Capacity Scaling of SDMA in Wireless Ad Hoc Networks

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by  Marios Kountouris

Capacity Scaling of SDMA in Wireless Ad Hoc
Networks
Marios Kountouris and Jeffrey G. Andrews
Abstract—We consider an ad hoc network in which each and each transmitter attempts communication with multiple
multi-antennatransmittersendsindependentstreamstomultiple receivers, each located at a fixed distance away from it.
receivers in a Poisson fieldof interferers. We providetheoutage
probability and transmission capacity scaling laws, aiming at A. Related work
9 investigating the fundamental limits of Space Division Multiple Therehasbeensignificantworkonfindingthetransmission
0 Access(SDMA).Wefirstshow thatsuperlinearcapacityscaling capacityofmulti-antennaadhocnetworksinaPoissonfieldof
with the number of receive/transmit antennas can be achieved
0
using dirty paper coding. Nevertheless, the potential benefits interferers.In[4]thespectralefficiencyofasingle-hopadhoc
2
of multi-stream, multi-antenna communications fall off quickly network with MIMO links and fixed density of interference

n if linear precoding is employed, leading to sublinear capacity is studied, and it is shown that the average per-link SINR
a growth in the case of single-antenna receivers. A key finding is increases with the number of receive antennas N as N2/α
J that receive antenna array processing is of vital importance in
whereαisthepathlossexponent.Basedontoolsfromstochas-
6 SDMAadhocnetworks,asameanstocanceltheincreasedresid-
1 ual interference and boost the signal power through diversity. tic geometry, the authors in [5] provide capacity scaling laws
when the N antennas are used to cancel the N −1 strongest

T] I. INTRODUCTION interferers.In[6]theperformanceofdiversity-orientedreceive
processing (e.g. maximum ratio combining) is analyzed Both
I The last ten years have witnessed the transition of multi-
. the above receive strategies achieve a sublinear density in-
cs antenna (MIMO) communication from a theoretical concept crease with N (of order N1−α2 and Nα2 respectively), which
[ toapracticaltechniqueforenhancingperformanceofwireless corresponds to only a logarithmic increase in per-link rate.
1 networks. Point-to-point (single-user) MIMO communication Interestingly, it have been recently shown that using partial
promises large gains for both channelcapacity and reliability,
v interference cancellation combined with array gain allows for
exhibiting linear capacity scaling with the minimum of the
8 lineardensityscalingwith N [7].SpatialmultiplexingMIMO
3 number of receive and transmit antennas [1], [2].
techniques with perfect channel knowledge at the transmitter
5 Fundamental information theoretic results advocate spatial
is analyzed in [8], showing the benefits from adaptive rate
2
sharingofthechannelbytheusers.Inspacedivisionmultiple
control.
.
1 access(SDMA),theresultingmultiuserinterferenceishandled
0 by the multiple antennas which in addition to providing per- B. Contributions
9
link diversity also give the degrees of freedom necessary for Inallpriorwork,eachtransmittersendsasingleormultiple
0
spatial separation of the users. SDMA schemes, also known streams to only one receiver, whereas here we study multi-
:
v asmultiuserMIMO,allow fora directgainin multipleaccess stream, multi-receiver transmission. A key finding is that
i
X capacityproportionalto the numberof transmitantennas.The the transmission capacity under dirty paper coding scales as
r capacity-achieving strategy for MIMO broadcast channels is N1+α2. To the best of our knowledge, this is the first result
a dirty paper coding (DPC) [3], which is a theoretical pre- in either centralized or ad hoc networks, where super linear
interference cancellation technique that requires perfect chan- capacity growth with the number of antennas is obtained.
nel knowledge at the transmitter. However,theinterferenceboostduetoaggressivelinearmulti-
In this paper we are interested in the throughput gains stream transmission techniques leads back to a sublinear
that multiple antennas and multiuser MIMO may provide in scaling. We also highlight that receive antenna processing is
uncoordinated ad hoc networks. Our focus is to determine able to restore the linear capacity scaling in ad hoc networks
the transmission capacity and outage probability scaling with by utilizing a significant fraction of the available degrees
the number of antennas as a function of network parameters. of freedom to cancel the inter-node interference. A novel
We aim at characterizing the fundamental limits of space and general methodology for quantifying ad hoc capacity
division multiple access (SDMA) for both linear and non- is developed when the total interference consists of inter-
linear precodingcombinedwith variousreceive antenna array user(SDMA)andPoissonfieldinterference.Interestingly,this
processingstrategies.Forthat,weconsideranetworkinwhich frameworkcan be used to analyze the performanceof SDMA
transmitters are randomly distributed on an infinite plane with limited feedback.
accordingtoa2-DhomogeneousPoissonpointprocess(PPP),
II. NETWORK MODELAND PRELIMINARIES
We consider a network in which the transmitting nodes
M.KountourisandJ.G.AndrewsarewiththeUniversityofTexasatAustin.
