Table Of ContentA General Method of Applying
Error Correction to Synchronous
Digital Systems
yD. B AKMSERONG
1 pour tha preted for plying err eoretion to synchronous
ary gia ras amproe rele, 1 cet fae eceme
tf riplicton oud “vate tli" a9 spect eon We pines, he mest
pvt hr sso to operate sonny, cs tle fault de praent oF
Imetnkucece isi enfore. An eon vetonnne sont, eatukng
Tov ropes of jn iv stad agin foe miei the pte’
lange novo aby made an hy crm areca be reat
Tie erento doce recta to are the ateme decreas 08 te
comp fhe ao te wih ype orca, ed ay amount
to teplcain nf egvhnment ens for inary re estemt. The paper
Asribs sneer race oes te Spent he sean, sewsee
Covenreting cievite 1 a gerst vey, Hein how t eeinate th
ane net vst Sef torn he by pone
mien oSteinabe with stealer wninenance rine. Ine companion
ppopert De Ky Bey Chandhurs dvetoga a graceal Weary af msnsnally
rfuatant waa for Ds aplication
“This yuper deserts gener! metal of epplving eror cretion ta
syihromons gta dato ster T lade, ava special ease te wel
Sowa orhente of taipentnn ih vot rag Since the seer mpgs
fron erecting ens eels eroeting eons well a corer
ing Kiem, Hen, matinteaante prevanned can bo alerted as e00m ne a
Fant cove Ao, fe bus he pepors of ezabbng the system tn which
5 is plied to eomsimae to letion comeethy even whem falls are
cent en risneenanr is ving performed, provides all the faults
fre confina to any ne a he savers whieh eompeign the
BTS mR NRL svSCaNE sHeRIEAt SORIENKE, ARETE Ht
vem, Therofor, if faults ace found and repaired euik'y, eo that they
‘lo not secumlate to tan ine hese the remlting crocs ate. beyond
ths erruronmectng exputlites of the webeme, the stem mey be kee
in centiouous operasia for ach lengor podods tha cox the equiva-
Tent system withow err detection wad eorretin.
Tr Us gas ie tied Tatyana ae aster
with error recon ean be tte ose wad ia a8 Tae
{hal of he equcvaeot uoneeluadeal nysect pvided Toul tne re
Diva siientiy avon tee their oeorsrenes, Sieh. patently oa
Treas fy eelablty depen ol sure ow the aveilaliy wf rapid
Uisgmostie a auliceepair fuililie, Coaverely, in the alwence of
‘maintenance the mon fe of ke reduaant systema il genera fe
less than thot of the aoneduncast systems, Henpe the stem it unt
ueully aplissble to m sytem hich must operate in sn euviromuen,
‘sere opi! faut repair i iemysibio-~ in ech situtions some other
thethod of building n mlibiliy sly wy nerolewel redundaney, would
In eomparison with teptration and ste aking mur procedure will
peamit mine yee lealisation of ul. At, for lange esters it
Shonld ret is lave ever al epson: rodudaney. Por mal esters
however, an eqnipment advantage may it always be reali Sin
the triplet ree i firly well Roos, we bullae by deer
ing i bu fom a slightly dterent point of ie, whieh shows bow sh
pear a aspera case of our sreeedur.
Fig. I ehowsa aystem, a with m ampzta end wontpnts, and uvo exant
replieas of the ayater, w sad e. Convsponding out wires from 3,
soul @ ave fd *0 “taajority” ciruits, or vote takers, each of wh oh-
Ds agrees with the mnjrity, ie, ith any be ara Dee a the
Jute ich ase sn agrorment. Ths, che sytem sorte foe ren
‘hie ae soi Uo the on puts ol sy oe of nse 4, Ra
We may consider the watt vers tn aeey formation bile aud
host from and ¢ to rary chek Ite, which geuerste Hemising*
tingle erzor-eorecting cos, inthe follwing manner. The vat he
Iw splays the ons bie rom 3, 8s © in 8 mils ising
ip ss eal 007 having u enti
pat outpot olga slg
ree) so
Pot. L
Ad oO 6 (0 otonts trom syst
Box x x 2X ontpnts from system 9
Hox xX x Xe ulpute tom eps
Mlieruaively, the mattis maybe thoaht of as displaying eos,
ech with Three entaeseonsisting of one nfm hit sl Low eben
ites Far explo, the fst em stains the ‘aertiny Bey al
chk bite By aad C., Uwo pity ews une eeteted Seu he
ova bie Ay wed By say te pry eatin
AB =O,
were 2 represents tho sts di 2. Bile A al Cy slily the parity
elation
AoC =6.
