Capacity Bounds and Lattice Coding for the Star Relay Network

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by  Hamidreza Ebrahimzadeh Saffar

1
Capacity Bounds and Lattice Coding for the Star
Relay Network
H. Ebrahimzadeh Saffar and P. Mitran
Department of Electrical and Computer Engineering
University of Waterloo, Waterloo, Ontario, Canada
Email: {hamid, pmitran}@ece.uwaterloo.ca
Abstract—Ahalf-duplexwirelessnetworkwith6lateralnodes, a1 b3
3transmittersand3receivers, andacentralrelayisconsidered.
Thetransmitterswishtosendinformationtotheircorresponding
r
receivers viaatwophasecommunicationprotocol.Thereceivers
b a
decodetheirdesiredmessages byusingsideinformation and the 2 2
signalsreceivedfromtherelay.Wederiveanouterboundonthe
0 capacityregionofanytwophaseprotocolaswellas3achievable
1 regions by employing different relaying strategies. In particular,
0 wecombinephysicalandnetworklayercodingtotakeadvantage a3 b1
2 of the interference at the relay, using, for example, lattice-based
codes. We then specialize our results to the exchange rate. It Fig.1. Starrelaynetwork withthetwophase (MBC)protocol. Single and
n is shown that for any snr, we can achieve within 0.5 bit of the doublearrowsshowthetransmissions inphases1and2,respectively.
a
upper bound by lattice coding and within 0.34 bit, if we take
J the best of the 3 strategies. Also, for high snr, lattice coding is while the half-duplex relay is at the center and each node
can only hear its immediate neighbor nodes. This network,
2 within log(3)/4≃0.4 bit of the upper bound.
2 referred to as the star network, is a constructive component
Index Terms—Star network, interference, lattice coding, side
of the triangular lattice with error free links addressed in [2],
information, network coding, exchange rate.
] where the network code design is based on the fact that each
T
node receives side information from its neighbors at the end
I
. I. INTRODUCTION of the first phase. Although we consider the use of phase 1
s
sideinformation,unlike[2],weassumenon-idealchannelsand
c Relaying in wireless communications has emerged to be a
[ also let the central relay perform joint physical and network
major subject of research in network information theory. In
layer coding.In phase 2, the relay broadcaststhe result to the
2 particular, bidirectional communication between two terminal
receivers and finally, each receiver extracts its own desired
v nodes a, and b, with the aid of a relay node r has been
message, using the side information from the first phase and
8 of interest to improve communication quality [7], [11], [12].
reception from the second phase.
0 These works use network coding at the relay to increase the
7 Thecontributionsofthispaperareasfollows.Wederivean
information exchange rate between a and b by XORing the
3 outer bound on the capacity region and also achievable rate
packets from a and b at the relay and broadcasting the result
. regions for three different communication strategies, namely,
1 to the terminal nodes. An exception is [5], where a natural
decode-and-forward(DF), amplify-and-forward(AF), and lat-
0 combination of signals at the physical layer is considered.
0 ticecoding,ofthestarnetwork.Wethenexaminetheexchange
Unlike[5],wherenocodingisperformedattherelaynode,
1 rate and compare the performance of the three strategies.
