Capacity Bounds of Half-Duplex Gaussian Cooperative Interference Channel

0.13 MB·English

by  Yong Peng

Capacity Bounds of Half-Duplex Gaussian
Cooperative Interference Channel
Yong Peng and Dinesh Rajan, Senior Member, IEEE
Abstract—Inthispaper,weinvestigatethehalf-duplexcooper- schemesarealsoproposedin[10],wherethetransmittershave
ativecommunicationschemeofatwouserGaussianinterference additional flexibility in choosing the order of DPC.
channel. We develop achievable region and outer bound for
In this paper, we compute bounds on the capacity of two
the case when the system allow either transmitter or receiver
user Gaussian IC in two differentscenarios: i) transmitter co-
9 cooperation. Weshow that byusingourtransmittercooperation
0 scheme, there is significant capacity improvement compare to operation(TC)andii)receivercooperation(RC). Specifically,
the previous results [9], [10], especially when the cooperation weallowallnodestooperateinhalf-duplexmodeonly,which
0
2 linkisstrong.Further,ifthecooperationchannelgainisinfinity, requires simpler and cheaper hardware.
both our transmitter and receiver cooperation rates achieve
In TC, the two transmit nodes serve as relays to each
n their respective outer bound. It is also shown that transmitter
a cooperation provides larger achievable region than receiver other. We assume that the channel gain between the two
J cooperation under the same channel and power conditions. transmitters is much higher than the others. In this case, it

7 is well known that DF strategy is superior [11], [12]. Thus,
1 I. INTRODUCTION in this paper we derive the achievable region with TC using

In wireless ad hoc networks, spatially dispersed radio ter- only the DF strategy. We show that the achievable region of
]
T minals can exploit cooperative diversity [1], [2] by relaying the proposed TC strategy is strictly larger than the results
I signals for each other. With cooperation, different clusters in [9], [10], especially when the cooperation link is strong.
.
s of terminals can act like transmit/receive antenna arrays and In case when the cooperation channel gain is infinity, the
c
achieve increased spatial diversity and throughput by joint proposed achievable region achieves the system upper bound.
[
encoding and/or decoding. In contrast, for the schemes in [9], [10], there is a large

1 The capacity of the two-user Gaussian interferencechannel performance gap between the lower and upper bounds.
v
(IC) is an open problem for many years and is completely In RC, the two receive nodes serve as relays to each
6
known only in some special cases (e.g., in the strong in- other. In this case, we assume that the relay to destination
0
6 terference case [8]). The capacity region has been studied channel is strong for RC, and CF [11] is preferable at the
2 under various cooperative strategies. Most of these schemes relays.Thus,toderivetheachievableregionwithRC,weonly
. assume that nodes operate in full-duplex mode. A coding consider the CF strategy. The proposed scheme achieves the
1
0 scheme for transmitter cooperation using decode-and-forward correspondingMIMOmultipleaccesschannel(MAC)capacity
9 (DF) for relayingand dirty papercoding (DPC) for codeword [13] whenthecooperationchannelgainisinfinity.To thebest
0 generationis proposedin[3]. Compress-and-forward(CF) and of our knowledge, the achievable rate with RC has not been
:
v DF relaying strategies for receiver cooperation are proposed studied under the half-duplex assumption. We also show that
i in [4] and generalized to both transmitter and receiver coop- under identical channel conditions and equal transmit power
X
eration in [5]. A comparison of different coding schemes for constraints on all nodes, TC achieves larger rates than RC.
