DTIC ADA506493: Stable Transmission in the Time-Varying MIMO Broadcast Channel

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B o t h li near and non-linear transmit precoding strategies based on accurate channel state information (CSI) can significantly increase
av a i l a b le throughput in a multi-user wireless system. With propagation delay, infrequent channel updates, lag due to network layer
ov e r h e a d, and time-varying node position or environment characteristics, channel knowledge becomes outdated and CSI-based
tr a n s m i ssion schemes can experience severe performance degradation. This paper studies the performance of precoding techniques
fo r t h e multi-user broadcast channel with outdated CSI at the transmitter. Traditional channel models as well as channel realizations
m e a s u r ed by a wideband channel sounder are used in the analysis. With measured data from an outdoor urban environment, it is
fu r t h e r shown the existence of stable subspaces upon which transmission is possible without any instantaneous CSI at the

transmitter. Such transmissions allow for consistent performance curves at the cost of initial suboptimality compared to CSI-based

schemes.

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HindawiPublishingCorporation
EURASIPJournalonAdvancesinSignalProcessing
Volume2008,ArticleID617020,14pages
doi:10.1155/2008/617020
Research Article
Stable Transmission in the Time-Varying MIMO
Broadcast Channel
AdamL.Anderson,1JamesR.Zeidler,1andMichaelA.Jensen2
1DepartmentofElectricalandComputerEngineering,UniversityofCalifornia,SanDiego,CA92093-0407,LaJolla,USA
2DepartmentofElectricalandComputerEngineering,BrighamYoungUniversity,Provo,UT84602,USA
CorrespondenceshouldbeaddressedtoAdamL.Anderson,[email protected]
Received1June2007;Revised28September2007;Accepted19December2007
RecommendedbyChristophMecklenbra¨uker
Both linear and nonlinear transmit precoding strategies based on accurate channel state information (CSI) can significantly
increase available throughput in a multiuser wireless system. With propagation delay, infrequent channel updates, lag due to
networklayeroverhead,andtime-varyingnodepositionorenvironmentcharacteristics,channelknowledgebecomesoutdated
and CSI-based transmission schemes can experience severe performance degradation. This paper studies the performance of
precodingtechniquesforthemultiuserbroadcastchannelwithoutdatedCSIatthetransmitter.Traditionalchannelmodelsas
well as channel realizations measured by a wideband channel sounder are used in the analysis. With measured data from an
outdoorurbanenvironment,itisfurthershowntheexistenceofstablesubspacesuponwhichtransmissionispossiblewithoutany
instantaneousCSIatthetransmitter.Suchtransmissionsallowforconsistentperformancecurvesatthecostofinitialsuboptimality
comparedtoCSI-basedschemes.
Copyright©2008AdamL.Andersonetal. This is an open access article distributed under theCreative Commons Attribution
License,whichpermitsunrestricteduse,distribution,andreproductioninanymedium,providedtheoriginalworkisproperly
cited.
1. INTRODUCTION and receiver have channel estimation errors or partial CSI.
In [4] capacity upper and lower bounds are given for the
The time-varying, multiuser, multiple-input multiple- single-user,flat-fading,MIMOchannelwithGaussianinputs
output (MIMO) wireless channel promises significant and normal channel error statistics. The work in [3] uses
gain in performance over that offered by conventional measured channel responses for moving nodes to analyze
single-antennasystems[1].Multiplexingandmultipleaccess the capacity degradation caused by outdated CSI at the
gainsincreasesystemthroughputandareachievedthrough transmitter (CSIT) and receiver (CSIR). In an effort to
the use of multiple antennas and spatial reuse. Temporal reduce this sensitivity to CSI quality, recent research has
diversity gains enabled by channel-time variation further suggested the formation of transmit beamformers using
increase system performance. Unfortunately, this temporal channel distribution information (CDI) at the transmitter
variation typically implies that outdated estimates of the (CDIT) [8, 9], a strategy which is optimal in an ergodic
channel state information (CSI) are used to construct the capacitysenseundercertainantennacorrelationconditions.
signaling strategy, resulting in capacity degradation [2, 3] An adaptive beamformer that uses both CSI and CDI is
that is analogous to that created by channel estimation suggestedin[10]wherecapacitydegradationfromoutdated
errors[4,5].Thesesameeffectsofchannelestimationerrors CSIoccursinatimedivisionduplex(TDD)MIMOsystem
andDopplersensitivityinpracticalprecodingsystemswere withaspatiallycorrelatedJakes’channel.
shown in [6, 7] to contribute significantly to performance Similar work for the multiuser MIMO channel has
loss. These observations motivate the development of focusedmoreontheeffectsofchannelestimationerrorsthan
transmissionschemeswhicharerobusttophysically-realistic the impact of outdated CSI created by channel-time varia-
channelvariations. tion.Forexample,forthesingle-inputsingle-output(SISO)
Priorworkhasstudiedperformancedegradationsofthe broadcast channel, a scheduling strategy was proposed in
single-user, point-to-point MIMO link when transmitter [11] to combat the effects of channel estimation error.
