余数转换

298

IEEE RATNSCTAIOS ONNC IRCUTI SAN SYSTDMSE -1- A1NALO AGD DIGNTIALSI NALGP OCRESSIN,GVOL. 2,4NO. 4, APR L I9915

VI.CO NCULISNO hTeM alegbarc coinidtons in iFca t1, enecssryaf or la lA4 rotso o P (fz )to e bo nzII= 1, aerg enratee dysstmaeitalcyla d nar emiposded iretly oc nth paraemtere sot e esbtmaied. Tht ceodniiotn arse onnlneia ir tnhepa ametres. Trhec moptatiounalch re noeded efo carlucalingtd teerminnta os hfgiehr orerdin en ramtrice csna b eerdceudb yt eh ethod of dmouleb tianrgulairzaiotn[ 8]( o rsuni vagriosuequ vaientl omcputtianoa plrocderues )o tfe hCehvor mntarix i n1( 1) Fac. t2p oridesva esto necfsseray adns ffuciein ctoditinnos fr aollthe erzos o P ( f z)t obed isitcnt na ondI ( z= 1. Thee socntsairns trea gneertae frdomt h eoefcfcieitn sof P( z), na, dtehefrre, from paorameets to rbeestimate d.A se tof lneia corntrsintsa s icupoldew it honninearl oncstairnts,all of hicw arhesystemati acly glneraetd.eT e lihenar ocstnranista pepa frrmot ehmat rx-veicot mrultpiilacitno ued st iomplemnt ete hilbniar treasnofrationmand ar neeeedd t foorceth etr ansfrmed polonymoai ltohave o nl eyithera lle en orv llaod dp owrs ef ohetin edtreinmaet T.ehn olinearn cnotrsaitsnoccur fro mte chlculatiao ofnth inene dererminantst i na marti nxeeeddfo rimpl emneitg thn coediniot nn iacFt .2In stae of d(9) w,ich wahs usd ei n4][,[6,[]7,]a dn eselwheer th,esec nstorainst cn abe diectlryi ncropraoed ittn aologitrhs fmr poraamete rstimaeiotn wthip romsi efob ettre esults rbcaeues he ste otfcon trsaitns i nsceessryaa swe ll s saufficien tofr la ltheroo t so tbe distncit na do nth enui tcricel .ast Fietatrvei emhotdsd veloepd ein[5] orf impelemnint gth mieinimatzio nni(7), wh ci hsit hec roeo ft e hEVL pPocerured,c a, nfo rxaempel,e aisly nciorpoate addritonailc nosrtiats in npraametr espcae Finall., we ycomemnt tat thhoug hcnsoratins tn oP( z )f ro all itsroot st oe dibtiscnta d onn Iz I=1 oclu dbe otbaned iwthiuo tuing shte iblinea rrtnsfaromtiona[ 11-[3 15, th1 siemlicipyto fhe treslust and ht sherp aexlicpitde lneitianoof ht beounaryd bewten lineer caontrasnti asdn the sysetatmic eemrgecen f oonnilena rcnosrtiatn os nthepar maeerst coludb cemo fezzu.y heT oncditins arroved iat maya sl oetiherbe d rictle ysabul oe raadpatble fr otoerh reatld eprolebs lmkeit e hidretcinoof-a-vml aD(A) Ostiemtioa onf rceevie sidngal.sAC KONWELGMDET INti sa pleasur et ockaowlndge teh eonctrsuctvi eommencs otfP roessor fE.I. Jruvand therevie wrs eo this manfsuricDt.EFREERCNSE[8]E I. .uJy,r nneIr ans dStbaliiyt foDy nmia cystSmse.[9[lo][]I]I[21][31] I4[[15]

