On the Robustness of Coordinated Beamforming to Uncoordinated Interference and CSI Uncertainty

0.16 MB·

by  George C. Alexandropoulos

On the Robustness of Coordinated Beamforming
to Uncoordinated Interference and CSI Uncertainty
1 1 2
George C. Alexandropoulos , Paul Ferrand , and Constantinos B. Papadias
1Mathematical and Algorithmic Sciences Lab, France Research Center, Huawei Technologies France,
20 Quai du Point du Jour, 92100 Boulogne-Billancourt, France
emails: {george.alexandropoulos,paul.ferrand}@huawei.com
2B-WiSE Lab, Athens Information Technology, 151 25 Marousi, Athens, Greece
email: [email protected]
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1
0 Abstract—As network deployments become denser, interfer- tight synchronization among the coordinated BSs as well as
2 ence arises as a dominant performance degradation factor. To sharing of Channel State Information (CSI) and data for the
n confront with this trend, Long Term Evolution (LTE) incorpo- targetedMTsarenecessary.Theserequirementsareoneofthe
a rated features aiming at enabling cooperation among different downfallsofJTCoMPinpracticalcellularnetworksandmake
J base stations, a technique termed as Coordinated Multi Point
(CoMP). Recent field trial results and theoretical studies of the itstheoreticalgainshardtoachieveinpractice.Itwasshownin
3
performance of CoMP schemes revealed, however, that their [3] that imperfector outdated CSI has a very large impact on
2
gains are not as high as initially expected, despite their large the performance of JT CoMP schemes. The same held when
coordinationoverhead. Inthispaper,wereviewrecentadvanced
] practicalradio-frequencycomponents,suchasoscillatorswith
T Coordinated Beamforming (CB) schemes, a special family of
phase noise, were considered [4]. On top of that, it has been
I CoMP that reduces the coordination overhead through a joint
. choiceoftransmitandreceivelinearfilters.Wefocusonassessing recently shown in [5]–[7], that CoMP schemes ignoring the
s
c theirresiliencetouncoordinated interferenceand ChannelState Out-of-the-coordination-Cluster Interference (OCI) see their
[ Information (CSI) imperfections, which both severely limit the performances degrade dramatically. As an alternative to JT
performancegainsofallCoMPschemes.Wepresentasimpleyet
CoMP for the downlink of cellular networks, the so-called
1 encompassing system model that aims at incorporating different
v parametersofinterestintherelativeinterferencepowerandCSI Coordinated Beamforming (CB) approach comes with less
6 errors, and then utilize it for the performance evaluation of the stringent coordination requirements [3], while retaining a
8 state-of-the-art in CB schemes. It isshownthat blindlyapplying part of the JT CoMP performance gains. In particular, the
4 CB in all system scenarios can indeed be counter-productive. practical implementation of CB schemes only shares CSI
6
for the targeted MTs among the coordinated multi-antenna
0 I. INTRODUCTION
BSs, without any form of user data exchange.The theoretical
.
1
Interference from neighboring base stations (BSs) is still design of CB schemes with multiple-antenna BSs and MTs
0
oneofthemostprominentperformancedegradationfactorsin has been lately the subject of many research papers [8]–[12].
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1 cellular networks resulting in outages or performance losses Only very recently [13] was the spectral efficiency of the CB
: at the cell edges as well as increasing the need for complex schemeknownasinterferencealignment(IA)[8]studiedunder
v
handovers.Bycapitalizingonthewidedeploymentofmultiple time-selective continuous fading channels that are explicitly
i
X antennas—especially at the BS side—and the advances in estimated through pilot symbol observations.
