The Multiplexing Gain of MIMO X-Channels with Partial Transmit Side-Information

0.17 MB·English

by  Natasha Devroye

1
X
The Multiplexing Gain of MIMO -Channels
with Partial Transmit Side-Information
Natasha Devroye and Masoud Sharif
Harvard University, Boston University
7
0
Abstract—THIS PAPER IS ELIGIBLE FOR THE STU- isillustrated2.Thetwotransmitters,Tx1andTx2operate
0
DENT PAPER AWARD. In this paper, we obtain the scaling independently, and each wishes to send messages to each
2
laws of the sum-rate capacity of a MIMO X-channel, a
of the two non-cooperating receivers Rx 1 and Rx 2.
n 2 independent sender, 2 independent receiver channel with
The messages are numbered 11, 12, 21 and 22, with the
a messages from each transmitter to each receiver, at high
J signal to noise ratios (SNR). The X-channel has sparked conventionthatindiceswillfollowtheform(Tx#Rx#).We
5 recent interest in the context of cooperative networks and it note that the MIMO interference channel is embedded in
1 encompasses theinterference,multipleaccess, and broadcast theMIMOX channelandmaybeobtainedbyeliminating
channels as special cases. Here, we consider the case with the cross-over messages 12 and 21. The messages 11, 12,

] partially cooperative transmitters in which only partial and
T asymmetric side-information is available at one of the trans- 21 and 22 will be simultaneously transmitted at the rates
I mitters. It is proved that when there are M antennas at all R11,R12,R21andR22respectively.Thesum-rateachieved
s. four nodes, the sum-rate scales like 2MlogSNR which is in bythistupleisdefinedtobe R=△ R +R +R +R .
11 12 21 22
c sharpcontrastto ⌊4M⌋,4M logSNRfornon-cooperativeX-
3 3 InFig.1(2),theMIMOcognitiveX-channelisdepicted.
[ channels[14],[11ˆ].Thisfurth˜erprovesthat,intermsofsum-
We define this to be a channel in which Tx 2 knows
rate scaling at high SNR, partial side-information at one of
1 thetransmittersandfullside-informationatbothtransmitters message 11, but not message 12, of Tx 1. Tx 1 knows
v
are equivalent in the MIMO X-channel. none of Tx 2’s messages. We use the term partial side-
2
information to refer to the the fact that Tx 2 knows
9
0 only one of Tx 1’s messages, and the term asymmetric
1 I. INTRODUCTION side-information to refer to the fact that Tx 2 has side-
0 information regarding Tx 1 but not vice versa. We note
7
Cooperation in wireless networks has sparked much that the side-information could have been both message
0
recentinterestin theresearchcommunity.Havingwireless 11 and 12 or just message 12 at Tx 2. In the latter event,
/
s nodes cooperate at the transmitting and/or receiving end, analogous results are obtained by permuting the indices
c
: can,forexample,improverates,diversity,orpowerutiliza- in our results. In the former, the rates achieved could be
v
tion. How much gain cooperation provides depends on a improvedatlow-mediumSNR,butaswewillsee,theextra
i
X numberoffactors,includingthesignaltonoiseratio(SNR) information contained in knowing the message 12 as well
r and the amount of side information at the transmitters. as message 11 is unnecessary as SNR → ∞, where we
a
In this work, we look at the value of partial transmitter will show that the partial information is sum-rate scaling
cooperation,in terms of the scaling law of the sum-rate in optimal. In Fig. 1(3) shows a MIMO broadcastchannelin
the MIMO X-channel at high SNR .1 which both Txs share all their messages (full, symmetric
The multiplexing gain of the sum-rate of the MIMO side-information)[15], [1] with 2M transmitantennasand
X-channel has been recently studied [14], [11], [12]. The two independentRxs with M antennas each.
MIMO X-channel is a simple 2 transmitter (2 Tx), 2 The multiplexing gain of the channel depicted in Fig.
receiver (2 Rx) channel in which each Tx has a message 1(1) is defined as the limit of the ratio of the maximal
for each Rx. Its study is of theoretical interest, as it achieved sum-rate R in the capacity region C , to the
i i
encompasses classical channels such as the interference, log(SNR)3 as the SNR →∞, or
the multiple access and the broadcast channels. While the
maxR (SNR)
classical MIMO X channel forbids cooperation between M = lim i .
i
the two Tx, in this work we allow for transmitter cooper- SNR→∞ log(SNR)
ation, in the form of transmitter side-information. ThemultiplexinggainofthechannelofFig.1(1)hasbeen
The three channels shown in Fig. 1 illustrate three
multiuserMIMOchannelswithdifferentamountsoftrans- 2Throughout this work the term MIMO will denote that there are M
mitterside-information.InFig.1(1),theMIMOX-channel antennas ateachofthefournodesTx1,Tx2,Rx1andRx2.Inorder
tosimplifyresultswewillsometimessetM =1,inwhichcasewedrop
thewordMIMO.