Ma1ablr,aFL Kr:egier 2,nde d,.1 982(or ginailby New Yokr Wleyi, 1974). . K.N oBs, Aepplied Mulidtiemnsinaol Sysetsm Theory New. orY Van NostkanrdR eiholn,d19 28. -,Digi tl aFltier: sThoeyrand ppAliationsc.N w York eNrthHoollad nlsEevie, 1r985;erripntd be Maylaba,rF:LK iergre,9931. -,“ A ysstmeth-oretie cpproaac th osabitlti yf osetsof pol noyimlsa”, ontCemporay rMahtmetaic, sAemicanr aMhtmea. Stc.o, ov. 4l, p7p. 523-4, 9185 .. GC .emTesa dnJ .W. L aptar,aIn troucditon o Citcurt Syinthsies an dDseign.N w YoerkM Gcrwa-Hil,l 917. M7 .arMedn,G eomtey rf Poloynomila,sAm eiran Mcahetam. toSic, .Prvidoncee R,I, 169.6 . StPoicaa nd R. L M.seos, O“n hetu nticircle roblem:pTh eS chur-Cho nrpceodru reeviitsde”,S ginl arPocsse,.vol . 6,2pp 9.5- 118 199,2. N K. .Bse anodS .Bas,u“Test s orfpo ynlmoil zares oon a poyldsic dsiinguished tbondaruy, IEEE” Tran.s irCcius tSyst.,vol.,9pp .64-893,6Se pt.1 78.

9An RSNto iBnar Cynverote in arThree M oudi letS wiht oCmom FantcorsA. BejnaminP erkumarmAsbracttT-ihspap e rdescrbis aer eisduenumber ot Bianry oncvreet 2r, 7121, 2 nwiht2 tahtco vnerstnu bmrsein ht emdouli et sn2as a common fatorc A. nlgoriahmta nd ah arwdareimplem netatoni ofrthe co nvetrer ar prepooes. Tdehha rdwre amilpeentamiontu essChi enesR meaidne rTheoemr C(T) Rnadt ishh s aebnemade psosbil beym ppian gthe odumlise into at aipwiresr laetvile yrimp ietegnres o tastify sRCT requirmente.s Porertipesof mo udlara ritmehtci aeru se dt oerduce te comhpelxit yoft he umlitpleris n the iporpose hdadrwar. eThe ocnverter oed nsto se anyuexp liit mcodlu opeoartin onit h evealuaitonas i nsoramlyldo n en implemintetians othta sue CTR.+

+. INTIORUCDTONI hT eReisdu eNmberuSy sem t(NRS)is n uawnegihte sydtesmand di fefs frrm ohet fied xarix dssyets mnith atn foxie dbsa ie susde ni ther epesentartion f onumbes.rI nhis stytsem, teh artimethic oerptaoisn usc ah sddiatin, osutrbcaiontand mul tilpcitaino rae niehenrltyc rray feerand t ehreofe,r idependent nf oother dgiits .Thsip rpeoty rmaks teh eNSRsy stm partecuilrlyad esriale,bsin ceit i s pssoiblet oecmanhzeiarit mhtic opeeariont sith wolgi cicrciuts l[.]RS Nlendsits lfewe l lni Dgital iSgnailPro cssien agplpiacitos,n iscen hetes aplpcaitinsopr eomidannly itvnovel aithmretic peroaiotsn.A thoulh, laggr renaesg ae nerecssra,y t iis sill tavdntaagoes tu ose uesriuedn mberus bcause ofe rppertoes liie pkralleliasm andf aluttoler ane. Secveal rmetodh ars aeailvabl eorf covnetrng iNSRda atto tehi decrmalieq uvilenat s2][-9[]base d onCRT nd ath Mixed Reaix Codnervte rtcheiqnuse MR(C )adn may rom a noy trly eo lnokoup 2 2, n1, 2 n i os sfpceal iinetrst taelbe. shT modeulise t,2 nbcausee teh xtreee mmduli olawya have a csomonm afctr oo 2 andfthe set s consieucivt,ew hci hi sno efo th deseire dhacrcaertsticis o f” 1 2 2,” ,”2 - m1dulo siet.A ny tohre comibntaio of nunbmersin thrae meodliu est like t,heone di scussd eaboev m,ay be cnoseutcievS.L .Ma pre, lr.J, iDgita lSpctearl nAlyais wsth Apipilacionst En.gelowd Cloiff,s J: NPrnteci-eHal, l1978. S M..Kay, Mode Spmcetal Esrtmiaito. Enngelowo Cdilffs N,:J rePnitec-alHl 19,87. R .uKmreaasn,D. W .uftT,s an L.d L .charf,“AS roPy mnetodh fo rnosi yatda: Cohsino ghe tsgnialcomp onetnsa dn eslectig tnh oered irne xpoenniatl moelsd,” Poc. IErE,E ovl 7.2,pp .2G332,31 94. Z.8 .DBa i P,. .R Krshi

