ovsf
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ieee paperTRANSCRIPT
Nonblocking OVSF Codes for 3G Wireless and Beyond Systems
HasanCamComputerScienceandEngineeringDepartment
ArizonaStateUniversityTempe,AZ 85287
AbstractOrthogonal variable spreadingfactor (OVSF) codesareemployedin thethird generation (3G)widebandcodedivi-sionmultiple access(WCDMA) wirelesssystemaschan-nelizationcodes. Any two OVSF codesof different treelevelsareorthogonal if andonly if oneof two codesis notmothercodeof theother. This resultsin a majordrawbackof OVSF codes, calledblocking property: whenanOVSFcodeis assigned,it blocksall of its ancestoranddescen-dantcodesfrom assignmentbecausethey arenot orthogo-nal to eachother. This paperpresentsnonblocking OVSF(NOVSF) codes thatareorthogonalto eachother. There-fore, no NOVSF codeblockstheassignment of any otherNOVSF code.
1 IntroductionThe third generation (3G) wireless standards
UMTS/IMT-2000 use the wideband CDMA (WCDMA)to support high data rate and variable bit rate serviceswith different quality of service(QoS) requirements. InWCDMA, all userssharethesamecarrierunderthedirectsequence CDMA (DS-CDMA) principle [5]. In the3GPPspecifications[6], orthogonal variable spreading factor(OVSF) codes [4] are usedas channelization codesfordataspreading on bothdownlink anduplink. OVSF codesalsodetermine the dataratesallocatedto calls. BecauseOVSF codes require a single RAKE combiner at thereceiver, they are preferable to multiples of orthogonalconstant spreading factor codes which need multipleRAKE combinersat thereceiver.
In OVSF, oncea particularcode is used,its descen-dantandancestorcodescannot beusedsimultaneouslybe-causetheir encoded sequencesbecomeindistinguishable.Thus,the OVSF codetreehasa limited number of avail-ablecodes.BecauseoneOVSF code tree,alongwith onescrambling code,is usedfor transmissionsfrom a singlesourcethat may be a basestationor mobile station, the
sameOVSF codetreeis usedfor the downlink transmis-sionsandtherefore the basestationmustcarefullyassigntheOVSFcodesto thedownlink transmissions.Theuplinktransmissions do not suffer from this limitation dueto thateachmobilestationasasinglesourceusesauniquescram-bling codewith thespreading codesof its OVSFcode tree,which makessignalsfrom different mobile stationssepa-rablefrom eachother.
All OVSF codescanbe generatedrecursively usingabinary complete tree. OVSF codesarevery valuable re-sourcesin a CDMA-basedwirelesssystemand, therefore,should bemanagedveryefficiently. Any two OVSF codesof different treelevelsareorthogonalif andonly if oneoftwo codesis not mother codeof theother. This resultsina major drawback of OVSF codes,calledblocking prop-erty: whenan OVSF codeis assigned,it blocks all of itsancestoranddescendant codesfrom assignmentbecausethey are not orthogonal. To alleviate the effects of theblocking property of OVSF codes,various schemeswereproposedsuchascode reassignmentschemes[7, 8, 9, 10],time sharingof channels,statisticalmultiplexing of burstydatatraffic [11]. To thebestof ourknowledge,noimprove-mentotherthantheimprovement of FOSSILcodes[12] inonebranch of the code-treewasmadeon the orthogonal-ity propertiesof OVSF codes. FOSSILcodessatisfy theorthogonality propertiesof OVSF codesplusanadditionalpropertythatthosecodesof only onebranch of theOVSFcode-treeareorthogonalto eachother.