1Althoughtheachievablerateregionwederiveisforgeneralchannels, 3Note that the usual factor 1 is omitted in any rate expressions, but
themultiplexinggainresultsholdonlyforGaussiannoisechannels,hence rather the number of times th2e sum-rate looks like log(SNR) is the
wewilloftenomitthetermGaussianforbrevity. multiplexing gain.
2
Message 11 Message 11 Message 11 II. THEMIMO COGNITIVE X CHANNEL
Tx 1 Rx 1 Tx 1 Rx 1 Tx 1 Rx 1
Message 21 Message 12 Message 11M essage 21 Message 12 Message 21 Message 12 chaInnntehliswsitehctiMonawnetensnhaoswattheaatcthheofMtIhMeO4 cnoogdneistivheasXa
Tx 2 Message 22 Rx 2 Tx 2 Message 22 Rx 2 Tx 2 Message 22 Rx 2 multiplexing gain of 2M, that of the broadcast channel
(1) X Channel (2) Cognitive X Channel (3) Broadcast Channel of Fig. 1(3). The MIMO cognitive X channel is shown
Fig. 1. Three2×2MIMO X-channels whosemultiplexing gains are in Fig. 2(a). The word cognitive stems from the work [5]
contrasted here. All Rxs decode independently. (1) MIMO X-channel in which an interference channel with asymmetric side-
withonemessagefromeachTxtoeachRxandnoside-information. (2) information between Txs is called a cognitive radio chan-
MIMOcognitiveX-channelwithpartialasymmetricside-information:Tx
nel (also known as an interference channel with degraded
2 knows message 11 but not message 12, and Tx 1 knows none of Tx
2’s messages. (3) MIMO Broadcast channel: 2M transmitting antennas message sets in [13], [16]). In our figures, we denote
delivering messagestotwoindependent M antenna Rxs. side-information using a double arrow accompanied by
the message that is known a priori. In the MIMO X-
channelthereare fourmessages:11, 12,21 and 22,which
are encoded as M ,M ,M and M respectively.Our
11 12 21 22
recently studied [11], [14]. The MIMO interference chan-
channel is a standard interference channel [2] with direct
nel embedded in channel (1) is known to achieve a mul-
channelcoefficientsof1andcross-overcoefficientsofα
12
tiplexing gain of M [12]. This channel has no cross-over
andα .TheM-dimensionaltransmittedrandomvariables
21
messages(messages12and21).Interestingly,whencross-
X ∈X andX ∈X arereceivedasthesignalsY ∈Y
1 1 2 2 1 1
over information is present, as in the MIMO X-channel
and Y ∈ Y in the sets according to the conditional
of (1), the multiplexinggain lies in the range[⌊4M⌋,4M] 2 2
3 3 distributions p(y1|x1,x2) and p(y2|x1,x2). We consider
[11],[14],possiblyimprovingupontheMIMOinterference
an additive white Gaussian noise channel,
channel. The MIMO broadcast channel of Fig. 1(3) is
known to have a multiplexing gain of 2M, equal to the Y1 =X1+α21X2+N1 (1)
numberof transmit,andreceiveantennas.Inthe broadcast Y =α X +X +N , (2)
2 12 1 2 2
channel, full, symmetric Tx side-information is used, that
where N ∼ N(0,N ) and N ∼ N(0,N ) are inde-
is, both Tx 1 and Tx 2 know all messages of the other. In 1 1 2 2
pendentand we assume individualaveragetransmit power
thiswork,weshowthatsuchfulltransmitside-information
constraints of P (P ) on X (X resp.). The directed
is not strictly necessary to achieve the same sum-rate 1 2 1 2
double arrow from X to X denotes partial asymmetric
scaling of 2M. We show that the partial side-information, 1 2
side-information: the encoding M is known to Tx 2.
shown in the MIMO cognitive X-channel also achieves a 11
Standard definitions of achievable rates and regions are
sum-rate scaling of 2M as SNR →∞.
