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# Class Note for ECE 274 at UA 2

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This 12 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 40 views.

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Date Created: 02/06/15

ECE 274 Digital Logic Lezluve ii Lattulellrthsmal DuxlllunalNumbevkevmxedzllun Mdllmufunxlg edNumber ECE 274 Digital Logic shading Mimi inputs D dlultal systems nut limited In aslrlule i ui I lnlegevV2luelemvenlurellme twenty mot Plevluus v We said am digital system deal With is and nsunly HM do wt mind We isz isms Lmkzlenmdlngmmber b si 4 zmnqe i ion l ECE 274 Digital Logic shading rumba Deamzl Numbev aasctcnmctimal ad mum vevvexenl s QuznllW Svmbd in Damion mm how my cilhzlmzrliw Yenwwbal nlZ icsnas Nhrelhzni mamquot mm EDNA slam m m uxedbeuuxe i m m nger 523 5mma39mma39m39 5 innisa mos i ECE 274 Digital Logic Enood Numbers Binary Numbers Base 2 Base two Binary Twosymbols Oarid 1 More than 1 77 next posinon SD 22m Dusmm DUNE Bf 2 Q How much7 ini i 27m zii 2n vowm zwm i une nunureu une l lm ini lt une zeru one mi ECE 274 Digital Logic Enoodng Numbers Binary Numbers Range of integer binary number can represent depends on blB used n bits can represent integers in me raggeofOtOQMD l ll lllz183m 2371255 112 3m 11215n Mostsigm cant bit MSB I Leftrmostbit Assocategmmmgpestpoyerorz mum teastsigmicant bit LSB I Rig trmostbit Mm LSE Assooamd With lowest power of 2 Convenient to group blB together Abitsrnbble mnmu lullnlll 5 bits 7 byte 39 39 mbble byte ECE 274 Digital Logic Dec mai Ecuiyaient of a Binary Number rEXample 1 wnat is the decimal eguiyaient7 11027Dec1mal 1 1 o 22 21 z 11024122121ozu 1412o1 5m ECE 274 Digital Logic Decimal E uivaient ofrar Binary Number 7 Example 2 What is me decimal equivaiew 001101102 7 Declmal nunn11u2u27u261251 2U 201 1 1 1 1 gt u128u541321mn814gt1quotZgt quot 1 54 ECE 274 Digital Logic Decimal Equivalent of a Binary Number 7 Example 3 What is me decimal equivaiew 0010 00012 7 Declmal Choose your answer A 100110 B 17 c 33 ECE 274 Digital Logic comerurig from Deamal to Binary Numbers Shblracnon Memod How do We go from decimal to binary7 Desired decimal number 12 Subtracum Method 32 mm much 3 E 77 T Goal 7 get me binary weighs 0 1 15 mm to add upto medecimai 338 77 quariuty Work from leltto rim 0 o 1 8 wk imng Rigwtmle rmay llinls 22 m 2 4 2 i matshouch thave been lhererlryit 41zooNE ECE 274 Digital Logic Comerun from DeomaLto Binary Numbers subiracnorr Memod Remainder I Subtraction method To make me pb easier espeoally for big numbers We can Just subuact a seiecued binary Weight from me remainng quarmy Thm wemyee nemmirme Elusrmtvr and we scan 293m cm Q i if 7 1615mm much the szmtbinavv Dusmm 32 16 8 4 2 1 5cm Min vernalan Elusrmtv is u Remaining quantity g 32 is ma much 2 m 2 4 2 answa 1n ECE 274 Digital Logic Comerungfrom Deomai to Binary Numbers Divlslol i Memod Declmleumher BmzlyMurrter Dlvlsol i Memod Divide decimal number by Mew 2 2 and insert remainder 72 H39VEO new binary number connrrue dividrvg quonem by 2 urmi me quonentis o Huet rem a We I T mummy mm mu keep dwmmvbyz men 2 remmeer he Example convert 12 into bwarv quatieMF Jis mam mu keep mmyz 2 men We 3 uuwemm r5 mm mu keep dwmmvbyz mer my I l l 2 amemmeu Wecan canclude mm 511mm W