Thisresearch hasbeensupportedbytheDARPAIT-MANETprogram. are distributedaccordingto a stationaryPoisson pointprocess
(PPP) with intensity λ on the plane. This is a realistic model successfullydecodedifSINR≥β andforafailureprobability
assuming that the transmitting nodes in the network are ǫ, we have
randomlyandindependentlylocatedanddonotcooperate.The P =P(SINR≤β)≤ǫ. (3)
out
signal strength is subject to pathloss attenuation model, |·|−α
The SINR statistics are a function of the interferers density
for a distance d with exponent α > 2 as well as small scale
λ and P is clearly an increasing function of λ. In this
unit-mean Rayleigh fading. The interferingnodes constitute a out
paper, multiple streams are sent by each transmitter, thus
markedPPP,denotedasΠ={X ,I },whereeachX ∈R2 is
i i i differentSINRstatistics maybeseenonthedifferentstreams,
the location of the i-th interferingtransmitter,and with marks
resulting on a per-stream outage constraint ǫ . If the target
I thatdenotethefadingfactorsonthepowertransmittedfrom k
i SINR on stream k is given by β , the maximum density of
the i-th node and then received by a typical receiver. k
concurrent (single-stream) transmissions λ per m2 such that
Let all nodes transmit with the same power ρ and H be ǫ
0 (3) is satisfied, is defined as
the signal fading between a typical receiver and its intended
transmitter,labeledT0.Assumingthatallreceiversarelocated λǫ =max{λ:P(SINRk ≤βk)≤ǫk, ∀k}. (4)
atafixeddistanceD awayfromtheirtransmitter,theresulting
Inouranalysis,weassumethattheSINRstatisticsareidentical
signal-to-interference-plus-noiseratio (SINR) is given by
for each receiver. In general, if target SINR and outage
SINR= ρH0D−α (1) constraints are specified for each stream (subchannel), then
ρ I |X |−α+η the weakest stream will become the limiting factor in the
i∈Π(λ) i i
P optimalcontentiondensity,andhencetheoptimalareaspectral
where η is the background noise power and I is the fading
i efficiency. Assuming transmission at the Shannon target rate
coefficient from the i-th interferer (with H ,I ,I ,... inde-
0 1 2 b=log (1+β)bps/Hz,the area spectralefficiencyis defined
pendent). In contrast to the above related work, we explicitly 2
as
considernoise, whoseeffectis importantin the power-limited
C =Kλ (1−ǫ)b bps/Hz/m2 (5)
regime. ǫ ǫ
and depends on the number 1≤K ≤M spatial streams sent
A. MIMO SDMA Channel Model
by each source node.
We consider a network in which each transmitter with M
III. DIRTY PAPERCODING
antennas communicates simultaneously with |K| = K ≤
M receivers, each with N receive antennas. In this point- In this section, we derive upper and lower capacity bounds
to-multipoint context, each of the K streams sent by the whendirtypapercodingisemployed.Despitebeingcapacity-
transmitter contains a separate message destined to different achieving for MIMO broadcast channels, DPC is not neces-
receivers. By the stationarity of the Poisson process, it is sarilyoptimalinadhocnetworks,whichcanbeconsideredas
sufficient to analyze the the performance of a typical TX compound interference channels. However, joint optimization
- multi-RX link, which we refer to as TX and RX(k), for ofallprecodingmatricesisa challengingtask,whichrequires
0 0
k = 1,...,K. From the perspective of each typical receiver, global CSI of the instantaneous channel conditions of all
thesetofinterferers(whichistheentiretransmitprocesswith transmitting nodes (interferers). Furthermore, such optimiza-
the exception of TX ) also form a homogeneous PPP due to tion induces heterogeneous and not necessarily Poisson point
0
Slivnyak’s Theorem [9]. processesforwhichclosed-formresultsarehardorimpossible
The received signal y at reveiver k ∈K is given by to obtain.