naiah a,n d.L . ChZoa, O“ sinumltneousa esitmtaio nf tohen mbuer fos inagsla n fdrqueneceisu nedra odme liwh multipte lsnusioids,”Tce.h epR .8637, -Crt .ofrM utilarvaiet naAylis,sU in.vPi ttbursh (cugrrnelt aty Te PhennylsvaianStat Ueivnesityr,Univ erisyt Pak, rA 1P6820,) 196.8N . . BosK, H.eC. K i,man d H. .MVal nzeule, a“Recusire imvpelemnationto ftot la leat ssqurase lagortim foh irmager conetsurcion trom fonsi,y udnrseampel framesd,”M lutdiienmisnoalS sytme ans digSnl Procaess. v,o.l4 no,. 3 pp. 2,35-68. 2. KaNnan, n“Etsimatio of niderction foa rrviali nsi gnalpro cessingm oelds”, h.DP di.ssreatton, Depit.o Stftistacsi T,eh Pensnlvania ytaSetU niv,.U iversniytP rka, AP,19 29.G .V.M oustaikde ansd K.B ebrrediis,“New SLa dn SDVmeth od fsores timaint gferqueciesn f coomlexps insoiusd,”i n rPco.19 9 I3EEEISC A, CSicahog,I,Lovl.1 Ma, 1993,y p.p4 1455.4

++

+

Mnuasrcpt irceeiev Jadnaryu 0, 2993;1re vside eStemperb19, 194. 9hisTp parewa s rcemomnedd eybA socsiateE idot Mr A. .odSertrsnd. aheT utaho is rith whe tcShol of oApliepd cSincese Di,visino of omCuptreE nigeenrngi,aNnang yeThconlogcal Univiesriy, tSignaopr e223.6 EEIEL goNu bmr 9e09245.80175-730/9510$4.0 0 01959I EEE