n ECE 274 Digital Logic Comerungfrom Deomai to Binary Numbers Divlslol i Method sample 2 Declmleumher BmzlyMurrter Let s try anomer example using me Dlvlslol i Meme may 5 mm 2an m e n Convert 10 mm binary mime r5 mm mu keep dwmmvbyz nalien ZJis mm mu keep dwmmvbyz 1 men 2 12 mamas 1 A 2 1 uuwemm r5 mm mu keep dwmmvbyz U 4 2 my ECE 274 Digital Logic Enoodin Numbers Ocigl Represermuon Base 5 Other bases ayaiiabie mm m 12 3 4 5 a 7 Mi him I Base E 7 Octal Base 15 e Hexadeo mai hex u i 2 3 41m 5 a 7 Octal Symbols Range fitm m7 Q Wrivell 1n inncvzl Useful snamand in binary Binary to Octal oomersion 3 5 quot x Gimp binary mmbas in 35 Rwlate mmtaiespunang Q Write 15 inhinary anal digit i Octal to Binary oomersion 111 m1 mu 1 Rwlace anal diuitwith taiespnnain 3 bits cerium same yaiue Q Write 27 in dacimal 2 t x 1 t xquot 23quot 13 ECE 274 Digital Logic Encoding Numbers Hexadeomal Numbers Base 15 smie ieeeeeiae eeper Hex V m A Eadn posinon represem four i i L base two posmms Used as compact means to write binary rumbers Symbols range from o to F w my hex niraiy Biriary to Hex conversion H mm S mm GYCUD lZIlnaW mmbas in 4 s 1 mm 9 mm 2 m A m Rwiacemmtaizpunangnex 3 mm B W 11191 4 mun MUD Hex b Binary eonyersien 5 mm D iiui Rwlace hex digitwith a DMD E mu mzpur im 4 bits mnm 7 D111 r 1111 same value Qwrive 1i11g1vn3inhex Q WrivexAinnhinary Q Write Acninrlecimal i H F e 6 l l 12 l quotl72m mm mm 1111 1 14 ECE 274 Digital Logic Number Represenmnon for Computers ys for Humans Modem computer represent unnnnu l m l l l m m llH numbers using 32 or 54 bits 1 Binary represenianon can be t comfuslng for numans Daiasbeet documemznol i etc c3 81 An mas generally uses hex inuoducedsirrplenumbers only positive yaiues runsigwd numbers Later We39ll consider Signed numbers posinye amp neganye integers Fixedrpoint Floaungpoint ECE 274 Digital Logic Addmon ofunsi med Numbers Commonoperauon performed x y s e bbeedem byeompdtereaddmon of u u u u mmmm unsignednumbers u 1 1 u l I l I l l I l Add Mopbimdmbere what are me possible values9 mm W mzhx 1 1 n n 1 l n 1 H c S 1 H H 1 1 H H m i 15 ECE 274 Digital Logic HalfrAdder Let s create Circuit to perform addiuon sum xxoRy carry xv Special tvpe of Circuit 7 half adder I Adds WobiB x v x v D c c s Civmt Giauhital Smbul 17 ECE 274 Digital Logic Addmm ofMu rBlt Unsigned Numbers What if We Want to bigger nurrbers i e rndiupie bit numbers7 I Add on of two Zrtxtnumbers AIAH a an c 5 sn AM A mu A create mnh table create camspundmg cmmlfaxeachantpm ECE 274 Digital Logic E nemai Growth ofTWprLevel Adder What if We Want to bigger nurrbers 1 e muluple b1 numbers7 I Add on of two rtxtnumbers I Add on of two 4rb1triumbers I Add on of two l l39blt mmbers A2 A1 An 132 1313n gt finding111 C 51 51 A3 A2 A118 mm m a a a a gt mu m 21mm 255mm 0 s3 5 5 sn Eanmnal Gmwlh in Tmtevd Adda Imulemmlanm 1g ECE 274 Digital Logic Addmon of Unsigned Numbers Wim Carry instead We can consider each b1315221111115 column separately mmm m 11m 4 medlacamdzx 1 1 1 1 pus blzcmybnfmm 1n1u prevhmsmw 111n1 mum upm1tl I U U U 1 1 1 1 w mn2x n n 1 1 n n 1 1 n1n1u1u1 1 us an 1 11 1m 11 1m 1m 11 l ECE 274 Digital Logic Full Adder Let screatecircuittoperform n x y s c addmon u n n n n sum xxoRyxoRe n n 1 1 u n 1 n 1 n Carr xygtltqyq U11 U l 1 n n 1 n Special typeofcircuitrfulladder 