CODES
Lie-Liang Yang and Lajos Hanzo
Dept. of ECS, University of Southampton, SO17 1BJ, UK.
Tel: +44-703-593 125, Fax: +44-703-594508
Email: lh@ecs.soton.ac.uk, http://www-mobile.ecs.so ton. ac. uk
ABSTRACT
Inthispaper residue number system(RNS)
arith-meticandredundantresiduenumbersystem(RRNS)
basedcodesaswellastheirpropertiesarereviewed.
We propose a number of applications for RRNS
codes and demonstrate how RRNS codes can be
employed in global communication systems, in
or-derto simplify the associated systems by unifying
theentire encoding and decoding procedure across
globalcommunication systems.
1. INTRODUCTION
RNS-based arithmetics exhibit a modular structure that
leads naturally to parallelism in digital hardware
imple-mentations. TheRNS has twoinherent features that are
attractiveincomparisonto conventional weighted number
systems,suchas for examplethebinaryweighted number
system representation. These two features are [1{ 4]: 1)
thecarry-freearithmeticand2)thelackofordered
signi-canceamongsttheresiduedigits. Therstpropertyimplies
thattheoperationsrelatedtothedierentresiduedigitsare
mutuallyindependentand hencethe errorsoccurring
dur-ingaddition,subtraction andmultiplicationoperations,or
dueto the noise induced by transmission and processing,
remainconnedto their originalresidue digits[1,5{7]. In
otherwords, these errors donot propagate and hence do
notcontaminateother residuedigitsduetotheabsenceof
acarryforward. Theabove-mentionedsecondpropertyof
the RNS arithmeticimplies that redundant residue digits
canbediscardedwithoutaectingtheresult,providedthat
asuÆcientlyhighdynamicrange is retainedby the
resul-tantreduced-rangeRNSsystem,inordertounambiguously
describethenonredundantinformation symbols.
As it is well known in VLSI design, usually so-called
systolicarchitectures[1] areinvokedtodivideaprocessing
taskintoseveralsimpletasksperformedbysmall,(ideally)
identical,easily designed processors. Eachprocessor
com-municatesonlywithitsnearestneighbour,simplifyingthe
interconnection designand test, while reducing signal
de-laysandhenceincreasingtheprocessingspeed. Due toits
Thisworkhasbeenperformedintheframeworkofthe
Pan-EuroepanISTprojectIST-1999-12070(TRUST),whichispartly
fundedbytheEuropeanUnion. Theauthorswouldliketo
ac-knowledge the contributions of their colleagues, although the
viewsexpressedarethoseoftheauthors.
Thenancial supportof theEPSRC,Swindon,UK isalso
gratefullyacknowledged.
carry-free property, the RNS arithmetic further simplies
the computationsby decomposinga probleminto asetof
parallel, independentresiduecomputations.
The properties of the RNS arithmetic suggest that a
redundantresiduenumbersystem(RRNS)canbeusedfor
self-checking, error-detection and error-correction in
digi-tal processors. TheRRNStechniqueprovidesauseful
ap-proachtothedesignofgeneral-purposesystems,capableof
sensing and rectifying their own processing and
transmis-sionerrors. Forexample,ifadigitalreceiverisimplemented
using theRRNShavingsuÆcient redundancy, thenerrors
in theprocessedsignals canbedetectedand corrected by
the RRNS-based decoding. Furthermore, the RRNS
ap-proach [1] is the only one, where it is possible to use the
verysamearithmeticmoduleintheverysamewayforthe
generationofboththeinformationpartandtheparitypart
ofaRRNScodeword. Moreover,duetotheinherent
prop-erties of the RNS, the residue arithmetic oers a variety
of new approaches to therealization of digital signal
pro-cessing algorithms,suchasdigitalmodulationand
demod-ulation as well as the fault-tolerant design of arithmetic
units [2{ 4,7{ 9]. It also oers new approaches to the
de-sign of error-detection and error-correction codes. Let us
rstreviewthebasicmathematicalmodelrequiredforthe
conversion of operands fromthe weighted numbersystem
to the RNS, orconversely,from theRNS to the weighted
numbersystem.