employed[4],[5].Althoughourachievablerateregionwill
In this work, we assume non-causal side-information,
bedefinedforfinite alphabetsets, inordertodeterminean
whichmaybeeithersymmetricorasymmetric,butrequires
achievable region for the Gaussian noise channel, specific
that all messages are known fully before transmission
forms of the random variables described in Thm. 1 are
starts. This is in sharp contrast to the work [10], in which
assumed. As in [3], [7], [9], Thm. 1 can readily be
similar 2 Tx, 2 Rx channels with causal side-information
extended to memoryless channels with discrete time and
are considered. In [10] it is shown that even if both the
continuousalphabetsbyfinelyquantizingtheinput,output,
Txs and Rxs may cooperate using noisy links, in a causal
and interference variables (Gaussian in this case).
fashion(althoughfullduplextransmissionispermitted)the
We now outline an achievable rate region for the Gaus-
multiplexinggainisalwayslimitedto1whenM =1.This sian MIMO cognitive X-channel, which will be used to
isin sharpcontrastto whennon-causalside-informationis
demonstrate a sum-rate scaling law of 2M. The capacity
present, resulting in a multiplexing gain of 2.
region of the Gaussian MIMO broadcast channel [15] is
This paper is structured as follows. In section II we achieved using Costa’s dirty-paper coding techniques [3].
define the MIMO cognitive X channel and demonstrate Inthe MIMO cognitiveX-channel,atTx 1,the encodings
an achievable rate region, which we to show that the M11 andM12 maybejointlygenerated,forexampleusing
multiplexing gain of the sum-rate is 2M. In section III a dirty-paper like coding scheme. That is, one message
we explore the effect of cross-over information on the may treat the other as non-causally known interference
multiplexing gain: we define and demonstrate that the and code so as to mitigate it. At Tx 2, not only may the
multiplexing gain of the cognitive interference channel encodingsM21 andM22 bejointlydesigned,buttheymay
(wherenocross-overmessagesarepresent)is1,incontrast additionallyuseM11 asa-prioriknowninterference.Thus,
to the multiplexing gain of 2 seen in the cognitive X Tx 2 could encode M22 so as to potentially mitigate the
channel. These results allow us to compare the achievable interferenceY2 will experiencefrom M11 as well as M21.
rate regions of the cognitive and cognitive X channels in LetR11 betheratefromM11 →Y1,R12 fromM12 →Y2,
section IV at various SNRs. We conclude in section V. R21 from M21 →Y1 and R22 from M22 →Y2.
△
Due to space limitations, most of the proofs are deferred Theorem 1: LetZ =(Y ,Y ,X ,X ,M ,M ,M ,
1 2 1 2 11 12 21
to [6] which may be found online. M ), and let P be the set of distributions on Z that can
22
3
N1 N1
M11 Message 11 Message 1 lim maxR11+R12+R21+R22 =2M, (4)
M12 X1 Tx 1 Rx 1 Y1 X1 Tx 1 1 Rx 1 Y1 P→∞ logP
MP2o1wer PM1essage 11Message 21 Message 12 Power P1Message 1α21 1 α12 CwXh−ecroeg,thwehemreaxCXis−ctaokgeinstohveecraaplalc(itRy1r1e,gRio1n2,oRf2th1e,RM2I2M) O∈
MPo22werX P22Tx 2 Message 22 Rx 2 Y2 PowerX P22Tx 2 Message 2 Rx 2 Y2 cognitive X-channel.
Proof: We sketch the proof for M = 1, and defer
N2 N2
a) Cognitive X Channel b) Cognitive Channel details, as well as the more involved proof for general M
to[6].Roughlyspeaking,the generalM caseisprovenby
Fig.2. AdditiveGaussiannoiseinterferencechannelswithcross-
over parameters α ,α , transmitted encodings X ,X with evaluating the same mutual information terms of Thm. 1,
12 21 1 2
expected transmit power limitations P and P , and received
1 2 as done for M = 1, with the added complications (such
signals Y and Y . (a) Cognitive X channel: four messages
1 2
encoded as M , M , M , M . M is the partial and as matrix inversions) that arise from considering vectors
11 12 21 22 11
asymmetricmessageknowledgeatX2.(b)Cognitivechannel:two rather than scalars.
