Parallel Algorithms for Residue Scaling and Error Correction in Residue Arithmetic

Transcription

1 Weless Engneeng Technology 8- htt://ddoog/6/wet Publshed Onlne Octobe (htt://wwwscog/ounal/wet) Paallel Algoths fo Resdue Scalng Eo Coecton n Resdue Athetc Hao-Yung Lo Tng-We Ln Deatent of Electcal Engneeng Natonal Tsng Hua Unvesty Hsnchu Cty Chnese Tae Eal: hylo@eenthuedutw Receved August d ; evsed Setebe th ; acceted Setebe th Coyght Hao-Yung Lo Tng-We Ln Ths s an oen access atcle dstbuted unde the Ceatve Coons Attbuton Lcense whch ets unestcted use dstbuton eoducton n any edu ovded the ognal wo s oely cted ABSTRACT In ths ae we esent two new algoths n esdue nube systes fo scalng eo coecton The fst algoth s the Cyclc Poety of Resdue-Dgt Dffeence (CPRDD) It s used to seed u the esdue ultle eo coecton due to ts aallel ocesses The second s called the Taget Race Dstance (TRD) It s used to seed u esdue scalng Both of these two algoths ae used wthout the need fo ed Rad Conveson (RC) o Chnese Resdue Theoe (CRT) technques whch ae te consung eque hadwae colety Futheoe the esdue scalng can be efoed n aallel fo any cobnaton of odul set ebes wthout usng loou tables Keywods: Chnese Ree Theoe (CRT); Eo Coecton; Eo Detecton; Paallel Resdue Scalng; Resdue Nube Systes (RNS); Taget Race Dstance (TRD); Taget Resdue-Dgt Dffeence Intoducton Because the esdue nube syste (RNS) oeatons on each esdue dgt ae ndeendent cay fee oety of addton between dgts they can be used n hghseed coutatons such as addton subtacton ultlcaton To ncease the elablty of these oeatons a nube of edundant odul wee added to the ognal RNS odul [RRNS] Ths wll also allow the RNS syste the caablty of eo detecton coecton The ealest wos on eo detecton coecton wee eoted by seveal authos [-] Waston Hastng [] oosed the sngle esdue dgt eo coecton Yau Lu [] suggested a odfcaton wth the table loous usng the ethod above elbau [-6] oosed coecton of the AN code Raachan [] oosed sngle esdue eo coecton Lenns Altan [8-] aled the concet of odulus oecton to desgn an eo chece Etzel Jenns [] used RRNS fo eo detecton coecton n dgtal fltes In [-6] an algoth fo scalng a esdue dgtal eo coecton based on ed ad conveson (RC) was oosed Recently Katt [] has esented a esdue athetc eo coecton schee usng a odul set wth coon factos e the odul n a RNS need not have a awse elatve e In ths study we develoed two new algoths wthout usng RD (ed-ad dgt) o CRT (Chnese eaned Theoe) fo seedng-u the scalng ocesses slfyng the eo detecton coecton n RNS The fst algoth s used fo these uoses though the esdue dgt dffeence cyclc oety (CPRDD) wthn the ange of t whee t n n n wth addtonal odul The odul n ae called the nonedundant odul; n n n ae the edundant odul The nteval s called the legtate n ange whee the nteval t s the llegtate ange whee t n t s the total ange Ths ae s oganzed as follows: Secton II wll descbe the schee the cyclc oety of esdue dgt dffeence (CPRDD) Secton III descbes the Taget Race Dstance (TRD) algoth followed by soe eales Secton IV dscusses esdue scalng eo coecton usng the TRD CPRDD algoths Fnally the concluson s gven n secton V Eo Detecton Coecton Usng Resdue Dgt Dffeence Cyclc Poety Any esdue dgt eesentaton n odul set n has ts cyclc length wth esect to ts odule nube Fo eale f the odul set s ( Coyght ScRes