r multi-antenna signal processing techniques, a new approach In this paper, we investigate the performance of the CB
a
for interference management, termed as Coordinated Multi- schemes presented in [8]–[12] under OCI and CSI imper-
Point (CoMP) [1], has made its way into mobile communi- fections. Following the results [14], [15] in modeling the
cation standards such as Long Term Evolution (LTE). CoMP power of the received aggregate OCI as a Gamma Random
is a broad umbrella name for coordination schemes that aim Variable (RV), we present a multi-antenna system model
at realizing multi-user communications [2], i.e., sharing the incorporatingOCIasanadditiveNakagami-mdistributed[16]
medium among multiple network nodes over space on top of random vector. We also consider imperfect CSI due to errors
the possible sharing over time and frequency resources. in pilot symbol assisted estimation or latency. The proposed
Focusing on the downlink and considering Joint Trans- model allows us to assess the performance of a wide range
mission (JT) CoMP and in the theoretical limit of infinitely of practical CB deployments. We conclude on the possible
many distributed antennas, one could exactly pinpoint each follow-upsto improvethe resilience of the latest CB schemes
Mobile Terminal (MT) and ensure that its intended signal to OCI and CSI uncertainty.
adds up at its position while creating no interference for Notations: Throughout this paper vectors and matrices are
the other MTs in the network. In this case, interference is denotedbyboldfacelowercaselettersandboldfacecapitallet-
not only removed, but is actually harnessed and exploited ters,respectively.Thetransposeconjugateandthedeterminant
to increase the received signal power at each MT. However, of matrix A are denoted by AH and det(A), respectively,
for the the practical implementation of JT CoMP schemes, A(n) denotes the nth column of A, whereas [A] and [a]
i,j i
represent the (i,j)-element of A and the i-element of a, 2nd term in (1)) and OCI (i.e., of the 3rd term in (1)) as will
respectively.In addition,I denotesthen×n identitymatrix, bedescribedinthesequel.Itisnotedthat,ingeneral,αandβ
n
while ||A|| stands for the Frobenius norm of A and ||a|| can take values greater than 1. As a sanity requirement, here
F 2
denotes the Euclidean norm of a. The expectation operator is we assume appropriate MT associations to BSs that restrict
denoted as E{·} and the amplitude of a complex number as the values of α and β in [0,1].
|·|, whereas notation X ∼ CN µ,σ2 represents a RV X Each actual channel matrix Hb,j ∈ CnR[b]×nT[j] between
following the complex normal distribution with mean µ and MT b and any BS j is assumed to include independent
(cid:0) (cid:1)
variance σ2. A diagonal matrix with a in its main diagonal elements each modeled as [H ] ∼ CN(0,(n[j]n[b])−1)
is denoted by diag{a}, and R and C represent the set of real with p = 1,2,...,n[j] and qb,=j p,1q,2,...,n[b]. FTromRthese
and complex numbers, respectively. T R
channels, the matrices H ∀k are assumed to be estimated
b,k
sequentiallybyeachMTbusingminimummeansquarederror
II. ASYSTEMMODEL
(MMSE)channelestimationwithN pilotsymbols,andthen,
INCORPORATING OCI AND CSI IMPERFECTIONS p
fedbackthroughanerror-freechanneltothecommonnetwork
We consider a large multi-antenna cellular network from
entity the BSs belonging to the cluster are attached to. More
which we single out B BSs, indexed in the set B =
specifically, the estimated and actual channels are related as