2. RESIDUENUMBER SYSTEM TRANSFORM
ANDITS INVERSE
Forconvenience,wedenetheresiduenumbersystem
trans-form(RNST)asthealgorithmthattransformsany
conven-tional weighted numbersystem, such as for example the
natural binary codeddecimal (NBCD) numbersystem to
the RNS [3,7]. The inverseRNST (IRNST) is denedas
the algorithm, whichtransforms theRNS tothe weighted
numbersystem,anissuetobeelaboratedonbelow.
Let X be a non-negative integer numberin the range
of0X <% n
,where%isabaseandXisrepresentedinthe
weightednumbersystemasX =fbn
1 bn
2
::: b2 b1b0g=
P
n 1
j=0 b
j %
j
, where b
j
2 f0;1;:::;% 1g . Specically, a
weightedbinaryrepresentationisconsidered,when%=2.
Let f m1;m2;:::;mug , the set of so-called moduli
in-volvedintheRNS,beasetofpairwiserelativeprime
num-bers,wherenotwomodulihaveacommonintegerdivisor
greaterthan1. If M = Q
u
i=1 m
i %
n
,thentheRNSTis
therangeof0X <% -asau-tupleresiduedigitsequence
(r1;r2;:::;ru),whereri=X (modmi)=X b X
m
i cmifor
i=1;2;:::;u,bzcdenotesthelargestintegernotexceeding
z,and0r
i
m
i
1. Furthermore,[0;M 1]isdened
asthedynamicrangeoftheRNSandanyintegerXinthis
rangeisuniquelyrepresentedbyau-tupleresiduesequence,
whichisexpressedas:
X,(r1;r2;:::;ru); 0X<M;
ri=X (modmi); i=1;2;:::;u: (1)
AssumingthattheintegersX1andX2haveRNS
repre-sentationsofX1=(r11;r12;:::;r1u)andX2=(r21;r22;:::;
r2u),respectively,thenX1X2,wheredenotesaddition,
subtractionormultiplication,yieldsanotheruniqueresidue
sequenceX
3
,providedthatX
3
remainswithinthedynamic
rangeof[0; M). ThearithmeticoperationsovertheRNS
canbecarriedoutonaresidue-by-residuebasis,whichcan
beexpressedas:
X1X2,[(r1ir2i)(modmi)] u
i=1
; (2)
whereonthelefttheoperationrepresentsordinary
addi-tion,subtractionor multiplicationof two integers,and on
therightitrepresentsthesameoperationsperformedonthe
basisoftheappropriateresiduedigitsr1i andr2i,with
re-specttotheircorrespondingmodulusm
i
. Thisallowsusto
convertbinaryarithmeticoperationscarriedoutwithlarge
integerstoresiduearithmeticoperationsinvolvingahigher
numberofsmallerresiduedigits,where theoperationscan
beexecutedinparallelfashionandthereisnocarryforward
between the residuedigits. Thelack ofcarry forward
be-tweentheresiduedigitspreventsthepropagationoferrors
betweenresiduedigits.
Incontrast totheRNST, theIRNSTis denedasthe
algorithmthattransformstheoperandsfromtheRNS
do-maintotheconventionalweightednumbersystem
represen-tation. A well-knownimplementationoftheIRNSTisthe
so-calledChineseremaindertheorem(CRT)[5]. According
to the CRT, for any given u-tuple (r
1 ;r
2 ;:::;r
u
), where
0ri<mifori=1;2;:::;uthereexistsoneandonlyone
integerX suchthat0X<M andri=X (modmi). It
canbeshownthat the numerical value ofX can be
com-putedby[5]:
X= u
X
i=1
riTiMi (modM); (3)
where M
i
= M=m
i
and the integers T
i
are computed a
prioribysolvingtheso-calledcongruenceT
i M
i
=1(modm
i ).
ThecoeÆcients Ti are alsooften referred to asthe
multi-plicativeinversesofM
i .