2 Paallel Algoths fo Resdue Scalng Eo Coecton n Resdue Athetc ) then the cyclc lengths of any esdue dgts ae esectvely Snce these cyclc lengths ae not equal they ae vey dffcult to use as tools fo eo detecton coecton Actually thee ests the oety of coon (unfo) cyclc length n RNS between esdue dgtal-dffeences (RDD) Consde thee odul set The esdue eesentatons the coesondng dgt-dffeences ae shown n Table defned as the dffeence n value between two dgts d whee d s ae all odulo to ostve values wth esect to f the cycle length of s assgned Note that the esdue dgt-dffeences d n Table ae obtaned fo fo f f Ths dffeence of o n values ay be ostve o negatve deendng uon o o esectvely All negatve values ust be odulo to ostve values Fo eale on staed ow 8 as shown n Table the dgt dffeence n value fo s d It esults n d Fo the cyclc oety of esdue-dgt dffeence (CPRDD) n RNS we now have the followng theoe Theoe Fo a odul set n esdue eesentaton fo n n RNS thee ests a cyclc oety n dffeences between two esdue dgts d o The cyclc length can be assgned ethe to o deendng uon odulo oeaton wth esect to o Poof: Consde the case esectve to the esdue-dgt dffeence (RDD) between two dgts n X n can be n geneal eessed by the equaton d q (-) whee q q ae nteges Fo slcty we only consde the case of assue the case of can be obtaned n a sla way The elated theoe algoth ae descbed as follows ) In cycle (the ntal cycle) we have X wth q d? As Table Cyclc oety of Resdue Dgt Dffeence Decal d d * wth cycle Out of Range we have d wth s s n cycle whee eans the lagest Coyght ScRes

3 Paallel Algoths fo Resdue Scalng Eo Coecton n Resdue Athetc ntege less than o equal to Thus the RDD has s n the ntal cycle fo each odulus e n cycle d fo all ) Net consde each odulus Snce X X q then d q whee Fo RDD = (fo cycles then ) d wth s s Fo RDD = (not necessay n cycle ) d wth s s Fo RDD = d wth s s cyclc length s Thus the nube of cycles wthn Coollay Fo the above theoe we can ed- ths cyclc length fo N s N fo ately obtan that each cycle n the esdue-dgt dffeence of wll stat at locaton end at locaton N Coollay It s easly shown that thee ests nube of cycles wth esect to the cyclc length of Theoe The algoth of theoe ts coollaes can be etended to two o oe a-wse esdue- Poof Snce the esdue-dgt dffeence of dgt dffeences n eesentaton s a-wse Poof: consde a thee odul set we have two athe legtate ange of ths a-wse RDD s wse odul sets whose RDD (Resdue Dgtal Dffeence) (fo though ) Fo coollay the s d X X q s whee s agan the efeenced odule Assue also a-wse nubes Follow the sae ocedue as ste () as above ) Fo q s d thus d s s Ths shows that d has also s n cycle of The cyclc length s the nube of cycles fo s o ) Fo q s d h (a constant fo any RDD) f s h d h h h h Ths shows that the d h n any locaton has also s h n cycle of The nube of cycles fo Cobnng these thee odul nto one set we have cyclc s stll Coyght ScRes