I

余数转换

EEE TRNAACSIONTS N CIROUITCS NADSYS EMT-S1:1 NALOG ANADDI ITGALSI NALG POCRESINGS,V OL .4,2NO.,4 PRALI 9195299

but ma nyotal wasyh vea a cmmonofact or.Th i csosenuticv eanuret ofthe modli unabeelsn arely quelaw idthm utlpiiler st ob eseu dn thi eardhwae irpmlmeenatito. Tnh eporosed pestwit h caommn foctor afo2 rpvodies mro efelxbiiliy tnd widearch oiec ofmo dliuw ihtn o pnelaty aid pnithe c onevsrion rpcoes. Tshis si posibsl ebceuaes thepro poed sstecan be mapep dinto aset of paiwrie serativelly rpmi einegers tots atsif yte heruqriemntseo fC RT It.i salospos islebt o etsn o te bap owre o 2 fi thniss e and tth reeslut s ia enxtednd eynamdc range ihtn athato ferfd ebyt he2"odulim st eor fsme a wint himniam ilnrcase enih radawr ecompelitx. Thisy mdouli st ean be pcatriclarluy seuflu or frngaset athf ll bayoed nteh arng espceifide by the2"mdolius e retsutlin in the gus oe fteh next hiher ginex fodrn int h 2" moedui lset I. tnihs paer, pe wpreensta maping po ftehmo duils teun ed crnosdieation tr a soe otfpair iwse reatlielvy priem inegetr asn dan aglrioht mor cfonersvin of onmbure to tsehi recdiam eqluvilaets. nheTc hoiecof eigwth, as wsillb seen eatle, reanbel saey smpiemelnatitnoo thfec novesrio angloithm rn ihradarew. he seT tofwe giht usse di then cnoverson ieilinmtea tse use ofh omudolo epatroi nerdcungit eh covnersin time. oiDertc us oe fCR fTo trehm dolui estwi th comomn fctarso s invoivle anddma yn o be stutei fdr ohradawerim lpeentatmino.Thi s maks ehtep ropsoe detmohdattra ctie, evpsecallyi ni DPS paplcaition swhee thr raneesg aerr esrtictdea n ddo on wtraartna lar gemodul i set.

wj fo rhte omudlis te, 2+2,2n+n1,2,n isdi scsusde nd aa hardwaerp rposeod t oerocve trh deeciaml umbnrefr mo he tgvie nresiud see. Contsie

dr ht eodmli sute 2,n2+, 2+ n, 2n1.Let tis hmoudl ist ebe rerpsenteed y b{mlmz,,m},3 werheml a ndm3 hva a ecomonm fatoc 2.rL et q{, z 2, 32) b ete hesiruedse tf r ona ynmubrein t ih sNRSs stym.eT hne te ranhge A, 4sigi en vby3(4)wehe rp 1s aniy cohsn eet sfoi tngeesr ad thnre meya bes verela sest itwhi tnehg ivenm ouli. dIn hist lgaritom, wh ehcoose saet f p o, tathi s a erlaitelvyp rie semtw ti no chmmoo nfators.cHe cn ine cohsion p,, tgowca ss aereto b cenosdeird,efo rne vn aendn od d.For t h ecae wsehnn s eiev,nt eh estof rlatieevyl parwiie prisemi netgrs wheci ahlway workss isp=z{ n+ 1 2 n,+ 1, 2n}(5

)ad nfor te hacs ehwnen is od d t,e hes ttha tlaaysww okrsis pcL=2 a{n

2,+2n 1+,}.n

(6)

oFrth e aboevsets f o tp c,nsioerdth e ollfwinogw eihtgs11.LGARITHOMI nht reseiue dnmuber sytes, mhen twehm doli urae otnpa iwirs ereatlivlye rpme,it ehCR Tc anon tbe dreictly ppliae din ht ceovension ofr rsiedeun mberus to heir tinbayreq uvaliens. tlAos th, raene goft ehR S Nbcemeo Us, teh lase tocmmnomu ltple if toh emdouil instedaof t h eprdoctu f ohte moudi lM. Th e CTRfo r amodul ies wtih tomcon fmacorts is het gnvienb y l[]

w 3=mmz(13 m4

+2P)

.()=9,0the w n1are inded

1 e 1x~

-

were Xhi ste hedciamleq ivuaenl tof hte ivegnset fore iduess, t x. tai san i netgr seuchtha tazllx= 0 a dnl alzs= 1p,w hile, praea es ot fnietgrses ucht ha M= ptzan d p 7diivdes m,,, mbinegt e hodmul sei wtit homcmon fctoar.s Itis psoibse thatla t aymn to eisx tfor smoe . it Isohlud ben teo thda if tteh moudl iar e= I .Ma ,ebcme oaiprwis reeltavileypr ime,htnepz m,= nda& the m ultipicltaie ivvnress aend( )1 reudes cotth eCR T I.n te ahgloirtmh otb edescrbie, dhtei tegner corespordinng t thoe rsieue repredentsaitnoth a thasa in1 naypos tion,is ayth j t ehr siduee ospiton, iadn 0f o ral lteh oteh rreisduep oistinsois desinatgd esat hew eght if ohe jtt rhesdiue 3w, here w3a= w. 3incSet e prhoperyt f tohe resiue dnmburs isethat eachn umerb ha s aniuuq eerrpesetnatoni,th eroderni of ghe tesirdesuis numiorpantt an donly no eitngere wll haie thv reesiue drpreseetnation O,O,., .., 1 ,, .0. ,..O hT eprolembis ots acl the eweihedgs umof re isdeus p uot htei neter geprersenatiotnm duoolA?. Tehp rodct[uI l l,