“These matinee sate thot, when the complete sen: operating
comely both By and Cw hve Ue re value as Ay
Hwang I the ablity to delet any sing err iv oslo
solver el np which bit fe eos. so Ubet eorrecions ex bs
Derformed, Therefote, in prea, the seme re ce eneevtion
Sony paracn of errors which se eonfved tun ingle nea le male's.
Fie lait whieh aro confined te wie af the eynlnse A, By ot ©
fase enove on the suis af het stem aloe, this ero enecting
Scheme wil rit the nverll ssh to epetateeoereti eve whem
fn one of the rer eyateme comprising Wi Faulty, a bs dsl fur
Dison, (hi partir coding i neisient,neeause two check Bits
560 awe ana susan mucusicat suuUsA, MANE 10
ace neode foreach iagormaton bit Te Wo be opal that the use of
snore niet ones will esl les cup rduidaney, print
case ewer shove is will Ease toe prnerat. "Te pate then
45 fo ind w way of enzmizing n system pots to permit error eomeetion
wrth mare niente. A nel odoin hit wll nw be deer,
Suppose systema of ig. 1 ie designed oy ue i Bay done Sb
saber, say of electscally independent subaits, ewe eubunt eacey~
fg not rote tn p ofthe sate outpate, ae etoma ix Tig
‘Tal as hat fe eonfiel bos etait ea wa ee
errors Uh nto of tt bait Theory, eraser the Fallon
rut in whic se wate of ear subi are dp jn separate
wih yeti po. Tas a
(q— A alsinelsabunite si
igs 2 lve: provide fae sts
[0 Oe onkpe Fem soi 1
«} Ce
ros XX ss Xe outyaty from subunit © = 1
@-a4: i
mas (XX e+e Xoutymts from wal g
Since fants na sine scbunit eflce! only single row of tho rats,
‘ve may, locesanpe, apply Hamming gle eons ng ees
mt iar
RUE CORSETNON FOR AYSCHMONDEN HHGTAL AYRTEMA SRI
petoluma nase, Ths fr — 4, wb need only cre cheek bite per
foun te provide Tnutsing sngleernircurortion, and the code w=
endancy i ral se than a tke plication hemo ( instead of 3)
“Retualy, Timing coder s2e aot te us, efficent that eau he
tse dor thi trroesrteting scheme, becouse they da more than ie
cesved. Speeifecily, ey seers envstin of any stile erie
ou ofthe mais even sug these emore rete is wine
{ingle row. Il os pply the fttheraeteeton that ll ern re
fined tow raw, shen more eisionr codes aee posible aad ure died
lotr.