4 Paallel Algoths fo Resdue Scalng Eo Coecton n Resdue Athetc length (fo eale ) The nube of cycles fo ae N N N 6 esectvely As shown n Table the RDD as of d d ae All as n each cycle ( ) In geneal n N wth ows n RDD n each ow Ths coletes the oof Eale - Consde a odul set X ts coesondng esdue dgts eesentaton set s The cyclc length s the nube of cycles fo ae N N 8 N esectvely Eo detecton coecton: Befoe the CPRDD algoth used fo eo detecton coecton s descbed soe basc tes n use ust be defned Defnton : Stde dstance S : It s the nceental o deceental dstance between odul n absolute value fo th cycle to th cycle Fo eale: S () Eo detecton Let the odul set be whee ae the nonedundant odul ae the edundant odul Snce the cyclc lengths of CPRDD d s ae constant t s thus easly found that the nube of cycles on tac L fo the statng ont (o othe d ) to ts taget oston In tun the dstance of RDD s can also be found Theoe The nube of cycles on tac L (colun d ) fo any statng ont (say dˆ ) to ts taget oston d can be found usng the equaton below; dˆ S d whee S the stde dstance between odul = the nube of cycles assng though fo statng ont dˆ to the destnaton d on tac L If dˆ then the nube of cycles ae equal to the total cycles fo the statng ont to ts taget oston d Poof: Snce s the nube of cycles fo to d wth esect to odule s the cyclc length thus s the total dstance fo the statng ont ˆd to ts taget oston d The eanng dstance fo on tac L n the th cycle ust d be on the sae ow of on tac L Thus RDD RDD d Once the RDD s of n ae found the eo detecton coecton fo odul can be found ust by coang the calculated cycles o RDD wth the ognal esdue eesentaton a-wse so that the eo odule can be detected The ocedue fo eo detecton by usng CPRDD algoth s suazed as follows ) Choose two ost sgnfcant (lagest) odul as the efeed odul aong the n odul say n n ) Fnd the s dstance of a cycle Sn n n n ) Fnd the dgt dffeence d n n fo X n n n n n ) Ceate the equaton of RDD d o n n n n S d RDD n n n n n n n n n n n ) Solve fo fo Equaton (-) as the n n S d n n n n n n n n (-)? ae nown The value of ust be less than o equal to 6) Fnd the coesondng RDD fo the statng ont to n ) Calculate n n n dstance fo RDD RDD chec the values of n n If these sets nubes ae equal then no eo occus; othewse eo ests We tae the sla nuecal as eale - to vefy ths algoth (CPRDD) Eale - Assue that a odul set nube X whose es- the eo detecton can be descbed as follows Let us begn ou ocedues fo the RDD d Snce S s dstance of a cycle d d due eesentaton s 6 If an eo occus at X 6 d 6 8 Then Nd S 8 Solve fo let wthn legtate ange then Coyght ScRes

5 Paallel Algoths fo Resdue Scalng Eo Coecton n Resdue Athetc The coesondng RDD fo these two ay dstances ae esectvely RDD 6 RDD 6 Thus the geneated esults of the esdue eesentaton fo RDD ae esectvely RDD 6 X 6 X Snce the calculated esults of X X ae not dentcal thee ust be eos n one of these odul We cannot detene whch one s eoneous To locate the odule whee the eo ests at least one addtonal (edundant) odule ust be used The ocedue fo eo coecton by usng CPRDD algoth s essental the sae as the eo detecton Howeve two addtonal edundant odul ust be added fo one eo coecton Note that only one edundant odulus added fo eo detecton ) Choose o as a efeed odulus ) Fnd as the sae oce- dues of eo detecton stes - ) Eane the values of If coon value ests aong then no eo occus If thee s one only one say that has no coon value wth all othe then an eo ets n odulus Ths coletes the eo coecton ocedues The followng eale s llustated hee to vefy ths algoth Eale - Eo coecton As befoe we can futhe locate coect a sngle eo by addng two edundant odul Let us use the sae eale The odul set whee ae edundant odul the esdue X eesentaton 6 If a sngle eo occus at eg X s assgned as a efeence odule then d 6 d d d Fo CPRDD algoth we can fnd the nube of cycles fo these RDD s S S S S Snce the cycle length s all above be less than 6 Thus we have 6 6 values ust If no eos occu all s ae equal e Coaed to the above esults wth awse odul only eets ths condton Thee ests no such value n Ths shows that the odule s faulty theefoe we can coect t as follows: snce the RDD cycle length Thus 6 Ths coletes the eo coecton Note that the above CPRDD s fo each esdue-dgt dffeence d can be ocessed n aallel In addton f the efeenced odule s assgned to the eoneous odule by chance eg ths algoth wll fal to locate the eo In ths case thee ae no s values that can be found to atch ths condton The way to solve the oble s of couse to assgn any othe odul eg o The hadwae desgn fo the oosed algoth n Eale - s shown n Fgue The Taget Race Dstance (TRD) Schee The conveson o decodng technque fo esdue eesentaton to X n bnay s usually accolshed usng the ed-ad dgt (RD) o Chnese eaned theoe (CRT) An otal atched aallel convete of ths nd can be seen n [] The RD s shown by the followng eesson wth weghted nubes: a a a a n n n wth n whee n s the ed-ad conveson (RC) of Otzaton can be obtaned usng ths ethod as the accessed table loou te s eactly equal to the ght addton te afte edate colun stage fo the tee netwo of the addes Coyght ScRes