k
Wefirstderivetwogeneralboundsontheoutageprobability
y = P D−α/2H x + P |X |−α/2H x +n when the channel gain follows a chi-square (χ2) distribution.
k rM 0k k rM i ik i LetL denotetheLaplacetransformoftheprobabilitydensity
Xi∈Π Y
(2) function (pdf) of the interference Y, defined as LY(s) =
where H0k ∈CN×M is the channel between T0 and receiver 0∞e−syfY(y)dy = E e−sY . The Laplace transform of the
k, Hik ∈ CN×M is the channel between receiver k and nRoise is defined as LN((cid:2)s)=e(cid:3)ηs
interferingtransmittersT ,x isthenormalizedtransmitsignal Lemma 1: The success probability in a random access
i k
vector, and n is complex additive Gaussian noise. Unless network,wherethechannelfadingfollowsachi-squaredistri-
otherwise stated, we assume that each transmitter has perfect bution with 2d degrees of freedom, i.e. H0 ∼ χ2(2d), is upper
knowledgeofthechannels(CSI)ofitsintendedreceivers,and bounded by
eachreceiverhasperfectknowledgeofitsownchannelmatrix. P(SINR≥β)≤L (s)L (s/ρ) (6)
Y N
For exposition convenience, we drop the index 0 as all the
subsequent analysis is performed on the above typical link. with s= βDα.
4d
Proof: See [10].
B. Key Performance Metrics
The above lemma provides a large-deviation bound on the
A primary performance metric of interest in an uncoordi- successprobabilitybyboundingthe tailof thereceivedsignal
nated,randomaccessadhocnetworkistheoutageprobability (i.e. a χ2 random variate). In other words, a concentration
P withrespecttoapre-definedtargetSINRβ.Amessageis inequality of a central χ2 variate is exploited to upper bound
out
the probability the received signal power falls away from its in ad hoc SIMO channels using partial zero-forcing (PZF),
mean d. Although Lemma 1 captures the essential capacity thus orderwise we have CDPC = O(N2/α). In order words,
CPZF
scaling,thefollowingresultoughttobeusedformoreaccurate the interference pre-substractioncapability of DPC allows for
bounds on the success probability. N2/α moreconcurrenttransmissions(streams)perunitareain
Lemma 2: The outage probability in a random access arandomaccessnetworkascomparedtosingle-streamMIMO
network where H ∼χ2 is bounded by communications.
0 (2d)
For small outage constraints ǫ, by expanding L (s) using
A (ckβDα)≤P(SINR≤β)≤A (kβDα) (7) Y
d d first order Taylor series around zero we have that
where A (ζ) = d d (−1)kL (ζ)L (ζ/ρ) and c = Lemma 4: The optimal contention density λmimo when
d k=0 k Y N DPC
(d!)−1/d. P (cid:0) (cid:1) DPC precoding is employed to M multi-antenna receivers is
Proof: See [10]. given by
The above lemma relies on tightest known bound for the λmimo = FMNǫ e−ηβρDα (11)
distribution function of χ2 distribution, and is more accurate DPC IMβ2/αD2
yet more involved than (6). where
−1
A. DPC with Multi-antenna Receivers MN−1 k k−j j−1
k η 1
The exact received signal power at each receiver is hard FMN = (m−2/α) .
(cid:18)j(cid:19)(cid:18)ρ(cid:19) j!
to derive since no closed-form solution for the optimal DPC kX=0 Xj=0 mY=0
 (12)
transmit covariance matrices is known. However, an upper
Proof: See [10].
bound on the SINR can be derived by bounding the received
Note that for large number of antennas, F increases as
signal power and considering H ∼ µ2 (HHH ), where MN
µ denotes the maximum eigen0value mofaxHHH0 .0Since the O((MN)α2), which is consistent with the scaling given in
max 0 0 Lemma3.Similarly,applyingLemma2,thefollowingbounds
distribution(cdf)ofthesquareofthemaximumsingularvalue
can be derived:
of the user channel is involved (sums of exponentials and
Proposition 2: If DPC is employed, the maximum density
Laguerre polynomials), closed-form expressions yield little
under an outage constraint ǫ is lower bounded by
intuition. For that, the largest squared singular value is upper
bounded by µ2max ≤ kH0k2F ∼ χ2(2MN). The marks of the ǫ−(1−eηρζ)d Sd−,11 ǫ−(1−eϑρηζ)d Sd−,ϑ1
PPP interference are sums of the interference from the M (cid:16) (cid:17) ≤λ ≤ (cid:16) (cid:17)
I β2/αD2 DPC ϑ2/αI β2/αD2
independent messages transmitted by each interfering node, M M
(13)
i.e. the sum of M independentχ2 (due to the independence
(2) with d=MN, ζ =βDα, and ϑ=Γ(d+1)−1/d.