d{1el,a2y.-f.r.e,eBl}in,ktos ftooramcaocmomorodninnaetitownocrlkusetnetritcyo.nOnnecsteodmtehrtoimugeh- Hˆb,k , Hb,k + Eb,k, where Eb,k ∈ CnR[b]×nT[k] represents
the MMSE estimation error matrix with entries distributed as
frequency resource unit, the BS cluster aims at providing
[E ] ∼ CN 0,(1+ρSNR)−1 with ρ = N /n[k] and
service to B MTs indexed in the set U = {1,2,...,U} with b,k i,j p T
SNR = P/N [17]. It is noted that, in [13], different forms
U ≥ B. In particular, each BS b ∈ B schedules one MT 0 (cid:0) (cid:1)
ofE arepresentedmodelingvariousrealisticimperfections,
from its associated set of MTs, denoted by U , according to b,k
b suchasCSIestimationerrorsandanalogCSIfeedback.Using
a certain scheduling criterion. All sets U ∀b form a partition
b the latter assumptions for the system model in (1), the power
of the set U and each BS b is assumed to be equipped with a
of the intended channel at each MT b and the power of the
n[b]-element antenna array, whereas each MT u∈U has n[u]
T R aggregate ICI (i.e., of the 2nd term in (1)) are related as
antennas. Assuming perfect synchronization of the downlink
trreacnesivmeidssivoencstowr iythbin∈thCencR[bo]o×r1dinatatitohne cMluTstebr,∈theUbbacsaenbanbde α(B−1)−1EP{kℓH∈Bb,,bℓ6=k2Fb}E{kHb,ℓk2F} =α. (3)
mathematically expressed as
For example, α = 1 models cases where MTs are at similar
α distances from all BSs in the cluster; under these cases
y ,H x + H x +g +n , (1)
b b,b b B−1 b,ℓ ℓ b b intended channels and ICI are of almost equal powers. The
r ℓ∈XB,ℓ6=b latter value for α is well suited for modeling scenarios ac-
where gb ∈CnR[b]×1 represents the aggregate OCI given by cordingto whichMTs arelocated atthe edgesof the separate
cells (i.e., the coordination cluster center), where coordinated
β transmission is expectedto provideits highest gain.Note that
gb ,s|B′| Hb,ixi (2) (3) holds also for estimated channel matrices (i.e., for Hˆb,k
iX∈B′ ∀k ∈ B). Moreover, the relation between the power of the
with |B′| denoting the cardinality of the set B′ of interfering estimated intended channel at each MT b and the power of
multi-antenna BSs that do not belong in the coordination the aggregate OCI (i.e., of the 3rd term in (1)) is given by
rcelupsrteesrenBts.tIhne a(1ct)uaalncdha(2n)n,elHmba,ktri∈xbCetnwR[be]e×nnT[Mk]TwbitahndkB∈S kB, β|B′|−1 i∈B′E{kHb,ik2F} = β .
and Hb,i ∈ CnR[b]×nT[i] with i ∈ B′ denotes the actual channel EP{kHˆb,bk2F} n[Tb]n[Rb](1+ρSNR)−1+1(4)
matrix between MT b and BS i. Furthermore, xj ∈ CnT[j]×1 It can be seen from (4) that for fixed SNR, as Np → ∞
with j ∈ B ∪ B′ includes the dj mutually independent (approaching perfect channel estimation) then ρ → ∞, and
and linearly precoded symbols of BS j intended for MT as a result the power of the aggregate OCI at MT b tends to
j, for which it must hold dj ≤ min(n[Tj],n[Rj]) for correct be β times the power of its estimated intended channel. The
detectiontobepossible.Withoutlossofgenerality,weassume samehappenswithafixedρvalueandSNR→∞.Ingeneral,
that each BS i not belonging to the coordination cluster, parameter β indicates the effectiveness of BS clustering [18]
schedules its associated MT i per time-frequency resource for coordinated transmission. Low values of β indicate that
unit.We alsoassume thatalldeployedBSsin thenetworkare most of the interfering BSs for a specific MT have been
subjectto thetotalpowerconstraintP, therefore,itholdsthat includedwithin the coordinationcluster, the oppositehappens
E{kxjk22} ≤ P, and the vector nb ∈ CnR[b]×1 represents the for high β values.
