 43 ECE 274 Digital Logic CLblCal Represermorr lereerdi mensional cube Ill w MEI 1XEI W nxnxlnlxnxnnz2nbefunhevmmbred m xx rm Fzze ufzube 1m FarrbemVexerledindKerenlwivi in Fltnnnrnmmnmxnnl nxn1xn1 l x xl l ll xnnxlnl x Lemexvemivevepveenl2lionewiv2lerllo ltxx l xl mar ECE 274 Digital Logic CLblCal Represermorr Fourrdi mensional cube ore cube li islde arroiher irrrrer cube represent d1 outer cube representd o 45

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