2 EURASIPJournalonAdvancesinSignalProcessing
Furthermore, capacity regions for the MIMO broadcast The examination of different precoding strategies per-
channel with erroneous CSIT and CSIR are found in [12] formed in this paper considers both modeled channels,
usingthedualitybetweenthebroadcastandmultiple-access which allow a parametric evaluation over a variety of
channels (MAC) [13]. The work in [5] uses error statistics channel conditions but may not accurately represent the
for the sum-capacity-optimal dirty-paper coding (DPC)[1] physicaltime-spaceevolutionofthesubspace,andmeasured
to determine when time-sharing outperforms DPC in the channels which allow performance quantification over a
multiple-input single-output (MISO) broadcast channel. A limitedsetofrealisticenvironments.Thissectiondetailsthe
similar study for erroneous CSIT was also performed for modelsandmeasurementsusedtofacilitatethisstudy.
thecomputationallysimplerzero-forcingDPC(ZF-DPC)in
[14]usingcapacityboundssimilartothosepresentedin[4]. 2.1. Channelmodels
Thisworkbuildsontheexistingunderstandingtostudy
the behavior of different CSIT-based transmit precoding Because effective multi-antenna transmit precoding strate-
gies exploit spatial structure in the channel, it is important
techniques [15] in the time-variant multiuser broadcast
that the channel model used accurately reflects this spatial
channel (BC). The study considers DPC, linear beamform-
information. The spatial correlation of the transfer matrix,
ing,andtime-divisionmultipleaccess(TDMA)techniques.
which is created by the angular properties of the multipath
While numerous beamforming algorithms exist for various
propagation as well as the antenna configuration, is a
design criteria [15–17] we focus on the beamforming
common mechanism for capturing this spatial structure in
algorithm that maximizes capacity for a MIMO broadcast
the model. To aid in the analysis, the correlation matrices
channel(forlinearprecoding)asdefinedin[15]whichisan
at the transmit and receive ends of the link are assumed
extensionofthealgorithmin[18]for(MISO)channels.The separable,resultinginaKroneckerdescriptionoftheoverall
TDMA scheme removes multiple accessinterference(MAI) spatial correlation [20]. Using this mechanism, the channel
andtheneedforCSITbyassigningeachuserauniquetime matrixatthenthtimesampleis
slot for channel access and by using the optimal signaling (cid:3) (cid:3)
strategyforanuninformedtransmitter.Theanalysisofthese Hj(n)= Rr,jHw,j(n) Rt,j, (2)
schemesbeginswithsimulationsbasedonacceptedmodels
where H (n) is anN × N matrix with zero mean, unit
w,j r t
for the spatially-correlated time-variant channel [19, 20].
variance,i.i.d.complexGaussianrandomvariablesatsample
However, since these models may not capture the complex indexn,R ,andR aretheN ×N andN ×N receiveand
r,j t,j r r t t
physical structure of the multiuser time-variant MIMO transmit correlation matrices, respectively, for the jth user,
channel [21], the results obtained using the models are andthesquarerootop√erat√iononsomepositivesemidefinite
reinforced using simulations with experimentally obtained matrixZisdefinedas Z Z = Z.Thereissomedebateon
channels [22] taken in an outdoor environment on the theaccuracyoftheKroneckermodelbecausethemodelhas
BrighamYoungUniversity(BYU)campus[3,23].Motivated been verified for a small number of antennas in [24] while
by the performance degradation observed for the existing deficiencies in the model for a larger number of antennas
signaling schemes, the paper finally develops and analyzes have been identified in [25]. The application of interest in
aniterativebeamformingalgorithmthathassimilarperfor- thisworkisamobileadhocnetwork(MANET)inwhichall
mancetothecapacityoptimalbeamformerwhenusedwith nodesareequallyequippedwithasmallnumberofantennas,
justifyingtheuseoftheKroneckermodelassumptionin(2).