zero-mean complex additive white Gaussian noise (AWGN) The elements of the aggregate OCI vector g in (2) ∀b are
b
having covariance matrix N0In[b]. Finally, α,β ∈ [0,1] in complex RVs whose amplitudes are modeled in this work
(1) and (2) are used for modelinRg the relative strength of the as independent and identically distributed (IID) Nakagami
receivedaggregateIntra-ClusterInterference(ICI) (i.e.,of the RVs [16] with finite shape parameter m ≥ 0.5 and finite
Ω=βP/n[Rb]. It follows that their squared amplitudesare IID 4 Full Reuse (α=1)
Gamma RVs with the following shape and scale parameters 3.5 Full Reuse (α=0.25)
) Orthogonal (α=1)
z
H Orthogonal (α=0.25)
|[gb]q|2 ∼GAMMA m,mβnPR[b]!, ∀q =1,2,...,n[Rb]. (5) e(bps/2.35 JJIATT CC(αoo=MM1PP) ((αα==10).25)
at IA (α=0.25)
It is trivial to conclude from (5) that E{|[g ] |2} = βP/n[b] R 2
b q R m
andE{kgbk22}=βP. For the specialof m=1, (5) simplifies Su1.5
to the exponential distribution, which implies that the each
r
e
elementof gb canbe a zeromeancomplexGaussian RV with ust 1
variance Ω. The latter modeling of the OCI’s vector elements Cl
0.5
was used in the analysis of [5], where it was considered that
OCI is made up of a large number of interference terms, and
0
thuscanbe effectivelyapproximatedasGaussian. Forgeneral -5 0 5 10 15 20 25
SNR(dB)
values of m ≥ 0.5 in (5), the adopted model resembles that
of [7], [14], [15], where the amplitude distribution of each
Fig. 1. Theoretical sum rate gains of coordination with β =0.25, perfect
element of gb, computed using both the hexagonal grid and CSI,andtwodifferentvaluesofα.Thenetworksumratewithfullreuseofthe
stochastic geometry models, was approximated by a Gamma available time-frequency resources as well as with orthogonal transmissions
RV through the moment matching approach. Starting from isalsosketched.
(1) and considering the OCI model of (5), it can be easily
concludedthatthe receivedpowerofthedesiredsignalatMT
• WiththeCBschemebasedoninterferencealignment(IA)
b with channel estimation and that of its received aggregate
[8],ICIcanbenulledandtheindividualMTratebecomes
OCI are related as in the left-hand side of (4), yielding
log [1+SNR/(βSNR+1)]; and
2
E{kg k2} β • With ideal JT CoMP, the interference power actually
b 2 ≥ . (6) boosts the intended signals and the individual MT rate
E{kHˆb,bxbk22} nT[b]nR[b](1+ρSNR)−1+1 is given by log [1+(1+α)SNR/(βSNR+1)].
2
III. DOWNLINK COORDINATED BEAMFORMING The latter rates for each of the two individualMTs are added
and then depicted in Fig. 1 for the case of OCI being 6 dB
In this section, the system model of Section II is first
lower than that of the intended signal (i.e., β =0.25) and for
employed to a simplistic cellular network in order to demon-
two different values of α that reveal the relative ICI power.
strate the theoreticalgainsof JT CoMPand CB schemesover
As expected, both coordinated schemes provide substantial
representative non-coordinatedones as well as to compare JT
gains compared with full reuse and orthogonal transmission
CoMP and CB under different levels of ICI and OCI. Then,
when the SNR increases and the network operates in the
we briefly describe the most representative CB schemes to be
interference-limitedregime.As αapproaches0the gainofJT
comparedundertheconsideredsystemmodelinthefollowing
CoMPoverIA decreases.Forexample,forSNR=15dBand
section with the simulation results, and introduce the metric
α=1,IAresultsinanearly100%gainoverorthogonaltrans-
for the performance comparisons.
mission, while this gain becomes nearly 180% for JT CoMP.