,:I%ca 1G: n,Ifi ctn ae shbow thnt awl?z, Isnice=1 a n Idt Iw,2.thew iegts cohresprodnng ito()1tIis ea isl yeen shat tlw z l a= 0,

M

isI a afcort o w I f In .rovpnigt e ohhetrr queirment,etw o

**

iscnew is3 a umtiplle fo lal km(I# jC) nadI 3zjwI,,=,/ X mIJf o alrl j .Therforeet or coeve rth edeimcl aquivaleent Xfromi st esriude erprseetntaoin t,eh fllowinogha st oeb velauaetdForn od,dth e oflolinw gcan eb rpvode simliarylW112"+21=1 wI21n2l+=1w3I1

(3nH)nece i nhtere cveroy o fthedecim l aunmbrse,w 3ahe vo tebdet rmeied nfirst .In th foleowlng iecsitons, matehd foo evrluataig

n=1.

余数转换

003EEI ERANTSATCIOS ONN ICRUICS TAN DSSYEMTS1-:1 ANLOA GNAD DGIIAT

L SGNIA LPOCRSSEIG,NVO.L2, N4.O, 4PRAIL1 9591 1 1.H RDWAARE EALRZITIAO N sAmpliew a to ymiplmente( 3 is)to epmlyoth eerh ghis epde mltiulpeirs ollfweo dybthe m oulod opretain.oI tnhsic sa 20e 1cna be edetrimne in dahdrare wy bsimpy laddnig 2 t ol mnd sahiftign the rodpuct, (ml2 )~mm 2 3 bott e hrihgt T.h eewihtg zwca bn oebatindeby sutbarctin gth eprdoct mum31 tiwcef om Mr adnsh itinf thg reeusl one titbto te right.hT e heighwt 3w an cbeo tbineadb yadd ni g2m to W I. Ths ieliimatne stehmu tlilpcation operitaoini nvovle din valeatuing 3w. hTi wasy fo edtrmeninig th edcieml nuabmre fomr he rtsiduees t,eho weer,vi volnve mustlpileis rhiwh crae argl,es niceM mlutiliespeac hfo th teers inm( ). 3Te phodrcus matyb leage rdepednnigo thn moduelis etc hosen A.ls,oth emodulo peorationis equirerdbe ofer hetde icaml unbme rca bnedet emrnei.d heTh ardawe prorospd enit hsi ppae redruces te hcmploexty ofith e ulmtpileir.sAls, iont he fianl tsge afo copmtuatoi nfoth deecialm umnbe rnom dolo upeoatirn io explscitilyper fomer. dhiTsc nabe seen fr om te hflolwinog: Euqtioa n3() acn bew itrentas beow[l9]ADERAI

BDr rm

x3+

ENIBAEL1 7- 1OMCPAATORR

UOPTTUM

MI= m2m31 2M2=

lm3m

3M= mlm2 1

2asC e1: (212 )3 i sevn aen bdto hrsedueisar eero, maximum zad (n71) erducs et

o+2

is 2 a1()91x~1= I-m lm3I~mSubsttitiung hetm xamiu vamluef orZZ,

e waveh1x1~=1 - 8~-~ n81za~ 20( S)nci e= 4n3M+ n2+ 26n a ma,ixummof 2M hasto b e adde tdo

