Table Of ContentOn Four-group ML Decodable Distributed Space
Time Codes for Cooperative Communication
G. Susinder Rajan, Anshoo Tandon and B. Sundar Rajan
Department of Electrical Communication Engineering
Indian Institute of Science, Bangalore, India
Email: susinder, anshoo, bsrajan @ece.iisc.ernet.in
{ }
7
0 Abstract—Aconstructionof anewfamilyofdistributedspace different from the traditional Space-time codes for colocated
0 time codes (DSTCs) having full diversity and low Maximum MIMO systems, which are designed without respecting such
2 Likelihood (ML) decoding complexity is provided for the two constraints.
phase based cooperative diversity protocols of Jing-Hassibi and
n The ML decoding of a STBC in K complex variables
the recently proposed Generalized Non-orthogonal Amplify and
a x ,x , ,x is, in general, joint decoding of all the K
Forward (GNAF) protocol of Rajan et al. The salient feature of 1 2 K
J ···
the proposed DSTCs is that they satisfy the extra constraints variables. However, for the Alamouti code K = 2 and the
9 imposedbytheprotocolsandarealsofour-group MLdecodable variables x and x can be decoded independently for ML
1 2
which leads to significant reduction in ML decoding complexity decoding.In general,if K =gλ and the variablescan be par-
] compared to all existing DSTC constructions. Moreover these
T titioned into g subsets each containing λ number of variables
codes have uniform distribution of power among the relays as
I wellasintime.Also,simulationsresultsindicatethatthesecodes andtheMLdecodingcanbedoneforthevariablesofasubset
.
s perform better in comparison with the only known DSTC with independentlyofthevariablesofothersubsetsthecodeissaid
c the same rate and decoding complexity, namely the Coordinate tobeg-groupML decodableorλ-symboldecodable[9],[10].
[
Interleaved Orthogonal Design (CIOD). Furthermore, they per- The Alamouti code is single-symbol decodable or two-group
1 formveryclosetoDSTCsfromfieldextensionswhichhavesame ML decodable. Following the work of [1], several distributed
v rate but higher decoding complexity.
spacetimecodes[6],[7],[8]wereproposed.However,mostof
7
I. INTRODUCTION thesecodeconstructionsdidnotconsidertheimportantaspect
6
0 Cooperative diversity is a technique by which multiple of ML decoding complexity at the destination. This problem
1 terminals(usersor relays)cooperateto forma virtualantenna gainssignificantimportanceespeciallyifthenumberofrelays
0 arraytherebyleveragingthespatialdiversitybenefitsevenifa in the network is large. An initiative in this direction was
7
local antenna array is not available. A cooperative diversity first taken in [11] wherein two-group ML decodable DSTCs
0
protocol dictates how the users would actually cooperate wereproposedusingaconstructionprocedurecalled’doubling
/
s among themselves to achieve the required diversity order. construction’. Later in [3], a class of rate one, full diversity,
c
: Severalcooperativediversity protocolshave been proposedin four-groupMLdecodableDSTCscalledPrecodedCIODs(Co-
v
the literature [1]-[6]. In thispaper,we focuson the two phase ordinate Interleaved Orthogonal Designs) were proposed for
i
X based protocols of Jing-Hassibi [1] and the GNAF protocol arbitrarynumberof relays.Howeverthese DSTCs hada large
r of Rajan et al [2] for three reasons- (i) the operations at numberofzeroentriesinthedesignwhichledtoalargePeak
a
the relay nodes are considerably simplified, (ii) we can avoid toAveragePowerRatio(PAPR).In[10],aclassoffour-group
imposing bottlenecks on the rate by not requiring the relay ML decodable STCs were proposed for the colocated MIMO
nodes to decode and (iii) the framework of distributed space- systems. However these STCs fail to satisfy the additional
time codes allows for more flexibility and higher spectral constaints on the code structure imposed by the cooperative
efficiency[1],[2],[4].Transmissioninthisprotocolcomprises diversityprotocol.Henceitisimportanttoaddresstheproblem
of two phases- broadcast phase and cooperation phase. In the of constructing a new class of four-group ML decodable
broadcast phase, the source broadcasts its information to the DSTCs which have low PAPR and uniform distribution of
relaysandthedestination.Inthecooperationphase,eachrelay power among the relays.