CSIT and provides stable throughput performance when
constructed with CDIT. The stable performance offered Any model must also ensure that the channel samples
possesstheproperrelationshipintime.Thiscanbeaccom-
by this algorithm implies the existence of slowly varying
plished by properly representing the temporal correlation
subspacesinthetime-varyingmultiuserMIMOchannel.
between channel samples for sample spacings smaller than
thechannelcoherencetime.Thisworkassumesthetemporal
2. SYSTEMANDCHANNELMODELS
correlationfunctionmodelsuggestedbyJakes[19]whichis
The MIMO broadcast channel communication scenario of givenby
(cid:4) (cid:5)
interestconsistsofasingletransmittingnodeequippedwith ρ(τ)=J 2πf τ , (3)
0 d
N antennas and K receiving nodes (users) each with N
anttennas.TheN ×1receivedvectorforthe jthuserattimer where J0(·) is the zeroth-order Bessel function of the first
r
kind and f is the normalized Doppler frequency. For
samplencanbeexpressedas d
simulation purposes a sum of eight weighted sinusoids is
(cid:2)K usedinJakes’modelwiththespecifiednormalizedDoppler
y (n)=H (n)x (n)+ H (n)x(n)+η (n), (1)
j j j j i j taken into account to produce the temporal correlation in
i=/j (3).
where Hj(n) is the Nr × Nt matrix of channel transfer Thechannelmodelrealizesbothspatiallyandtemporally
functions for user j, xi(n) is the Nt × 1 signal vector correlated channel coefficients by filtering Hw,j(n) in (2)
destinedfortheithuser,andη (n)isadditivewhiteGaussian with the time-varying coefficients generated from Jakes’
j
noise(AWGN).Equation(1)presumesnospecifictransmit model with the Doppler frequency f in (3) chosen to
d
precodingandisthereforeappropriatelymodifiedlaterinthe match that of the measured channel. Further, the spatial
discussionofspecifictransmissionschemes. correlationmatricesusedin(2)areeithermodeledwithan
AdamL.Andersonetal. 3
exponentiallydecayingfunctionorestimatedfrommeasured where{·}H isthematrixconjugatetranspose.Similarly,the
data as explained in Section2.2. The channel matrices for receivecorrelationmatrixestimateis
different users are realized independently. Throughout this
paper, the term “modeled channel” refers to a sequence of (cid:4) (cid:5) 1 N(cid:2)−1 (cid:4) (cid:5) (cid:4) (cid:5)
channelmatricesgeneratedusingthisprocedure. Rr,j n0,N = NN Hj n0+n HHj n0+n . (5)
tn=0
2.2. Channelmeasurements The fact that the correlation matrices are functions of the
startingchannelindexn andlengthoftheestimateN sug-
0
The test equipment at BYU allows sampling of a single- geststhatthechannelisnot stationary.Thisnonstationarity
user point-to-point MIMO link with Nr = Nt = 8. is a mathematical manifestation of physical changes in the
The measurements can accommodate up to 100 MHz of propagationenvironmentcreatedbychangesintheangular
instantaneous bandwidth at a center frequency between 2 characteristics of the propagation environment due to such
and 8 GHz. Specific details of the measurement equipment effectsasanodemovingaroundacornerortheintroduction
areavailablein[26]. ofamobilescatterer.However,drasticnonstationaryeffects
Prior to data collection, calibration measurements were occuronatimescalelargerthanthechannelcoherencetime,
taken with the transmitter “off” to measure background and therefore values of N are chosen to remain within the
interference. At the chosen carrier frequency of 2.45 GHz,
channelstationaritytime.
the external interference was found to be below the noise
floor in the environment considered. A second calibration
3. TRANSMITPRECODINGWITH
performed with both the transmitter and receiver “on” but
TIME-VARYINGCHANNEL
stationary revealed that the time variation of the channel
causedbyambientchangessuchaspedestrians,atmospheric
Transmitprecodingtechniquesattempttomanipulateinput
conditions,andothernaturaldisturbanceswasinsignificant
data signals to achieve a specified design criterion for the
fortheenvironmentsexaminedinthispaper.
overall system. The types of precoders can be classified as
The channel coefficients used in this analysis were
either linear or nonlinear [15] while system optimizations
measured with a stationary transmitter and a receiver
rangefrommaximizingthroughputtominimizingtransmit
movingataconstantpedestrianvelocity(30cm/s).Sincethe
power for a given signal-to-interference plus noise (SINR)
channelishighlyoversampled,withsamplestakenevery3.2
requirement.Regardlessoftheschemeoroptimizationused,
milliseconds, data decimation or interpolation can be used
tocreateanyeffectivenodevelocity.Foragiventransmitter most algorithms require instantaneous CSIT. This require-
location,measurementsfordifferentreceiverlocationswere ment suggests that as nodes move and CSIT becomes out-
taken (using the same receiver velocity), with each location dated,theperformanceguaranteeofanalgorithmnolonger
producing the channel matrix for one user in (1) for the holds. This section describes the optimal sum capacity
simulated multiuser network. Since it was observed that technique (DPC), linear transmit beamforming (BF), and
channel-timevariationresultsalmostexclusivelyfromnode time division multiple access (TDMA), and evaluates their
movement, the superposition of these asynchronous mea- performancedegradationduetonodemotion.