Thispaper presentsnonblockingOVSF(NOVSF)codessuchthat they do not have any blocking at all. Hence,allNOVSF codesareorthogonalto eachotherand,therefore,canbe assignedsimultaneously asfar asorthogonality isconcerned.In this paper, four differenttechniquesaredis-cussedto obtain the proposedNOVSF codes. Two tech-niquesareinvolvedwith therearrangement of OVSF codetreesas follows. Initially, thereare X orthogonal codeswith the spreadingfactorof X , whereX is eitherfour or
eight. Eachof theseX orthogonalcodesmay first gener-ateY orthogonalcodeswith spreading factorof Y andthenthe generatedcodesmay be placedon a layer of NOVSFcodetree,whereY is a power of two. WhenX is equalto four, thecodetreehasfour layerswhosespreading fac-tors range from 4 to 32, which could be a very desirablecodetreeespeciallyfor broadbandfixedwirelessnetworkswherethe highestspreading factor is desirablenot to ex-ceed32. The third technique proposesa ratherdifferentapproach,that is, it proposesto sharethe initial X OVSFcodesin time whenever it is predictedto result in betterspectralefficiency and throughput. Thus, this techniqueis involvedwith time multiplexing andOVSF codes.Thefourth techniqueintroducesaverystructuredwayof gener-atingNOVSFcodesstartingwith spreading factorof 4 andending with any spreadingfactor. Thetree-structuredgen-erationof NOVSFcodesin this techniqueis accomplishedin two stages.In thefirst stage,baseOVSF codesaregen-eratedandthenNOVSF codesaregeneratedin thesecondstage.
Theremainderof this paperis organizedasfollows. InSection2, we review the tree-structured generation andblocking properties of OVSF codes. Section3 describestheproposedNOVSFcodes.Concluding remarksaremadein Section4.
2 OVSF Code Generation and BlockingProperty
Let Cn � i denote the ith memberof the sethaving n bi-naryspreading codesof n-chip length,for n � 2k andk is apositive integer. Cn � i canbeusedto generatetwo orthogo-nal binary codesequencesof length2n, C2n � i � �
Cn � i � Cn � i �andC2n � j � �Cn � i ��� Cn � i � , where � Cn � i denotestheinvertedsequence (or binary complement)of Cn � i [4].
Recursive generation of higher-dimensional OVSFcodesfrom lower-dimensional OVSFcodescanalsobede-pictedusingacode-treestructure[4] asshown in Figure1.All higher layercodesspannedfrom alowerlayercodearedefinedasdescendant codes.All low layercodeslinking aparticular codeto therootcodearecalledits mothercodes.Two sibling codesare thosegenerated from their imme-diatemothercode. All codesin eachlayer aremutuallyorthogonal.Any two codesof different layersareorthogo-nalexcept for thecasethatoneof thetwo codesis amotheror ancestorof anothercode. Thisexception in OVSFcodesleadsto theblocking property thatoncea particular OVSFcodeis assigned,it blocks theassignment of its all descen-dantor ancestorcodesbecausethey arenot orthogonaltoeachotherand,therefore,their encodedsequencesarein-distinguishable. For instance,whencodeC4 � 1 is assignedas shown in Figure1, all of its ancestorand descendant
codes areblocked.
2,2C = (1, −1)
2,1C = (1, 1)
4,2C = (1,1,−1,−1)
4,3C = (1,−1,1,−1)
4,4C = (1,−1,−1,1)
4,1C = (1,1,1,1)
8,1C = (1, 1, 1, 1, 1, 1, 1, 1)
8,2C = (1,1,1,1,−1,−1,−1,−1)
1,1C = (1)
SF = 8SF = 4SF = 2SF = 1
layer 1 layer 2 layer 3 layer 4
assigned code
Figure1: OVSF codeblocking.
3 Nonblocking OVSF CodesThebasicideasbehindtheproposedNOVSF codesare
discussednext by describing four differentcases.In everycase,all thecodesareorthogonal to eachother. They dif-fer from eachother in the rangeof SF andwhethertimemultiplexing is applied.
Technique 1: NOVSF codes for 4 � SF � 32.In this case,asshown in Figure2, thereareinitially fourorthogonal codes,namely, A, B, C, and D. Using thesefour orthogonal codes,a binarycodetreeis constructedasfollows. CodeA is madethe root codewith SF � 4 inthe layer1 of the tree. For the treelayer2, the followingtwo orthogonalcodeswith SF � 8 aregeneratedfromcodeB: � B � B and � B �� B . Similarly, four codesaregeneratedfrom codeC andareplacedon layer3 of thetree.Finally,eightgeneratedcodesfrom D areplacedon layer4 of thetree.All thecodesof thetreeareorthogonal to eachotherand,they canbevery desirablecodes for broadbandfixedwirelessnetworks wheremaximum SF shouldnot exceed32. Indeed,eachoneof four codesA, B, C, andD canhaveany spreading factordependingontherequesteddatarates.