r
a transmitter cooperation in terms of the relative geometry of
II. SYSTEM MODEL
transmit and receive clusters is given in [6]. The sum rate ca-
pacitywithtransmitteronly,receiveronlyandbothtransmitter Consider a two-transmitter two-receiver network shown in
and receiver cooperation is studied in [7]. By using DF and Fig. 1, where node 3 is the intended receiver of node 1 and
DPC at the cooperative transmitters and Wyner-Ziv CF at the node 4 is the intended receiver of node 2. The independent
receiversand assuming equalpower gain for all channels, the messages transmitted by node i, i 1,2,3,4 are encoded
∈ { }
proposed scheme in [7] is shown to have significant capacity into N complex symbols x [1],x [2],...,x [N], under the
i i i
gain over strong IC [8]. While full-duplexcooperative IC has power constraint 1 N x [n]2 P . If the messages
N n=1 i ≤ i
been significantly studied, only limited results are known in transmittedbynode1and2hasatotalalphabetofM andM
1 2
P
thehalf-duplexscenario.Cooperativediversitywithtransmitter respectively, their respective rates are then R = logM /N
1 1
cooperation for fading channels is considered in [2]. A 2- andR =logM /N bits/transmission.Thechannelgainfrom
2 2
phase transmitter cooperation scheme using DF and the so node i to node k and k > i, is represented by a complex
called recycling DPC (RDPC) is introduced in [9]: Similar constant hik = cikejθik. It is assumed that all nodes have
perfect knowledge of the channel gain and all the phase
1TheauthorsarewiththeDepartmentofElectricalEngineeringatSouthern offsets can be perfectly synchronized. Let z denote the i.i.d.
i
Methodist University, Dallas, Texas. Email: {ypeng,rajand}@engr.smu.edu.
complexcircularly symmetric Gaussian noise process at node
This work has been supported in part by the National Science Foundation
throughgrantCCF0546519. i, with the nth element zi[n] C (0,1). We assume that
∼ N
Transmitter Cooperation Receiver Cooperation
can also send w and v in phase 3, respectively. Due to the
d d
w ,v 11111 3 w 11111 3 limited space, we only outline the results at each phase.
1r 2r 1r
1
e Transmission Scheme: The transmission is divided into 3
s
a phases as shown in Fig. 1-(a), with time portion λ λ and
h 1 2
P v
2 4 1r 2 4 λ3.InPhase1,node1isintransmitmodeandallothernodes
are in receive mode. The received signal at node i y [n] =
w i
2 11 3 d 11 3 h1ix1[n]+zi[n], n 1,2,..., λ1N ,i = 2,3, and 4. In
∈ { ⌊ ⌋}
es Phase 2, node 2 is in the transmit mode and all the other
a
hP v1r,w2r 2 4 v2r 2 4 ws,w2r tnroadnesms aitremiondreecaenidvenmodoedse.3InanPdha4sear3e,innodthees 1reacneidve2maroedien.
The receivedsignal in phases2 and 3 can easily be expressed
wd,w3r,v3r 11 3 w2r 11 3 vs,v2r similar to phase 1.
3
es Outline of Achievability:
a
hP v ,v ,w 2 4 v 2 4 1) Phase 1: If c13 >c14, generate codeword X1(v2r) with
d 3r 3r d length λ N, N and power α P(1), P(1) = κ P /λ .
(a) (b) 1 → ∞ 2 1 1 1 1 1
Given X (v ), use DPC to generate X (w ) with length
1 2r 1 1r
Fig. 1. System models of half-duplex Gaussian cooperative interference λ1N and power α1P1(1). Otherwise, do DPC with the reverse
channel with(a)transmitter cooperation and(b)receiver cooperation. order. Since v is known to node 2, it can subtract X (v )
2r 1 2r
and decode w if the rate of w satisfies [9]
1r 1r
RTX λ C c2 α P(1) . (3)
the communication is in a half-duplex fashion, i.e., each of 1,r1 ≤ 1 12 1 1
the nodes can be either in the transmit mode or the receive Node 3 can decode v if the(cid:16)rate of v s(cid:17)atisfies
2r 2r
mode.ForTC,onlythetwotransmitnodes(node1and2)can
λ C c2 α P(1)/(1+c2 α P(1)) , if c >c
cooperate with each other while for RC, only the two receive 1 14 2 1 14 1 1 13 14
RTX .