transmits a linearly processed version of the received vector. The main contribution of this paper is in providing a new
For this purpose,each relay is equippedwith a unitarymatrix construction of DSTCs with the following salient features.
which we call ’relay matrix’. It was shown in [1], [2] that Source can transmit 1 complex symbols per channeluse
• 2
to the destination it would appear as if a space-time code Full diversity
•
was transmittedfromcolocatedmultiple antennas.Furtherthe Four group ML decodable
•
design criteria to achieve full diversity also remains to be the Unitary relay matrices. (This eliminates the need for a
•
wellknownrankcriteriafor colocatedMIMO (MultipleInput whitening filter at the destination thus further reducing
MutipleOutputSystems).Howeveritisimportanttonotethat decoding complexity.)
DSTCs need to satisfy many additional constraints on the Unitary weight matrices together with unitary relay ma-
•
code structure (for example, unitary relay matrices) and are trices makes the power distribution uniform among the
relays and in time and hence results in low PAPR. total average power spent by the source and the relays
together.
The paper is organized as follows: We briefly describe the
notion of four-group ML decodable codes and also state the The received vector at the destination can be written in
DSTC design constraints which are due to the protocol in matrix form as follows
Section II. In Section III, we describe the code construction
procedure and show that they possess the salient features y = yD,1 = π2π1P2SH +W (2)
stated above.Few illustrative examplesof DSTC construction (cid:20) yD,2 (cid:21) sπ1P +1
are also provided for 4 and 16 relays. Simulation results where S,H and W are as shown in (3) at the top of the
are presented in Section IV. Conclusions and discussions on next page. The DSTC in this case is the collection of all the
further work comprise Section V. (2T) (R+1)matricesS.ObservingthestructureofS in(3),
Notation: For a complex matrix A, A , AT and AH denote we se×e that it is sufficient to design the submatrix of S given
∗
the conjugate, transpose and conjugate transpose respectively. by SER = A1s ... ANs AN+1s∗ ... ARs∗ . We
AI denotestherealmatrixobtainedbytakingtherealpartsof areinterestedin thecasewhenthesubmatrixSER isobtained
(cid:2) (cid:3)
allthe entriesof thematrixA andAQ denotesthe realmatrix from a linear dispersion STBC say S(X). Let S(X) =
obtainedbytakingtheimaginarypartsofalltheentriesofthe K x A where, x ,x ,...,x are the K real variables of
i=1 i i 1 2 K
dmeatetrrimxiAna.nFtoarnad strqaucaereofmtahterimxaBtr,ix|BB| arensdpeTcrti(vBel)y.denote the Ptthhee ’liwneeiagrhStTmBaCtriSce(sX’,)daenfidnethtehemcaotdriec.es Ai ∈CT×Nt, called
ThevectorX =[x1,x2,...,xK]T A RK iscalledthe
II. FOUR GROUP MLDECODABLE DSTC DESIGN information symbol vector. ∈ ⊂
PROBLEM Suppose we partition the information symbol vec-
tor as XT = XT,XT,...,XT , where XT =
In this section, we briefly describe the two phase based 1 2 g k
cooperative diversity protocol of [1], [2] and introduce the [xjk+1,xjk+2,...,xjk(cid:2)+nk], j1 = 0 and(cid:3)jk = ik=−11ni for
problemstatement.Considera wirelessrelaynetworkconsist- k =1,2,...,g andtheircorrespondingsetofweightmatrices
ing of a source node, a destination node and R other relay also into g groups Lk,k =1,...,g, the kth grouPp containing
nodes which aid the source in communicating information to nk matrices, then S(X) can be written as
the destination.Thechannelpathgainsfromthesource to the g nk
ith relay, denoted by fi and those from the jth relay to the S(X)= Sk(Xk),where Sk(Xk)= xjk+iAjk+i.