Technique 2: NOVSF codes for 8 � SF � 512.In this case,asshown in Figure3, thereareinitially eightorthogonal codes,namely, A, B, C, D, E, F, G, and H.Using thefirst sevenorthogonalcodes,a binary codetreeis constructedas follows. CodeA is madethe root codewith SF � 8 in the layer 1 of the tree. For the treelayer
4,1C = (A)
8,1C = (B,B)
8,2C = (B,−B)
16,1C = (C,C,C,C)
16,2C = (C,C,−C,−C)
16,4C = (C,−C,−C,C)
16,3C = (C,−C,C,−C)
32,2C = (D,D,D,D,−D,−D,−D,−D)
32,1C = (D,D,D,D,D,D,D,D)
32,3C = (D,D,−D,−D,D,D,−D,−D)
32,4C = (D,D,−D,−D,−D,−D,D,D)
32,5C = (D,−D,D,−D,D,−D,D,−D)
32,6C = (D,−D,D,−D,D,−D,D,−D)
32,7C = (D,−D,−D,D,D,−D,−D,D)
32,8C = (D,−D,−D,D,−D,D,D,−D)
SF = 8
D =(1,−1,−1,1)
C =(1,−1,1,−1)
B =(1,1,−1,−1)
A =( 1, 1, 1, 1)
Orthogonal Codes: A, B, C, D
SF = 32SF = 16SF = 4
Figure2: Thebinarycodetreefor NOVSFcodeswith 4 �SF � 32.
2, the following two orthogonal codeswith SF � 16 aregeneratedfrom codeB: � B � B and � B �� B . Similarly, fourcodesaregenerated from codeC andareplacedon layer3 of the tree. As illustratedin Figure3, codesD, E, F ,andG generate8, 16, 32, and64 codes,respectively, andareplacedon layers4, 5, 6, and7, respectively. CodeHcanbeusedasa standbycodein any treelayerwhenevermorecodesareneeded. Indeed,eachoneof theeightcodesA, B, C, D, E, F , G, andH. canhave any spreading factordependingontherequesteddatarates.For instance,if thereareeight usersrequestingcodes with SF � 8, theneachlayeris assumedto beassignedacodewith SF � 8.
Technique 3: NOVSF codes with SF � 4 and time multi-plexing.In this case,asshown in Figure4 thereare initially fourorthogonal codes of SF � 4, namely, A, B, C, and D.Eachcodeis associatedwith atime-slotnumberandcycle-length,in addition to the SF of the code. Cycle-lengthissimply thesumof the time slotsin a cycle. Thetime-slotnumber is thelabelof thetimeslot in acycle.Whenacodeis notsharedin time,its cycle-lengthbecomesequaltoone.Thus,acodeis assignedto acommunicationchannel alongwith its time-slotandcycle-length.
Therearemainlytwo reasonswhyacodemaybesharedin time. Onereasonis to have betterutilization of codes,which leadsto an improvement in spectralefficiency ofWCDMA. Anotherreasonto shareacodein timeis to helpratematching techniquessuchasrepetitionor puncturingto achieve therequesteddatarates.Notethatrepetitionorpuncturingis usedtoadjustthechannel-codingrateof each
8,1C = (A)
16,1C = (B,B)
16,2C = (B,−B)
32,3C = (C,−C,C,−C)
32,4C = (C,−C,−C,C)
64,2C = (D,D,D,D,−D,−D,−D,−D)
128,2C = (E,E,...,−E,−E)
256,2C = (F,F,...,−F,−F)
32,2C = (C,C,−C,−C)
32,1C = (C,C,C,C)
512,2C = (G,G,...,−G,−G)
SF = 8 SF = 16 SF = 32 SF = 64 SF = 128 SF = 256 SF = 512
layer 3 layer 4 layer 5 layer 7layer 6layer 2layer 1
2 codes 4 codes 8 codes 16 codes 32 codes 64 codes
H =( 1,−1, 1,−1,−1,1,−1,1)
G =( 1,−1, 1,−1,1,−1,1,−1)
F =( 1,−1, 1,−1,−1,1,−1,1)
E =( 1,−1, 1,−1,−1,1,−1,1)
D =( 1, 1,−1,−1,−1,−1,1,1)
C =( 1, 1,−1,−1,1,1,−1,−1)
B =( 1, 1, 1, 1,−1,−1,−1,−1)
A =( 1, 1, 1, 1, 1, 1, 1, 1)
Orthogonal Codes: A, B, C, D, E, F, G, H
1 code
Figure3: Thebinary code treefor NOVSF codeswith 8 �SF � 512. Only oneNOVSF codeis illustratedin layers 4to 7 dueto spacelimitations.