6 Paallel Algoths fo Resdue Scalng Eo Coecton n Resdue Athetc ' ' ' ' ' d d ' d d d s s s ' s sp s s 6 q * 6 * 6 Coae atch ccuts * ultle adde = 6 Fgue The hadwae leentaton fo the oosed eo coecton locaton algoth can be accolshed wthout usng loou tables Howeve te s stll consued eadng a lage nube of loou tables Addtonal hadwae colety s equed by the adde-tee netwos An algoth called the taget ace dstance was wth a sle stuctue was develoed fo hgh-seed conveson TRD algoth Suose each esdue nube n the RNS X has ts own tac L the dstance ove tac L fo (statng ont) to X (end ont) though cycles can be eessed usng D Obvously the ay (no ultles of ) dstance of s D To obtan the X fo ts n esdue eesentaton of we ust fnd a taget such that tavesng the sae dstances ove tacs l l l esectvely e when the TRD dstance of each taget s eached then D D D The TRD dstance of X can be found fo the followng theoe: Theoe Consde the sle case of two odul sets Its esdue eesentaton tagets ae esectvely Let D be the ay dstance of esdue fo to on the tac L D be the ay dstance of fo to on tac L Then the TRD dstance fo these two esdues that have the sae TRD dstances can be obtaned by the followng equaton TRD (-) In addton can be calculated fo the equaton D Coyght ScRes

7 Paallel Algoths fo Resdue Scalng Eo Coecton n Resdue Athetc whee s the cyclc length of s nube of cycles all of the nteges Poof: It s easy to show that the above TRD s the coon taget dstance of Snce And X thus TRD X s the TRD dstances fo both of Coollay: It s evdent that the above theoe can be etended to n odul set n esdue nube n The coesondng TRD of n ae theefoe TRD n n n n In addton can be solved fo the followng equatons whee Note that n ae the tagets of odul n esectvely the TRD n s the dstance that has equal tac lengths e L L Ln L That s; L L L L n n Eale - Let the odul set be the esdue eesentaton be The ocedues to fnd the TRD dstance can be descbed as follows: ) Fnd the ay dstance D D of esdue snce s equed thus TRD ) Reeat the ocedue to fnd the nube of cycles the last TRD dstances (destnatons) TRD TRD Snce ˆ ˆ thus TRD TRD ˆ 6 6 thus TRD 8 TRD The fnal TRD dstance s the coon dstncton of ths syste fo tagets e TRD X Ths esult can be vefed as follows: Fgue Shows the TRD s on tacs l l l l esectvely Eo detecton coecton by TRD algoth A edundant esdue nube syste wth edundant odul wll allow detecton of any sngle eo [] Consde the odul set the coect esdue e- X 6 Let us esentaton Tac L L L L ( = ) ( = ) ( = ) ( = ) K = K = K = L L = = 6 = TRD( ) TRD( ) 8 = TRD( ) = TRD( ) = TRD( ) 8 = TRD( ) TRD( ) TRD( ) Fgue TRD s on tac L L L L Coyght ScRes