1 u 1 u 1 Addsd39reebiB 11m 1 1 1 1 1 1 1 l x v c Gvauhical Smbul 0mm 2 ECE 274 Digital Logic Create component for each colurm Adds thatmlumn s blE genaaces am and any bIE AIuernauve Memod to Design an Adder Innate AddII Ig m Hand 22 ECE 274 Digital Logic carrwlipple Adder Called a carryrrIpple adder AbItadder snown Adds Mo Artxtnumbers generates Ertxtoucput Ertxt output can be cmsIdered 4m Sum plus 17bit carry out Can east buIld any Size adder Away W W Anew ll ll l lll IIIIIIII x y E X Y DI X V 393I X V 393I AJA AlA Bu z i n as as as as mm D l l I I I I I a 53 s s s cam swam 23 ECE 274 Digital Logic CarwRIpple Adder39s Behavch u Assume all inputs Imually n 0111 0001 answersnqud be mum ompm aner 2n I FA delav Wmng answer n sumemmg Wrung397 Nu quotjustneed muretlme fur canth npple Lhmughthe chain uffull adders 24 ECE 274 Digital Logic can Rl e Adder s Behavlq 01 M 0001 answer snuuld be uluuu aupu mains 2 m deliV aupu iieiens a m deliV Caneet answer appears after 4 FA delays aupu melSnS 4 m deliV Z ECE 274 Digital Logic Composll lg Adders We can build larger adders from smaller ones Bulld and Erblt carrynrlpple adder using 443W carry nppl eaddeis AskASA aiaeasa AiAaAiAn Bu z l n lll lIIllIlI AJAzAlAn Ha z l n AqAaAiAu Bu z l n rhltaddev ci behnaddev cii m Saszslsn cii essasisn l l l l 515555 sssasisn AiAeAsA AaAaAiAu Bi e s Ha z l n arm addev ci Gmphlcal Svrllhal 25 ECE 274 Digital Logic Design Example Muluple by 3 Design Example I Bulld a circuit that Will mul ply an mslgl39led x number m 3 input Enbltul39slgled numberi 107lgtltprodlalctlgt fin p o Output lonbltprodalct WW a Mmun anneal rlvul nu mu mi 255 a zssa7ssaunmmnm leaves a slhm m at ECE 274 Digital Logic Desi n Exam le mule by 3 Note3llll 11111151 galliumquot llllllll llllllll AVASASA AaAszAu 5156555 BJBszBn arm addev or m Sissssssyszsi l Hilllllll il l l ll lll AxAyAsAsA Mam 1 B i s s Basiaan am mm by u my mmssasmsn l l l l l p9 pxmspsppjmypn ECE 274 Digital Logic Design Exarrple Muluply by 3 Using 5m Left Alternauyeiy We should shift biB m i one posuon to left 5m Len Operanon ltlt effecnvely multiplies value by 2 u I ltlt 2 l 2 bub lquot mini 2quot 21 l mnni m muni 2quot bxygnmmnu u Llyilli ilxin HlHHH HHHHl AxAvAsAsA AJAZAlAV B v s s BJBZBlBD m 3 quot Ermaddev D u nub 6quot min I lllllllll p9 pxpypspspmpmpn 31 ECE 274 Digital Logic 5m Right Same as Divide by 2 Shifting Works other way too 5mm ml me posmon m right 5m mgbt Opera m gtgt effectively divides value by 2 u I gtgt 2 1 2 Emmi n mini 2quot mnni 4 many 2quot mini 2quot mini 6 may 12quot ECE 274 Digital Logic 025i nmm 2 Cum gpsgtinustaie Desiunmmpie umpensatingstaie i OvaAimexzziezmuutvdelenazle outgmmtuiimiwuwmn tutu mu mu m zuiuum ham 2 w input m Mun manta w 5mm zruuuttimiuum output X39bilxum invulx 3i ECE 274 Digital Logic DES EH Exam D E CD7 DEHSELWE SEE E mmquot Mus i 39 ma MN mm mm a at W HIIIHI MW Hum mm 32 ECE 274 Digital Logic Desiunmmpie Erbittaiguiatuv DES EH EXEMD E Erbil CE EU ELDY i Wimumummutuwp mum i oulpu mui m m LE input Hi 37mm 7 xrbiiinpuia Output mums ECE 274 Digital Logic Desw nEXam e Srbxtqaku ator D P swncnes 00000000 00000005 AVASASA AaAzA Au 5156555 El z n arm addev m m mm 3 mm Cauamr 00000 LEDs Srbxt DIP sthchbased addmg calculator The addmon Z3 15 shown

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