We: ms wish ta shone how es poe cok dow spatem into
eerily neencnt nbunis, Digital systems nay bs lai into
“ren expos thoes which percorm only combinational Togie (hax 00
fetoory), sae Fuss whieh priors seghential loge chive menus)
‘The Later expt mune intern, I iia soa bea with the former
fret We sreume Chuonghons thst the dats othe ipl wise re
ier, woud that “atin th com do oF ecass erro on the np
‘Suppose Mer that the + sobowis in Fige 2 whidh prods che
soln ons, oases ectzely of eetfeatial “ages Hi evident
Tiht he myetem eam he broken dows into such ubuaite beeease, for
fstample, each sutpel ea evel hy designing eepazats eutins
Tonal hye sieeait wih gervss ex Hie appropriate Lielean tio of
"ime we npn saiables, Alvernativels, some ravings in loge elements
maybe posible by designing lliuntnnal lg eens, eseh generat
ing oa the pest ofa sing sua
‘Fe provide he's sutpte, cational susuite are reeled, aed ane
Asya (e-F 1)tcongh gin Hig. "To design Ue, Iie ops
lorie able to express cnc ebele tou a Pecan Toctnn of the me
iron vasiablee "This car fr dene note the savotane of the e208
turn cv wikspeily tach ther ont to be the sume mageulo 2
Mss set of inermation vate, wed ince the Later ave ose
Fruntions of the pts, me «an therone expres fie ebesk ontpate
Aiea fuer of thems, Wee may of come, work wit uth
tahis insted offen! wepireentesione
I the ener of sequent log, 8 cempticatin ie inceeducal whch
smurf expsi ith the ale of Fa. Te this Sgurey w mental
‘ystom is repreeae se ronssting of to major units, Cait 1 eonsite
Seinlyo mambinetoneh Inge wt 2 ears indy of eas
+ Bt phos plac he temo denon ee
Soa eatfigsfeatstoncs teeter he ype decgmted FE ee Caton
(Gar D
‘Te combinational loge generates tw se of output:
(@) The a system outputs
() « "Teedlose" outputs which provide the inputs to the memory
it,
‘The outputs of the memory unit, in eonjunetin with them ystems
ints somite the npnte 1 the makina Tego uni
appt wit 1 desig a lertuesyiepenens subunits
In gener Uh Gn 3) inputs do wo wil Fes all rata, A fll
ina single munities ror om the cunp of that ani, an
Hes el fe bck via the aemay unit to the pate a se
of the ocher subse, Fons a few eyeles of operation it epee
hae he ougputs of all eubunts wil be in eer Beaune of a ene in
inst on suinrit, Thissinetion ean be remedied Ly soplying crm:
verti to snne ot all ef thn # feedback wire in addition to the
jeter ott wes, These ational eomvtione shuld be mad be-
‘cen the qutpnts of wait 2 aod te inputs of uns I in ore to cornet
fore raused by ota ie unit 2us well as i ust
"Auteruatvaly, tie pombe te design Uke apse wo as Uo old come
reoting the intemal feedbuck wes and yet insure aw fault acts
mae tama pet then eystem eutpes For example, instead of eal
ing down the atom ino suv ae eould replete ds apeomn +
times and liao omit 1, 2,9 Prom the Fe rp, ea
pute p+ 1, 92". 2p, ftom the eond replica ~~~ and eurpst
wa ph A p 2,200 oom the ch sop
Nivdoult thisalseaatve raistion could be acbiored without meng
+ complete replece of the spetem, However, the neceeusy deri pr
Sousa note formulaced and the malting equipmene redundanes
is dificult eatin. To onan the design procera forthe Gr
mentioned method ie stvaighsforwed, ie redundszry is easier 2 eh
‘hate amd, at lent sith preset dewives anv tecique, ie appeces to
Fost i eonsidernbly Ire ovemall redundaney. ‘heseore, ithe 1
buainde*f the paper we coll assume that th at method iso be usd
Lepnos eens HoH EYSERHUNDIS FICIAL s¥MENI 8
sig, toe comostou of aecneatia eater eauire that ni.