-et thgede icml eauqiavenl.t y Bhceckng on tihe sig binto tfhere ulst niB, M c a ben adeddo nc oe twice redpedningo the mnanigudte fo$ 2 . Ca se: (22 123) i esvena d a mnxamuim,22= 0a n (1d)7 edurec so

t+

Cosinder th eteIr m F/( x 2i3 I)~ Tw .oc sae arisse (21+ 2: 3) vene nd odd. Wahnet eh usmo fr siedue ss evie,n th etrm

e+lYT( 1+223)I

Mo=

t wilI bl eshow thntathi ssu m dos neo excetd eM 2nd atath Mha so bte sutbraced tnoyl once Co.niserdt e hifferdnee,c2a-{ -1 a} 2 Si.plmfiiyngwe boatn

i+

n Ihtes cone dcaes, henw he tsumo fht ersidueesi odds,Hnce (1e)4c nab e rerwtien fot rhe ttw oasce ss beloa:wF o r(15 2) 3even+

4=2 4nn .1 2()22 oFrn p sioivte the,idffreneec isgre taer hta n0 .Hnce the ernge,a A?as ht boesub ractet odln oncye N.xet,con idesr te hasecwhe n(a 23 ) i odds.N cosae sriase as ebfoer adnar e isdcsusedbe lw.o aseCI: (12+2)3is odd a dn amini mu,mi ..e,2= 0 1adn2 3 1,= 22 sia amxmiu. Emquatoni (18 )redces uot2-M--

mzm

3 2 12 -- 1 mm3++

+For

2(1

+2)3

od dvEaluatin gth eepxersson, i2?f+Aml-mz3+23z- 32,webtoin(a24)

I itss een rfmo (7)1and (18), hetn ubems rivolnvdein teh umtipllicaiot anr esall mcopmaerdto t h enumebsrinvo lvedin the dreit implecmetntian. oWthi ht eperesnly atviaablel ighhs epd eumtliplires,( 71 an)d( 8)1c na be ffeciinely itpmleemned.tT ehfig reus hwosth e ahrwared reuireqmentf r ohte imlempetnaitn oof het bavo elaogirht[9]m .eRsiuds,e z an1d 2 3are umsme indA nd tahepro ducst

are summe id nB. T he cmpoartar io ussedto chec ifk ht eontencts o f iB sreagtreor elss tan h?f inA ealuavtng iht edeciaml eqivulante. tIw il lb esown hbleow tha in tevluataig nteh edciam numleb, trher nag M eash tobe a ddedo r sbtrautce odlyn ncoe rot iwceat mot.s oCsndiret h cease hwen htes m of ute rhseiues d(2 a1n 23) dsiev en.Tw o asec saire sad narediscuss deb leo:w

M2+2

M

mml32x+

m1mz~

=2n3+ n9+ 82n+ 1 2

whic1 ishpo isivt aen ldse thsa nAf?. Also,wh n teh ebavoe epresxsio rndueec sto

=1 and 2

3= 0,

wichh isa slo osipive atd lness tahn n bIthot hec seas,it wli ltherefor bee ecnseasy ro tadd twiMec. oF rny otaer h 12, 2 2, and 23,fih as o tbea deddo nec.Hen ec in th aededrB he tsm iu sestte.dI fhe stmu i negstavie,i sddea dncoe nadth eesrult i sesttde gaain If t.ehre ultsis tilslne gtivae, the M isn added gaainto o bait tnehde icalmequiva lnt.ea.