destinationdenotedbygj areallassumedtobei.i.dCN(0,1). Xk=1 Xi=1
Thechannelpathgain,g0 fromthesourcetothedestinationis Now if the information symbols in each group take values
alsoassumedtobe (0,1).Allthenodesareequippedwith independent of information symbols in the other groups and
CN
a single antenna and are subject to the half-duplexconstraint, if the weight matrices satisfy
i.e., a node cannot transmit and receive simultaneously. Fur-
AHA +AHA =0, i L , j L ,p=q. (4)
ther,weassumethatthenodesaresynchronizedatthesymbol i j j i ∀ ∈ p ∀ ∈ q 6
level. Each transmission from source to destination comprises
then,itcanbeeasilyshown[9],[10]thatS(X)isg-groupML
of two phases- broadcast phase and cooperation phase. In
decodable.Inotherwords,theMLdecodingcanbeperformed
the broadcast phase, the source transmits a T length vector
by minimizing the metric
s taken from a codebook consisting of information vectors
C = {s1,...,sL} satisfying E sHs = 1 to all the relays y π2π1P2S (X )H 2 (5)
and the destination. In the coo(cid:8)perati(cid:9)on phase, all the relay k −sπ1P +1 k k k
nodes are scheduled to transmit together (assuming symbol
for each 1 k g individually instead of minimizing the
level synchronization) a DSTC. For this purpose, each relay
≤ ≤
computationally more intensive metric
is equipped with a unitary matrix A which we call ’relay
i
matrix’.To beprecise,the ith relaytransmitsa scaled version
π π P2
(tosatisfypowerconstraint)ofA r orA r ,wherer denotes y 2 1 S(X)H 2 (6)
the receivedvector at the relay.Tihie signiali∗modelis ais shown k −sπ1P +1 k
in (1) at the top of the next page, where Hence the ML decoding complexity is reduced to a large
t denotes the vector transmitted by the ith relay and v extentdependingonthevalueofg.Inparticular,thedecoding
i i
• denotes the additive noise at the relay whose entries are complexity is reduced from 2λ to g2λg where, λ depends
assumed to be i.i.d (0,1). on the rate of transmission as measured in bits/s/Hz. In this
- y and y denoCtNe the received vector at the desti- paper, we consider only the case of g =4. Combining all the
D,1 D,2
nation during the broadcastphase and cooperationphase given requirements, the 4-group ML decodable DSTC design
respectively. w and w represent the additive noise at problem is thus to find space-time block codes satisfying the
1 2
the destination whose entries are i.i.d (0,1). The following three constraints.
CN
quantities π and π are the power allocation factors 1) Any column should contain only the variables of the
1 2
satisfying π1 + π2TR = 2T so that P represents the design or only their conjugates.
yD,1 = √π1Pg0s+w1
ri = √π1Pfis+vi, i=1,...,R
∀
t = π2P A r , i=1,...,N
i π1P+1 i i (1)
t = q π2P A r , i=(N +1),...,R
i π1P+1 i i∗
yD,2 = qRi=1giti+w2
P
π1P+1I s 0 ... 0 0 ... 0
S = π2P T1
" q 0 A1s ... ANs AN+1s∗ ... ARs∗ #
(3)
HT = g0 g1f1 ... gNfN gN+1fN∗+1 ... gRfR∗
w
1
W = (cid:2) (cid:3)
" π1πP2P+1 Ni=1giAivi+ Ri=N+1giAivi∗ +w2 #
q (cid:16) (cid:17)
P P
2) All the relay matrices should be unitary. In other words, codewords arising from design C or C
1 2
3) The weight matrices of the code should satisfy the commute.Thuswecanconstructa4 4designbysubstituting
×
conditions for 4-group decodability. either C1 or C2 for A and B in (7) as shown below.