8 Paallel Algoths fo Resdue Scalng Eo Coecton n Resdue Athetc assue that s the edundant odul wth a sngle eo X F 6 esdue eesentaton The TRD theoe can be used to detect ths eo We fnd that fnal TRD fo does not fall nto the legtate ange as follows e R F TRD TRD TRD TRDTRD ˆ 6 TRD 6 ˆ 6 TRD 6 8 The fnal TRD dstance TRD 8 8 If we need to locate coect ths odule eo anothe edundant odule ust be added Let us assue that fo ths equeent n the above esdue eesentaton The cuent edundant odul set s the coect esdue eesentaton s 6 Let us assue that ae the edundant odul Wth a sngle eo F 6 The TRD theoe can agan be used to locate coect ths eo We fnd that fnal TRD s fo dose not fall n the legtate ange but othe fnal TRD s fo 6 do falls n the legtate ange: ) TRD fo TRD TRD TRD ˆ 8 TRD 6 out of legtate ange ) TRD fo TRD TRD 6 ; ˆ 8 TRD 8 ˆ TRD wthn legtate ange Thus the eo s located at odule ust be coected to Ths algoth can also be used fo ultle eo coectons Howeve at least thee edundant odul ae equed The ocedues ae sla Scalng wth Eo Coecton The above oosed algoth used fo eo detecton coecton has the advantage of not equng loou tables No CRT (Chnese esdue theoe) decodng ocesses ae equed Howeve t s stll te consung eques etensve hadwae colety fo each odule havng ultle-value nuts to the atch unt selectng a coect one as a outut To ove ths dawbac an otal atchng algoth s oosed hee fo the eo coecton The followng two theoes wll be used an eale follows Theoe Let be two elatve e nubes n RNS fo odule odule esectvely Then thee ust est the elaton eesented by the equaton whee so assung The that ae estcted to nteges Poof: As a fst ste let It s easly seen that wll be satsfed Net consde Snce thee ae two dffeent a cobnaton thus the dffeence between of wll always be satsfed fo whee s estcted n nteges Theoe 6 If the values of n the equaton ae nown then can always be detened fo equaton whee o ae wthn the ange: o Poof: Let the dffeence value of be equal to d then d wll be the nteges wthn the ange between e o These two eessons show that we can always select an ntege value wthn the nteval between o to satsfy the Coyght ScRes

9 6 Paallel Algoths fo Resdue Scalng Eo Coecton n Resdue Athetc condtons d o d Eale - Let Fnd the nu values of esectvely fo the followng equaton : Snce we have d o (-) (-) fo Equaton (-) so fo Equaton (-) so fo Ths esult can be vefed by substtutng nto the above equaton Theoe 6 s vey useful as shown n the followng eale In Theoe of Secton III the nube of cycles on tac L fo the statng ont to ts taget oston d can be eessed by settng dˆ e s d o s d (-) whee s s the odule stde dstance efeng to odule Slaly the nube of cycles on tac l fo the statng ont to ts taget oston can be eessed by settng dˆ e; s d o s d (-) Snce fo theoe the cyclc length of the esdue dgts dffeences efeence to odule s constant (unfo) then thee ust est a condton c s c s Elnatng the above tes fo Equatons (-) (-) c c c d c d D D S o (-6) S o (-) S o (-8) Elnatng fo Equaton s (-) (-6) solve fo fo (-) o 8 Chec fo Equaton (-) 6 Ths shows that the eo occus at odule Fo ths esult we can edately obtan 6 Notng that t ay haen that the assgned efeenced eoy odul falls concdentally wth eo eoy odule In ths occuence we cannot fnd the coect (nteges) values of P P wthn the legtate ange It sees that ths algoth can only detect eo To colete the eo coecton ocedue we can sly change the efeenced odule to any othe follow the sae ocedue as befoe Ths guaantees that the oosed algoth n Theoe wll also wo well n ths case The hadwae stuctue fo llustatng ths algoth s shown n Fgue The oosed TRD (taget Race Dstance) schee used fo eo coecton can be used fo scalng assgnng nubes n a esdue nube syste A edundant esdue nube syste (RRNS) s defned as befoe n an RNS wth addtonal odul The odul ae called the nonedundant odul whle the eta odul ae the edundant odul The nteval s called the legtate ange whee the nteval s the llegtate ange whee whee c c s the total ange In the D c d c d RRNS the negatve nubes wthn the dynac ange Eale - ae eesented as states at the ue etee of the total Let the odul set ange whch s at of the llegtate ange The os 6 the eo the eo occus at tve ebes ae aed to the nteval Follow the sae ocedues of the Eale - to use ths algoth f s odd o f s even The negatve S o (-) nubes ae aed to the nteval Coyght ScRes