some recent works perform coding at the physical layer by
:
v using structured codes and in particular codes defined on
i lattices [9], [10], which can be considered as joint physical II. PRELIMINARIES
X
layer and network coding. A common aspect of these works A. Network Model
r
a is the exploitationof interferenceat the relay to obtain higher Weconsiderawirelesshalf-duplexcommunicationnetwork
rates, based on either an amplify-and-forward (AF) relaying with six terminal nodes that are located around a central
strategy [5], or using lattice-based codes, [9], [10]. For codes relay. The terminal nodes are divided into three transmitter
with group structure (e.g., lattice-based codes), the modular (source)andreceiver(sink)pairs{a ,b }3 .Eachtransmitter
i i i=1
sum of the codewords are themselves codewords, thus the node a requires to send information with rate R to the
i i
decoder should search a single codebook, as opposed to a corresponding receiver b with the aid of the relay. In case
i
product codebook, resulting in higher rates. where R=R =R =R , we call R the exchange rate. We
1 2 3
Besidesrelayingandtheuseofcrosslayercoding,therehas assume that each node can overhear the signals only from its
also been interest in taking advantage of side information in immediateneighbors.Itis also assumedthatthenetworkuses
the wireless environmentsto improveperformance[2], [6].In a two-phase communication protocol. In the first phase, the
practicalwirelessnetworkswithalargenumberofusers(e.g., lateral sources transmit their messages simultaneously and in
mesh networks), overhearing nodes (opportunistic listening, the second phase, the relay broadcasts some function of the
[6])in differentphasesof a communicationprotocolbecomes superposedsignals received in the first phase. It is sometimes
vital. However, exploiting side information for lattice coding relevantto call phases1 and 2 of the protocolmultiple access
is not addressed in the literature. channel (MAC) and broadcast (BC) phases respectively, and
Motivated by these observations, we consider a problem in thus refer to the communication protocol as MBC. The sink
whichthreetransmitterterminalsdesiretounicastinformation nodes have to estimate their desired messages by use of
to three receiversby meansof a centralrelay in two consecu- the signals they received after both phases. Fig. 1 depicts a
tivephases.Inphase1,thetransmittersbroadcasttheencoded symmetric schematic of the network, transmissions, and the
messages. The terminals are located on vertices of a hexagon relativepositionoftheterminalnodesandtherelay.Thelateral
2
nodes and the relay form a hexagon with the length of the ∆ I(X(2);Y(2),Y(2),Y(2)), ∆ I(X(1),Y(1)|X(1),X(1)),
2 r a2 a3 b1 1 a1 r a2 a3
edges being less than the transmission ranges. The topology
∆ I(X(1);Y(1),Y(1)|X(1),X(1))+∆ I(X(2);Y(2) ) .
of the system, thus, is not necessarily symmetric. 1 a1 b2 b3 a2 a3 2 r {a1,r}c
For the sake of practical comparison between different o
There are six similar upper bounds on R and R , derived
strategiesanddesignoflatticecodes,weconsidertheGaussian 2 3
from correspondingcut-sets. The minimum of the six bounds
case in this paper. In this scenario, all of the channels in
on R and R are denoted by C and C , respectively.
the system are assumed to be additive white Gaussian noise We2also bo3und the two-term2sums s3uch as R + R by
1 2
(AWGN). making appropriate cut-sets. By using cut-sets {a ,a ,b ,r},
1 2 3
{b ,b }c, and {a ,a ,b }c, respectively, we derive the upper
1 2 1 2 3
B. Notations and Definitions bound R +R ≤C , where
1 2 1,2
For the communicationnetwork shown in Fig. 1 and every
nodei∈{a1,b1,a2,b2,a3,b3,r},wedenotethechannelinput C1,2 =minn∆1I(Xa(11),Xa(12);Yb(11),Yb(21)|Xa(13)) +
and output signals in phase m by random variables X(m) ∆ I(X(2);Y(2),Y(2),Y(2)),∆ I(X(1),X(1),X(1);Y(1),Y(1))+
i 2 r b1 b2 a3 1 a1 a2 a3 b1 b2
iannpduYtid(mist)r,ibreustpioenctpiv(emly)(.xX)i(.mB)oilsdcfahcoesevnecftroorms Xalp(mha)baentdXYi a(mnd) ∆2I(Xr(2);Ya(12),Yb(22)),∆1I(Xa(11),Xa(12);Yr(1),Yb(11),Yb(21)|Xa(13))o.
i i i
represent the sent or received vectors of node i, respectively, By using symmetric cut-sets, similar bounds are found on
indexedby phase numberm. The message that nodei wishes R2+R3 (denoted by C2,3) and R1+R3 (denoted by C1,3).
to send to the corresponding receiver is denoted by wi. It is Finally, by considering the cut-sets {a1,a2,a3,r} and
also convenienttodenotetheadditivewhiteGaussiannoiseat {a1,a2,a3}c, we derive two upper bounds on the sum-rate
node j, in phase m, by Zj(m) for a given time and by Z(jm) R1+R2+R3 and we choose the tighter one as the ultimate
foravectorreception.Hence,fortheGaussiancase, we have upper bound on the sum-rate which can be written as
Yb(i1) =Xa(1j)+Xa(1k)+Zb(i1),(i,j,k)∈P3, (1) C1,2,3 =min ∆1I(Xa(11),Xa(12),Xa(13);Yb(11),Yb(21),Yb(31))
Y(1) =X(1)+X(1)+X(1)+Z(1) (2) +∆ In(X(2);Y(2),Y(2),Y(2)),
r a1 a2 a3 r 2 r b1 b2 a3
Yb(i2) =Xr(2)+Zb(i2),i=1,2,3, (3) ∆1I(Xa(11),Xa(12),Xa(13);Yr(1),Yb(11),Yb(21),Yb(31)) .