The CRTis a classical algorithm for the
implementa-tionoftheIRNST.However,thereal-timeimplementation
oftheCRTisnotpractical,sinceitrequiresmodular
oper-ationswithrespecttoalargeintegerM. Inordertoavoid
processinglarge integers,especiallyinthe contextoferror
control invokingtheRRNS, afrequentlyusedapproachis
theso-calledbaseextension(BEX)operationinconjunction
withthemixedradixconversionapproach[7].
3. REDUNDANTRESIDUENUMBER
SYSTEM ANDRRNSENCODING
If a residue number system is designed not only for the
representationofdata,butalsofortheprotection ofdata,
usuallywedesigntheRNSusingso-calledredundant
mod-uli, in order that the system has the capability of
self-checking, error-detection and error-correction[1]. Inthis
case the operand X is limited to the so-called
informa-tion dynamic range of
0; M= Q
v
i=1 mi
, where v u,
andm
1 ;m
2 ;:::;m
v
arereferredtoastheinformation
mod-uli, while mv+1;mv+2;:::,muare theso-called redundant
moduli. We express the product of the redundant
mod-uli as M
R =
Q
u v
j=1 m
v+j
. The interval[0;M) constitutes
the legitimate range of the operand X, and the interval
[M;MM
R
) is the associated so-called illegitimate range.
Any integer belonging to the legitimate range will be
la-belledaslegitimateandthosebelongingtotheillegitimate
rangeasillegitimate,sinceitdoesnotrepresentalegitimate
numberor operand,directlyaccruing fromthe
nonredun-dantinformation-bearingresiduedigits.
AnimportantpropertyoftheRRNSisthataninteger
representedbytheresiduesoftheRRNScanberecovered
byanygroupofvnumberofmoduliandtheircorresponding
residue digits. This property constitutes the basis inthe
designofRRNSbasederror-detectionanderror-correction
schemes.
A RRNShaving v numberof information moduliand
u vnumberofredundantmoduliisdenotedasaRRNS(u;v)
code. Theoperationassociatedwithgeneratingthe
redun-dant residue digits of an integer operand X can also be
viewedorinterpreted,asanencodingoperationgenerating
theparityresiduedigitsofaRRNS(u;v)code. TheRRNS
encodingoperationcanbeimplementedbasedonagroupof
codewordshavingtypicalintegervalues,byinvoking
exclu-sivelyadditionoperations. Inordertoaugmentthis
state-ment,belowweprovideanexampleshowingtheassociated
groupsof codewords fortheRRNS(7,3)codehaving
infor-mationmoduliofm
1 =4; m
2 =5; m
3
=7andredundant
moduliofm4=9; m5=11; m6=13; m7=17. The
legit-imaterangeofthiscodeis[0,139],since457=140.
Ta-ble1portraysarangeoftypicaldecimalintegermessages,
whicharebasedontheintegers`1,2,5,10, 20,50,100'
of-tenusedinmonetarysystems. Itiswellknownthatonthe
average any integer inthe legitimate range of [0,139] can
beexpressedby thesumofasubsetofthelowestpossible
numberofthespecicintegersgivenabove. InTable2we
summarised a range of binary integer numbers and their
correspondingRRNScodewords.
Example 1 Forthe encoding of the decimal integer X =
123, since X can be expressed as 123=100+20+2+1,
the RRNS codeword of X can be simplyobtainedfrom the
codewordscorrespondingtoX
7 , X
5 ,X
2
inTable1andX
1
onthebasisofmoduloaddition. Itcanbereadilyshownthat
thecodewordofX=123is(3;3;4;6;2;6;4). Similarly,for
a binary integer X =1011011, the associated codeword is
simplyconstituted bythecorrespondingmoduloadditionof
X
7 ; X
5 ; X
4 ; X
2 andX
1
of Table 2, where the subscripts
7;5;4;2;1 are simplythe indices of the binary 1 positions
in X. The associated codeword of X =1011011 is hence
messageX residuedigits residuedigits
r1 r2 r3 r4 r5 r6 r7
X
0
=0 0 0 0 0 0 0 0
X
1
=1 1 1 1 1 1 1 1
X
2
=2 2 2 2 2 2 2 2
X3=5 1 0 5 5 5 5 5
X4=10 2 0 3 1 10 10 10
X
5
=20 0 0 6 2 9 7 3
X
6
=50 2 0 1 5 6 11 16
X
7
=100 0 0 2 1 1 9 15
Table 1: RRNScodewords ofsometypicaldecimalinteger
messagesX in the RRNSwithmoduli m
1
=4; m
2 = 5,
m3 =7,m4 =9; m5 =11, m6 =13and m7 =17, where
r
i
=X (modm
i
)andM =457=140.