nodes(node3 and 4) can cooperatewith each other. It is also 2,1 ≤λ C(cid:16)c2 α P(1) , (cid:17) otherwise
assumed that the cooperation nodes are close together, i.e.,  1 14 2 1
c12 and c34 are large compared to the other cik’s. Further, 2) Phase 2:(cid:16)If c24 > (cid:17)c23, generate codeword X1(w2r)
we define the following non-negative parameters satisfying with length λ N and power β P(1), P(1) =γ P /λ . Given
2 2 2 2 1 2 2
α1 + α2 = 1, β1 + β2 = 1, κ1 + κ2 = 1, γ1 + γ2 = 1, X1(w2r), use DPC to generateX2(v1r) with lengthλ2N and
µ1+µ2+µ3 =1, η1+η2+η3 = 1 and λ1+λ2+λ3 = 1. power β P(1). Otherwise, do DPC in the reverse order. Node
1 2
Also defineg1 =[c13 c23], g2 =[c14 c24], h1 =[c13 c14]and 1 can decode v if the rate of v satisfies [9]
1r 1r
h =[c c ]. Let C(x)=log(1+x).
2 23 24
RTX λ C c2 β P(1) (4)
III. TRANSMITTER COOPERATION 2,r1 ≤ 2 12 1 2
(cid:16) (cid:17)
and node 3 can decode w if the rate of w satisfies
A. Achievable Rates 2r 2r
Theorem 1: For the half-duplex Gaussian interference λ C c2 β P(1)/(1+c2 β P(1)) , if c >c
2 23 2 2 23 1 2 24 23
channelwhere the transmitterscan cooperatewith each other, RTX .
all rate pairs (RTX,RTX) satisfying 1,2 ≤λ2C(cid:16)c223β2P2(1) , (cid:17) otherwise
1 2
(cid:16) (cid:17)
RTX min RTX +RTX ,RTX +RTX (1) 3) Phase 3: After phase 1 and 2, v1r and w1r have been
1 ≤ 1,d 1,r1 1,d 1,r2 exchanged between the sources. Node 1 and 2 can then sent
RTX minnRTX +RTX ,RTX +RTX o (2) messages jointly using the coding scheme of a two user 2-
2 ≤ 2,d 2,r1 2,d 2,r2 transmit-1-receiveantennaMIMOBC [14]. Theproblemnow
n o
are achievable, where RTX is given by (5) and (7), RTX is is to find the optimal covariance matrices of the two transmit
i,d i,r1
given by (3) and (4), and RTX is given by (6) and (8). signalsforbothreceivenode3and4.In[15],asimplemethod
Proof: We construct a 3i,-rp2hase transmission strategy as of generating MIMO BC covariance matrices is proposed by
shown in Fig. 1-(a), to show the achievability. Let w ’s and transforming the covariance matrices from its dual, MIMO
i
vi’s be the messages intended to node 3 and 4 respectively. MAC. We use this method to find the covariance matrices Σi
The specific message sent in each phase is detailed in Fig. 1- and Σ′i in our coding scheme.
(a).Inphase1 and2,the two sourcenodestransmitmessages If c13+c23 > c14+c24, generate codeword X2(vd) with
w1r and v1r to each other, and w2r and v2r to the receive length λ3N and power η1P2(2), P2(2) = γ2P2/λ3 at node 2.