of the channels and the precoders of interfering transmitters
twraitnhsftohremreiscegiviveenfilbteyrsL).YT(hs)us=, Iek−λ∼β2χ/α2(2DM2I))Man[8d]twheheLreaplace Sd,ϑ =nX=d1(cid:18)nd(cid:19)(−1)n+1n2/αeϑρηζ (14)
2π M−1 M Forasymptoticallylarged,Γ(d+1)α2d ∼d2/α(usingStirling’s
I = B(m+2/α,M −(m+2/α)) (8) formula) and I ∼ M2/α. The asymptotic capacity scaling
M α (cid:18)m(cid:19) M
mX=0 CDPC depends on the scaling of SMN for large number of
with B(a,b)= 1ta−1(1−t)b−1dt being the Beta function. antennas.
0
Based on (8), LRemma 1 applies as follows: B. DPC with Single-antenna Receivers
Proposition 1: The maximum contention density of an ad
When each transmitter communicates with M single-
hoc network in which each transmitter communicateswith M
antenna receivers, no receive antenna processing/combining
receivers using dirty paper coding, is upper bounded by
can be performed; therefore H is at best distributed as
0
(4MN)2/α ηβDα χ2 , whereasthe interferencemarksremainunchanged,i.e.
λ ≤ −log(1−ǫ)+ . (9) (2M)
DPC IMβ2/αD2 (cid:20) 4MNρ(cid:21) Ii ∼χ2(2M) Following the analysis in [6], we can show that
Thesecondtermin(9)capturestheeffectofbackgroundnoise Proposition 3: TransmissiontoM single-antennareceivers
andhasaadditivecontributiononthecontentiondensity.Note (MISO)usingDPCprecodingresultsinamaximumcontention
that the noise effect falls off to zero for large number of density λmiso of
DPC
transmit/receive antennas.
SinceIM ∼πΓ(1−2/α)M2/αforlargeM,itcanbeeasily λmDPisCo = I Fβ2M/αǫD2e−ηβρDα. (15)
shown that: M
Lemma 3: The transmission capacity employing dirty pa- where F is given in (12).
M
percodingscalessuperlinearlywith the numberofantennas, Since for large M, FM = O(1), the transmission capacity
IM
i.e. exhibits linear scaling, i.e. Cmiso = O(M). The lack of
DPC
C =O(MN2/a) (10) receive antennas leads to no receive diversity or interference
DPC
For M = N, the ASE scales as C = O(N1+2/a). As cancellation gain, thus the per-user outage probability is of
DPC
shown in [7], a linear scaling of O(N) can be achieved orderO(1). Thelinearscalingin the ASE withthe numberof
transmit antennasis mainly due to the fact that M concurrent where S′ = d n n d η (n−j)(−1)j+1j2/α and
streams per transmitter are sent. N (cid:18)j(cid:19)(cid:18)j(cid:19)(cid:18)ρ(cid:19)
nX=1Xj=1
IV. LINEARPRECODING WITHANTENNACOMBINING IM is given by (8).
Therefore, the capacity scaling depends on the scaling of
Sincethecomplexityofdirtypapercodingisveryhigh,lin-
earprecodinghasattractedwideattentionasalowcomplexity SN, i.e. CZaFs = O(SN−1M1−α2). For M = N, we have that
Cas = Θ(M), due to the fact that selection improves the
(butsuboptimal)techniquewithcomplexityroughlyequivalent ZF
typical channel without amplifying interference. Since order
to point-to-point MIMO systems. Since linear precoding is
statistics (dueto selection)providesan M2/α-foldincreaseof
able to transmit the same number of data streams as a DPC-
the received signal power, a linear capacity growth with the
basedsystem,itthereforeachievesthesamemultiplexinggain
number of antennas can be achieved.
in a MIMO broadcast channel, but incurs a power offset
3) ZFBFwithSingle-antennaReceivers: Thebeamforming
relative to DPC. In this section, we aim at deriving the
achievable throughput of SDMA transmission when linear vectorofreceiverk,denotedaswk,ischosentobeorthogonal
to the channel vectors of all other intended receivers, i.e.