Binary Nonredundant Redundant
messageX residuedigits residuedigits
r
1 r
2 r
3 r
4 r
5 r
6 r
7
X0=0000000 0 0 0 0 0 0 0
X1=0000001 1 1 1 1 1 1 1
X
2
=0000010 2 2 2 2 2 2 2
X
3
=0000100 0 4 4 4 4 4 4
X4=0001000 0 3 1 8 8 8 8
X5=0010000 0 1 2 7 5 3 16
X6=0100000 0 2 3 5 10 6 15
X
7
=1000000 0 4 1 1 9 12 13
Table2: RRNScodewordsofsometypicalbinarymessages
X inthe RRNSwith moduli m1 = 4; m2 = 5, m3 = 7,
m
4
= 9; m
5
= 11, m
6
= 13 and m
7
= 17, where r
i =
X (modm
i
)andM =457=140.
4. ERROR-DETECTIONAND
ERROR-CORRECTION IN RRNS
Inthissection,webrieysummarisesomepropertiesofthe
associated error-detection and error-correctionprocedures
inthecontextoftheRRNS.Error-detection/correction
de-coding of RRNS(u;v) codes has been discussed in depth
in [3,5{7]. Let us invoke two simple examples, in order
togaininsightintotheerror-detectionanderror-correction
mechanismoftheRRNS.
Example2 Letusconsiderthemoduli3;4;5;7,where3;4
and 5 are the information moduli and 7 is the redundant
modulus. Theinformationdynamicrangeis[0;M =345)
=[0;60). Uponconsideringan integer decimalmessage of
X=21,thecorrespondingresiduevaluesareX =(0;1;1;0).
IfthereisanerrorintheRNSrepresentationdueto
trans-missionorprocessing, forexampler
3
ischangedfrom 1to
3,thenthereceivedRNSrepresentation becomes(0;1; ~
3 ;0).
Uponfollowingthegeneralapproach ofthe CRTinEq.(3)
andusingtherstthreeresiduedigitsandtheirmoduli,we
obtain:
M1=45=20; M2=35=15; M3=34=12;
T1=2; T2=3; T3=3;
X =[0220+1315+3312]mod60
=33:
However, where X = 33 (mod 7) =5 6= r4 =0, and we
canconclude thattherewereerrors intheRNS
representa-tion. Therefore,upondesigningtheRNSusing one
redun-dant modulus,theresiduedigiterrorof r3 canbedetected.
Example 3 Letusnowinvokeanadditionalredundant
mod-ulus,namely11intheaboveexample,whichresultsina
to-taloftworedundantmoduli,namely7and11intheRRNS.
Letusalso considertheinteger messageX =21,now
hav-ing corresponding residue digits of X =(0;1;1;0;10) and
that r3 is inerror andit waschanged from 1to 3,i.e the
receivedRNSrepresentationbecomes(0;1; ~
3;0;10).