where P ,{(1,2,3),(2,3,1),(3,1,2)},and all of the nodes Therefore, an outer bound on the capacity region oof this
3
have the same power constraint P and the complex circular network, denoted by C is described by
symmetric noise has variance N. The signal-to-noise ratio
NP is denoted by snr. Furthermore, we denote the set of C = (R1,R2,R3)∈R3+ : R1 ≤C1, R2 ≤C2, R3 ≤C3
transmissionsandreceptionsbyallnodesinasetS atagiven
nR +R ≤C , R +R ≤C , R +R ≤C
time, in phase m, by X(m) and Y(m) respectively. We also 1 2 1,2 2 3 2,3 1 3 1,3
S S
define RS , i∈SRi. R1+R2+R3 ≤C1,2,3 .
For a given communication strategy, ∆ ≥ 0, denotes the
m o
relativeduratiPonofthemthphaseoftheprotocolandwehave Gaussiancase:ComputingtheouterregionC fortheGaussian
∆ = 1. In particular, for the present MBC protocol, case and dropping degenerate inequalities, we have that the
m m
∆ +∆ =1. For a given phase m, the error events {wˆ 6= exchange rate R is bounded by
wP1} bet2ween encoder a and decoders are denoted by Ea(im),
ai i ij R≤min ∆ log(1+snr),∆ log(1+3snr) , (4)
if b is the decoder, and by E(m), if the relay is the decoder. 1 2
Finajlly,wedenoteǫ-weaklytyipricalsequencesoflengthn∆m, 2R≤min(cid:16) ∆1log(1+4snr+3snr2), (cid:17)
according to distributions of phase m, by T(m).
ǫ ∆(cid:16) log(1+snr)2+∆ log(1+3snr) , (5)
1 2
III. OUTERBOUND 3R≤min ∆ log (1+snr)2(1+7snr) , (cid:17)
1
To derive the outer bound on the three rates of the star
network with the MBC protocol, we use the variation of the ∆ log (1(cid:16)+snr)2((cid:2)1+4snr) +∆ log(1(cid:3)+3snr) , (6)
1 2
cut-set bound given in [8]. By looking at all cut-sets and
dropping those implied by the other ones, we find six upper where all l(cid:2)ogarithms are in base(cid:3) 2. It is easy to see(cid:17)that (4)
bounds on each of the single rates R , R , and R , three implies (5) and (6), thus (4) represents the tightest cut-set
1 2 3
upper bounds for two-term sums, and two upper bounds for upper bound on the exchange rate of the star network for the
the sum-rate. Gaussian case within a two-phase protocol.
The cut-sets that give the upper bounds on R are {b }c,
1 1
{a3,b1}c, {a2,b1}c, {a2,a3,b1}c, {a1,b2,b3} and {a1,r}. IV. LISTENINGAND RELAYING STRATEGIES
Based on these respectively, R ≤C¯, where
1 1 A. Decode-and-Forward (DF)
C =min ∆ I(X(1),X(1),X(1);Y(1))+∆ I(X(2);Y(2)), Inthedecode-and-forwardprotocol,therelaytriestodecode
1 1 a1 a2 a3 b1 2 r b1 the messages sent by the transmitters separately and then
∆ I(X(1)n,X(1);Y(1)|X(1))+∆ I(X(2);Y(2),Y(2)), forwardsacombinationofthemtothereceivers.Eachreceiver
1 a1 a2 b1 a3 2 r a3 b1 node also decodes the messages of the other two pairs at the
∆ I(X(1),X(1);Y(1)|X(1))+∆ I(X(2);Y(2),Y(2)),
1 a1 a3 b1 a2 2 r a2 b1 end of the first phase, using the side information received