s s s s
III. CODE CONSTRUCTION PROCEDURE 1 2 − ∗3 − ∗4
s s s s
Having described the problemstatement, in this section we S = 2 1 − ∗4 − ∗3 (9)
s s s s
3 4 ∗1 ∗2
explicitlyconstructanewclassofrateone,fulldiversityfour-
group ML decodable DSTCs. s4 s3 s∗2 s∗1
It can be easily verified that S is indeed 4-group ML decod-
Consider the following design of size R R,
× able. The four-groups are s ,s , s ,s , s ,s
1I 2I 1Q 2Q 3I 4I
A BH and s ,s respectively{.Toexte}nd{thisappro}ach{formor}e
S = B −AH (7) num{be3rQof 4reQl}ays, we need to find designs analogous to C
(cid:20) (cid:21) 1
and C for higherdimensions. Towardsthat end, considerthe
where, A and B are identical R R designs in different 2
2 × 2 R R design
variables.IfthecodewordsinAcommutewiththecodewords 2 × 2
W X
in B, then we have D =
X W
(cid:20) (cid:21)
AHA+BHB 0
SHS = . (8) where, W and X are identical R R designs in different
0 BBH +AAH 4 × 4
(cid:20) (cid:21) variables. We call this construction as the ’ABBA’ construc-
Thisensuresthatthe codeS is two-groupML decodable,one tion. It is easy to check that if the design W or X is g-
group involving variables in A and the other group involving groupMLdecodable,thenthedesignDobtainedusingABBA
thevariablesinB.Thisfactwasexploitedin[11]toconstruct construction is also g-group ML decodable. Further, we have
two-groupMLdecodableDSTCs.Thisconstructionprocedure
was namedas ’Doublingconstruction’in [11]. However,note W X W′ X′ WW′+XX′ WX′+XW′
that in addition if A and B are individually two-group ML » X W –» X′ W′ –=» XW′+WX′ XX′+WW′ –
decodable,thenitcanbeeasily shownthatthe codeS willbe
W′ X′ W X W′W +X′X W′X+X′W
four-groupMLdecodable.Inthispaper,weessentiallyexploit » X′ W′ –» X W –=» X′W +W′X X′X+W′W –.
this fact. Hence we require two-group ML decodable codes
Thus we observe that codewords obtained from design D
whose codewords commute among themselves. Towards that
commute if W W = WW , X X = XX , W X = XW
end, consider the following 2 2 designs. ′ ′ ′ ′ ′ ′
× and X′W =WX′. Since W and X are taken to be identical
s1 s2 s1 s2 designs, this simply means that if codewords in W commute,
C1 = s2 s1 and C2 = s2 s1 thencodewordsofDalsocommute.Usingtheabovefacts,we
(cid:20) − (cid:21) (cid:20) (cid:21)
now give our construction procedure for any R = 2a relays
We can show that both C and C are two-group ML decod-
1 2 for a>2.
able codes. For the design C , the two groups are s ,s
2 { 1I 2Q} Construction 1: Thestepsintheconstructionprocedureare
and s ,s respectively. More importantly, we have the
{ 1Q 2I} enlisted as below.
following identity true for any complex number γ.
Step 1: Starting with either C or C , keep applying ABBA
1 2
s s s s s s s s constructioniterativelyonittilla R R designC isobtained.
1 2 ′1 ′2 = ′1 ′2 1 2 2× 2
γs s γs s γs s γs s ThedesignC willbetwo-groupMLdecodableanditwillalso
2 1 ′2 ′1 ′2 ′1 2 1
(cid:20) (cid:21)(cid:20) (cid:21) (cid:20) (cid:21)(cid:20) (cid:21)
s s s s s s s s −s∗ −s∗ −s∗ −s∗ −s∗ −s∗ −s∗ −s∗
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
s s s s s s s s −s∗ −s∗ −s∗ −s∗ −s∗ −s∗ −s∗ −s∗
2 s2 s1 s4 s3 s6 s5 s8 s7 −s1∗0 −s∗9 −s1∗2 −s1∗1 −s1∗4 −s1∗3 −s1∗6 −s1∗5 3
3 4 1 2 7 8 5 6 11 12 9 10 15 16 13 14
666 ss45 ss36 ss27 ss18 ss81 ss72 ss63 ss54 −−ss∗1∗123 −−ss∗1∗114 −−ss∗1∗105 −−ss∗1∗96 −−ss∗1∗96 −−ss∗1∗150 −−ss∗1∗141 −−ss∗1∗132 777
6 s s s s s s s s −s∗ −s∗ −s∗ −s∗ −s∗ −s∗ −s∗ −s∗ 7
6 6 5 8 7 2 1 4 3 14 13 16 15 10 9 12 11 7
6 s s s s s s s s −s∗ −s∗ −s∗ −s∗ −s∗ −s∗ −s∗ −s∗ 7
6 7 8 5 6 3 4 1 2 15 16 13 14 11 12 9 10 7
6 s s s s s s s s −s∗ −s∗ −s∗ −s∗ −s∗ −s∗ −s∗ −s∗ 7
S=6666666666 sssss1111809123 sssss1111021479 sssss1111120569 sssss1111251069 sssss1111344569 sssss11111436503 sssss11111526341 sssss11111615432 sssss1∗1∗2∗3∗4∗56 sssss1∗2∗1∗4∗3∗65 sssss1∗3∗4∗1∗2∗74 sssss1∗4∗3∗2∗1∗83 sssss1∗5∗6∗7∗8∗12 sssss1∗6∗5∗8∗7∗21 sssss1∗7∗8∗5∗6∗30 sssss9∗8∗7∗6∗5∗4 7777777777 (10)
6 s s s s s s s s s∗ s∗ s∗ s∗ s∗ s∗ s∗ s∗ 7
6 14 13 16 15 10 9 12 11 6 5 8 7 2 1 4 3 7
6 s s s s s s s s s∗ s∗ s∗ s∗ s∗ s∗ s∗ s∗ 7
6 15 16 13 14 11 12 9 10 7 8 5 6 3 4 1 2 7
6 s s s s s s s s s∗ s∗ s∗ s∗ s∗ s∗ s∗ s∗ 7
4 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 5
have commuting codewords. Therefore the condition for full-diversity is ∆s1 = ∆s2.