precoding techniques are employed.
h w = 0, ∀j ∈ K,j 6= k. By construction the distribution
j k
A. Zero-forcing Beamforming of H = |h w |2 is χ2 , whereas the interference marks I
0 k k (2) i
When zero-forcing beamforming (ZFBF) is employed, the remain gamma distributed, i.e. I ∼ χ2 . The following
i (2M)
beamforming vectors are chosen such that no inter-user proposition characterizes the performance of ZFBF in MISO
(SDMA) interference is experienced at any of the receivers. SDMA ad hoc networks:
1) ZFBF with Multi-antenna Receivers: For M ≥ KN Proposition 4: For a random access wireless network in
with N > 1, the precoding matrix {Wj}Kj=1 is chosen such which the transmitters spatially multiplex M single-antenna
that at each receive antenna n, the zero inter-stream interfer- receivers using ZFBF, the maximum density under an outage
ence constraint imposes: hk,nwj,l =0, ∀j 6=k,∀n,l∈[1,N] constraint ǫ is given by
andh w =0,∀l 6=n.Therefore,thetheeffectivechannel
k,n k,l
gain at each receive antenna is given by |hk,nwk,n|2, which −log(1−ǫ) ηβ1−α2Dα−2
λ = + (18)
followsaχ2(2(M−KN+1)) distribution.SimilarlytoProposition ZF IMβ2/αD2 ρIM
3, we can show that the maximum contention density in the
small outage constraint regime is given by where IM is given in (8).
Proof: See [10]
λZF = I Fβc2ǫ/αD2e−ηβρDα. (16) Thus, for large M, the interference power increases with
KN M2/α, whereas the signal power does not increase with the
withc=M−KN+1.Therefore,thecapacityscalesasC =
ZF number of antennas, resulting in asymptotic ASE scaling of
O((M(K−KNN)2+/α1−)21/α),whichequalsO((KN)2/α)forM =KN. CZF = O(M1−2/α). Note that the same orderwise capacity
If N > M, the extra degrees of freedom available at scaling is achieved by interference-aware beamforming [5],
the receiver side can be exploited to eliminate the inter- where a N-dimensional received array is used to cancel the
nodeinterference(e.g.employingazero-forcinglinearreceive nearestN−1interferers,undertheassumptionthatthereceiver
filter). In that case, the received signal power is distributed has knowledge of the interferers’ channels. Furthermore, it
as H0 ∼ χ2(2(N−M+1)), whereas the interference marks are can be shown that regularized channel inversion (MMSE
sums of M χ2 random variables. This results in C = precoding) provides the same O(M1−2/α) scaling, however
(2) ZF
O(M(N−M+1)α2) = O(M1−2/α) < O(N1−2/α). The fact higherSINRtargetβ peruserstream maybeachievedforthe
M
that multi-stream transmission boosts the interference coming same outage constraint ǫ.
from interfering transmitters, zero-forcing linear processing
fails to provide capacity scaling N1−2/α with the number of B. Block Diagonalization
receive antennas, as in [5].
If M ≥ KN, SDMA inter-user interference can also be
2) ZFBF with Receive Antenna Selection: Consider now a
eliminated by using block diagonalization (BD). The precod-
system that performs antenna selection at the receiver side,
ing matrix is chosen to be H W = 0, for k 6= j, thus
k j
i.e. each destinationnode selects its best receiveantenna.The
converting the system into K parallel MIMO channels with
channel vector h fed back from receiver k to the transmitter
k effectivechannelmatricesG =H W .Sincethenetworkis
k k k
corresponds to the antenna that has the best instantaneous
interference-limited, equal power allocation is asymptotically
channel (best among N i.i.d. channel vectors). Therefore,
(inSNR)optimal,thusanupperboundontheSINRunderBD
H ∼ max h(n), while the interference marks remain the
0 k can be found by considering that the received power is equal
1≤n≤N
same as inSection III. Fora suchnetwork,asF (x)=(1− to the square of the maximum singular value µ2 (G GH)
H0 max k k