surements into a single-synchronized multiuser broadcast It is important to choose a measurable quantity such
channel seems reasonable. Throughout this paper the term that comparison between different algorithms can be per-
“measured channel” refers to channel coefficients acquired formedinameaningfulmanner.Forthiswork,weconsider
inthisfashion. performancemetricsbasedonmaximizingthetotalmutual
The statistical space-time-frequency structure of the information between transmitter and receiver for all users
experimentalMIMOchannelshasbeenwellanalyzedinthe given the theoretical concept of Gaussian input signals [4].
literature [2, 27] with ensemble averages over a variety of
For a given transmit precoding algorithm with fixed input
locationsshowingthecoefficientstoobeyazero-meancom-
parameters, the total mutual information is referred to as
plex Gaussian distribution (Rayleigh channel magnitudes)
either the expected sum rate or throughput of the system
with spatial and temporal correlation functions that closely
measuredinbits/sec/Hz.
resemblethosegeneratedusingtheclassicJakesmodel[23].
Tocalculatethesummutualinformationforthebroad-
Becausethetransmitandreceivespatialcorrelationmatrices
cast channel with outdated CSIT, a general procedure is
are used in the development of the transmit precoding
followed for all transmit precoding techniques. First, the
strategy introduced in this paper as well as the generation
CSIT (assumed perfect) obtained when the receivers are at
ofmodeledchannelmatrices(seeSection2.1),estimationof
aninitialpositionisusedtogeneratetheprecodedtransmit
thesematricesfromthedataisanimportantconsideration.
vectors over a range of receiver motion, meaning that the
Thetransmitcorrelationmatrixestimatedusing N samples
CSIT used is outdated except at the initial position. The
startingatsamplen canbewrittenas
0
received vector for each user is then determined using (1),
allowing computation of the mutual information for each
(cid:4) (cid:5) 1 N(cid:2)−1 (cid:4) (cid:5) (cid:4) (cid:5) user, and the sum mutual information is computed as the
R n ,N = HH n +n H n +n , (4)
t,j 0 NNrn=0 j 0 j 0 sumofeachindividualmutualinformationvalue.
4 EURASIPJournalonAdvancesinSignalProcessing
3.1. Optimaltransmitprecoding (i.e., inaccurately assumes n = n ), then the “optimum”
0
inputcovariancesarefoundusingthedualityoftheMAC/BC
Thenonlineardirty-papercoderisoptimalinthesensethat
anditerativewater-filling[1].
itmaximizesthesummutualinformation(andthereforesum
capacity) when the receivers are at their initial position.
3.2. Lineartransmitprecoding
ConsiderthecaseofstrictDPCatthetransmitter,whereuser
1 is encoded first, user 2 second, and so on. The iterative
Linear transmit precoding, or beamforming, is a technique
implementation of the algorithm results in received vectors
thatuseslinearpreprocessingtomitigatemultiuserinterfer-
at sample n given CSIT Hj(n0) obtained at sample n0 ≤ n ence. Different types of BF algorithms are used to optimize
givenby different communication parameters [15]. Because we are
(cid:4) (cid:5)
considering techniques which maximize the sum mutual
y n ,n
j 0 information,weadoptarate-maximizingBFtechnique[18].
⎧
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨H1(n)x1(n)+H1(n)(cid:2)i=K2xi(n)+η1(n), j =1, Ibunpj(otnhn0i)stchfaoesreC,uStshIeTerNHjt×a(rne1)cc.aoApmaspscuiutmyteoidnpgatitmasmaalmabxpeialmemuifnmodremoxferno0wneebiadgshaettdas
j 0
=⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩Hj(n)xj(n)+Ej(cid:4)n0,n(cid:5)(cid:2)ij=−11xi(n)+Hji=(cid:2)Kj+1xi(n)+ηj(n), sfotrreuamserisjtartasnasmmpitlteednt≥onea0cihs user,thereceivedsignalvector
2≤ j ≤K, (cid:4) (cid:5) (cid:4) (cid:5) (cid:2)K (cid:4) (cid:5)
y n ,n =H (n)b n x (n)+H (n) b n x(n)+η (n).