3
during the first phase. Consequently, at the end of the second Pr(wˆ 6=w ),Pr(wˆ 6=w )→0, if
a2 a2 a3 a3
phase, each receiver has a combination of all three messages
as well as the two messages of other two pairs. Thus, the R2 <∆1I(Xa(12);Yb(11)|Xa(13)), (8)
receivers can extract the desired messages. R <∆ I(X(1);Y(1)|X(1)), (9)
Theorem 1: An achievable region for the star relay net- 3 1 a3 b1 a2
work with the two phase DF MBC protocol is the closure of R +R <∆ I(X(1),X(1);Y(1)). (10)
the convex hull of all points 2 3 1 a2 a3 b1
Finally, the receiver b estimates its desired message
1
w , by looking for a unique wˆ such that
n(R1,R2,R3)∈R3+ : (Xa1(2)(wˆ ,wˆ ,wˆ ),Y(2)) ∈ T (1). Therae1fore the error
r a1 a2 a3 b1 ǫ
Ri<min(cid:16)∆1I(Xa(1i);Yb(j1)|Xa(1k)),∆1I(Xa(1i);Yb(k1)|Xa(1j)), event E1(2) ={wa1 6=wˆa1} can be written as
∆1I(Xa(1i);Yr(1)|Xa(1j),Xa(1k)),∆2I(Xr(2);Yb(i2))(cid:17), E1(2) =E1(2)∩ E¯r∩E¯s,1 ∪(Er ∪Es,1) ,
Ri+Rj <min(cid:16)∆1I(Xa(1i),Xa(1j);Yb(k1)), where Er = E1(1,r)∪E2(1(cid:2),r)(cid:0)∪E3(1,r) an(cid:1)d Es,1 = E2(1,1)∪(cid:3) E3(1,1) are
the events of decoding error at relay and b after phase 1, re-
∆ I(X(1),X(1);Y(1)|X(1)) ;(i,j,k)∈P , 1
1 ai aj r ak (cid:17) 3 spectively.Consequently,bytheAEPproperty,theprobability
R +R +R <∆ I(X(1),X(1),X(1);Y(1)) , of the error event E(2) can be upper bounded by
1 2 3 1 a1 a2 a3 r o 1
Pr E(2) ≤Pr E(2) E¯ ∩E¯ +Pr E ∪E
over all joint distributions 1 1 r s,1 r s,1
p(1)(xa1)p(1)(xa2)p(1)(xa3)p(2)(xr), over the alphabet (cid:2)≤2−n(cid:3)R12n(cid:2)∆2(cid:16)I(X(cid:12)(cid:12)r(2);Yb(12))−2(cid:3)ǫ(cid:17)+P(cid:2)r Er∪Es(cid:3),1 . (11)
X ×X ×X ×X .
Prao1of Oua2tline:aE3ncodring: For i = 1,2,3, the transmitters By choosing R1 < ∆2(I(Xr(2);Yb(12))−2(cid:2)ǫ), and app(cid:3)lying all
ignegnetroateth2enRdiistrraibnudtoiomnsn∆p(1i)-(lxeng)thresespqeucetnivceelsyXto(a1i)coancsctorurdc-t i∞ne.qBuyaliftoiellso(w7i)n,gthteherigsahmtheaanrdgusimdeenotff(o1r1d)ewcoildlivnagnaisthnoasdens→b2
theircodebooks.Then,{a }3 apiicktheir messages{w }3 and b3, we can derive two similar set of inequalities to (8)–
at random from the indexi is=et1s { 1,2nRi }3 , respecatiivei=ly1, (10) andalsoconcludethatR2 <∆2(I(Xr(2);Yb(22))−2ǫ)and
where [1,M] := {1,2,··· ,M},(cid:2)and sen(cid:3)di={1X(a1i)(wai)}3i=1 Rth3e<con∆d2it(iIo(nXsr(o2f);TYhb(e13o)r)e−m2ǫ1).hFoilnda.lly,sinceǫ>0isarbitrary,
respectively.The relay nodealso generates2nP3i=1Ri random Gaussian Case: For the case, where all the channels are
n∆ -lengthsequencesX(2)(w ,w ,w ). The relaynoder
estim2 ates w˜ , w˜ , andrw˜ aat1thea2endao3f phase 1 and then AWGN, and we have R = R1 = R2 = R3, it can be easily
a1 a2 a3 seen that the conditions of Theorem 1, will reduce to
broadcasts the signal X(2)(w˜ ,w˜ ,w˜ ).