e−x)N = N N (−1)n+1e−nx, the maximum contention scaled by the path loss and the transmit power. Since G
n=1 n k
density (foPr ǫ→(cid:0)0)(cid:1)is given by is a Wishart matrix with N × (M − (K − 1)N) degrees
λaZsF = S′ I βǫ2/αD2e−ηβρDα (17) oapfpflryeiendgomTheaonrdemµ2m1aixn≤[6]k,Gitkckan∼beχe2(2a(sNilMy−sh(Kow−n1)Nth2a)t) [11],
N M
Proposition 5: For small outage constraints, the maximum x 10−3
4.5
density using BD is upper bounded by
DPC upper bound 1 − eq.(13)
λBD ≤ I βF2r/ǫαD2e−ηβρDα. (19) 3.45 DDDPPPCCC ucslipompsueelr.d bp−oefourrfnomdr m −2a e−nq ce.e(q1.1(9))
K
iwFFsohgreir∼lvePaerrnrg(oNeob=fyMn:Nu(8mS−M)ebee(−rK[1o(0f−K].tr1−a)nN1sm)2N)it2/2/r,αec,Fetrihvuiessaganistveseunnmnabisny,g(w1M2e),h=aavnedKtIhNKat, Transmission capacity Cε 12..2355
r
we have that λBD ≤ O(Nα4K−α2). Therefore, the capacity 1
scaling when block diagonalization is employed scales as
C ≤ O(N4/αK1−2/α), which is a decreasing function 0.5
BD
with K. This implies that capacity growth is maximized if 0
1 2 3 4 5 6 7
K =1 receiveris served, leadingto a super linear scaling for Number of Tx antennas
α < 4. The scaling law is orderwise equivalent to N ×N
Fig.1. Capacity scalingCDPC ofdirtypapercodingversusthenumberof
eigenbeamforming transmission (spatial multiplexing) with transmitantennas (M =N)forα=4.
interference cancellation of the N −1 closest interferers.
V. NUMERICAL RESULTSAND DISCUSSION 1.8x 10−3
DPC with single−antenna Rx
Inthissection,weassesstheperformanceofSDMAadhoc 1.6 ZFBF with Rx Antenna Selection
ZFBF with single−antenna Rx
networks with default parameters D = 10m, ǫ = 0.1, α = 4, 1.4 ZFBF with multi−antenna Rx
and β = 3 as a means to verify our theoretical analysis. We
atpihndeetorIefnprdofteFremrtriihgevaenuencrdpceere-u1alpicvmtptheiecrietsraeuDlbdlsyPorCuterhnegetdliremsavnnea(usc.nmmftA.bia(sses9srsi)pouoranmefndcpditatcrip(ato1enand3cs,)miMt)ydiatiirn=stadyncNottpehman.epnpaeaanrsreacdlioyndwtiitcinhtahgel Transmission capacity Cε 001...1682
exhibits a super linear scaling behavior with the number of
0.4
antennas,alsocapturedbythederivedbounds.Thetightnessof
0.2
theupperboundsdependsonthepathlossexponentαandM,
being tighter for α decreasing. Furthermore, for small outage 0
1 2 3 4 5 6 7 8 9 10
constraint, (11) accurately predicts the SDMA transmission Number of Tx antennas
capacityofdirtypapercoding.Notealsothatsubstantialgains
Fig. 2. Transmission capacity versus the number of antennas for different
appear even when only a few streams are transmitted. SDMAprecoding techniques andα=4.
InFigure2weverifythelinearscalingofDPCwithsingle-
antennareceiversaswellasthesublinearcapacitybehaviorof
linear precoding. It should be noticed that diversity-oriented [5] K. Huang, J. G. Andrews, R. W. Heath, Jr., D. Guo, and R.
Berry, “Spatial interference cancellation for multi-antenna mobile ad
receive processing combined with linear transmit processing
hoc networks,” IEEE Trans. on Info. Theory, submitted, available at
isnotsufficienttoachievelinearscaling,ascomparedtoDPC
arxiv.org/abs/0804.0813.
evenwithonereceiveantenna.Thisismainlytothefactmulti- [6] A.Hunter,J.G.Andrews,andS.W.Weber,“Thetransmissioncapacity
stream transmissions boost the aggregate interference seen of ad hoc networks with spatial diversity,” IEEE Trans. on Wireless
Communications, toappear, 2008.
at a typical receiver, resulting in similar spectral efficiency
[7] N. Jindal, J. G. Andrews, and S. W. Weber, “Rethinking MIMO for
performance as in the point-to-pointMIMO case. wireless networks: linear throughput increases with multiple receive
antennas,” IEEEICC2009,toappear[http://arxiv.org/abs/0809.5008].
[8] A.HunterandJ.G.Andrews,“Adaptiveratecontrolovermultiplespatial
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