nodes by broadcasting their signals using DPC. In phase 3, GeneratecodewordX1(v3r)and X2(v3r) with lengthλ3N at
after the sources exchanged their information, the system is node 1 and 2 respectively with covariance matrix Σ2, where
equivalent to a two user 2-transmit-1-receive antenna MIMO Σ2 can be found by using the results given in [15]. Let
BC. The source nodescan then jointly broadcastw3r and v3r B1 = I+hT2h2(µ3P1(2) +η2P2(2)), P1(2) = κ2P1/λ3, then
tothereceiversusingDPC[14].Further,thetwosourcenodes Σ = B−1(µ P(2) +η P(2)). Let A = 1+h Σ hT, then
1 1 2 1 3 2 2 2 1 2
Σ = A (µ P(2) + η P(2))I. Given X (v ) and X (v ), After decodingv and v , node4 can decodev if the rate
2 2 3 1 2 2 2 d 1 3r 2r 3r 1r
use DPC to generate codewordX (w ) with length λ N and of v satisfies2
1 d 3 1r
power µ P(2) at node 1. Generate codeword X (w ) and
1 1 1 3r RTX RTX +RTX +RTX (8)
X2(w3r) with length λ3N at node 1 and 2 respectively with 2,r2 ≤ 2,1 2,2 2,3
covariance matrix Σ . If c + c c + c , do DPC where
1 13 23 14 24
≤
with the reverse order. Note that in this case, the covariance
matrix becomes Σ′ = B′−1(µ P(2)+η P(2)), where B′ = λ2C c224β1P2(1) , if c24 >c23
1 1 3 1 2 2 1
wI+hehreT1hA1′2(µ=2P11(+2)h+1ηΣ31Ph2(T12)1).aNnoddΣe′23=firAst′2(dµe2cPod1(e2)s+wη3r3,Pi2(t2)c)aIn, R2T,X2 ≤λ2C (cid:16) 1+c224cβ21βP2(cid:17)(P1)(1)!, otherwise .
do so if the rate of w satisfies 24 2 2
3r
B. Outer Bound
g Σ gT
λ3C c2131µ11P1(12)!, if c13+c23 >c14+c24 2-tFraonrsTmCit,-1w-rheecneivc1e2an→ten∞na, MtheIMsOystBemC.bTehceomcaepsaacittyworeguisoenr
R1T,X3 ≤λ3C 1+g Σ′gT +gc21Σµ′2gP1T(2)+c2 η P(2)!,. oFfurtthhiesr,MwIMheOn BonCe[u1s4e]riissasnileonutt,erthbeouanchdieovnabaclehireavtaebfleorratthee.
1 1 1 13 1 1 23 1 2 active user is bounded by the single user half-duplex relay
Node 3 then decodes wd, if the rate of wd satisfioetsherwise csehtanonfealchmieavxa-flbloewr-amteinp-aciurtsb(Rou1+n,dR[2+12)]s.aHtisefinecse, with TC, the
λ3C c213µ1P1(2) , if c13+c23 >c14+c24 Ri+ ≤0≤mρai≤x1min{Ri+,1(ρi), Ri+,2(ρi)}, i=1,2 (9)
R1T,Xd ≤λ3C (cid:16) 1+g1Σc′121(cid:17)g3µ1T1+P1(c222)3η1P2(2)!,otherwise . R1++R2+ ≤∀P1+[P2<PC(g1TP1g1+g2TP2g2). (10)
(5) where C(x)=log I+x and is the unionof all rateswith
After decoding w and w , node 3 can finally decode w any power allocatio|nsP |and P that satisfies the total power
2r 3r 1r 1 2
S
if the rate of w satisfies constraint P, and
1r
RTX RTX +RTX +RTX (6) R+ (ρ )=α C c2 +c2 P +α C (1 ρ )c2 P
1,r2 ≤ 1,1 1,2 1,3 1,1 1 1 12 13 1 2 − 1 13 1
where R1+,2(ρ1)=α1C(cid:0)c213P1 +α2(cid:1)C c213P1(cid:0)+c223P2+2ϕ(cid:1)1
λ1C c213α1P1(1) , if c13 >c14 RR2++,1((ρρ2))==αα1CC(cid:0)cc2122P+(cid:1)c22+4Pα2 C+(cid:0)cα22CP (+1c−2ρP2)c+2242Pϕ2 (cid:1)
R1T,X1 ≤λ1C (cid:16) 1+c213cα2131αP21(cid:17)(P1)1(1)!, otherwise . wher2e,2ϕ12= ρ11c21(cid:0)(cid:0)3c22243P21(cid:1)P2 an2(cid:1)d ϕ(cid:0)21=4 1(cid:0) ρ2c22144c2224P1P2(cid:1)2, (cid:1)
Similarly, node 4 first decodes v3r, if the rate of v3r satisfies IpV. RECEIVERCOOPERpATION
A. Achievable Rates
g Σ gT
λ C 2 2 2 , Theorem 2: For the half-duplex Gaussian interference
R2T,X3 ≤ 3 1+g2Σ1g2T +c214µif1Pc11(32)++c2c3224>η1cP124(2+)!c24 . cahllarnanteelpwaihrser(eRt1Rhxe,rRec2Rexi)vesrasticsfaynincgooperate with each other,