Decoding and error arnalysai1s: Tah2e deac3oding at the receiver R<min(∆1,∆2)log(1+snr), (12)
nodes {b }3 is done in two steps. Each receiver node (e.g., 2R<∆ log(1+2snr), (13)
i i=1 1
b1)decodesthemessagesoftheothertwopairs(e.g.,wa2 and 3R<∆1log(1+3snr). (14)
w ) at the end of the first phase, using the side information
a3
receivedduringthefirstphase.Second,usingthisinformation,
B. Amplify-and-Forward (AF)
the receiver node tries to find its own desired message (e.g.,
w ), after the second phase. The detailed decoding process In this section, we consider a two phase amplify-and-
a1
anderroranalysisatnodesr andb isfurtherexplainedinthe forward (AF) strategy, in which the relay simply forwards a
1
sequel. Note that the analysis is similar for nodes b and b , scaledversionofthesignalitreceivesafterphase1,imposing
2 3
due to the symmetry of the network. the equality ∆ = ∆ = 1/2. Thus, we assume continuous
1 2
Relay: The relay decodes w˜ , w˜ , and w˜ at the end of input and output alphabets for the terminals and the relay,
phase 1, using joint typical decao1dinga2, if this tari3ple is the only which is inherent to an AF strategy.
one satisfying (X(1)(w˜ ),X(1)(w˜ ),X(1)(w˜ ),Y(1)) ∈ Theorem 2: An achievable region for the star relay net-
a1 a1 a2 a2 a3 a3 r work with the two phase AF MBC protocol is the closure of
Tǫ(1). Indeed, during phase 1 with a block length of n∆1, the convex hull of all points
a MAC is formed from a , a , and a to r. The er-
1 2 3
rPorr(wa˜naly6=siws of),tPher(Mw˜AC6=iws kn)o,wPrn(w[˜1] t6=hawt w)e w→ill0h,avaes n(R1,R2,R3)∈R3+ : Ri < 12I(Xa(1i);Yb(i2)|Xa(1j),Xa(1k)), (15)
a1 a1 a2 a2 a3 a3
n→∞ if 1
Ri+Rj <min(cid:16)2I(Xa(1i),Xa(1j);Yb(i1),Yb(i2)|Xa(1k)),
RS <∆1I(XS(1);Yr(1)|XS(1c)), (7) 12I(Xa(1j),Xa(1i);Yb(j1),Yb(j2)|Xa(1k))(cid:17); (i,j,k)∈P3
1
for all S ⊆{1,2,3}, where Sc ={1,2,3}−S. R1+R2+R3 <i∈m{1,i2n,3}(cid:16)2I(Xa(11),Xa(12),Xa(13);Yb(i1),Yb(i2))(cid:17) o,
Receiver b : Terminal node b , decodes wˆ and wˆ after
1 1 a2 a3 over all joint distributions p(1)(x )p(1)(x )p(1)(x ), over
phase1fromthereceivedsignalY(1),if thereexistsaunique a1 a2 a3
Tpaǫi(r1)w.ˆaU2sainngd wtˆhae3,esrurocrhathnaatly(sXis(a12o)f(b1wˆtah2e),MXA(a13C)(wˆ[1a]3,),wYeb(11h))av∈e PtahireogoaeflnpOehruaabttleeint2eXn:RaE1in×rcaonXddiano2gm×:Fn2Xo-rlae3in.=gth1,s2eq,3u,enthceestrXan(as1im),itatcecronroddinegs
4
to the distributionsp(1)(x )andbroadcastthe sequencesthat Proof:Thisfollowsby(1)–(3),Theorem2andX(2) =αY(1),
correspondto themessageasi w duringphase1.Atthe endof where α2 = snr , due to the node power conrstraint, Pr =
ai 1+3snr
phase 1, the relay scales (or amplifies) its received signal by E|X(2)|2 =α2E|Y(1)|2.
r r
a coefficient α and generates the n-length sequence X(2) =
2 r C. Lattice Coding
αY(1). The relay then broadcasts X(2) to the side nodes.
r r
Decoding and error analysis: The decoding at receiver Theorem 3: An inner bound to the exchange capacity of
nodes {b }3 is done after phase 2. After phase 1, each the star relay network with the MBC protocol and synchro-
i i=1
receiver node (e.g., b ) buffers the sum signal it has received nized AWGN channels is given by
1
tfiroonminitsconneijguhnbctoironnowdietsh(teh.ge.,sYigb(n11a)l).iIttruesceesivtehsisfrsoidmetihneforremlaay- R<∆1log(1/3+snr), (18)
afterphase2(e.g.,Y(2)),toperformajointlytypicaldecoding provided that ∆ log(1/3 + snr) < ∆ log(1 + snr). By
b1 1 2
anddecodeitsowndesiredmessage(e.g.,w ).Decodingand maximizing the right hand side of (18) over ∆ , subject to
a1 1
error analysis at b is explained in the sequel. Because of the this condition, we find every R<C is achievable where
1 Latt.
symmetry, the analysis is similar for nodes b and b .
