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Optimized Power-Allocation for Multi-AntennaSystems impaired by
Multiple Access Interference
and Imperfect Channel-EstimationEnzo Baccarelli, Mauro Biagi,
Cristian Pelizzoni, Nicola Cordeschi
{enzobac, biagi, pelcris, cordeschi}@infocom.uniroma1.it
Abstract— This paper presents an optimized spatial signalshaping
for Multiple-Input Multiple-Output (MIMO) ”ad-hoc”-like networks.
It is adopted for maximizing the informationthroughput of
pilot-based Multi-Antenna systems affected byspatially colored
Multiple Access Interference(MAI) and channelestimation errors.
After deriving the architecture of the MinimumMean Square Error
(MMSE) MIMO channel estimator, closedform expressions for the
maximum information throughputsustained by the MAI-affected MIMO
links are provided. Then,we present a novel power allocation
algorithm for achieving theresulting link capacity. Several
numerical results are providedto compare the performance achieved
by the proposed power-allocation algorithm with that of the
corresponding MIMOsystem working in MAI-free environments and
equipped witherror-free (e.g.,perfect channel-estimates). So doing,
we are ableto give insight about the ultimate performance loss
induced inMIMO systems by spatially colored MAI and imperfect
channelestimates. Finally, we point out some implications about
SpaceDivision Multiple Access strategies arising from the
proposedpower allocation algorithm.
Index Terms— Multi-Antenna, MAI, imperfect channel estima-tion,
signal-shaping, space-division multiple-access.
I. I NTRODUCTION AND GOALS
Due to the current fast increasing demand for high-throughput
Personal Communication Services (PCSs) basedon small-size
power-saving palmtops, the requirement for”always on” mobile data
access based on uncoordinated ”ad-hoc” and ”mesh” type networking
architectures are expected todramatically increase within next few
years [18,20,27,28]. Inorder to increase the channel throughput,
the spatial dimensionis viewed as lowest cost solution for wireless
communicationsystems. As a consequence, in these last years
increasingattention has been directed towards designing
array-equippedtransceivers for wireless PCSs [25,27]. Moreover,
such tech-nological solution suitably addresses those
energy-constrainedapplication scenarios in which wireless ad-hoc
and meshnetworks are though to be applied, by providing
adequatediversity and coding gains. This is justified by
considering thatboth ad-hoc and mesh networks are typically
characterized byusers equipped with battery-powered terminals. So
the MIMOcapability to offer same performances of SISO systems
with
Enzo Baccarelli, Mauro Biagi, Cristian Pelizzoni and Nicola
Cordeschi arewith INFO-COM Dept., University of Rome ”Sapienza”,
Via Eudossiana 18,00184 Rome, Italy. Ph. no. +39 06 44585466 FAX
no. +39 06 4873300.
This work has been partially supported by Italian National
project:Wire-less 8O2.16 Multi-antenna mEsh Networks (WOMEN)under
grant number2005093248.
a considerable gain in terms of power consumption, makesthe
Multi-Antenna approach suitable for wireless ad-hoc andmesh
networks [30,31,32].
A. Related Works
In this respect, current literature mainly focuses
ontransceivers working under the assumption of MIMO
channel’sperfectestimation. Specifically, in [1,2] the capacity of
MIMOsystems under spatially colored MAI is evaluated when theMIMO
channel is assumed to be perfectly known at receiveand transmit
sides, while in [23] the MAI is assumed stillspatially colored but
the channel is assumed perfectly knownonly at the receiver. The
above assumptions may be consid-ered reasonable when quasi-static
application scenarios areconsidered (e.g., Wireless Local Loop
systems, [1]), but mayfall short when emerging applications for
high-quality mobilePCSs [9,10,27] are considered. Finally, some
recent worksaccount for imperfect channel estimation [5,10], but
they donot analyze the effect of spatially colored MAI on the
resultingchannel throughput.