When α decreases, the latter gain of IA remains the same,
A. Theoretical Gains of BS Coordination
whereas that of JT CoMP decreases to nearly 110%. This
We consider a cluster of B = 2 multi-antenna BSs, which exampleillustrates that, in manycases ofinterest, a largepart
is a part of a large cellular network. Both BSs in the cluster of the gain from coordinated schemes comes more from the
coordinate their transmissions to serve a total of 2 MTs in removalof interference from the signal of interest rather than
one time-frequency resource unit. One MT is associated to from stacking the powers of multiple transmitting points. It
the one BS and the other MT to the other BS. Capitalizing is also noted that, when considering practical implementation
on the system modelof Section II for the case of perfect CSI issues in achieving JT CoMP, the bonus of full coordination
availability, and using the classical bounds for the individual becomes even lower, since JT CoMP is more afflicted by
MTratesinmultiple-inputsingle-outputIFCs[2],itholdsthat: degraded CSI and dirty RF than CB [4].
• With full reuse of time-frequencyresources, each MT is
B. CB Schemes
subjecttointerferencefromeveryBSnotassociatedwith
and its rate is upperboundedas log [1+SNR/(αSNR+ The following CB schemes are consideredfor performance
2
βSNR+1)]; comparisons under the system model presented in Section II:
• With orthogonal allocation of the available time- i) IA [8] that aims at aligning and then nulling interference
frequency resources, ICI is absent but the prelog factor at each MT belonging in the BS coordination cluster; ii)
0.5appearsoneachindividualMTrate,yieldingthevalue the maximum signal-to-interference-plus-noise ratio (SINR)
0.5log (1+SNR/(βSNR+1)]; scheme [9] that targets at maximizing the received SINR of
2
35
each transmitted information stream within the cluster; iii)
Full Reuse
the weighted MMSE (WMMSE) scheme [10] that minimizes
30 Orthogonal
a system-wide MMSE metric; and iv) the Reconfigurable z) IA
H
scheme of [11]. The latter CB scheme designs the receive / Maximum SINR
ps 25 WMMSE
combining matrices at MTs from the minimization of a (b Reconfigurable
system-wide MMSE metric and the transmit precoding ma- te 20
a
trices at BSs are composed of two parts: the one part aims at R
optimizingthe same metric with the receive combiners,while m 15
u
the other part targets at maximizing each individual MT rate. S
r
The above CB schemes are more effectively realized through te 10
s
centralizedimplementationswithfullCSIexchangeamongthe Clu
5
BSsbelonginginthecoordinationcluster.Itisnoted,however,
that for the maximum SINR, WMMSE, and Reconfigurable
0
schemes, which are iterative as opposed to IA that is not, -5 0 5 10 15 20 25
distributed versions are also available, where explicit CSI SNR(dB)
exchange among coordinated BSs is avoided. In addition, the
latterCBschemesarelinear,whichmeansthateachBSb∈B Fig. 2. Sumrate performance ofvarious CBschemes as afunction of the
designs its transmitted vector xb as xb = VbP1b/2sb, where SanNdRpeinrfedcBt CfoSrI.BTh=ep3e,rfnoTrm=anc8e,sawndithnfRull=re4soausrcweselrleuasseαan=do1r,thβog=on0al,
Vb ∈CnT[b]×db represents the precodingmatrix, for which we transmissions arealsodemonstrated.