2 3 log(1+snr)log(1 +snr)
Receiver b1: After phase 2, terminal node b1, C = 3 . (19)
looks for the triples wˆ , wˆ , and wˆ , for which Latt. log(1+snr)+log(1 +snr)
a1 a2 a3 3
(X(1)(wˆ ),X(1)(wˆ ),X(1)(wˆ ),Y(1),Y(2)) ∈ T(2).
a1 a1 a2 a2 a3 a3 b1 b1 ǫ Proof Outline: The idea of the proof is an extension of the
It then claims the index wˆ , as the decoded message, if the
a1 one in [9], which uses nested lattices to encode and decode
first element of all such triples is identically wˆ . The error
a1 the information. The difference is that we also exploit the
event E(2) is thus the union of four events
1 side information available to the terminals after phase 1. In
E(2) =E(2)∪E(2) ∪E(2) ∪E(2) , (16) particular, we consider the star relay network with perfectly
1 ∅,1 {2},1 {3},1 {2,3},1 synchronized AWGN channels. We use a fine lattice, nested
where we define in a coarse lattice with second moment P, with its points
E(2) = X(1)(wˆ ),{X(1)(w )} , locatedin the basic Voronoiregionof the coarse lattice asthe
S,i ai ai aℓ aℓ ℓ∈Sc codewords. We denote the coarse lattice by L and the fine
wˆawℓˆ6=ai[w6=awℓ:aℓi∈S(cid:0) {X(a1ℓ)(wˆaℓ)}ℓ∈S,Yb(1i),Yb(2i) ∈Tǫ(2), rlaetgtiicoenonfesltaetdticiensiatrebydeLnoft,etdhubsyLS(L⊆)Lanfd. TSh(Le b)a.sSiceeV[o9r]onfoori
forbrevity,whereSc ={1,2,3}−{i}−S.From(1(cid:1)6) andby more details on lattices. f
applying the union bound and the AEP property, we have Encoding: The encoders {a }3 map their messages
i i=1
Pr E1(2) ≤2nR12−2n(cid:16)I(Xa(11);Yb(12)|Xa(12),Xa(13))−2ǫ(cid:17) {owf atih}e3i=b1asoicn Vtoorothneoinc∆e1ll-dLime∩nsiSo(nLal).pToihnetse{nccio(dwearsi)}th3i=en1
+(cid:2)2n(R(cid:3)1+R2)2−2n(cid:16)I(Xa(11),Xa(12);Yb(11),Yb(12)|Xa(13))−3ǫ(cid:17) generaterandomn∆1-lengthdfithervectors{di}3i=1 according
+2n(R1+R3)2−2n(cid:16)I(Xa(11),Xa(13);Yb(11),Yb(12)|Xa(12))−3ǫ(cid:17) tmoutthuealulyniifnodrmepednisdterinbtuatinodnkonvoewr nS(tLo)t.hTehreeldaiythaenrdvercetcoerisvearrse.
+2n(R1+R2+R3)2−2n(cid:16)I(Xa(11),Xa(12),Xa(13);Yb(11),Yb(12))−4ǫ(cid:17). ThetransmitsignalsX(a1i)(wai)arethenconstructedaccording
to the following rule
By following the same argument for decoding at nodes b
2
andb3,wecanderivesimilarupperboundsonthecorrespond- X(a1i) =ci−di mod L, i=1,2,3. (20)
ing error probabilitiesPr E2(2) and Pr E3(2) . Since ǫ>0, is Decoding:The relay decodesc =(c +c +c ) mod L,
chosen arbitrarily, by choosing R1, R2, and R3 according to afterphase1andthentransmitsthreindex1 ofc2 ,us3ingrandom
the conditions of Theore(cid:2)m 2,(cid:3)we can (cid:2)upper(cid:3)bound the error r
coding.Thereceivers(e.g.,b )decodethemodularsumofthe
probabilities by 0 as n→∞. other pairs codewords (e.g.,1c +c mod L), after phase 1.