sf Vig. bedesgnad yer cerdent smite and tha ce ayersere
by a confonatonal Inge wal whi yeaeraten & eek oxtputs, where
Fis lenge rsh so provide the neesrary parity checks for entoetng
fof stir te cette teat all» fran fo may neque comee=
Vin), Asin Fhe pute eomsinitional vare, rach choek bit can be es
resent na tielo 2 of an sppropriste subset of chen outpuls af
fmt Laut the eatpte of writ 2, wd since Bese are known. Boul
Faetioe a the Gm 2} inser nei 1, each ebook bic output emt
Tikewisw bw expressed ae eoleafimetion of these ae inputs
Ti of court necessary tut ths ses bit lie sizes also We ds
sige vlopendene subs wit oa mere Laan outputs por sabe
it
[Before disoussing nite ores, me ah to esa ower bousds em
the nunsber of seek its, #, nero! Lo TO ont eprorcoxeeting
Tiiremats Sportal, eefecine (she mats abv for he output
itu rsinnitsa system 4, es wht! nian Yaloea ba yeu
*> per. cotesinn of every pssihis pater of erm in aay singe
rent the rows
‘chaly, tn Inver bowls ane appisabl, "be fst own whith ie
ap te arg fae ele g/> 22 beg he nu ee ee
vies per ow, emily derives fll bere hot ie me a
posible crrorpaeras in a sng ow ix (2" ~ T) af we elude the no-<e
fos pales, Psa, the toa! mia of ero patter in al gros ie
(fot = VoOiavena, ust hele amg bo for sn "pacity
‘hile patesue ue hots sre cer pater Tas rues Una sy
the insti
ye yen
That is
© 2 om ir — e+ Oh a
‘vere ihe spate Dakss estes che sl inlngor wih
Blogs 2 — 4+ 0.
“Th sop lower Boe), whic ie Mewes cass the fish when
“Fyn whieh in tora of geet practical synifeace, fe
beep 8
Tie desived br devrnsg the masizeume rarer of eae words et
BML e oma, sverme rreutest soWSAty MANA te
fon be chee ont ofa et of 2 hincry word of onus pg a which
Fan? ue specie’ comecommesing teem Me iention
road I the Append,
Surprising, it waa found act too eicul: to construct codes for
ovat valnes ofp and g in the range 2 ¢ p< 10,2 <q <9 whieh
arvievetl the appropriate ower hours
‘Suomen: tothe wor deere heve, Kay-Chaudbar® developed
ever Tomy of rsnitally edna coder for che applieston, Man
sen ihr wa lv en of plans vo esa ere anew hes pe
Sinwly desved, snes they see also minimally nundant, an sine the
ror comesting equipment requizd to implement either thera oF the
Ley-Chandbus exit ie of she eam general ehuruetr and cooler.
‘Peace fakes fee" ane ened belom in tates foes, mormon
ing to there values ofp, flows
amity a= 8 tends
Fouls 2 ST Saad 9.
Fasnl 3 = 15, ana
Peasly 8 {9 ~ 3} Pram 2 = 4)
fe er rr eT
x 0 G0 xX x G0
ee
xo 0 xX xX 9 90
ee es
x 0 0 X x a 0
4 18 Bow
x Xx x oo 0
am + sae
red a o 0
B 6 58 a8
° oO oO
18 1B an 8
a) 0 o 0
Ting onde ve oowtseta ancy Aim at al, Tepe
soa CORRECTION Pom SYRICHHONULS MaeNAT AYRES 585
“The so of gis beside mul bil position indiates the parte groupe
which ht it eucers Par example, im Say 1 he eg so) altel)
{the tna bit the tnt roe 25, cating that this il ees parity
‘heck gene 12am 3, See baer iiastetion, eho bits i fay
hae abled 3, 11, 1 12 sd 2 enter parity goon i thir em
Inula 2s oro nln Ube ate no eur, The i pons which eee
tly single gt ne «bck hit positions. They a2e denoted by X°> aa
the ifoemation its ae dented ips. The mate depled for fuaily
ion 2.5 5 malin, however, ila 2X orn 2 x 4 mates ie dered,
tae nica respectively the lat cow of the lat to owe Sins mre
Epply to fasion? nnd 8.