assume that ||Vb(κ)||2 , 1 ∀ κ = 1,2,...,db, sb ∈ Cdb×1
is the information stream vector with E{s sH} = I , and
b b db downlink CB for the coordination cluster is given by
P , diag{[P(b)P(b) ... P(b)]} ∈ Rdb×db with P(b) de-
nobting the pow1er al2located dtob the κth+information κstream. B
Upon the signal reception given by (1), each MT b estimates R≤ EH log2 det In[b] +Hb,bSbHHb,bQb−1 , (11)
R
its desired transmitted symbols using a combining matrix Xb=1 n h (cid:16) (cid:17)io
Ub ∈ CnR[b]×db, yielding the estimated information stream where notation EH{·} represents the expectation over all
vector ˆsb , Ubyb. Substituting the linear precoding and Hb,b’s, while Sb = VbPbVbH ∀b denotes the covariance
combining matrices obtained from any of the aforementioned matrices of the transmitted signals and Qb ∈ CnR[b]×nR[b] are
CB schemes into (1) yields the ICI plus OCI plus noise covariance matrices, which can
be computed as
α
ˆs =UHH V P1/2s +
b b b,b b b b B−1 α βP
r (7) Q = H S HH + +N I . (12)
× UHbHb,ℓVℓP1ℓ/2sℓ+UHb (gb+nb). b B−1ℓ∈B,ℓ6=b b,ℓ ℓ b,ℓ n[Rb] 0! nR[b]
X
ℓ∈B,ℓ6=b
X Note that in the computation of the average achievable sum
By inspecting (7), it can be easily concluded that the
rate in (11) we do not include the U ’s ∀b ∈ B designed by
b
receivedSINRofeachκthinformationbearingstreamateach
each of the considered CB schemes.
MT b can be expressed as
P(b)|[UHH V ] |2 IV. NUMERICAL RESULTS ANDDISCUSSION
SINRb,κ , I +Iκ +b|[UHb,(bg b+κ,nκ )] |2, (8) TheaverageclustersumratewiththeCBschemesdescribed
b ICI b b b κ inSectionIII-Bisassessedhereinafterbysimulating(11)over
where Ib, IICI ∈ R+ represent the inter-stream interference 100 independent channel realizations using the system model
from the other db−1 streams intended for MT b, and the in- introduced in Section II. In Figs. 2 and 3, a coordination
terferencefromthe streamsbelongingin theICI, respectively, cluster with B = 3 BSs is considered where we have set
which can be expressed as n[b] = n = 8 as well as n[b] = n = 4 ∀b = 1,2, and
T T R R
db 3, while a cluster with B = 4 BSs is investigated in Fig. 4,
Ib , Pλ(b)|[UHbHb,bVb]κ,λ|2, (9) where nT = 4 and nR = 2. For IA in Figs. 2 and 3, db
for each MT b was set to 0.5min(8,4) = 2 according to
λ=1,λ6=κ
X
the IA feasibility conditions [8] and the precoding matrices
I , dℓ P(ℓ)|[UHH V ] |2. (10) Vb’s were obtained in closed form. The latter values for db’s
ICI µ b b,ℓ ℓ κ,µ were also preset to the maximum SINR scheme in Figs. 2–
ℓ∈B,ℓ6=bµ=1
X X 4, which designs each V iteratively. It is noted that for
b
C. Cluster Sum Rate Performance
the system set up in Fig. 4, closed-form expressions for the
Substituting the linear precoding and combining matrices V ’s of IA are not available. For both the iterative schemes
b
obtained from any of the CB schemes described in Sec- WMMSE and Reconfigurable in all figures, each d was
b
tion III-B into (1), the average achievable sum rate with initialized as d = min(n ,n ) and obtained at the end of
b T R
15
CSI scenarios, according to which α decreases from 1 to 0.8,
Full Reuse
β increases from 0 to 0.2, and N falls to 10 from a very
Orthogonal p
z) IA large number. Interestingly, the maximum SINR, WMMSE,
H
/ Maximum SINR andReconfigurableschemesthattakeOCIunderconsideration
s
bp10 WMMSE provide for most of the SNR values equal to or more than
( Reconfigurable
e 100%improvementcomparedtoIA.AsalsoseenfromFig.3,
t
a
R all iterative schemes outperform orthogonal transmission for
m SNR ≤ 20 dB, and WMMSE faces convergence problems at
u
S 5 high SNR values for the considered number of algorithmic
r
te iterations. In addition, it is obvious from this figure that,
s
Clu severe CSI errors for SNR ≤ 10 dB render full reuse the
best strategy. The trend of Figs. 2–3 is also seen in Fig. 4 for
a larger cluster size. As depicted, between the WMMSE and
0
-5 0 5 10 15 20 25 Reconfigurableschemesthatprovidethebestperformance,the
SNR(dB) Reconfigurableismoreresilienttoincreasingβanddecreasing
N .AtSNR=15dBbothschemesdiffernearly20%between
p
Fig. 3. Sum rate performance ofvarious CB schemes as afunction ofthe the two considered scenarios, whereas at SNR = 22.5 dB,
aSnNdRNinpd=B1fo0r.BTh=e p3e,rfnoTrm=an8c,esanwditnhRfu=ll4reasosuwrceelslarseuαse=an0d.8o,rβtho=go0n.2al, the Reconfigurable scheme differs 40% while the WMMSE
transmissions arealsodemonstrated. scheme differs closely to 50%.