(6) j 0 j j 0 j j i 0 i j
i=/j
whereE (n ,n)=H (n)−H (n ). (10)
j 0 j j 0
Themutualinformationforthe jthreceivedvectorin(6)
Becausethedistributionsofboththedesiredsignaland
givenknowledgeatthereceiverofbothH (n)andH (n )is
j j 0 the interference plus noise given H (n) are Gaussian, the
j
(cid:10) (cid:4) (cid:5) (cid:4) (cid:5)(cid:11)
mutualinformationforuser j isfoundinamannersimilar
I x (n);y n ,n |H (n),H n
DPC j j 0 j j 0 tothatoutlinedinSection3.1,resultingin
(cid:10) (cid:4) (cid:5) (cid:11) (cid:10) (cid:4) (cid:5) (cid:4) (cid:5)(cid:11)
=h y n ,n |H (n) −h y n ,n |x (n),H (n),H n , (cid:4) (cid:5) (cid:4) (cid:5)
j 0 j j 0 j j j 0
I [x (n);y n ,n |H (n),H n ]
(7) BF j j 0 j j 0
(cid:12)(cid:12) (cid:13)(cid:14) (cid:4) (cid:5)(cid:15) (cid:12)(cid:12)
twhheeernetxroj(pny)faurnecatisosnu.mWedithtopberefeGcatuCsSsiIaRn,tinhpeuetrsroarnbdehco[·m]eiss =log(cid:12)(cid:12)(cid:12)I+Hj(n)(cid:13)(cid:14)Ki=1Qi(cid:4)n0(cid:5)(cid:15)HHj (n)(cid:12)(cid:12)(cid:12), (11)
(cid:12)I+H (n) K Q n HH(n)(cid:12)
deterministic at the receiver (i.e., the receiver is aware j i=/j i 0 j
that CSIT is outdated), and therefore both [y (n ,n) |
j 0 where Q (n ) = E[b (n )bH(n )] and E[xH(n)x (n)] is
H (n),H (n )] and [y (n ,n) | x (n),H (n),H (n )] are j 0 j 0 j 0 j j
j j 0 j 0 j j j 0 unity. If the optimization results in the zero matrix for
Gaussian distributed. As a result, the upper and lower
Q (n ), then user j is excluded from access to the channel.
boundsonmutualinformationwitherroneousCSIfrom[4] j 0
Forcompleteness,wecanwritethetotalexpectedrategiven
areequivalentandreduceto
theoutdatedinputcovariancesQ(n )forbeamformingas
i 0
(cid:10) (cid:4) (cid:5) (cid:4) (cid:5)(cid:11)
I x (n);y n ,n |H (n),H n
DPC j j 0 j j 0 (cid:4) (cid:5) (cid:2)K (cid:10) (cid:4) (cid:5) (cid:4) (cid:5)(cid:11)
(cid:12)(cid:12)Z +H (n)Q (cid:4)n (cid:5)HH(n)(cid:12)(cid:12) CBF n0,n = IBF xj(n);yj n0,n |Hj(n),Hj n0 .
=log j j (cid:12)(cid:12) j(cid:12)(cid:12) 0 j , j=1
Zj (8) (12)
(cid:2)K (cid:2)j−1 Some comments are necessary regarding the capacity
Z =I+ ΨHi(n)+ ΨEi(n0,n),
j j j maximizing MISO beamforming algorithm [18]. In [15],
i=j+1 i=1 this technique was used with multiple receive antennas by
where for some matrix V, ΨV = E[VHQ (n )V] and iteratively performing the algorithm while updating the
j j 0 receiver beamformer with minimum mean squared error
Q (n ) = E[x (n )xH(n )] represents the input covariance
j 0 j 0 j 0 (MMSE) weights, although no proof of optimality was
matrixcalculatedatsamplen .Aftercomputing(8)foreach
0 made.Sincethebeamformingweightsare,inform,capacity
user,thesummutualinformationbecomes optimal(CO)fortheMISOchannelandhavethestructure
ofaregularizedchannelinversion(RCI),itisreferredtohere
(cid:4) (cid:5) (cid:2)K (cid:10) (cid:4) (cid:5) (cid:4) (cid:5)(cid:11)
C n ,n = I x (n);y n ,n |H (n),H n , asCO-RCI[15].
DPC 0 DPC j j 0 j j 0
j=1
(9) 3.3. TimedivisionmultipleaccesswithoutCSIT
where C (n ,n) is implicitly a function of the input Time division multiple access (TDMA) is a transmit pre-
DPC 0
covariance matrices Q(n ). If the transmitter assumes that coding technique that ideally creates an interference-free
i 0
thereisnolagbetweenacquisitionofCSITandtransmission environment. Furthermore, since it does not require CSIT,
AdamL.Andersonetal. 5
it can provide stable throughput in a time-varying channel 20
althoughitsignificantlylowerstheoverallthroughputsince
it does not accommodate simultaneous channel access for use)
multipleusers.ThisformofTDMAisachievedbyemploying el 18
n
time sharing at the transmitter and optimal coding with an
h
nscoheCmSeI,Tthaessburmoaidncgasat cRhaaynlenieglhrefdaudcinesgtochaanvniretlu.aFlosringthlei-s bits/c 16
(
user channel (xi is zero for i=/ j) where each user has a put
receivedvector ugh 14
o
y (n)=H (n)x (n)+η (n). (13) thr
j j j j d
e
ct 12
e
Sinceeachuserisonlyaccessingthechannelafractionof xp
E
thetime,themutualinformationforuser jwillsimplybe
(cid:12) (cid:12) 10
(cid:12) (cid:12) 2 4 6 8 10
1 (cid:12) P (cid:12)
ITDMA[xj(n);yj(n)|Hj(n)]= K log(cid:12)(cid:12)I+ N Hj(n)HHj (n)(cid:12)(cid:12), Numberofusers(K)
t
(14) DPC
CO-RCI
where P is the total available power at the transmitter TDMA
and the optimal input covariance reduces to the scaled Figure 1: Expected throughput versus number of users for fixed
identity for all n given channel coefficients which satisfy a N =N =4antennasandP =10inRayleigh,flat-fadingchannel
0 r t
spatiallyuncorrelatedGaussiandistribution.Thestabilityof model.Allnodeshaveperfectchannelknowledgeforallrealizations
TDMA without CSIT is manifest in the total sum mutual ofthechannel.