B. Proposed Contributions
Therefore, motivated by the above considerations, in thiswork we
focus on the ultimate information throughput con-veyed by
pilot-based wireless MIMO systems impaired byspatially colored MAI
whenimperfectchannel estimates areavailable at transmit and receive
sides. Specifically, maincontributions of this work may be
summarized as follows.First, after developing the optimal MMSE
channel estimatorfor pilot-based MIMO systems impaired by
spatially-coloredMAI, we derive the closed form expression for the
resultingsustained information throughput. Second, we propose an
iter-ative algorithm for the optimized power allocation and
signal-shaping under the assumption ofimperfectchannel estimates
attransmit and receive sides. Third, we provide numerical
resultsand performance comparisons for testing the effectiveness
ofthe proposed spatial-shaping and power allocation algorithmwhen
”ad-hoc” networking architectures are considered. Final-ly, we
point out some (novel) guidelines about the optimizeddesign of
Space Division Multiple-Access strategies arisingfrom the proposed
power allocation algorithm.
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C. Organization of the work
The remainder of this paper is organized as follows. Thesystem
modelling is described in Sect.II and the MIMO chan-nel MMSE
estimator is developed in Sect.III. The informationthroughput
evaluation and the resulting optimal power alloca-tion algorithm
are presented in Sect.IV. In Sect.V a model forthe spatial MAI
arising in Multi-Antenna ”ad-hoc” networks isdescribed. Numerical
plots and performance comparisons fortesting the proposed power
allocation algorithm are presentedin Sect.VI. Finally, Sect.VII is
devoted to discuss some generalguidelines for the overall design of
MAI-impaired Multi-Antenna pilot-trained transceivers.
Before proceeding, let us spend few words about the adopt-ed
notation. Capital letters are for matrices, lower-case under-lined
symbols denote vectors, and characters with overlinedarrow→ denote
block-matrices and block-vectors. Apexes∗,T , † are respectively
meant as conjugation, transposition andconjugate-transposition,
while lower-case letters are used forscalar values. In addition,det
[A] andTra[A] mean determi-nant and trace of matrixA , [a1 ... am],
andvect(A) denotesthe (block) vector obtained by stacking theA’s
columns.Finally, Im is the (m × m) identity matrix, ||A||E is
theEuclidean norm of the matrixA, A ⊗ B is the Kroneckerproduct of
the matrixA by matrixB, 0m is the m-dimensionalzero-vector,0m×n
stands for(m×n) zero-matrix,lg denotesnatural logarithm andδ(m,n)
is the (usual scalar) Kroneckerdelta (e.g.,δ(m,n) = 1 for m = n and
δ(m,n) = 0 form 6= n).
II. T HE SYSTEM MODELING
The application scenario we consider refers to the emerginglocal
wireless ”ad-hoc” networks [18,20,27,28] where multipleautonomous
transmit-receive nodes are simultaneously activeover a limited-size
hot-spot cell, so that all transmissions areaffected by MAI [18].
The (complex base-band equivalent)radio channel from a transmit
node Tx to the correspondingreceive one Rx is sketched in Fig
1.
xT xR
11h
r th
21h Space Time Encoder and
Modulator
with
t
Antennas
Demodulator,
Channel
Estimator and
Decoder with
r
Antennas
1 1
2 2
t rMIMO FORWARD CHANNEL
FEEDBACK LINK
� ��
Source
MessageDetected
Message
�
H�
H
dKdK
Fig. 1. Multi-Antenna system equipped with imperfect (forward)
channelestimateŝH and impaired by MAI with spatial covariance
matrixKd.
Transmit and receive units are equipped witht andr anten-nas,
respectively. The MIMO radio channel should be affectedby
slow-variant Rayleigh flat fading1 and multiple accessinterference.
Path gains{hji} from i-th transmit antenna toj-th receive one may
be modelled as complex zero-mean unit-variance random variables
(r.v.) [5,6,7,8], and they may beassumed mutually uncorrelated when
the antennas are properlyspaced2.
Furthermore, when low-mobility applications are considered(e.g.,
nomadic users over hot-spot cells), all path gains maybe assumed to
change every T≥ 1 signaling period at newstatistically
independentvalues. The resulting ”block-fading”model may be used to
properly describe the main featuresof interleaved frequency-hopping
or interleaved packet-basedsystems [7,18,19]. MAI affecting the
link in Fig.1 depends onthe network topology [1,2,20].
Specifically, we suppose thatit is at least constant over a packet
period. Anyway,{hji}and MAI statistics may be different over
temporally adjacentpackets, so that Tx and Rx nodes in Fig.1 do not
exactly knowthem at the beginning of any transmission period.