Accord-ingtotheCRT'sapproach,theintegerXintherange[0;60)
canberecoveredbyinvokinganythreemoduliandtheir
cor-responding residue digits, if no errors occurred in the
re-ceivedRNSrepresentation. Letusnowconsiderallpossible
cases andattempt torecover the integer X represented by
(0;1; ~
3 ;0;10), upon retaining all possible combinations of
threeoutofve residuedigits,whichresultsin:
(r
1 ;r
2 ;r
3
)=(0;1;3) , X
123
=33(mod60);
(r1;r2;r4)=(0;1;0) , X124=21(mod84);
(r1;r2;r5)=(0;1;10) , X125=21(mod132);
(r1;r3;r4)=(0;3;0) , X134=63(mod105);
(r1;r3;r5)=(0;3;10) , X135=153(mod165);
(r
1 ;r
4 ;r
5
)=(0;0;10) , X
145
=21(mod231);
(r2;r3;r4)=(1;3;0) , X234=133(mod140);
(r
2 ;r
3 ;r
5
)=(1;3;10) , X
235
=153(mod220);
(r
2 ;r
4 ;r
5
)=(1;0;10) , X
245
=21(mod308);
(r3;r4;r5)=(3;0;10) , X345=98(mod385);
where X
ijk
represents the recovered result by using
mod-uli m
i ;m
j and m
k
as wellas their corresponding residue
digits ri;rj and rk. From these results we observe that
X
134 ;X
135 ;X
234 ;X
235 and X
345
are all illegitimate
num-bers,sincetheirvaluesareoutofthelegitimaterange[0;60).
In the remaining ve cases, except for X123, all the
re-sults are the same and equal to 21. Moreover, all these
results were recovered from three moduli without including
m
3
, i.e. from X
124 ;X
125 ;X
145 and X
245
, which are equal
to 21. Hence,we might conclude that thecorrect result is
21andthattherewasanerrorinr3,whichcanbecorrected
bycomputing^r
3
=21(mod5)=1.
RRNS codes exhibit similar coding properties to the
well-knownReed-Solomon (RS) codes [5,7]. RRNScodes
constituteaclassofmaximumminimum-distanceseparable
codes. An RRNS(u;v) code - where the information
dy-namicrangerepresentedbytheRRNS(u;v)is[0; Q
v
i=1 mi)
andtheRRNScode'sdynamicrangeis[0; Q
u
i=1 m
i )-hasa
minimumdistanceof(u v+1)andhenceitiscapableof
detecting(u v)orlessresiduedigiterrorsandcorrectup
tob(u v)=2crandomresiduedigiterrors. AnRRNS(u;v)
codeiscapable ofcorrectingtrandom residuedigiterrors
andsimultaneouslydetecting(>t)residuedigiterrors,
if and only ift+ (u v). Moreover, anRRNS(u;v)
codeiscapable ofcorrectingtrandom residuedigiterrors
andsimultaneouslycorrectingresidueerasures,provided
that 2t+(u v).
Accordingtothecarry-freepropertyandduetothelack
of weighted signicance of the residue digits in the RNS
arithmetic, in RRNScodes some of the channel-impaired
residue digitscanbediscardedasanerrorcorrection
re-ouslydecodetheresult. Theabovestatement canbe
aug-mentedasfollows. Letfm
1 ;m
2 ;:::;m
u
gbeasetofu
mod-uli ofan RRNS(u;v) code, where m1 <m2 <::: <mu.
Furthermore, let X be the integer message, which is
ex-pressedwiththeaidoftheresiduedigitsas(r
1 ;r
2 ;:::;r
u )
withrespecttotheabovemoduli. Ifthedynamicrangeof
X is [0; Q
v
i=1 m
i
), wherev u,thenX canbe recovered
fromany voutoftheunumberofresiduedigitsandtheir
relevantmoduli. Thispropertyimplies thatd(du v)
numberofresiduedigitscanbedroppedbeforetheIRNST
stage, while still recovering the message X using the
re-tained residue digits, provided that the retained residue
digitsarethecorrect,ieuncorruptedresiduedigits.
Moreover,ifweletdbethenumberofdiscardedresidue
digits,where d u v, then, aRRNS(u;v) codeis
con-vertedtoaRRNS(u d;v)afterdoutoftheuresiduedigits
arediscarded. Hence,thereducedRRNS(u d;v)codecan
detect up to (u v d) residue digit errors and correct
up to b(u v d)=2c residue digiterrors. This property
suggeststhattheRRNS(u;v)decodingcanbedesignedby
rstdiscarding d (d u v) outof the u residue digits,
whichis followed by RRNS(u d;v) decoding. Since the
discardedresiduedigitsandtheircorrespondingmodulido
nothavetobeconsideredintheRRNS(u d;v)decoding,
thedecodingprocedureisthereforesimplied.