A A A A
A A A A
Orthogonal Codes: A =( 1, 1, 1, 1) B =( 1, 1,−1,−1) C =( 1,−1, 1,−1) D =( 1,−1, 1,−1)
(B,−B)
(B,B) (B,B) (B,B) (B,B) (B,B) (B,B) (B,B) (B,B)
(B,−B) (B,−B) (B,−B) (B,−B) (B,−B) (B,−B) (B,−B)
C
C
C
C
C
C
C
C
D D D D D D D D
channel 2
channel 1
channel 3
channel 4
channel 5
channel 6
channel 7
channel 8
Code A, cycle−length=2, channel 1= slot 1, channel 2= slot 2
Code B, cycle−length=1, channel 3= slot 1, channel 4= slot 1
Code C, cycle−length=3, channel 5= slot 1, channel 6= slot 2, channel 7= slot 3
Code D, cycle−length=1, channel 8= slot 1
Figure4: TheNOVSF codeswith timemultiplexing.
transport channel to matchthe codedbit ratesto oneof alimited setof rateson thephysicalchannel.
In Figure4, code A with SF � 8 is sharedby two com-munication channels suchthat channels 1 and 2 employcodeA in time slots1 and2, respectively. No time multi-plexing is appliedfor code B. Codes� B � B and � B ��� B ofSF � 8 eachthat aregeneratedfrom codeB areassignedto channels 3 and4, respectively. Similary, codeD is notsharedin time either. But, codeC is sharedby threechan-nels.Sincethenumberof channelsthatshareC in differenttime slotsis equalto three,it maybeeasierto support dif-ferentdatarates.
Technique 4: NOVSF codes with SF � 4.In thiscase,weshow how to generateNOVSFcodeswhenthereis no limit on theupper bound for SF. We first defineBOVSF codesandthenNOVSF codes.BOVSF codes: 1) Let the codesA � [1, 1] andB � [1, -1] be two initial BOVSF codes.2) Usea BOVSF codeXof lengthk to generatetwo orthogonalcodes of length4kandlength2k:
�X � X � X � X � and
�X � � X � , where � X
is the invertedsequenceof X . Usingthis procedurerecur-sively, generateall BOVSFcodesthatcanberepresentedasnodes (otherthantheroot node) in a balancedbinarytree.BOVSFcodeshavethesamepropertyasOVSFcodes,thatis, i all BOVSF codesof the samelvel of BOVSF code-treeareorthogonalto eachother, andii any two codesofdifferent layersareorthogonal exceptfor thecasethatoneof thetwo codesis a parentcodeof theother.NOVSF codes: For a given BOVSF codeY of lengthk,generatethefollowingso-calledNOVSFcodeof length4k:�Y � Y ��� Y ��� Y � , where � Y is the invertedsequenceof
Y . By repeatingthis procedurefor eachandevery BOVSFcode,generateall NOVSFcodesthatcanberepresentedasnodes (otherthantheroot node) in a balancedbinarytree.NOVSFandBOVSFcodesarerepresentedbyCs
n � i andBsn � i,
respectively, wheres, n, and i standfor spreading factor,numberof codeswith thesamespreadingfactor, andindexof code,respectively.
Therecursivegenerationof BOVSF andNOVSF codesareshown by codetreesasin Figures 5 and6.
It is statedearlierthat the tree-structuredgenerationofBOVSF codesis very similar to that of OVSF codes.Toconsolidate their similarity, the following lemma showsthatBOVSF codeshave thesameorthogonality propertiesas OVSF codes. Then, the next theoremproves that allNOVSF codes areorthogonal to eachother.