6 ±
Step 2: Apply doubling construction (7) to C by substituting Thus, if ∆s1I = ∆s2I and ∆s1Q = ∆s2Q then full
C forAandB(albeitwithdifferentvariables).Thusweobtain diversity is guara6 nte±ed. Let us define p=6 √±12(s1I +s2I) and
a R×R design which will be four-group ML decodable. q = √12(s1I − s2I). Then the required condition is simply
Note thatthe codesdesignedusing Construction1 have the ∆p∆q = 0, which is nothing but non zero product distance.
property that any column has only the variables or only their So let p6,q take values from a rotated Z2 lattice constellation
conjugates. Moreover it can be shown that these codes have designed to maximize product distance [12]. Then let s1I =
unitaryrelaymatricesaswellasunitaryweightmatrices.This p√+2q and s2I = p√−2q. Similarly it is done for s1Q,s2Q also.
leadstouniformdistributionofpoweramongtherelaysaswell Thus we have
as in time and hence low PAPR.
1 1
s
Example 1: Using construction 1, we get the 1I = √2 √2 GZ2 (13)
s 1 1
16 16 design shown in (10) at the top of this page. (cid:20) 2I (cid:21) " √2 −√2 #
×
The four groups of variables that can be decoded 1 1
s
separately are s1I,s2I,...,s8I , s1Q,s2Q,...,s8Q , s1Q = √12 √21 GZ2 (14)
s ,s ,...,s { and s ,s },...{,s respectivel}y. (cid:20) 2Q (cid:21) " √2 −√2 #
9I 10I 16I 9Q 10Q 16Q
{ } { }
Observe that any column has only the variables or only its 0.5257311121 0.8506508083
where, G= − − .
conjugates.Furthernotethatalltherelaymatricesareunitary. 0.8506508083 0.5257311121
Itcanalso becheckedthatalltheweightmatricesare unitary. In this ma(cid:20)nn−er, we can show that it is possible(cid:21)to achieve
full diversity for all the codes in this paper.
A. Diversity of constructed codes
IV. SIMULATION RESULTS
Theconstructedcodescanbemadetoachievefulldiversity
In this section, we compare the performance of the newly
by letting the variables in each of the four groups to take
proposedDSTCfor a 4 relay network(9) with thatofDSTCs
values from an appropriately rotated ZR2 lattice constellation. from CIOD [2] and field extensions [6], [7], [8]. The power
We wouldliketo emphasizeherethatthe variablesina group
allocation factors used for the simulation are π1 = T, and
sinhotuhledobtheearllgorwoeudpst.oAtalgkeebvraailcuensuimndbeepretnhdeeonrytopfrothveidveasrieafbfelecs- π2 = R1. The signal sets chosen for the different DSTCs are
as follows:
tive means to construct rotated Zn lattices with full diversity
1) CIOD - QPSK rotated by 31.7175
and large minimum product distance [12]. Such lattices can ◦
2) Field extension - QPSK
be used to obtain full diversity for our codes. We will now
3) Newly proposed code - QPSK appropriately rotated as
illustrate how this can be done for the 4 4 design shown in
× explained in subsection A of Section III.