Therefore,according to Fig.2, the packet structure is composed by
T≥ 1slots: the first TL ≥ 0 ones are used by Rx for learning theMAI
statistics (see Sect.II.A); the second Ttr ≥ 0 ones areemployed for
estimating the (forward) MIMO channel pathgains{hji} (see
Sect.II.B) and, finally, the last Tpay , T −Ttr − TL ones are
adopted to carry out payload data (seeSect.II.C).
TL(learning) Ttr(training) Tpay(payload)
Fig. 2. The packet structure (T, TL + Ttr + Tpay)
As consequence, after denoting as RC (nats/slot) the space-time
information rate, the resulting system spectral efficiencyη
(nats/sec/Hz) equates
η =TpayT
RC∆sBw
, (1)
where∆s (sec.) and Bw (Hz) denote the slot duration and
RFbandwidth of the radiated signals, respectively.
A. The Learning Phase
During the learning phase (see Fig.2), Tx in Fig.1 is offand Rx
attempts to ”learn” the MAI statistics. Thus, allreceive antennas
are now used to capture the interfering signals
1The flat fading assumption is valid when the radiated signal RF
bandwidthBw is less than the MIMO forward channel coherence
bandwidthBc.Furthermore, we anticipate that the effects of
Ricean-distributed fading onthe system performance are accounted
for and evaluated in the followingSects. V, VI.
2For hot-spot local area applications, proper antenna spacing
may beassumed of the order ofλ/2 [15]. However, several measures
and analyticalcontributions estimate (very) limited throughput loss
when the path gains’correlation coefficient is less than 0.6 [4 and
references therein].
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emitted by the interfering transmit nodes3. So, after denotingby
ẏ(n) , [ẏ1(n)...ẏr(n)] the r-dimensional column vector ofthe
(sampled) signals received at the n-th ”learning” slot, thislast
equates
ẏ(n) ≡ ḋ(n) , ẇ(n) + v̇(n), 1 ≤ n ≤ T. (2)The overall
disturbance vectoṙd(n) , [ḋ1(n)...ḋr(n)]T in(2) is composed by
two mutually independent components,which are denoted bẏw(n) ,
[ẇ1(n)...ẇr(n)]T and v̇(n) ,[v̇1(n)...v̇r(n)]T , respectively.
The first component takes intoaccount for the receiver thermal
noise and then it is modeled asa zero-mean, spatially and
temporally white Gaussian complexr-variate sequence, with
covariance matrix
E{
ẇ(n)(ẇ(m))†}
= N0I rδ(m,n), (3)where N0 (watt/Hz) is the power spectral
density of thethermal noise. The second component in (2) takes the
MAIinto account. It is modelled as zero-mean, temporally
white,spatially coloredGaussian complex r-variate sequence,
whosecovariance matrix
Kv , E{
v̇(n)(v̇(n))†}≡
2664c11 ... c1rc∗12 ... c2r...
......
c∗1r ... crr
3775 , (4)is supposed to be constant over a packet transmission4
(atleast). Since its value may be different over temporally
adja-cent packets, we assume that both Tx and Rx nodes of Fig.1do
not exactly know the overall disturbance covariance matrix
Kd , E{
ḋ(n)(ḋ(n))†}≡ Kv +N0I r, (5)
at the beginning of any new packet transmission period. Sincethe
received signals{ẏ(n)} in (2) equate MAI{ḋ(n)} ones,from the Law
of Large Numbers [26] we obtain the followingunbiased and
consistent (e.g, asymptotically exact) estimateK̂d for the MAI
covariance matrix:Kd
K̂d =1
TL
TL∑n=1
ẏ(n)(ẏ(n))†. (6)
Concerning the accuracy of the estimate in (6),
analyticalresults (see [3 and references therein]) show that the
relativesquare estimation error||Kd − K̂d||2E/||Kd||2E vanishes as
atleast1/TL. So, in principleTL = 10 suffices for achieving
3In principle, some system synchronization should be assumed to
guaranteethat the learning procedure is carried out by only one
user at time. However,under the (milder) assumption that each user
actives his learning procedureat randomly selected times, it is
likelihood to retain negligible the probabilitythat more users are
simultaneously in the learning phase. Anyway, weanticipate that the
numerical results of Sect.VI.C support the conclusion thatthe
performance of the optimized power allocation algorithm we propose
inSect.IV, is quite robust against errors possibly present in the
estimate of actualMAI covariance matrixKd in (5).