10 Paallel Algoths fo Resdue Scalng Eo Coecton n Resdue Athetc ' ' ' ' ' - d s - d ' 6P P s d d s s' d - s' RO X s s s s *+ X= d d ' d d ' d 6 d ' d d ' Fault syndos Fgue In the bloc daga usng otal atchng between ultles P P the esdue dgts ae coected by - = d f s odd o f s even [] The one-to-one coesondence between the nteges of the dynac ange the states of the legtate ange n the RRNS can be establshed usng a olaty shft [] The olaty shft s defned as below X X fo even X fo odd whee X denotes the value X afte a olaty shft X f s odd so that X a olaty shft needs to be efoed o to coectng o scalng snce X belongs to the legtate ange If a e e s n- sngle esdue dgt eo toduced coesonds to odules then afte a olaty shft X E X we whee X e w s the ultlcatve nvese of e w e w e The odul denotes a sngle esdue dgt eo ust fall wthn the llegtate ange [] Snce can be ee- Coyght ScRes

11 8 Paallel Algoths fo Resdue Scalng Eo Coecton n Resdue Athetc sented unquely by a a a whee a s ae the coeffcent fo the Chnese Ree Theoe (CRT) e a whee a Note that the edundant dgts a a a ae zeos f no eo s ntoduced whle at least one edundant dgt s not equal to zeo f a sngle eo s nto- duced Theefoe t has the sae eanng that o a a a s used to be the entes of the eo coecton ) s an d ) Although the eos detecton coecton descbed n secton II have been slfed the ocesses due to no need of CRT conveson It s stll hadwae cole te consung fo the esdue scalng oeaton To ove ths dawbac a dect esdue-scalng algoth can be used It s fleble dect to detect event the eos The fleblty eans that the scalng facto can be abtay chosen any sngle odule such as e not necessaly begnnng fo to n ode The dect caablty eans no equeent fo CRT etenson ocesses fo decodng o loou tables The followng theoe (theoe ) eale ae clafed Theoe If the scalng facto K s one of the odule set the esdue dgts ae esectvely then the esdue dgt scaled by a facto y can be obtaned usng the equaton y (-) Poof: It s easy to show that when Equaton (-) s dvded by on both sde we have y y (-) Eale - Fo convenent coason of the oosed TRD algoth to othe schees such as aeaed n [] we tae the sae nuecal eale n [] Let the odul set 6 whee ae egula odul 6 ae edundant odul Then 6 6 X The suffcent cond- tons fo coectng sngle esdue dgts eos ae ) s o s o a The au s 6 ) { } a o The a Thus the odul set satsfes the necessay suffcent condtons fo coectng sngle eos dgt Assue X 8 a sngle dgt eo e s ntoduced then X 8 Afte a olaty shft X X Follow the sae ocedues as shown n Eale - CPRDD s aled fo coecton wthout the need fo usng a table ) Assgn the odul as the efeence odul the followng esdue dgt efeences ts coesondng CPRDD equatons: s d ae obtaned d d d d ; ; ; ; d 6 6 ) Choose two hghest dgt dffeence as one a fo equal taget ace dstance eg 6 Then the tue ay RDD equatons ae (-) And 6 (-) whee ae selected so that the two RDD ae equal dstances ) Elnatng tes n Equaton s (-) (-) by uttng 6 whee then ) Substtutng nto equatons (-) (-) esectvely we have then 6 also 6 Coyght ScRes