(9).
Fig.1 shows the codeword error rate comparison of the
Note that the matrix S is invertible if either A or B is
newly proposed DSTC with that of DSTCs from CIOD and
invertible. This is because,
field extensions. Observe from Fig.1 that the newly proposed
S = SHS 21 = (AHA+BHB)(AAH +BBH) 12 code performs slightly better compared to the 4-group ML
| | |max |A2,|B 2 | decodableCIOD[2].Moreoveritsperformanceisonlyslightly
≥ | | | | (11) inferior compared to the 1-group ML decodable DSTC from
(cid:8) (cid:9)
where, the second inequality was obtained on application of field extensions [6], [7], [8].
Corollary 4.3.3 in [13]. Thereforeit is sufficient to show how
V. DISCUSSION
C can be made fully diverse. We have
2
A class of full diversity 4-group ML decodable DSTCs
∆C2 =(∆s1+∆s2)(∆s1 ∆s2). (12) with unitary relay matrices and unitary weight matrices were
| | −
0
10
CIOD
Field extension
−1 Proposed code
10
−2
e10
at
R
r
o
r
Er −3
10
d
r
o
w
e
d
Co10−4
−5
10
0 5 10 15 20 25 30
P(dB)
Fig.1. Performancecomparison oftheproposedcodewithDSTCsfromCIODandfieldextension
explicity constructed in this paper for number of relays of [4] J. N. Laneman and G. W. Wornell, “Distributed Space-Time Coded
the form R = 2a,a N. The effect of power allocation on Protocols for Exploiting Cooperative Diversity in Wireless Networks,”
∈ IEEETrans.Inform.Theory,Vol.49,No.10,pp.2415-2525,Oct.2003.
the error performance was not considered in this work and
“Cooperative Diversity in Wireless Networks: Efficient Protocols and
is an interesting problem for further work. Further study on Outage Behavior,” IEEE Trans. Inform. Theory, Vol. 50, No. 12, pp.
the codes of this paper has revealed an interesting algebraic 3062-3080,Dec.2004.
[5] K. Azarian, H. El Gamal, and P. Schniter, “On the achievable diversity
structurewhichgivesaunifyingtheoryofseveralknowncodes
multiplexing tradeoff in half-duplex cooperative channels,” IEEETrans.
in the literature including the ones in this paper. An algebraic Inform.Theory,Vol.51,No.12,Dec.2005,pp.4152-4172.
approachtodesigningthesecodesiscurrentlyunderprogress. [6] PetrosElia,P.VijayKumar,“Diversity-Multiplexing OptimalityandEx-
plicitCodingforWirelessNetworksWithReducedChannelKnowledge,”
Presented atthe 43rdAllerton Conference onCommunications, Control
ACKNOWLEDGMENT andComputing, Sept.2005.
[7] Frederique Oggier, Babak Hassibi, “AnAlgebraic Family ofDistributed
ThisworkwaspartlysupportedbytheDRDO-IIScProgram Space-Time Codes for Wireless Relay Networks,” Proc. IEEE Inter-
onAdvancedResearchinMathematicalEngineering,partlyby national Symposium on Information Theory, Seattle, July 9-14, 2006,
pp.538-541.
the Councilof Scientific & IndustrialResearch (CSIR), India,
[8] Petros Elia, Frederique Oggier and P. Vijay Kumar, Asymptotically
through Research Grant (22(0365)/04/EMR-II)to B.S. Rajan. OptimalCooperativeWirelessNetworkswithoutConstellationExpansion,
The authors are gratefulto the anonymousreviewers for their Proc.IEEEInternationalSymposiumonInformationTheory,Seattle,July
9-14,2006,pp.2729-2733.
valuable comments which helped improve the clarity of the
[9] Md. Zafar Ali Khan and B. Sundar Rajan, “Single-Symbol Maximum-
paper. Likelihood Decodable Linear STBCs,” IEEE Trans. on Inform. Theory,
Vol.52,No.5,pp.2062-2091, May2006.
[10] ChauYuen,YongLiangGuan,TjengThiangTjhung,“Aclassoffour-
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