4The assumption of temporally white MAI sequence{v̇(n)} may
beconsidered reasonable when FEC coding and interleaving are
employed [11].In addition, by resorting to the Central Limit
Theorem, the overall disturbance{ḋ(n)} in (2) may be considered
Gaussian distributed. Since the Gaussian pdfmaximizes the
differential entropy [12], by fact we are considering a worst-case
application scenario. Finally, since the network topology for
servingnomadic users is slow-variant [20], it can be reasonable to
supposeKv in(4) to be constant (at least) over each packet
transmission period.
mean square estimation errors under 10%. Furthermore, sincethe
numerical results in Sect.VI.D confirm that throughputloss, due to
imperfect MAI covariance matrix estimate, maybe neglected forTL
exceeding 10, we assume that, at the endof the learning phase
(e.g., at stepn = TL), Kd is perfectlyestimated by Rx node and then
it is transmitted back to Txvia the ideal feedback link of Fig.15.
This assumption will berelaxed in Sect.VI.C, when we will test the
sensitivity of theproposed signal-shaping algorithm to errors
possibly affectingthe estimated̂Kd.
B. The Training Phase
Based on the MAI covariance matrixKd, Tx node can nowoptimally
shape the pilot streams{x̃i(n) ∈ C1, TL +1 ≤ n ≤TL + Ttr}, 1 ≤ i ≤
t, which are used by Rx to estimatethe MIMO forward channel path
gains{hji, j = 1, ..., r, i =1, ..., t}. Specifically, when the
pilot streams are transmitted,the sampled signals{ỹj(n) ∈ C1, TL+1
≤ n ≤ TL+Ttr}, 1 ≤j ≤ r, received at the output of j-th receive
antenna are
ỹj(n) =1√t
t∑
i=1
hjix̃i(n) + d̃j(n), TL + 1 ≤ n ≤ TL + Ttr,1 ≤ j ≤ r, (7)
where the overall disturbances
d̃j(n) , ṽj(n) + w̃j(n), TL + 1 ≤ n ≤ TL + Ttr,1 ≤ j ≤ r,
(7.1)
are independent from the path gains{hji} and still de-scribed by
(4) and (5). Hence, by assuming the (usual) powerconstraint
1t
t∑
i=1
||x̃i(n)||2 = P̃ , TL + 1 ≤ n ≤ TL + Ttr, (8)
on the average transmitted power̃P , the resulting signal
tointerference-plus-noise ratio (SINR)̃γj at the output of
j-threceive antenna equates (see eqs.(7), (8))
γ̃j = P̃ /(N0+cjj), 1 ≤ j ≤ r, (8.1)where N0 + cjj is j-th
diagonal entry ofKd. All the
(complex) samples in (7) may be collected into the (Ttr ×
r)matrix Ỹ ,
[ỹ
1...ỹ
r
]given by
Ỹ =1√tX̃H + D̃, (9)
where X̃ , [x̃1...x̃t] is the pilot matrix,H , [h1...hr] is
the(t × r) channel matrix and̃D , [d̃1...d̃r] is the (Ttr ×
r)disturbance matrix. Since the pilot streams are power limited(see
eq.(8)), the resulting power constraint onX̃ becomes
Tra[X̃X̃†] = tTtrP̃ . (9.1)
5We remark that Time-Division-Duplex (TDD) WLANs, designed for
low-mobility applications, are usually equipped with (very)
reliable duplex chan-nels [15,18]. So the above assumption may be
considered well met. Anyway,the performance loss arising from noisy
feedback channels is investigated inSection VI.C.
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In Sect.III we detail how the training observationsỸ in (9)
areemployed by Rx in Fig.1 for computing the MMSE channelestimates
matrix̂H , E{H | Ỹ}. At the end of the trainingphase (e.g., atn =
TL + Ttr), Ĥ is transmitted by Rx back toTx through the (ideal)
feedback link of Fig.1.