12 Paallel Algoths fo Resdue Scalng Eo Coecton n Resdue Athetc ) Checng othe thee RDD s The only dffeent odule esdue occus on odule nube at e The thee taget dstances can be fo any odule esdue say (ecet ) The esdue eesentaton of X s theefoe X 8 If a sngle dgt eo e s ntoduced then X 8 The coesondng eo s theefoe 6 6 X X e Afte a olaty shft 8 8 the scalng facto K to s 8 6 The fnal ste ust K use a loou table to obtan the esult [] K Fo vefyng ou oosed algoth the table of the coesondng s not equed as n [] The o- K cesses fo fndng coectng a sngle eo based on ou ethod ae descbed below ) Fnd the esdue dgt dffeence to a selected odule say as befoe 8 Fo vefyng that ou oosed algoth detects coects sngle eo wthout usng a table the sae nuecal eale s used to descbe the ocedue as follows: d Then d d d d6 6 ) Choose two hghest dgt dffeences as one a fo equal taget ace dstances eg 6 the followng two equatons can be obtaned: (-a) (-b) 6 ) Elnatng tes n (-a) (-b) by uttng 6 then ) Substtutng nto Equaton s (-a) (-b) esectvely we have then 6 also 6 X X X X 8 Obvously the eo s located at thus e X X Futheoe the CPRDD algoth can be used dectly n aallel fo esdue scalng eo coecton Thus the ocess s geatly seeded u Eale - Fo convenent coason the sae nuec eale as n [] s llustated hee Consde 6 scalng facto K If an nut X a sngle esdue dgt eo e coesondng to Then X 6 Afte a olaty shft ) Dvdng by afte subtactng fo 6 ths leads Coyght ScRes

13 Paallel Algoths fo Resdue Scalng Eo Coecton n Resdue Athetc ) Dvdng by afte subtactng fo 6 6 ; ; ; 6 6 Snce fo above only does not atch wth all othe s e 6 6 Theefoe thee occus an eo at Once ths eo s detected t s easly found coected fo the above equatons 6 whch n tun 6 X that 6 6 Dvded by ; 6; ; ; 6 ; Dvded by 6 ; ; ; The hadwae stuctue of ths eale fo the esdue scalng s shown n Fgue Actually ths algoth can be dvded by any abtay odul Eale - Dvded by any abtay odul say t ust subtact fo X Then ; ; ; ; 6 6 ; chec 8 8 Ths esults It can be seen fo above that 6 6 whch ae equal each othe as eected Eale -6 Fo ocessng two esdue scalngs eo coectons n aallel we tae Eale - as an llustaton Let scalng facto K e the fst esdue scalng facto s the second one s o vese vesa It s easly shown that the etended CPRDD algoth s used can be coleted n one cycle That s ; 8 ; ; ; The esult s dentcal X e whch ae dentcal esults as shown n Eale - Eale - Fo eo coecton 6 * Coyght ScRes

14 Paallel Algoths fo Resdue Scalng Eo Coecton n Resdue Athetc RDD d Note:select = as odule efeence RDD d _ s 6 s s RO RO RO RO s 6 select coon value 6 < * ultl e 6 6 _ X= 6 Fgue Hadwae stuctue of the esdue scalng nube fo Eale - Ths shows 6 d Fo above esults ths checs that scalng S 6 6 ; d S 6; whch s wthn the accuacy of d S ; the esdue scalng facto d6 S 8; In a geneal case ths te we ust odfy the subtacton of fo the X befoe the ocess of the scalng If s the the coect RDD 6 scalng facto then the subtacton ust change to Theefoe the e- X X whee so that o coecton s ade by d 6 o Let us cons X d X 6 6 whch coesonds to the va- de the followng eale: lue n Eale - n scalng facto (dvdng Eale -8 by at) X of odul set Coyght ScRes