λ C g Σ′gT/c2 η P(2) , otherwise RRX RRX +RRX +RRX (11)
Node 4can3the(cid:16)n d2eco1de2vd24if 1the2ra(cid:17)te of vd satisfies R12RX ≤≤ R12R,,Xdd +R12R,,Xrr11 +R12R,,Xrr22 (12)
R2T,Xd ≤λ3C 1+g2Σc122g42Tη1+P2(c221)4µif1Pc11(32)+!c,23 >c14+c24 . gbariyvePe(ar2noc1hob)ifye:avntThadbhele(ei1n,38ew-)qp.huheaarlseietiReRsiRC,fdXrosimcshge(2imv5ee)ntiosbys(3h(1o1)w6,)naanindndRF(2iiRg,2rX.)2,1iR-s(ibRg,)rXi.v1eIinns
λ C c2 η P(2) , otherwise phase 1, the signals from node 1 and 2 are received at node
3 24 1 2
 (cid:16) (cid:17) (7) 3noadnedse4x.chRaantgheerintfhoarnmdateicoondiinngphtahsees2igannadls3, bthyesetwndoinrgeceeaicvhe
1It is easy to show that if we let Σ1 = Σ′2 = diagnµ2P1(2),η3P2(2)o otheracompressedversionofwhattheyreceived.Thereceive
andΣ′1=Σ2=diagnµ3P1(2),η2P2(2)o,theachievableratesofourscheme ncoomdepsrethsseendpseirgfnoarlmanddectohdeinsiggnbayludsirinecgtltyhereacgegivreedgaitniopnhoafseth1e.
reducestotheratesgivenbyparallelcodingDPCin[10](orRDPCin[9],if
wefurtherrestricttheconditiontoc13≤c14andc24≤c23).Notethatusing
theabovecovariancematricespairsisequivalenttothecaseassumingrandom 2Note that for the transmission order given in Fig. 1-(a), v1r is encoded
phase shifts for different channels, i.e., the received signal from different andtransmittedinphase2,thereceivercandecodeitonlyafterv2r andv3r
transmitters cannotbesynchronized. beendecoded atphase1and3ofthenexttransmissionblock.
Let w ’s and v ’s be the messages intended to node 3 and Finally, node 4 can decode v if the rate of v satisfies
i i d d
4 respectively. The specific message sent in each phase is
detailedinFig.1-(b).Weoutlinethecodingschemeasfollows. R2R,Xd ≤λ3C c224P2(3) . (22)
Transmission Scheme: In Phase 1, nodes 3 and 4 are in (cid:16) (cid:17)
Wenowconsiderthedecodingofw andv .Bydecoding
receive modeand nodes1 and 2 are in transmitmode. Again, 1r 1r
w and v , a compressed version of the signals received in
since the expressions of the received signals can be easily s s
phase 1 have been exchanged between the receivers. Let σ2
shown, we omit them due to limited space. In Phase 2, node 1
andσ2 bethecompressionnoiseofthereceivedsignalatnode
3 is in receive mode and all the other nodes are in transmit 2
3 and 4 respectively. Using similar derivations as in [7], σ2
mode.In Phase3, node4 is in receivemodeand allthe other 1
and σ2 are given by
nodes are in transmit mode. 2
Outline of Achievability: 1+g Σ gT 1+g Σ gT g Σ gT 2
X1P(hwa1sre)1:anAdtnXo2d(evs11r)anwdi2th,gpeonwereartseλP11(N1) le=ngtµh1cPo1dλe1woarndds σ12 = (cid:0) 12RxR2,1Xs(cid:1)/λ(cid:0)1 −12 1x+2g(cid:1)2−Σx(cid:0)g2T1 x 2(cid:1) (23)
P2(P1)ha=seη12P:2Aλt1nroedsepe1ctaivnedly2., generateλ N lengthcodewords σ2 = 1+g2(cid:16)Σxg2T 1+g(cid:17)1Σ(cid:0)xg1T − g1Σ(cid:1)xg2T 2 (24)
X1(wd) and X2(v2r) with powers P1(22) = µ2P1/λ2 and 2 (cid:0) 2RR1,Xs(cid:1)/λ(cid:0)1 −1 1+g(cid:1)1Σx(cid:0)g1T (cid:1)
lPe2n(2g)th=codηe2wPo2r/dλs2Xr2e(swpesc)tiavnedlyX. 2A(tw2nro)d3ew4it,hgpeonweerarteP4(λ1)2N= where Σx =dia(cid:16)g P1(1), P2(1)(cid:17)(cid:0)is a 2×2 dia(cid:1)gonal matrix.