2 3
Gaussiancase:Now,weassumethechannelstobeAWGN. Finally the receiversdecodec andfind the desiredcodeword
r
Tomaximizethemutualinformationmeasures,allofthenodes by a modular subtraction. The details of decoding at both
employ Gaussian complex codebooks with transmit power P phases are as follows.
for both phases. We consider the exchange rate and rewrite 1) Phase 1 Decoding: Since the channels are AWGN and
the conditionsof Theorem2, for AWGN channels. The result synchronized,afterphase 1, the relayreceivesthe sum ofsig-
is stated as a corollary. nals transmitted by {a }3 and the receivers {b }3 receive
i i=1 i i=1
Corollary 1: An achievable exchange rate of the star net- the sum of signal pairs broadcasted by their neighbor nodes.
work with the two phase AF protocoland AWGN channelsis The relay and terminals perform similar lattice decoding
given by R<CAF, where proceduresafterphase1.Thereceivedvectorattherelayafter
phase 1 is
1 snr
C =min log 1+ snr ,
AF 2 1+4snr Y(1) =X(1)+X(1)+X(1)+Z(1), (21)
( (cid:20) (cid:18) (cid:19) (cid:21) r a1 a2 a3 r
1log 1+ 1+6snr snr+ snr snr2 , from which the relay decodes cr, which is itself a codeword,
4 1+4snr 1+4snr dueto thegrouppropertyof theset L ∩S(L) underaddition
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19) (cid:21) f
1 2+11snr 2snr mod L. For this purpose, the relay forms the signal ˆcr =
log 1+ snr+ snr2 . (17) (γY(1)+d +d +d ) mod L, where γ ∈ R, is a scaling
6 1+4snr 1+4snr r 1 2 3
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19) (cid:21)) coefficient, determined to maximize the achievable exchange
5
two phase protocol with AWGN channels, in terms of the
6
exchange rate and their gap to the upper bound. From (12)–
5 (14), we findtheoptimizedover∆1 achievablerate forDFas
a function of snr as
)
R
(4 Lattice coding log(1+snr)log(1+3snr)
rate AF CDF = 3log(1+snr)+log(1+3snr). (23)
e3 Upper bound
g
n We also findtheglobalupperboundbyoptimizing(4)over
a
h
c2 ∆ to be
x 1
E
log(1+snr)log(1+3snr)
1 DF CUB = log(1+snr)+log(1+3snr). (24)
0
−10 −5 0 5 10 15 20 25 30 Fig. 2 illustrates the achievable curves (23), (17), and
snr(dB)
(19) for DF, AF, and lattice-based schemes respectively and
(24) for the upper bound. It can be seen that the lattice-
Fig.2. Achievable exchange rates fordifferent strategies andupperbound. basedstrategy outperformsotherschemesforhighsnr values,
where it is asymptotically within log(3)/4 ≃ 0.4 bit of
rate (i.e., maximize the number of fine lattice points in the
the upper bound, while at the worst case, it has a gap of
basic cell of the coarse lattice). From (20) and (21),
log(3)log(5/3)/log(5) ≃ 0.5 bit. If we plot the best of the
3 curves in terms of snr, the compound scheme will be within
cˆr = (γ(X(a11)+X(a12)+X(a13)+Z(r1))+ di) mod L, 0.34 bit of the upper bound for all snr values.
i=1
(cid:2) X (cid:3)
3 VI. CONCLUSION
= c +γZ(1)−(1−γ) X(1) mod L. (22)
r r ai We considered a wireless network with three source-sink
(cid:2) Xi=1 (cid:3) pairs of terminals that want to communicate with the help
Because {c }3 areindependentofthe noiseand{X(1)}3 , of a relay, using a two phase protocol. We derived an upper
wecanrewriitie=(122)ascˆ =c +Z˜(1),whereZ˜(1) =(γaiZ(1i=)−1 bound and three achievable exchange rates. As a main part,
r r r r r we proposed codes for the system based on high-dimensional
(1−γ) 3 X(1)) mod L,isanequivalentnoisetermadded
i=1 ai lattices and incorporated relaying, as well as joint physical
to the desired signal with power N˜ = γ2N +3P(1−γ)2.
andnetworklayercodingwiththeuseofsideinformation.We
P
The optimal value of γ to minimize the equivalent noise
showedthatthelatticecodingschemecanachieveanexchange
power is γ∗ = 3P , and the corresponding optimal noise
3P+N rate within 0.5 bit of the upper bound.
power is N˜∗ = 3PN . By the properties of lattices and
r 3P+N
error probability of lattice codes for AWGN channel [3], REFERENCES
[4], the second moment of the fine lattice should be chosen
[1] T.CoverandJ.Thomas,ElementsofInformationTheory,2nded. New
D(Lf) = N˜r∗ + δ, for arbitrary δ > 0, in order to have York:Wiley, 2006.