5. APPLICATIONSOFRRNS CODES
Inadditiontothe abovementionedapplications ofRRNS
codesforself-checking,error-detectionanderror-correction,
ausefulpropertyoftheRRNSisthatanRRNScodeword
doesnotchangeitserror-detectionanderror-correction
char-acteristics after arithmetic operations, such as addition,
subtraction and multiplication [7]. This property can be
employedinordertosupportfault-tolerantsignal
process-ing. Let us invoke an example, in orderto augment this
property.
Example4 Let us consider the RRNS codes based on a
RRNSusingnonredundant moduliof m
1 =4; m
2
=5and
m3 = 7 as well as redundant moduli of m4 = 9; m5 =
11, m6 =13 and m7 = 17. Table 1 summarised a range
of typical decimalintegers and their corresponding RRNS
codewords. Letusnowconsideranoperationinthedecimal
integereld:
Y =5Y1+Y2Y3; (4)
whereY1,Y2 andY3arepossiblyfromdierentsourcesand
mayincludethetransmission andprocessingerrors.
How-ever,duetotheinherentpropertiesoftheRRNScodes,the
operations inEq.(4) can be carriedout asfollows. Firstly,
before we evaluate Eq.(4), Y1, Y2 and Y3 are rst
error-correction decoded. Since four redundant moduli are
in-cluded, up totwo residue digit errors in Y1, Y2 or Y3 can
be corrected. Secondly, Table 1 is used in order to
eval-uate Eq.(4). Thirdly, since an RRNS codeword does not
change its error-detection/correct ion capability, when the
codewordsaresubjectedtothearithmeticoperationsof
addi-tionand multiplication, theresultantRRNS codeword
cor-responding toY in Eq.(4) can be further error-correction
decoded, inorder toremove theerrors that mayhave
hap-penedintheevaluationofEq.(4). Notethatnoconventional
codes, such as Hamming codes, BCH codes and
convolu-tional codes have this property, since all these codes will
subjectedtothemultiplicationoperation.
Using theoperand valuesof Table 1, let Y
1 =X
1 =1,
Y2 = X2 = 2 and Y3 = X6 = 50. For example, the
RRNS codewordcorresponding to5is (1;0;5;5;5;5;5)
ac-cording toTable 1. It can be readily shownthat the
code-word corresponding to the operation in Eq.(4) is given by
Y =51+250=105,yielding theresidue-sequence of
(1;0;0;6;6;1;3) - provided that no more than two residue
digit errors occurred in Y1, Y2 or Y3. Moreover, even if
tworesiduedigiterrorsoccurredduringthelastoperationof
Eq.(4),andhencetheabovecodewordchangedto(1; ~
1;0;6; ~
3;
1;3), i.e., r
2 and r
5
are in error, by using RRNS
error-correction decoding, thevalue ofY =105 can still be
cor-rectlyrecovered, sincefour redundant moduliwere invoked.
Let us nowdemonstrate anotherpotential application
of the RRNS codes inthe context of the generic
commu-nication system depictedinFig.1. Inthis framework,
re-dundancyhastobeaddedfortheimplementationof
fault-tolerantcomputing(orsignal-processing),protectionof
in-formation overboth wiredand wireless channels, inorder
to achieve reliable processing and communications.
Con-ventionally, the encoding and decoding at dierent stages
hasbeentreatedseparately. Theencodedcodewordsofthe
wired channels have beenhistorically decodedinorderto
remove the related redundancy, before forwarding the
in-formationtotheencoderoftheairinterfaceofFig.1. The
informationattheairinterfaceisthenoftenre-encoded
us-ing adierentchannelcode,inordertoimplementreliable
transmissionovertheassociatedwirelesschannels. Further
additionalencoding/decodingoperationsmaytakeplaceat
other interfacesof a globalcommunications system, as
il-lustratedinFig.1.