α P /λ and P(2) = α P /λ respectively. Node 3 first As discussed inn[7], since eaoch receiverhasa noisy version
1 4 2 4 2 4 2 of the received signal of the other receiver, the network is
decode w , if the rate of w satisfies
2r 2r
equivalenttoanICwithtworeceiveantennasateachreceiver.
c2 P(2) After normalizingthe noise power to 1 for all receive “anten-
RRX λ C 34 4 . (13)
1,2r2 ≤ 2 1+c2 P(2)+c2 P(2)+c2 P(1)! nas”, the equivalent channel gains between the transmit and
13 1 23 2 34 4
receive node pairs are given as c = [c √ζ c ]T, c =
Node 3 can then decode w , if the rate of w satisfies 13 13 2 14 23
s s [c √ζ c ]T,c =[√ζ c c ]T andc =[√ζ c c ]T,
23 2 24 14 1 13 14 24 1 23 24
R1R,Xs ≤λ2C c234P4(1)/(1+c213P1(2)+c223P2(2)) (14) cwhecrTePζi(1=),1S/N(1R+σ=i2)c. LceTt SPN(1R) a1n=d IcN13RcT13P=1(c1), cINT PR(11)=.
and decode v a(cid:16)nd w , if their respective rates sati(cid:17)sfy 23 23 2 2 24 24 1 2 14 14 2
2r d The capacity region of a 1-transmit-2-receive antennas IC
RRX λ C c2 P(2)/(1+c2 P(2)) (15) is not known except for the strong interference case [16]
2,2r1 ≤ 2 23 2 13 1 ( c 2 c 2 and c 2 c 2). In this case, the
14 13 23 24
R1R,Xd ≤λ2C(cid:16)c213P1(2) . (cid:17) (16) Rm||ResXsa|g|aensd≥wR1||RrXan|sd|atvi1srfiecsa||n[1b6e]||de≥cod||ed i|f|their respective rate
(cid:16) (cid:17) 1,r1 2,r1
Phase3:Atnodes1and2,generateλ N lengthcodewords
3
X (w ) and X (v ) with powers P(3) = µ P /λ and RRX λ C(SNR ) (25)
1 2r 2 d 1 3 1 3 1,r1 ≤ 1 1
Pco2(d3e)w=orηd3sPX2/λ(v3 r)esapnedcXtive(lvy.A)twniothdepo3w,geersnePra(1te)λ=3Nβ Plen/gλth R2R,Xr1 ≤λ1C(SNR2) (26)
1 s 2 2r 3 1 3 3 RRX +RRX λ min C(SNR +INR ) . (27)
and P3(2) =β2P3/λ3 respectively. Node 4 can decode v2r if 1,r1 2,r1 ≤ 1i=1,2{ i i }
c2 P(2) When c14 2 c13 2, c23 2 < c24 2, node 4 can
R2R,X2r2 ≤λ3C 1+c2 P(3)+34c23P(3)+c2 P(1)!. (17) eliminate|t|hein||ter≥fere||nce|b|yco||mpl|e|telyd|e|cod|i|ngthemessage
14 1 24 2 34 3 transmitted from node 1 and node 3 can decode by treating
Combining (15) and (17), node 4 can decode v2r if itsinterferenceasnoise.Thus,theachievableratesofw1r and
v are respectively
RRX min max RRX ,max RRX . (18) 1r
2,r2 ≤ 2,2r1 2,2r2 RRX λ log(I+SNR +INR /I+INR ) (28)
Node 4 can then dnecodes(cid:16)v , if t(cid:17)he rate(cid:16)of v sa(cid:17)tiosfies 1,r1 ≤ 1 | 1 1| | 1|
s s RRX λ C(SNR ). (29)
RRX λ C c2 P(1)/(1+c2 P(3)+c2 P(3)) . (19) 2,r1 ≤ 1 2