Pr[cˆ 6=c ]→0, as n→0. By letting δ →0, the maximum [2] M.Effros,T.Ho,andS.Kim,“Atilingapproachtonetworkcodedesign
r r
packed number of fine lattice points in the basic cell is for wireless networks,” in Proc. IEEE Information Theory Workshop,
PuntadelEste,Uruguay,2006,pp.62–66.
|Lf ∩S(L)| = [D(L)/D(Lf)]n∆1 = (1/3+snr)n∆1. Since [3] U.Erez,S.Litsyn,andR.Zamir,“Latticeswhicharegoodfor(almost)
the number of codewords2nR is at most the number of basic everything,”IEEETrans.Inform.Theory,vol.51,no.10,pp.3401–3416,
Oct.2005.
cellpointsforreliableMACdecodingat therelay asn→∞,
[4] U. Erez and R. Zamir, “Achieving 1log(1+snr) on the AWGN
we require R < ∆1log(1/3+snr). The phase 1 decoding at channel with lattice encoding and de2coding,” IEEE Trans. Inform.
the lateral receivers {b }3 follows the same outline as the Theory,vol.50,no.10,pp.2293–2314, Oct.2004.
i i=1 [5] S.Katti,S.Gollakota,andD.Katabi,“Embracingwirelessinterference:
relay. Noting that these nodes should decode the sum of two
analognetworkcoding,” inACMSIGCOMM,Kyoto, Japan,2007.
codewords mod L, ratherthanthree,weobtainthecondition [6] S.Katti,D.Katabi,W.Hu,H.Rahul,andM.Me´dard,“Theimportance
R<∆ log(1/2+snr)toguaranteecorrectdecodingatlateral of being opportunistic: practical network coding for wireless environ-
1
ments,” in Proc. of 43rd Allerton Conf. on Commun., Control, and
terminals as n→∞.
Computing, Allerton House,Monticello, IL,Sep.2005,pp.756–765.
2) BC Decoding: The relay uses a random code to convey [7] S. Katti, H. Rahul, W. Hu, D. Katabi, M. Medard, and J. Crowcroft,
ˆc to the nodes {b }3 in n∆ channel uses. The decoded “Xorsintheair:practicalwirelessnetworkcoding,”inACMSIGCOMM,
sirgnal of ˆc at eaichi=1b is ref2erred to as cˆˆ . The relay Pisa,Italy,2006.
r i r,i [8] S. J. Kim, P. Mitran, and V. Tarokh, “Performance bounds for bi-
codebookconsistsof|Lf∩S(L)| codewords.Therefore,from directional coded cooperation protocols,” IEEE Trans. Inform. Theory,
the classical AWGN channel error analysis, Pr cˆˆ 6=ˆc → vol.54,no.11,pp.5235–5241, Nov.2008.
r,i r [9] K. Narayanan, M. P. Wilson, and A. Sprinton, “Joint physical layer
0,asn→∞,aslongas∆1log(1/3+snr)<∆2log(1+snr). codingandnetworkcodingforbi-directional relaying,”inProc.of45th
(cid:2) (cid:3)
The proofis completeby notingthat, by theunionbound,the Allerton Conf. onCommun., Control, and Computing, Allerton House,
decoding error probability at each {b }3 will vanish if both Monticello, IL,Sep.2007,pp.254–259.
i i=1 [10] B. Nazer and M. Gastpar, “Compute-and-forward: harnessing interfer-
MAC and BC phases decodingsare separately asymptotically encethroughstructurecodes,”SubmittedtoIEEETrans.Inform.Theory,
reliable. Aug.2009.
[11] P. Popovski and H. Yomo, “Bi-directional amplification of throughput
in a wireless multi-hop network,” in Proc. IEEEVeh. Technol. Conf. -
V. COMPARISON AND NUMERICAL RESULTS Spring,2006,pp.588–593.
[12] Y.Wu,P.A.Chou,andS.-Y.Kung,“Informationexchangeinwireless
In this section, we compare the performance of three networks with network coding and physical-layer broadcast,” in Proc.
schemes examined in this paper (DF, AF, and lattice), for the Conf.onInf.Sci.andSys.,Baltimore, MD,Mar.2005.