evral remarks sil be made at this pot, Fie seve that in
‘the orginal nts we veprernted tho cher: bie ax boing oes tee
tire iu the Ist (n-— #} nr, However, hia not poessary: te cele
Tite may app oag with informatio bites in rom ofall ows, ix
Ihe casein Ps lhvor teatro: abe. The amagemeat je dicated by
ihe etrachre ofthe eae, bu Lite may be peraated hi one roe
with nyu
‘Seon, spovible to delve information bits fron any me wih
soa dastoging the utility esi anc suse. the eed its wT
Se omitin! fn the pariy checks in which they wow normaly pac
fisoate
Fall, we observe iat she three rors alaove provide naly for
voahowed p dnd 8 EOF Ray eet me wi Tore melices
whith > 4 or ¢ > Sy tia may le dane by building up the overall
Imatig, either versely oF horsntall oe both, from several ot the
Tice matric, Ths a wide sary af equipment arrangemen’s can be
Stonmatandated, However, ss bic up settielly, em wil ster
thn te vecunslaey. Fer examples if we form 9 muti wath
Toluca 15 roms by wsng tes natioes of iy 2 al Ba
ihelugod 2 X00 — 12 ehovk ica waereas he siuhmayFagiee by
send natney
Hoge 8 x 2° — 18-4) = 7 te
A ivennson of eroe-onrerting tui i eluded here to indite
roughly the amoun of uipent iwoved fn aor woreetion, sl W>
fhm bes for ao ance etn ick e peoposed Ir,
Twa eatin’ in Shevi Ist, with the prosnt. sem, ie
necessity to apy erm earteetion ether to them xysiem enpats a
Duelycorbitonal system, or tothe (n +f») outputs ad feedback
SSP ELE SARIHOE MEEMNTOAT JEST, WAMEIL ot
ouerione ofc semen agate. To da ts, &paty ches bits must
the georated, where is determied by te tren the como
reefing ele "explnye "Thotenee, the ins tthe ron meron
arse consist of (” }8}wlsee, whove 8 9 fe fe = fora picely
omnbisatioaal sem) and ies esray the chee hte, Th oth
the eeroenreeting sisuita consists of vies whith cary hw or
ete versions of the rreeeponding Ninpte
‘We shell wan: to drtnguis between the crzeeing eines ual the
Feeute waich are eomvseed carer, lata eupring Pio
ff g subunits when generate tho (¥" |. 6) outputs: Zor example the
eieeuite iu Fie 2, Thorofare, we abl borate fer ta the g scbuate
‘asthe system,” without mnaier.
hs venir eeerlingeeals nny Le conned peform the fol
wg Tn ene
1. Ievnstruel he pity ehoeks to termine hich, i any, ase
i
2. Pram Uae patton af parity er fies, deters which of the
{27 2 ing ios are oteryingerroucis es
"Comet that sit of the N wise hiv £86 catyng er-vaen,
tise, Rrroncensebee its de acs req escesin,
Cixcite to perfor the above tas ming he elite fn sovrm! was
sad wih varying degre uf renlney I aseire web AB ose
fstreme, the errarcorortng sets eoald be oovredadat, a whieh
base an eficr: preventive tarinteneaer rexsine would be seqhied
sarnse thie shay pevommt fr ong pers without ero Alert
‘hey eval ne hl wit minolevelrdneaey, in whieh ease preventive
Sentence sk ngs eessary Ie world he epi ee fee
Thevily, Atkin aller ive wim bo take seo al of Ihe econ
ering viva el serdeinting.” Thor parle whieh we rel
‘enorcdetcting would be aubjaced to mintenanoe omly whe 9 feult
‘ran deseco, the pts ich were not e-aeon-leeering woulel rine
Drevestive wate,
Ae fourth alesitive, it gt be stamped to make the enor
snesting rents eomplitely selperomeoresting, Mewever, 2 simple
Faiete gires a be given ic viene that kei
chews thie go!
Wig. d shows a blosediugram of propos errar-eorecting creat,
designed newarding tothe tind eItermtive above. Box Tin Pi con
‘ove Fie uuile whe rere Fats tnd 2 alse. Bas 2 peni
Iceti 3 ature, att al au sevarensing ad alae funetion. For
mpl, su af wine in thie ign are reprvenc by single dived