2,s ≤ 3 34 3 14 1 24 2 Similarly, when c 2 < c 2, c 2 c 2, the
14 13 23 24
(cid:16) (cid:17) || || || || || || ≥ || ||
After decoding v2r and vs, node 4 decodes w2r if achievable rates of w1r and v1r are respectively given by
R1R,X2r1 ≤λ3C c214P1(3)/(1+c224P2(3)) . (20) R1R,Xr1 ≤λ1C(SNR1) (30)
Combining (20) and (13)(cid:16), node 3 can decode w2(cid:17)r if R2R,Xr1 ≤λ1log(|I+SNR2+INR2|/|I+INR2|). (31)
RRX min max RRX ,max RRX . (21) When c14 2 < c13 2 and c23 2 < c24 2, node 3
1,r2 ≤ 1,2r1 1,2r2 and node||4 ca|n| deco|d|e w|| and v|| r|e|specti|v|ely|b|y treating
1r 1r
n (cid:16) (cid:17) (cid:16) (cid:17)o
interferenceas noise. The achievablerates of w and v are
3Note that w2r is the messagedecoded at phase 3ofthe previous block. 1r 1r
Itisre-encoded asX2(w2r)andrelayed totheintended receiver. then given by (28) and (31).
B. Outer Bound
5
The single user upper bounds in (9) are also upper bounds Outer Bound
4.5 c = ∞ (Tx coop.)
under RC. Further, if we let c = , the channel becomes 12
a two user 1-transmit-2-receive34ante∞nna MIMO MAC. Thus, 4 cc12 == ∞10 ( (RTxx ccoooopp..))
34
the achievable region is also bounded by this MIMO MAC 3.5 c = 10 (Rx coop.)
34
capacity, which is given by [13] 3 Interference Channel
R1++R2+ ≤C(hT1P1h1+hT2P2h2). (32) R22.5
2
V. NUMERICAL EXAMPLES
1.5
We compare our achievable region to some known results
through numerical examples. We focus on the symmetric 1
channelcase(similarresultscanbeshownfortheasymmetric 0.5
case). We set the direct channel gains as c = c = 1, the
13 24 0
crosschannelgainsasc14 =c23 =√2andtheaveragepower 0 1 2 R1 3 4 5
constraints P =5, i=1,2,3,4.
i
Fig.3. Achievable regions fortransmitter andreceiver cooperation.
5
Outer Bound
4.5 c = ∞
12 channelcapacityconstraints(see (9)), RC achievesless single
c = 10
4 c12 = 10, ∞ (RDPC) user rates under the assumed channel conditions.
12
3.5 Interference Channel Bridging the gap between the outer bound and the achiev-
able region for finite cooperative channel gains should be
3
considered in future work.
R22.5
2 REFERENCES
1.5 [1] A. Sendonaris et al., “User cooperation diversity–Part I and II,” IEEE
Trans.Comm.,vol.51,pp.1927–1948, Nov.2003.
1 [2] J.Lanemanetal.,“Cooperativediversityinwirelessnetworks:Efficient
protocolsandoutagebehavior,”IEEETrans.Inf.Thy.,vol.50,pp.3062–
0.5
3080,Dec.2004.