15 Paallel Algoths fo Resdue Scalng Eo Coecton n Resdue Athetc The scalng facto K s assued Then esdue can be found fo Thus 8 8 Altenatvely t could be fo othe odule whee whch has the sae nube to be subtacted Fo CPRDD algoth the scalng ocesses ae efoed as befoe we then have the followng esults by scalng facto K ; P P ; P P ; P P 8 ; P P ; P P Thus X whch s eactly the value s the ost closed to Ths esult can be checed usng sequental stes as follows: Fo 6; Dvded by : ; ; ; ; Dvded by : q q ; q q ; q q ; q 8 q ; q q 6 6 Ths esult of q shows that the CPRDD algoth has the caablty of aallel ocessng oeatons n esdue scalng eo coectons e any cobnaton odul scalng factos fo Ks of odul set { } can be efoed sultaneously Conclusons The athetc oeatons n the esdue nube syste fo addton subtacton ultlcaton can be seeded u by usng ts aallel ocessng oetes Howeve soe dffcult oeatons such as eo detecton coecton ust go though conveson o decodng ocesses fo the esdue eesentaton to the egonal bnay nube Ths s because the decodng technque s usually accolshed usng the ed-ad dgt (RD) o Chnese Reaned Theoe (CRT) whch ae te consung ocesses equng hadwae colety We oosed two algoths fo scalng eo coecton wthout the need fo loou tables o nceasng the encodng ocess The Cyclc oety of the Resdue-Dgt Dffeence (CPRDD) algoth can detect coect eos fo the RNS cyclc oety Any esdue odul set has a secfc cycle length whch can be obtaned fo the ndvdual esdue nube dffeence each a to a efeence eoy odule Once the cyclc length s nown then the ognal value s easly found n tun the eos can be detected coected The TRD (Taget Race Dstance) algoth cobned wth CPRDD s used fo scalng fo eo detecton coecton The scalng esults eo coecton can be dectly efoed by these two algoths wthout usng RD o CRT Thus the decodng ocess s sgnfcantly educed the hadwae stuctue s geatly slfed Seveal eales ae llustated vefed fo these two algoths REFERENCES [] R W Watson Eo Detecton Coecton Othe Resdue-Inteactng Oeatons n a Redundant Resdue Nube Syste Unvesty of Calfona Beeley Coyght ScRes

16 Paallel Algoths fo Resdue Scalng Eo Coecton n Resdue Athetc 6 [] R W Watson C W Hastngs Self-Checed Coutaton Usng Resdue Athetc Poceedngs of the IEEE Vol No 66 - htt://ddoog//proc66 [] S S S Yau Y C Lu Eo Coecton n Redundant Resdue Nube Systes IEEE Tansactons on Coutes Vol C- No - htt://ddoog//t-c [] D elbau Eo Coecton n Resdue Athetc IEEE Tansactons on Coutes Vol C- No 6 8- [] F Bas P aestn Eo Coectng Poetes of Redundant Resdue Nube Systes IEEE Tansactons on Coutes Vol No - htt://ddoog//t-c [6] F Bas P aestn Eo Detecton Coecton by Poduct Codes n Resdue Nube Systes IEEE Tansactons on Coutes Vol No - htt://ddoog//t-c [] V Raachan Sngle Resdue Eo Coecton n Resdue Nube Systes IEEE Tansactons on Coutes Vol C- No 8 - htt://ddoog//tc8666 [8] W K Lenns E J Altan Self-Checng Poetes of Resdue Nube Eo Checes Based on ed Rad Conveson IEEE Tansactons on Ccuts Systes Vol No 88-6 htt://ddoog// [] W K Lenns Resdue Nube Syste Eo Checng Usng Eed Poecton Electoncs Lettes Vol 8 No 8-8 htt://ddoog//el:86 [] W K Lenns The Desgn of Eo Checes fo Self- Checng Resdue Nube Athetc IEEE Tansactons on Coutes Vol C- No htt://ddoog//tc866 [] H Etzel W K Jenns Redundant Resdue Nube Systes fo Eo Detecton Coecton n Dgtal Fltes IEEE Tansactons on Acoustcs Seech Sgnal Pocessng Vol 8 No [] C C Su H Y Lo An Algoth fo Scalng Sngle Resdue Eo Coecton n Resdue Nube Systes IEEE Tansactons on Coutes Vol No 8-6 htt://ddoog// [] H Y Lo An Otal atched Paallel ed- Rad Convete Jounal of Infoaton Scence Engneeng Vol - [] A P Shenoy R Kuaesan Fast Base Etenson Usng a Redundant odus n RNS IEEE Tansactons on Coutes Vol 8 No 8-6 htt://ddoog//68 [] E D Dclauds G Ol F Pazza A Systolc Redundant Resdue Athetc Eo Coecton Ccut IEEE Tansactons on Coutes Vol No - htt://ddoog//68 [6] S S Wang Y Shau Sngle Resdue Eo coecton Based on K-Te -Poecton IEEE Tansactons on Coutes Vol No - htt://ddoog//68 [] R S Katt A New Resdue Athetc Eo Coecton Schee IEEE Tansactons on Coutes Vol No 6 - htt://ddoog//88 Coyght ScRes