information
(cid:2)K (cid:10) (cid:11) over an infinite set of channel realizations, which is not
C (n)= I x (n);y (n)|H (n) (15)
TDMA TDMA j j j possibleusingafinitesetofmeasureddata,andisnotstrictly
j=1
definedforoutdatedCSI.
whichisonlyafunctionofthecurrentchannelatsamplen. Giventhedifficultiesassociatedwiththeergodiccapacity
Performance comparisons between different transmit forthisapplication,metricsusedinthisstudyarebasedon
precoderscanbemadebyexamininghowtotalthroughput thesampleexpectedthroughput(SET)whichistheexpected
scaleswiththenumberofnetworknodes[15].Considerthe error-free throughput for the channel as a function of the
standard Rayleigh flat-fading channel scenario where there delay n− n0. We perform a time average over all possible
is no lag between CSIR and CSIT (i.e., n = n0) and all initial displacementsn0,sothattheSETfora displacement
nodeshaveperfectestimatesofthechannel.Figure1shows Δn=n−n0isdefinedas
throughputscalingasthenumberofusersincreasesforeach
ofthetransmitprecodingtechniquesdiscussed.Thesystem (cid:4) (cid:5) 1 Nm(cid:2)ax−Δn (cid:4) (cid:5)
isfixedatNt =4transmitantennas,Nr =4r(cid:14)eceiveantennas SX Δn = Nmax−Δn m=1 CX m,m+Δn , (16)
peruser,andatotalpowerconstraintofP= tr{Q(n )}=
i i 0
10.WhiletheseresultsrevealtheoptimalityofDPC,theyalso where N is the total number of samples in the dataset
max
showthatBFcapturesthemajorityofavailablethroughput and the subscript X is a member of the set of specified
for larger networks and that the TDMA performance does precoders {DPC, BF, TDMA}. Note that for Δ = 0, (16)
n
notscaleappreciablywithincreasingnetworksize. represents the time-average expected system throughput.
Since this study considers temporal channel variation and
4. PERFORMANCEMETRICS not coefficient estimation, it is assumed that the channel
estimates H (n) and H (n ) known, respectively, at the
j j 0
Assessing the performance of the algorithms under con- receiverandtransmitterareerrorfree.
sideration requires definition of meaningful metrics which It is noteworthy that CX(n0,n0 +Δn) is not necessarily
capture the performance degradation created by outdated adecreasingfunctionofΔ .Forexample,ifthechannelesti-
n
CSIT. Naturally, many different metrics could be defined, mateoccursattheendofafade,thesummutualinformation
with the conclusions drawn ultimately depending on these is likely to be greater as the nodes move and the channel
definitions.However,sincethegoalofDPCandCO-RCIisto improves. However, because the SET in (16) represents an
maximizethesummutualinformation,itislogicalthatthe average behavior, it generally decreases with increasing Δ .
n
performancemetricsusedinthisworkdependonthisquan- Figure2 plots the SET versus the spatial displacement Δ =
tity.Oneexcellentmetricwhichdescribesthemaximumrate Δ Tv, where T and v represent respectively the sample
n s s
atwhicherror-freetransmissionistheoreticallypossiblefor interval and the receiver velocity, for each of the transmit
a given channel type is the ergodic channel capacity [1]. precodersassumingJakes’channelmodelandanormalized
However, computing this quantity requires an expectation Dopplerfrequencyof f =0.0086chosenbaseduponT and
d s
6 EURASIPJournalonAdvancesinSignalProcessing
v.ThesystemparametersincludeK =5userseachequipped 18
withN = 4antennas,atransmitterwithN = 4antennas,
and a rtotal power constraint of P = 10.tThe maximum Hz) 16
tdhisrpoluagchempuetndtΔegirsaldimatiitoendhtoap3λpesninscwetihtheintratnhsisieinnttberevhaalv.iNorootef bits/s/ 14
(
that both DPC and CO-RCI experience a reasonably rapid ut
p
degradationinthroughputasaresultofoutdatedCSIT. gh 12
u TDMA
While plots of the SET such as that in Figure2 reveal hro
detailed information regarding performance degradation dt 10
due to outdated CSIT, it is useful to derive simple quan- cte CO-RCI
e
titative measures from the SET that allow single-number xp 8
e
comparisonofthebehaviorfordifferentenvironments.The ple
remainder of this section outlines two metrics based on am 6 DPC
S
the SET which help quantify the stability of the transmit
precoding algorithms and motivate the new algorithm 4
0 0.5 1 1.5 2 2.5 3
definedinSection5.