0 [3] A.Host-Madsen,“Anewachievablerateforcooperativediversitybased
0 1 2 3 4 5
R on generalized writing on dirty paper,” in Proc. of IEEE ISIT, p. 317,
1 June2003.
[4] A. Host-Madsen, “On the achievable rate for receiver cooperation in
Fig.2. Achievable regions fortransmitter cooperation andRDPC. ad-hocnetworks,”inProc.ofIEEEISIT,p.272,June2004.
[5] A.Host-MadsenandA.Nosratinia, “Themultiplexing gainofwireless
networks,”inProc.IEEEISIT,pp.2065–2069, Sept.2005.
Fig. 2 comparesthe achievableregionfromourTCscheme [6] C. Ng and A. Goldsmith, “Transmitter cooperation in ad-hoc wireless
with RDPC [9], [10]. It is shown that the achievable region networks:Doesdirty-papercodingbeatrelaying?,” inProc.IEEEISIT,
pp.2065–2069, Sept.2005.
usingourTC schemeis significantlylargerthanusing RDPC.
[7] C. Ng et al., “Capacity gain from two-transmitter and two-receiver
Further, the capacity gain of our TC scheme increases with cooperation,”IEEETrans.Inf.Thy.,vol.53,pp.3822–3827,Oct.2007.
the cooperation channelgain: As we increase the cooperation [8] H.Sato,“Thecapacityofthegaussianinterferencechannelunderstrong
interference,” IEEETrans.Inf.Thy.,vol.27,pp.786–788,Nov.1981.
channelgainfromc12 =10to ,theachievableregionmeets [9] K. Shum and C. Sung, “Transmitter cooperation by recycling dirty
∞
the outer bound. On the other hand, the achievable region of paper,”inProc.IEEEISIT,pp.777–781,July2008.
RDPC does not increase as long as the cooperation channel [10] N.Fawazetal.,“Whennetworkcodinganddirtypapercodingmeetin
acooperative adhocnetwork,” IEEETrans.Wireless Commun.,vol.7,
is not a capacity threshold (see equations (8) and (9) in [9]).
pp.1862–1867, May2008.
The achievableregionsare also comparedto the capacityof a [11] G.Krameretal.,“Cooperativestrategiesandcapacitytheoremsforrelay
standard strong IC (withoutnode cooperation).It is clear that networks,”IEEETrans.Inf.Thy.,vol.51,pp.3037–3063, Sept.2005.
[12] A.Host-Madsen andJ.Zhang,“Capacity boundsandpowerallocation
by allowingnodecooperation,the achievableregionincreases
for wireless relay channels,” IEEETrans. Inf. Thy., vol. 51, pp. 2020–
significantly. 2040,June2005.
Fig. 3 shows the achievable regions for both TC and RC. [13] E. Telatar, “Capacity of multi-antenna Gaussian channels,” European
Trans.Telecomm.,vol.10,pp.585–596, Nov.1999.
SimilartoTC,theachievableregionofRCalsoincreaseswith
[14] H. Weingarten et al., “The capacity region of the Gaussian multiple-
cooperation channel gain. When c = , the achievable re- inputmultiple-outputbroadcastchannel,”IEEETrans.Inf.Thy.,vol.52,
34
gionofRC overlapswiththe outerboun∞d.TheRCachievable pp.3936–3964, Sept.2006.
[15] S. Vishwanath et al., “Duality, achievable rates, and sum-rate capacity
region is also compared with TC. When c12 = c34 = 10, ofGaussianMIMObroadcast channel,” IEEETrans.Inf.Thy.,vol.49,
the achievable region of TC is strictly larger than RC. When pp.2658–2668, Oct.2003.
c = c = , both schemes meet their respective outer [16] S. Vishwanath et al., “On the capacity of vector gaussian interference
12 34 ∞ channels,” IEEEInf.Thy.Workshop,pp.365–369,Oct.2004.
bound. However, due to the single user half-duplex relay