However, with the advent of RRNS codes, the above
mentionedencoding/decoding procedures canbe
substan-tiallysimplied,sincetherequiredredundancyoftheentire
globalsystemcanbejointlydesigned. Letusinvokean
ex-ampletosupportthisargument.
Example 5 Withreference toTable1, letus assumethat
the fault-tolerant terminals use only one redundant
modu-lus forthe self-checking of theassociated arithmetic units,
employingforexampletheRRNS(4;3)code. Letusassume
furthermore thattworedundant moduliare requiredfor the
protection of information over the wired channel section
of Fig.1, employing the one-residue digit error-correcting
RRNS (5;3) code. Finally, four redundant moduli have
to be employed for the protection of information over the
low-reliabilitywirelesschannels,using thetwo-residuedigit
error-correction RRNS(7;3) code. Let m1, m2, m3, m4,
m5,m6andm7bethemoduliinvokedintherequiredRRNS,
where m
1 , m
2 andm
3
are thenonredundant moduli,while
m4,m5, m6 andm7 aretheredundant moduli.TheRRNS
codes protecting theentire system ofFig.1canbe designed
asfollows.
Step 1: The fault-tolerant terminalNo.1 uses moduli
m
1 ,m
2 ,m
3 andm
4
,inordertoformtheRRNS(4;3)
code-words,whichareexpressedas(r1; r2; r3; r4),wherer1,r2
and r3 arethe nonredundant residuedigits, while r4 isthe
redundant residue digit. Following all the associated
sig-nal processing operations tobecarried out bythe terminal
-whichmayinvolveadditions,subtractionsand
the fault-tolerant terminal forwards the RRNS(4;3)
code-wordstotheencoder of thewired channels. For simplicity
theoutputcodewordsofthefault-tolerantterminalNo.1are
still expressed as (r1; r2; r3; r4), which will also be used
throughoutourfurther discourse.
Step2:WiththeaidoftheBaseExtensionalgorithm[2],
thewiredchannel's encoder- namelyEncoder 1inFig.1
-computesanadditionalredundant residuedigitr
5
based on
(r1; r2; r3; r4)and(m1;m2;m3;m4;m5),inordertoform
theRRNS(5;3)codewords(r1; r2; r3; r4; r5). Then,these
codewordsaretransmitted overthewired channelinFig.1.
Step3: Thewiredchannel'sdecoder-namelyDecoder
1 in Fig.1 - receives the codewords (r
1 ; r
2 ; r
3 ; r
4 ; r
5 ),
whichareerror-correctiondecoded. Thecorrectedcodewords
are expressed as (r1; r2; r3; r4; r5), whichare forwarded
totheencoderof thewireless channel.
Step 4: Similarly to Step 2, with the aid of the base
extensionalgorithm[2],thewirelesschannel'sencoder
com-putestheadditionalredundantresiduedigitr
6 andr
7 ,based
on (r1;r2;r3;r4;r5) and (m1;m2;m3;m4;m5;m6;m7), in
ordertoformtheRRNS(7;3)codewords(r
1 ; r
2 ; r
3 ; r
4 ; r
5 ;
r6; r7). Then, these codewords are transmitted over the
wirelesschannelof Fig.1.
Step 5: After the wireless channel's decoder received
the codewords (r1; r2; r3; r4; r5, r6; r7), the codewords
areerror-correctiondecodedandtheredundant residue
dig-itsr6andr7areremovedfromthecorrectedcodewords. The
wireless channel decoder then passes the RRNS(5;3)
code-words to the wired channel's encoder, namely to Encoder
3in Fig.1. These RRNS(5;3) codewords are expressed as
(r1; r2; r3; r4; r5).
Step6:TheRRNS(5;3)codewordsaretransmittedover
thewiredchanneltothewiredchannel'sdecoder, namelyto
Decoder 3inFig.1.
Step 7: After thewired channel'sdecoder receivedthe
RRNS(5;3)codewords(r1; r2; r3; r4; r5),error-correction
decodingisinvoked,inordertocorrecttheresiduedigit
er-rors and then the redundant residue digit r5 is removed.