Lemma 3.1 BOVSF codes have the same property asOVSF codes, that is, i all BOVSF codes of the same levelof BOVSF code-tree are orthogonal to each other, and ii
4
4,4B = (C,−C)=(G)
8
4,2B = (B,−B)=(E)1
1,1B = (1) = (A)
16
4,1B = (B,B,B,B)=(D)
8
4,3B = (C,C,C,C)=(F)
64
8,1B = (D,D,D,D)=(H)
32
8,2B = (D,−D)=(I)
32
8,3B = (E,E,E,E)=(J)
16
8,4B = (E,−E)=(K)
32
8,5B = (F,F,F,F)=(L)
16
8,6B = (F,−F)=(M)
4
2,1B = (A,A,A,A) = (B)
2
2,2B = (A,−A) = (C)
16
8,7B = (G,G,G,G)=(N)
8
8,8B = (G,−G)=(O)
SF = 16,32,64
Root
SF = 2,4 SF = 4,8,16
Figure 5: BOVSF code tree.
4
1,1C = (A,A,−A,−A)
16
2,1C = (B,B,−B,−B)
8
2,2C = (C,C,−C,−C)
64
4,1C = (D,D,−D,−D)
32
4,2C = (E,E,−E,−E)
32
4,3C = (F,F.−F,−F)
16
4,4C = (G,G,−G,−G)
256
8,1C = (H,H,−H,−H)
128
8,2C = (I,I,−I,−I)
128
8,3C = (J,J,−J,−J)
64
8,4C = (K,K,−K,−K)
128
8,5C = (L,L,−L,−L)
64
8,6C = (M,M,−M,−M)
64
8,7C = (N,N,−N,−N)
32
8,8C = (O,O,−O,−O)
SF = 32,64,128,256SF = 8,16 SF = 16,32,64SF = 4
Figure6: Codetreeof NOVSF codes.
any two codes of different layers are orthogonal except forthe case that one of the two codes is a parent code of theother.
Proof. It follows from the definition of BOVSF codesthat a BOVSF codeBs
n � i generatestwo orthogonal codesB4s
2n � i andB4s2n � i � n.
B4s2n � i � �
Bsn � i � Bs
n � i � Bsn � i � Bs
n � i �and
B4s2n � i � n
� �Bs
n � i ��� Bsn � i � Bs
n � i ��� Bsn � i ���
NotethatB4s2n � i andB4s
2n � i � n areorthogonalto eachotherbe-causetheir innerproduct is is equalto
�Bs
n � i � � �Bsn � i �����Bs
n � i � � �Bsn � i � � 0 �
Notethat thefirst half of B4s2n � i is orthogonalto thefirst
half of B4s2n � i � n, andsimilarly thesecondhalf of B4s
2n � i is or-
thogonal to thesecondhalf of B4s2n � i � n. Also, notethat the
first half of B4s2n � i is thesameasits secondhalf, andsimi-
larly thefirst half of B4s2n � i � n is thesameasits secondhalf.
Furthermore,if thecodegenerationmethodof OVSFcodeswereimplementedin thegenerationof BOVSFcodes,onlythe first (or second) halves of B4s
2n � i andB4s2n � i � n would be
generated.Also, theinitial BOVSFcodesB22 � 1 � � 1 � 1� and
B22 � 2 � �
1 ��� 1� arethesameasthe two childrencodesoftheOVSF rootcode.Therefore,it maybethought thatthematrix of BOVSF codesat eachlayer of the code-tree isobtained by concatenatingtwo Walsh matricesof OVSFcodes.Therefore,BOVSF codeshave sameorthogonalitypropertiesasOVSFcodes. �Theorem 3.1 Any NOVSF code is orthogonal to all otherNOVSF codes.