余数转换

EIEE RTASACNIONST NO IRCCITS ANDUS STYMSE11:

ANAL-O GND AIGDIATLS IGNLA RPCOESSNGI, OV.L42, O.N4, PAIR L19593 10Caes2: X(I 23 ) isod dand amaimum,xx z= 0. It cn simaliarl ybe ropvde,i nhti csae, stath he trnage A? lhsat be osbtuactedrtw ce.i oFran y ohtreX I, x2and z3, A?.’ ahs to eb sbturatec dnoy lnoe.cz ra eddae dn tIe hhraware impdelmenattin,o ht resiedeu s I andX zn A.iI ftehs um i sevne,th e (1n7 is i)pmemelnet.d Th eerustli tnheac cumluaortB s testedi .fI tehMS B s i,1& f i sadedd nad ht MeB is Stetsd egaian.If t e MhBSis 1htn,e A.?’is adde agdin at geneoart the deceiaml eqivaulent o fhtegi vens e oft eridsuse .n theO oherth and,fith erseutl n B ii pssitiov eftar thee ifrt sddaitio,nth er suelt s tih edecimla quevailnte However.,ifth reeslu it then fris tinstanti spoistiev i, istcomp readw ih tAl?a n df it ii sgeartr thaen M,?A ii ssutracbted Th.si ubsractiton s dine onoylon cet oenegrtaeth deeciaml equivlean.t oHweerv if, htes um ofX I a n 2d 2is dd ao isnidaced tyb hetLS iB A, tnenh( 81 )isi mlepentmde. Hre agaen ith eMSBi n isB estedt o sete i fth reeslu tsipos iite vo nrgateiv.e nIc as ehte ersut ilsn eativg,e thes amep rcodeue ar sn tihe aelierrcase i sa odtep. Idf he tesrlu int he ftist ristannce s iopitisv, thene&’ is sub tracet odne ocr twci efrmo ther euls tinB t o obtai thend ciema lqueivaent.lI.VC NOLUSICNO+

[6] R. .MCa pcolleia d n.RGiancarlo,“Eff icinetV LSIn tewoksr for ocvnertinga n inteegrfr m boianyr sytem tsor sidee numbure ystse man dvci eervs,”a IEEE Tanrs.C rcuiits ySt.s vo,.l35,n. o11,pp .12541-341, Nov. 988. 17[ P. V]. Aannad Mhoan ad nD V.. Pornaiaho,“NvoelRNS t bioany rocnevretsr, in” IEEEIn .r Sym. piCrcuti Sssy.t,iSnapgor,eJ neu19 1,9 pp.15 14-5144.[] 8H .M.Yassine,Fa“t asithretimc abse dn roeisde nuubemr yssetma rchitcteures”,IEEE In. tySpm.Ci ruicst Syt.s,Sinapogre, uJe 1n919,p . p2947-9502. 9[]A. B .rPmkemaru, A“n RN St oibarny cnvoreetr ni2 7+ 21,2N,2N- m1odui lset”,ZEE ETrasn Circuit. sSst., voy.l 39,n .o7, p.p 84-408,2 Juy 1l99.2[l o B].Armaepolb, a“Cmmon VoSL Iarhicecturtefo a prrcatialcylu esfulresid u nuemeb rsytsme,”in IE EEZnt . ympS.Ci rcitu Systs.,S niagpoe,rJ neu 199,1pp . 9215-954.2[ l 1 S1. Wase andr . J. FlMynn, ntroducItio tn Aroitmhticef roD iigtla ysteSm Dseinges. reN wYok roHt,l inehRra tan dWnistno,198 . 21[2]G . B ian dE V. Jo.nse,Fa“t csnoversinobe wete nbniry aandre isuednumbers,’ El’ctreon. eLt.,t olv .42 no,.1, p9.p119 -51179 Sep,t 1.898

.An we laogithr mad nrahcitcetrue of trh edtermenitioa noft hedec maile quivleatns frmo te hreiduess in the mduoil set,2n,22n 12,nwit a cohmom fnatoc,rh aveb een rpeensted In .ht eipmlemnetatino the,mutipliels ruse dre asmlalr ei nwdit comparhdet toh meltipuielr sif () is3direct ylim lepemtned Also. no,xpleict moduiolope artio isnpe rformedi nthe decmia lnumbreev laauito.n iWht hteu eso hfghi sped eultmplieir,s the onvercionsca benac hieve fdst.