The resulting RRNS(4;3) codewords (r
1 ; r
2 ; r
3 ; r
4 ) are
thenforwardedtothefault-tolerant terminalNo.2.
Step8: Thefault-tolerantterminalNo.2nallycarries
outitstasksusingtheRNSrepresentationofitsoperandsin
theformoftheRRNS(4;3)codewords,invokesself-checking,
andnallyoutputs therecovered information.
In the upper branch of Fig.1 the wired channel's
en-coderonlyhadtocomputeoneadditionalredundantresidue
digit,incontrasttoinvokinganindependentencoder,asin
aconventional system, where eachsection of the
commu-nications links is protected independently. Similarly, the
wirelesschannel's encoderonly hadto computetwo
addi-tionalredundantresiduedigits,incontrasttocomplete
re-encodinginindependentlyprotectedconventionalsystems.
Inthe bottom branch of Fig.1, the encoder of the wired
channel,andtheencoderofthefault-tolerantterminalNo.
2-whichisnotshowninthegure-canbetotallyremoved,
while inconventional independentlyprotected systemsall
theseencoderswouldhavetobeincluded. Basedonthese
facts,wecan concludethat inconjunction with
appropri-ate joint system designusing RRNS codes, the
complex-ity of theerror-correction/detection sub-systems inglobal
telecommunicationsystems can bedecreased. Our future
workinthiseldwillbebasedondesigningsystems
invok-ingtheprinciplesproposedinthiscontribution.
Fault-tolerant
terminal
Air
interface
Wired
transmission
2
Decoder 2
Decoder 3
Encoder 3
Fault-tolerant
terminal
Wired
transmission
Air
interface
1
Encoder 1
Decoder 1
Encoder 2
channel
Wireless
Figure1: Atransmissionandcomputingsystemframework
includingfault-tolerantterminals,wiredtransmissionsand
wirelesstransmissions.
6. REFERENCES
[1] R.W.WatsonandC.W.Hastings,\Self-checked
com-putation usingresidue arithmetic," Proceedings of the
IEEE,vol.54,pp.1920{1931,December1966.
[2] W.K.Jenkinsand E.J. Altman,\Self-checking
prop-ertiesofresiduenumbererrorcheckersbasedonmixed
radix conversion," IEEE Transactions on Circuit and
Systems,vol.35,pp.159{167,February1988.
[3] L.-L.YangandL.Hanzo,\Performanceofresidue
num-ber system based DS-CDMA over multipath fading
channelsusingorthogonalsequences,"EuropeanTrans.
onTelecommunications,vol. 9,pp.525{536,November
-December1998.
[4] L.-L.YangandL.Hanzo,\Ratiostatistictestassisted
residue number system based parallel communication
systems," inProc.of IEEE VTC'99, (Houston,USA),
pp.894{898,May1999.
[5] H.Krishna,K.-Y.Lin,andJ.-D.Sun,\Acodingtheory
approachtoerrorcontrolinredundantresiduenumber
systems - Part I: theory and single error correction,"
IEEE Trans.CircuitsSyst.,vol. 39, pp.8{17, January
1992.
[6] J.-D.SunandH. Krishna,\A coding theoryapproach
toerrorcontrolinredundantresiduenumbersystems
-PartII:multipleerrordetectionandcorrection,"IEEE
Trans.Circuits Syst.,vol.39,pp.18{34,January1992.
[7] L.-L.Yang and L.Hanzo, \Coding theoryand
perfor-manceofredundantresiduenumbersystemcodes."
sub-mitted to IEEE Transactions on Information Theory,
1999.
[8] L.-L. Yang and L. Hanzo, \Residue number system
arithmeticassistedM-arymodulation,"IEEE
Commu-nications Letters,vol.3,pp.28{30,February1999.
[9] L.-L. Yang and L. Hanzo, \Residue number system
based multiple code DS-CDMA systems," in Proc. of
IEEE VTC'99, (Houston, USA), pp. 1450{1454, May
1999.
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