messages encoded as M ,M . X is the asymmetric side-
information known at X .11 22 1 First, note that the multiplexing gain of the MIMO
2
broadcast channel, whose capacity region outer bounds
ours, with 2 transmit antennasand single receive antennas
be decomposed into the form at the Rxs is 2. We will in fact prove that 2 is achievable
p(m |m )p(m )p(m )p(m |m ,m ) using the scheme of Thm. 1. To do so, we specify forms
11 12 12 21 22 11 21
p(x |m ,m )p(x |m ,m ,m ) (3) for the variables,and then optimize the dirty paper coding
1 11 12 2 11 21 22
p(y |x ,x )p(y |x ,x ), parameters,similar to Costa’stechnique[3].The Gaussian
1 1 2 2 1 2
distributions we assume on all variables are of the form
where we additionally require p(m ,m ) =
12 22
p(m12)p(m22). For any Z ∈ P, let S(Z) be the set M11 =U11+γ1U12 U11 ∼N(0,P11)
of all tuples (R11,R12,R21,R22) of non-negative real M12 =U12, U12 ∼N(0,P12)
numbers such that: M21 =U21, U21 ∼N(0,P21)
M =U +γ (U +a U ) U ∼N(0,P )
22 22 2 21 12 11 22 22
R11 ≤I(M11;Y1|M21)−I(M11;M12) X1 =U11+U12
R21 ≤I(M21;Y1|M11) X =U +U + (1−β)P2U
R11+R21 ≤I(M11,M21;Y1)−I(M11;M12) 2 21 22 q P11 11
Y = 1+a (1−β)P2 U +U +a (U +U )+N
R12 ≤I(M12;Y2|M22) 1 “ 21q P11 ” 11 12 21 21 22 1
R12+RR2222 ≤≤II((MM2122;,MY22|M2;Y122))−−II((MM2222;;MM1111,,MM2211)) Y2 =“a12+q(1−Pβ11)P2” U11+a12U12+(U21+U22)+N2.
where P =P +P and βP =P +P .
1 11 12 2 21 22
Any element in the closure of ∪Z∈PS(Z) is achievable. Here the variables U ,U ,U ,U are all indepen-
11 12 21 22
Proof: The codebook generation, encoding, decod-
dent, encoding the four messages to be transmitted. The
ing schemes and formal probability of error analysis are
β parameter divides power at Tx 2: βP is used in
2
deferred to the manuscript in preparation [6] which may
transmitting its own messages, m and m , while the
21 22
befoundonline.Heuristically,noticethatthechannelfrom
remainder of the power, (1 − β)P is used to reinforce
2
(M ,M )→Y isamultipleaccesswhichemploysdirty
11 21 1 the message encoded as m of Tx 1. Notice that because
11
paper coding [3], reducing the rate R by I(M ;M )
11 11 12 we only assume partial message knowledge (only one of
(like in Gel’fand-Pinsker [8] coding). Similarly, for the
the two messages of Tx 1 is known at Tx 2), only one
MAC (M ,M ) → Y the encodings M and M
12 22 2 12 22 of these messages gets reinforced at Tx 2. Also note
are independent (this is true in particular in the Gaussian
that p(m ,m ) = p(m )p(m ) and p(m ,m ) =
11 21 11 21 12 22
case of interest in the next subsection, and so we simplify
p(m )p(m ) as needed in Thm. 1. The rates R =
12 22 1
our theorem by ensuring the condition p(m ,m ) =
12 22 R + R and R = R + R to each Rx can be
11 21 2 12 22
p(m )p(m )) so that the regular MAC equations hold.
12 22 calculatedseparately.Eachcan bemaximizedwith respect
The rate R is subject to a penalty of I(M ;M ,M )
22 22 11 21 to the relevant dirty-paper coding parameter (γ for Tx
1
inordertoguaranteefindingann-sequencem inthede-
22 1, and γ for Tx 2). The bounds of Thm. 1 may be
2
sired bin thatis jointlytypicalwith anygiven(m ,m ).