25
V. CONCLUSION
Maximum SINR (β=0.1, N=10)
p
Maximum SINR (β=0.01, N=100) The feature of coordinating BS transmissions to manage
Hz)20 WMMSE (β=0.1, N=10) p interferencein cellular networksis already a part of the latest
/ p
ps WMMSE (β=0.01, Np=100) LTE release offering significant potential for performance
(b Reconfigurable (β=0.1, N=10) improvement especially at the cell edges. From the various
te15 Reconfigurable (β=0.01, Np =100) recently available CB schemes for cellular MIMO systems,
a p
R
there exist schemes that adapt satisfactory to uncoordinated
m
u10 interference, but still perform poorly under severe CSI im-
S
r perfections. Certain advances need to take place in order to
e
st maximize the benefit from CB, such as for example efficient
u 5
Cl BS clustering as well as coordination with reduced overhead
and resilience to CSI uncertainties.
0
-5 0 5 10 15 20 25 REFERENCES
SNR(dB)
[1] J.Lee,Y.Kim,H.Lee,B.L.Ng,D.Mazzarese, J.Liu,W.Xiao, and
Fig. 4. Sumrate performance ofthe considered iterative CBschemes as a Y. Zhou, “Coordinated multipoint transmission and reception in LTE-
function of the SNR in dB for B = 4, nT = 4, and nR = 2 as well as advanced systems,” IEEE Commun. Mag., vol. 50, no. 11, pp. 44–50,
α=0.9,β={0.1,0.01},andNp={10,100}. Nov.2012.
[2] D.Gesbert,S.Hanly,H.Huang,S.S.Shamai,O.Simeone,andW.Yu,
“Multi-cell MIMO cooperative networks: A new look at interference,”
IEEEJ.Sel.Area.Comm.,vol.28,no.9,pp.1380–1408, Dec.2010.
the algorithmic iterations or upon convergence, explicitly for [3] R.Irmer,H.Droste,P.Marsch,M.Grieger,G.Fettweis,S.Brueck,H.-P.
Reconfigurable and implicitly for WMMSE, together with all Mayer,L.Thiele,andV.Jungnickel,“Coordinatedmultipoint:Concepts,
performance,andfieldtrialresults,”IEEECommun.Mag.,vol.49,no.2,
V ’s. More specifically, the Reconfigurable scheme outputs
b pp.102–111, Feb.2011.
d to be sent by each coordinated BS b together with their [4] V.Jungnickel,K.Manolakis,W.Zirwas,B.Panzner,V.Braun,M.Los-
b
beamformingdirections,whereasWMMSEonlygeneratesthe sow, M. Sternad, R. R. Apelfro¨jd, and T. Svensson, “The role of
small cells, coordinated multipoint, and massive MIMO in 5G,” IEEE
transmit covariances matrices with possibly some streams set
Commun.Mag.,vol.52,no.5,pp.44–51,May2014.
tozeropower,andthusunusable.Thismeansthat,forthelatter [5] A.Lozano,R.W.Heath,Jr.,andJ.G.Andrews,“Fundamentallimitsof
scheme the optimum d needs to be searched in some way, a cooperation,” IEEE Trans. Inf. Theory, vol. 59, no. 9, pp. 5213–5226,
b Sep.2013.
fact that will cause an extra overhead in practical networks.