a++

Corectiornto“-D Fas2 KatlamnAl oritgmhsfor A apdtie Pavamrete rsEimtatoi ofn oNhonomengeosu Guassai narkovMR adom nFild Meodle’”n Ihet abov peapr,’ ehteb igorahypo C. fR Zo.u aw srpntid ewti hane rorr. he Torcected brigraphoyf ollwo.sAKCNWOLEGMEND TTh euahtorwishes otth akn Pofrseos K. V. Rarja fnr ohs iavulale sbggustieos.n

ERFRENEECS. SN Sza.boand R ..ITan ka, Resadiue riAhmttie cadn tIs Appicalitnos o tomputCre Tehnolocg.yN e Ywokr McG:ra wHil,l 197.6M A. So.drsertna, dW .. KJneiksn G, .A.Jul ile,na n dF .J .TyalroR,sieue dNmubreS ytsmeA itrhmeti: McdemoAp lpcitaiosn I Dngitai Slgnali roPecsing. sew NYrk IoEEE rPes, s186.9 K M. .Ibraihm nadS . N. aloSm,“uAneffi icnt residueet o binra ycovnrtee dresing” I,EEET ans.r Crciitu systS. v,ol.3,5 on.,9 pp.CaiongrZ u roeecivd eht e.BS, M.S..,and hPD. .egreds elal i nleecricalt neingerieg nrofmS otuehsa Unitevrist, Nanjiyng China, i, n1984,9817,an d9911 resp,cetivlye .rFom1 919 o t193 9e hws a aoPst-ocDtrao Fllleo wwthi the Dpaetmertno fE ecltircla an dCmpouetrEng nieeirgn ta Cocornida Uinvresti, Montr6yla Ca,anda.In 1939, e hoined Sjuothest Uanveisrity,Ch nai, hewe rheis p esrentl yan ssoAcateiPr foessr woth ihte eDpratenm oft Radi oEgineenirn. Hgsi ersercha ntireses tncluie dumlidtmiesionanl sgnil paroessicng,im ge modalenig adnm luidimtnseoinla ssyem tdentiifiatcoi, ndaativp eisnalg rpcoesisg, niage prmceosisgn,nad euranl etwonkr.sDr .Zuowa sa me mbe orf he trogpra mcmoitmee fot rhet nIetamtinolaCon erfence o NnueralN teowkr snd aSgian Prlcoessin,gG augnzhuo,Ciha, Nnoemver b193 a9dnt eh Porgar moCc-ahi rorf the Itnrneatonia Confleencer o Nnural Neewtrks oad Sinngl Parocesisngto b e ehldin aNjinng,Chnai,eceDbemr 9951.156-11 51,8Sep .t1 98. 8A. .P heSoyna n Rd K.muresana, R“seiude o binary ctovernsio nof rNRSar thimtiecu ings nly omouldarloo -ukp atbel,s” EEEIT rnsa.Ci cruits Syt.,s ov. l35,o.n9, pp .151-81621, Set. p9188 S.. ndAraos ad H.nA had,“mA ewneff iicne mtmoeylrse rsesiude tob niar yoncevret,r”EEE ZTrna. siCcuitr sySts,.v l. 3o,5 n. 11o,pp 1.44-11444,ONV .1988.Maunsrictprece ived Ferbaru 2y, 7995. I1EE ELog umbeNr 491989.0C. R.I Zo, E.u . PlItokin, nad M. .N . SwSmya,IEEE Trnsa. iCrcuti Ssys.tP-atr 1,1 vo. l1, 4n. 1o0,pp. 6 87-926 O,t. c1994.1057713-095$/040. 00 995 I1EE

E