11 21 evaluated by combining the appropriate determinants of
sub-matrices of the overall covariance matrix E[ΘΘT]
△
whereΘ=(M ,M ,M ,M ,Y ,Y ).Thesixbounds
A. MIMO Cognitive X-channel multiplexing gain is 2M 11 21 12 22 1 2
ofThm.1areevaluatedexplicitlyin[6]online;wesimply
Weusetheaboveachievablerateregiontoshowthatthe demonstrate the resulting expressions for the sum-rate to
sum-rateoftheMIMOX-channelwithpartialasymmetric Rx1,R aswellastoRx2,R intheexpressions(5)and
1 2
side-information has a multiplexing gain 2M. (6) resp. In order to simplify the expressions, we have set
Corollary 2: ConsidertheMIMOadditiveGaussianX-
channelwithpartialasymmetricside-informationdescribed θ =△ a (1−β)P2 and η =△ (a +θ).
in eqns. (1), (2) and Fig. 2(a) with P1 = P2 = P. Then 21s P11 12
4
1 P (P (1+θ)2+P +α2 (P +P )+N )
R +R ≤ log 11 11 12 21 21 22 1 (5)
11 21 2 2„γ2(P (P +α2 P +N ))−2γ P P +P (P +α2 P +N )«
1 12 11 21 22 1 1 11 12 11 12 21 22 1
1 (P η2+a2 P +P +P +N )(γ2P (P +γ2η2P )+P P )
R +R ≤ log 11 12 12 21 22 2 1 12 22 2 11 11 22
12 22 2 2„(P +γ2P ))γ2(P η2+P )(P +N )+γ (−2P (P η2+P )+P (P η2+P +N )«
11 1 12 2 11 21 22 2 2 22 11 21 22 11 21 2
(6)
We could search for the γ and γ that jointly optimize weinvestigatewhetherpartialasymmetricside-information
1 2
R +R . However, noticing that R depends only on γ , is always equivalent to full symmetric transmitter side
1 2 1 1
we heuristically select γ to maximize R , as information in terms of sum-rate scaling as SNR → ∞.
1 1
To do so we look at another channel with partial asym-
P (1+θ)
γ = 11 . (7) metric side information at the transmitters: the recently
1 P (1+θ)2+α2 P +N
11 21 22 1 exploredcognitiveinterferencechannel(alsoknownasthe
Whenwesubstitutethisγ intotheboundsonRx1’stotal interference channel with degraded message sets [13] or
1
sum-rateweobtainthebound(8).Noticethatanimportant the cognitive radio channel [5]), shown in Fig. 2(b). We
cancellation occurs in the denominator of (8) when the consider the same additive Gaussian noise channel as in
optimal γ is substituted in the sum-rate expression. (1), (2). The only differencewith the cognitiveX-channel
1
To maximize the R bound, we had set γ as in (7). is the absence of cross-over messages 12 and 21. We will
1 1
Although we could maximize R with respect to γ , we see that while partial asymmetric side information in the
2 2
use a simpler and more heuristic approach and simply X-channel results in the same sum-rate scaling as a fully
minimizethedenominatorofthesum-rateR =R +R cooperative (at the transmitters) X-channel, the opposite
2 12 22
with respect to γ , which yields γ = P22 . It is is true of partial asymmetric side information in the in-
2 2 P22+N2
interesting to note that this is exactly the same dirty paper terference channel: at high SNR its sum-rate scales like
codingparameterasCostaderives.Itisintuitivelypleasing, the interference channel. In other words, although partial
andalthoughitdoesnotstrictlymaximizeR withrespect side-information may help the interference channel in a
2
to γ , as we will see shortly, it performs sufficiently well medium SNR-regime [5], [13], at high SNR, one cannot
2
in the limit of large SNR, thus performs asymptotically improve the scaling law of the sum-rate. The Gaussian
optimally. If we fix P (does not scale with P) and let cognitiveinterferencechannelconsideredhere is the same
22
P =P =P allscalelikeP,subjecttoP +P =P channelasthatof[13],whereitscapacityregionisderived
11 12 21 11 12
andP21+P22 =P,thentheboundonthetotalsumrateto for the case α21 ≤ 1, and sum-rate capacity is found for
both Rxs R11+R12+R21+R22 scales like 2logP. This α21 > 1. The next theorem is a direct result from this
can be seen by noting that γ remains fixed and γ → 1 capacity region.
2 1
as P → ∞. Keeping P fixed was crucial for achieving Theorem 3: Considerthe Gaussianinterferencechannel
22
the logP scaling in R . Intuitively, this is because of where Tx 2 has non-causal knowledge of the message of
1
the asymmetric message knowledge; the interference Tx Tx 1, described in eqns. (1), (2) and Fig. 2(b) with P =
1
2 causes the Tx 1 is not mitigated. Keeping P constant P =P. Then
22 2
still allows Tx 2 to dirty paper code, or mitigate the
interference caused by M11 and M21 to Rx 2’s signal lim max(R1,R2)∈CR1+R2 =1, (9)
Y , while causing asymptotically (as P ,P ,P → ∞) P→∞ logP
2 11 12 21
negligible interference to Y . We note that the sum-rate
1
where R corresponds to the rates from the i-th source to
scaling does not depend on the power division parameter i
thei-thRx, andC isthecapacityregionofthechannel.