Proof. Wenow show thatany NOVSFcode,sayC4sn � i, is
orthogonalto its all descendant NOVSF codes.Note thatthetwo childrenof C4s
n � i are
C16s2n � i � �
B4s2n � i � � B4s
2n � i ��� B4s2n � i ��� B4s
2n � i �and
C16s2n � i � n
� �B4s
2n � i � n � � B4s2n � i � n ��� B4s
2n � i � n ��� B4s2n � i � n ���
Note thatC16s2n � i consistsof two copiesof B4s
2n � i followedbythe two copiesof its inverted sequences. Due to the factthat,if a codeα is orthogonalto anothercodeβ, α is alsoorthogonalto the invertedsequence of β (andvice versa),to show thatC4s
n � i is orthogonal to C16s2n � i it sufficesto show
thatC4sn � i is orthogonalto B4s
2n � i. Furthermore,becauseof the
generationmethod of BOVSFandNOVSFcodes,notethatall descendant NOVSF codesof C16s
2n � i alsoconsistof mul-
tiple non-invertedandinvertedcopies of B4s2n � i, in order to
show thatC4sn � i is orthogonal to its all descendant NOVSF
codes includingits child codeC16s2n � i it sufficesto show that
C4sn � i is orthogonal to B4s
2n � i. Similarly, in order to show that
C4sn � i is orthogonal to its all descendant NOVSF codes in-
cluding its child C16s2n � i � n it sufficesto show thatC4s
n � i is or-
thogonal to B4s2n � i � n. Hence,it will beshown next thatC4s
n � iis orthogonal to bothB4s
2n � i andB4s2n � i � n.
We shallfirst prove thatC4sn � i is orthogonalto bothB4s
2n � i.Let usfirst consider
C4sn � i � �
Bsn � i � Bs
n � i ��� Bsn � i ��� Bs
n � i �and
B4s2n � i � �
Bsn � i � Bs
n � i � Bsn � i � Bs
n � i ���Notethattheinnerproductof C4s
n � i andB4s2n � i is equal to
�Bs
n � i �����Bsn � i � � �Bs
n � i � � �Bsn � i � � 0
and,therefore, they areorthogonal. Similarly, we consider
C4sn � i � �
Bsn � i � Bs
n � i ��� Bsn � i ��� Bs
n � i �and
B4s2n � i � n
� �Bs
n � i ��� Bsn � i � Bs
n � i ��� Bsn � i ���
Notethattheinnerproductof C4sn � i andB4s
2n � i � n is alsoequalto �
Bsn � i � � �Bs
n � i �����Bsn � i � � �Bs
n � i � � 0
and, therefore, they are orthogonal, as well. Thus, C 4sn � i
is orthogonal to both B4s2n � i andB4s
2n � j. It follows that anyNOVSF codeis orthogonal to its all descendant NOVSFcodes. This implies all the NOVSF codesof the subtreewith topmost parentnode C8
2 � 1 (i.e., the right child of theroot node) areorthogonalto eachother. Similarly, all theNOVSFcodesof thesubtreewith topmostparentnodeC 8
2 � 2(i.e., the left child of theroot node)arealsoorthogonaltoeachother. Moreover, notethat the two childrenC 8
2 � 1 and
C82 � 2 of theroot nodearealsoorthogonalto eachotherbe-
causeB22 � 1 � �
1 � 1� andB22 � 2 � �
1 ��� 1� areorthogonaltoeachother. Therefore,all NOVSF codesareorthogonaltoeachother. �
When the root codeis usedin OVSF codes,no otherOVSFcodecanbeemployed becauseof theblockingprop-erty of OVSF codes. In addition, the root codehasthehighestdatarate.Therefore,whenOVSFcodesareusedintheforward link (or downlink) of 3G DS-CDMA systems,theaggregatedatarateof forward link channelsis at most
equalto the root coderate. (Indeed,the root codeis notusedin 3G DS-CDMA sincethe spreading factorshouldbe at least4). However, if NOVSF codes areusedin 3Gwirelessandbeyond systems,the aggregate data rate offorward link canbe the summationof all NOVSF codes’ratesbecauseall NOVSF codes canbeemployedsimulta-neously.
4 ConclusionWe have introduced nonblocking OVSF (NOVSF)
codesin the sensethat all codesare orthogonal to eachother and no code blocks the assignmentof any otherNOVSFcode.An immediateconsequenceof thispropertyis that the aggregatedatarateof all NOVSF codesis thesummationof all NOVSF codes’rates,asopposedto theOVSF property that the aggregatedaterate of all OVSFcodescanbeat mosttherateof rootcode.
AcknowledgmentTheauthor would like to thankHalim Yanikomeroglu andErtanOzturkfor having constructivediscussions.
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