Δ(wavelengths)
4.1. SETcrossoverdistance Figure 2: Sample expected throughput (SET) as a function of
delay for a network with parameters N = N = 4, K = 5, and
t r
As shown in Figure2, there is a displacement at which P = 10givenvarioustransmitprecodingschemesinthespatially
the expected throughput drops below that for TDMA. This white,Jakes’channelmodelwithanormalizedDopplerfrequency
displacement, denoted as d , is referred to as the SET of fd=0.0086.
T
crossover distance and quantifies the displacement beyond
which CSIT is no longer useful (i.e., beyond this displace-
ment,TDMAwhichusesnoCSIToffershigherthroughput). 5. STABLETRANSMISSION
Smallvaluesofd suggestthatagivenprecodingalgorithm
T
As shown in the previous sections, attempting to transmit
is highly sensitive to channel temporal variations and will
performpoorlyinpracticalsystems.InFigure2,d = 0for witheithertheoptimalnonlineartransmitprecodingscheme
T
TDMA, d ≈ 0.25λ for DPC, and d ≈ 0.4λ for CO-RCI (DPC) or linear beamforming on the optimal subspaces
T T
(CO-RCI) results in signficant performance loss with even
beamforming.
small node displacement. This observation suggests that a
signalingstrategywhichisinsensitivetonodedisplacement
4.2. Averagesampleexpectedthroughput(ASET)
mustusetransmissiononsuboptimalsubspacesthatremain
constantforlongerperiodsoftime.Motivatedbythisfact,we
WhiletheSETcrossoverdistancegivesanindicationofhow
presentaniterativebeamformingalgorithmthathassimilar
quickly the performance degrades with node displacement,
performancetoCO-RCIbeamformingwhenusedwithCSIT
it clearly provides only limited insight into the behavior.
and stable performance when used with CDIT. While the
This fact motivates another performance metric which
complexityofthisalgorithmishigherthanthatofCO-RCI,
incorporates the throughput over all displacements. The
it enables a significantly reduced frequency at which the
averagesampleexpectedthroughput(ASET)isdefinedby
transmitterBFweightsmustbeupdated.
1 M(cid:2)−1
SX(M)= M SX(Δn), (17) 5.1. MMSE-CSITbeamforming
Δn=0
Ourgoalistodefineabeamformingalgorithmthatachieves
whereMrepresentstheextentofdisplacementsintheregion
the capacity-optimal performance of CO-RCI when used
of interest. From Figure2, the normalized ASET values
withCSITbutcanbeextendedforusewithCDIT.Weapply
SX/STDMA are 1, 0.89, and 0.55 for TDMA, CO-RCI, and
thestandardcoordinatedtransmitter/receiverbeamforming
DPC,respectively.
algorithmsuggestedin[15]whereweightsatthetransmitter
Some important observations from Figure2 can be
made regarding the performance metrics and their effects andreceiverareupdatedinaniterativemanner.Tomotivate
the steps at each iteration of the algorithm, the following
on transmission stability. The distance d is meaningful
T
observationsareconsidered:
in that it defines the sensitivity of an algorithm to node
movement but is not practical as an optimizable variable.
(i)themetricofinterestismaximizingthetotalmutual
For example, maximizing d will not necessarily result in
T information (capacity) of the system with linear
a stable transmission policy since the majority of available
beamforming(Section3);
throughput may be lost in the first few fractions of a
(ii)MMSE beamforming at the receiver is capacity
wavelength. In fact, Figure2 indicates that the most stable
optimal[13];
transmission scheme is TDMAwhich maximizes the ASET.
These observations will be used to motivate a more stable (iii)there exists a duality between transmit and receive
transmitprecoderinSection5. beamforming[1,13].