β. That is, the sum-rate scaling remains 2 for any β > 0
(meaning some power is in fact allocated to transmit the The proof of this result is deferred to work [6] which
messages of Tx 2, and this power will then → ∞). The may be viewed online. It employs eqns. (24), (25) and
joint optimization of γ ,γ could lead to higher rates in Corollary 4.1 of [13]. This theorem shows that the cog-
1 2
lower SNR regimes. However, at high SNR our choice of nitive channel of Fig. 2(b) behaves like the interference
parameters leads to an optimal sum-rate scaling. channel at high SNR. Thus, when partial side-information
is available at the Txs, although achievable rates may be
enhancedin the low and medium SNR range as compared
III. THE COGNITIVE INTERFERENCE CHANNEL
totheinterferencechannel,atlargeSNR,thetwochannels
Intheprevioussectionwe demonstratedthatthescaling scale in the same manner. This is in sharp contrast to
law of the sum-rate of the MIMO cognitive X channel, the cognitive X channel, where partial asymmetric side
with partial transmitter side-information is 2M, which informationresultsinthesamesum-ratescalingasthefully
is optimal in the limit as SNR → ∞. In this section cooperative broadcast channel at high SNR.
5
1 (P (1+θ)2+P +α2 (P +P )+N )(P (1+θ)2+α2 P +N )
R +R ≤ log 11 12 21 21 22 1 11 21 22 1 (8)
11 21 2 2 (α2 P +N )(P (1+θ)2+P +α2 P +N )
(cid:18) 21 22 1 11 12 21 22 1 (cid:19)
Achievable rate regions for channels Achievable rate regions for channels Achievable rate regions for channels
1.4 at SNR 0, a21=0.2, a12=0.8 3 at SNR 10, a21=0.2, a12=0.8 10 at SNR 50, a21=0.2, a12=0.8
MIMO broadcast channel MIMO broadcast channel
1.2 Cognitive X channel 2.5 Cognitive X channel 9
Cognitive channel Cognitive channel 8
1
Sumrate R200..68 Sumrate R21.125 Sumrate R24567
0.4 3 MIMO broadcast channel
0.2 0.5 2 Cognitive X channel
1 Cognitive channel
00 0.1 0.2 Su0.3mrate 0R.4 0.5 0.6 0.7 00 0.2 0.4 0.6 S0.8umra1te R1.2 1.4 1.6 1.8 2 00 1 2 3Sum4rate5 R 6 7 8 9
1 1 1
(a) (b) (c)
SNR = 0dB SNR=10dB SNR=50dB
Fig.3. Comparisonofthecognitive interference andthecognitive X channels atvarious SNRs.
IV. COMPARISON OF COGNITIVE AND COGNITIVEX does not give any gain, in terms of scaling, over an
CHANNEL REGIONS ATVARIOUS SNRS interference channel in which there is no Tx cooperation.
In this section, we numerically evaluate the capacity
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theoreticperspective,”SubmittedtoIEEETrans.Inf.Theory,2006.
information at the Txs. This could lead one to think that [14] M. Maddah-Ali, A. Motahari, and A. Khandani, “Signaling over
partial asymmetric side-information, rather than full side- MIMOmulti-basesystems:combinationofmulti-accessandbroad-
castschemes,”in2006IEEEInternationalSymposiumonInforma-
information (or cooperation) between Txs always yields
tionTheory,July2006.
the optimal sum-rate scaling. However, we then showed [15] H.Weingarten,Y.Steinberg,andS.Shamai,“Thecapacityregionof
that this is not the case: in the interference channel with theGaussianMIMObroadcastchannel,”SubmittedtoIEEETrans.
Inf.Theory,July2004.
asymmetric side-information, where no cross-over infor-
[16] W. Wu, S. Vishwanath, and A. Aripostathis, “On the capacity of
mation is permitted, the scaling of the sum-rate for the interferencechannelwitdegradedmessagesets,”SubmittedtoIEEE
singleantennachannelis1,ratherthantheoptimal2.This Trans.Inf.Theory,2006.
demonstrates that at large SNR, asymmetric cooperation