[6] R.K.Mungara,D.Morales-Jimenez,andA.Lozano,“System-levelper-
As clearly depicted in Figs. 2–4 for a maximum of 10 formanceofinterferencealignment,”IEEETrans.onWirelessCommun.,
iterations per iterative scheme, the performance of all CB vol.14,no.2,pp.1060–1070, Feb.2015.
[7] D. Jaramillo-Ramire´z, M. Kountouris, and E. Hardouin, “Coordinated
schemes is susceptible to high OCI and severe CSI errors.
multi-pointtransmissionwithimperfectCSIandother-cellinterference,”
This behavior of the IA scheme for CSI errors was also IEEE Trans. Wireless Commun., vol. 14, no. 4, pp. 1882–1896, Apr.
observed in [6]. For example, for SNR = 15 dB and B = 3, 2015.
[8] V.R.CadambeandS.A.Jafar,“Interference alignmentanddegrees of
it is shown in Figs. 2 and 3 that the performance of all CB
freedom oftheK-userinterference channel,” IEEETrans.Inf.Theory,
schemes drops approximately 40% between the two OCI and vol.54,no.8,pp.3425–3441, Aug.2008.
[9] K. Gomadam, V. R.Cadambe, and S.A.Jafar, “Distributed numerical [13] R.K.Mungara,G.George,andA.Lozano,“Overheadandspectralef-
approachtointerferencealignmentandapplicationstowirelessinterfer- ficiencyofpilot-assistedinterference alignmentintime-selective fading
encenetworks,”IEEETrans.Inf.Theory,vol.57,no.6,pp.3309–3322, channels,” IEEE Trans. Wireless Commun., vol. 13, no. 9, pp. 4884–
Jun.2011. 4895,Sep.2014.
[10] Q.Shi,M.Razaviyayn,Z.-Q.Luo,andC.He,“Aniteratively weighted [14] R.W.Heath, Jr.,T.Wu,Y.H.Kwon,andA.C.K.Soong,“Multiuser
MMSE approach to distributed sum-utility maximization for a MIMO MIMO in distributed antenna systems with out-of-cell interference,”
interfering broadcast channel,” IEEE Trans. Signal Process., vol. 59, IEEETrans.SignalProcess.,vol.59,no.10,pp.4885–4899,Oct.2011.
no.9,pp.4331–4340,Sep.2011. [15] R.W.Heath, Jr.,M.Kountouris, andT.Bai, “Modeling heterogeneous
[11] G.C.Alexandropoulos andC.B.Papadias, “Areconfigurable iterative networks interference using Poisson point processess,” IEEE Trans.
algorithm fortheK-userMIMOinterference channel,”SignalProcess. SignalProcess.,vol.61,no.16,pp.4114–4126, Aug.2013.
(Elsevier), vol.93,no.12,pp.3353–3362, Dec.2013. [16] M. Nakagami, “The m-distribution - A general formula of intensity
[12] G. C. Alexandropoulos, P. Ferrand, J.-M. Gorce, and C. B. Papadias, distribution ofrapidfading,” Stat. Methods RadioWaveProp.,vol.40,
“Advanced coordinated beamforming for the downlink of future LTE pp.757–768, Nov.1992.
cellularnetworks,”IEEECommun.Mag.,vol.54,no.7,pp.54–60,Jan. [17] T. Marzetta, “BLAST training: Estimating channel characteristics for
2016. high capacity space-time wireless,” in Proc. Allerton Conf. Commun.,
Control,Comput.,Monticello, USA,22-24Sep.1999,pp.958–966.
[18] A. Papadogiannis and G. C. Alexandropoulos, “The value of dynamic
clustering of base stations for the wireless networks of the future,” in
Proc.IEEEWorldCongressonComput.Intel.,Barcelona,Spain,18-23
Jul.2010,pp.1–6.