AdamL.Andersonetal. 7
(1)AssumeaninitialsetofN randomtransmitweightsb withequalpowerallocationp =P/N
s i i s
(2)CalculatetheMMSEreceiverbeamformingweightsforallstreamstoallusers
(cid:14)
w =(I+H ( p b bH)HH)−1H bp
i,j j k k k k j j i i
(3)FindthesurvivorstreamsusingSINR
π(i)=argmax ρ
j i,j
(4)Numericallyoptimizethepowersp assignedtoeachstream
i
(5)UpdatetheMMSEtransmitterbeamformingweights
(cid:14)
b =(I+ p HH w wH H )−1H w p
i k k π(k) k,π(k) k,π(k) π(k) π(i) i,π(i) i
(6)Repeat(2)–(5)untilconvergence
(7)Repeat(1)–(6)forN =1,...,K
s
(8)Usew correspondingtothevalueofN thatmaximizes
i,π(i) (cid:14) s
C = Ns log(1+ρ )
MMSE-CSIT i=1 i,π(i)
Algorithm1:Iterativebeamformingformaximizationofsampleexpectedthroughput.
For the following, the optimization variable is indexed Oncestreamshavebeenmappedtousers,MMSEreceiver
over the current sample index n and the index n at beamformingweightsarecomputedusing
0
which the transmitter acquires CSI. This indexing is for
convenience when we address CDI beamforming, while for (cid:4) (cid:5) (cid:16) (cid:2)K (cid:4) (cid:5) (cid:4) (cid:5) (cid:4) (cid:5) (cid:17)−1
CSI beamforming the transmitter assumes n0 = n for all wi,j n0,n = I+Hj(n) pk n0 bk n0 bHk n0 HHj (n)
time (i.e., the transmitter only calculates a single set of k=1
beamformingweights). ×Hj(n)bi(n0)pi(n0).
Unit norm transmit beamforming weights b(n ) are (20)
i 0
initialized for a given number of data streams N using the
s
Eachreceiverthen“transmits”usingitssetofbeamforming
singularvectorsofarandommatrix,similartotherandom
weights over the reciprocal channel HH(n ), and for each
beamformingalgorithm[28],withthepowersforallstreams j 0
streamthetransmittercomputesupdatedMMSEbeamform-
initially equal. Given transmit weights and powers, each
ingweightsb(n ).Foragivensetoftransmitterandreceiver
receivercalculatesasetofN MMSEbeamformingweights, i 0
s
beamforming weights, the quasiconvexity of the single-
one for each of the N streams. For unit receiver noise
s
inputsingle-outputSINRfunctionenablesastraightforward
variance and assuming linear receiver processing (so that
numerical optimization of the power coefficients p(n ) to
multiple streams destined for the same user will interfere i 0
maximize the expected system rate. The sample expected
with each other), the resulting received SINR of the ith
throughput based on the beamforming weights and power
stream to the jth user for the MISO broadcast channel is
allocationsis
writtenas
ρ (cid:4)n ,n(cid:5)= (cid:14)pi(cid:4)n0(cid:5)(cid:4)bHi (cid:4)(cid:5)n0(cid:5)(cid:4)HHj(cid:5)(n)Hj(n)bi(cid:4)n0(cid:5)(cid:4) (cid:5), CMMSE-CSIT(cid:4)n0,n(cid:5)=(cid:2)Ns log(cid:4)1+ρi,π(i)(cid:4)n0,n(cid:5)(cid:5), (21)
i,j 0 1+ k=/ipk n0 bHk n0 HHj (n)Hj(n)bk n0 i=1
(18) where (18) is modified to include both transmit and
receive weights for the MIMO channel. The final solution
(cid:14)where pi(n0) is the power allocated to the ith stream and corresponds to the weights wi,j(n0,n) associated with the
ipi(n0)≤P. value of Ns that maximizes (21). The complete algorithm
The next step within the iteration is to assign a single for maximizing the sample throughput through linear
user to each stream. This is accomplished by sequentially processing, referred to as MMSE-CSIT, is summarized in
moving through each of the Ns streams and assigning to it Algorithm1.Notethatsincethealgorithmisperformedwith
the user which achieves the highest value of ρi,j(n0,n0). If n = n0,Algorithm1dropssampleindicesfromthevariable
π(i)representstheuserindexfortheithstream,thisprocess matrices.
isrepresentedmathematicallyas
Figure3 compares CO-RCI and MMSE-CSIT beam-
(cid:4) (cid:5) formingforN =6,N =1,P=10,andavariablenumberof
π(i)=argmaxρi,j n0,n . (19) receivernodestforperrfectCSI.Thechannelcoefficientswere
j
generated using the standard Rayleigh, flat-fading model
Itisimportanttonotethatwhilethestreammappingpolicy for the multiuser channel. Figure3 also shows the optimal
π(i) may result in nodes without an assigned stream at a nonlinear DPC precoder as a performance reference. Note
given iteration, these nodes may recapture a stream at a that, with power optimization, CO-RCI and MMSE-CSIT
futureiteration. perform almost identically, which is the intended result.