C. The Payload Phase
Based onKd and Ĥ, Tx node in Fig.1 may properly shapethe
(random) signal information streams{φi(n) ∈ C1, TL +Ttr + 1 ≤ n ≤
T}, 1 ≤ i ≤ t, to be radiated. Aftertheir transmission, the
resulting (sampled) signals{yj(n) ∈C1, TL + Ttr + 1 ≤ n ≤ T}, 1 ≤ j
≤ r, received by Rx are
yj(n) =1√t
t∑
i=1
hjiφi(n) + dj(n), TL + Ttr + 1 ≤ n ≤ T,1 ≤ j ≤ r, (10)
where the disturbance sequencesdj(n) , vj(n)+wj(n), 1 ≤j ≤ r,
are mutually independent from the channel coefficients{hji} and the
radiated information streams{φi}. As for thepilot streams, the
signals{φi(n)} radiated during the payloadphase are also assumed
power-limited as in
1t
t∑
i=1
E{||φi(n)||2
}= P, TL +Ttr +1 ≤ n ≤ T, (10.1)
so that the SINRγj at the output of the j-th receive
antennaequates6 (see eqs.(5), (10))
γj = P/(N0+cjj), 1 ≤ j ≤ r. (10.2)Now, from (10) we may express
(r × 1) column vector
y(n) , [y1(n)...yr(n)]T of the observations received duringn-th
slot as
y(n) =1√tHT φ(n) + d(n), TL + Ttr + 1 ≤ n ≤ T, (11)
where{d(n) , [d1(n)...dr(n)]T , TL + Ttr + 1 ≤ n ≤ T}is the
temporally white Gaussian MAI vector with spatialcovariance matrix
still given by eq.(5),H is the previouslydefined(t× r) channel
matrix7 andφ(n) , [φ1(n)...φt(n)]Tcollects the symbols transmitted
by the t transmit antennas.Furthermore, after denoting asRφ ,
E{φ(n)φ(n)†} thespatial covariance matrix ofφ(n) , [φ1(n)...φt(n)]T
, from(10.1) this last must meet the following power
constraint:
E{
φ(n)†φ(n)}≡ Tra[Rφ] = tP,
TL + Ttr + 1 ≤ n ≤ T. (11.1)Finally, by stacking the Tpay
observed vectors
in (11) into the (Tpayr × 1) block vector −→y ,6We point out
that our model explicitly accounts for the different power
levels eP andP that may be radiated by transmit antennas during
the trainingand payload phases, respectively.
7We anticipate that the combined utilization ofH in the model
(9) andHT in the relationship (11) simplifies the resulting
expressions for theMMSE channel estimates in (13) and the conveyed
information throughput
I
�−→y ;−→� |Ĥ
�in (24).
[yT (TL + Ttr + 1) ...yT (T )
]T, we arrive at the following
final observation model:
−→y = 1√t[ITpay ⊗ H]T −→φ +−→d , (12)
where the (Tpayr × 1) (block) disturbance vector−→d ,[dT (TL +
Ttr + 1) ...dT (T )
]Tis Gaussian distributed, with
covariance matrix given by
E{−→d (−→d )†
}= ITpay ⊗Kd, (12.1)
and the (block) signals vector−→φ ,[
φT (TL + Ttr + 1) . . . φT (T )]T
is power limited asin
E{−→
φ†−→φ
}= TpaytP. (12.2)
III. MMSE MIMO C HANNEL ESTIMATION UNDERSPATIALLY COLORED
MAI
Since in [9] it is proved that the MMSE matrix estimateĤ ≡
[ĥ1...ĥr] , E{H|Ỹ} of the MIMO channel matrixHin (9) is a
sufficient statisticfor the ML detection of thetransmitted messageM
of Fig.1, we do not lose informationby considering the receiver’s
architecture composed by theMIMO channel MMSE estimator cascaded to
the ML detectorof the transmitted message. Thus, before starting to
developthe MMSE estimator, let us note that theỸ’s columns in (9)
aremutually dependent, so that any estimated channel
coefficientĥji is a function of thewhole observed matrix̃Y.
However,the j-th columnĥj of Ĥ can be computed via an
applicationof the Orthogonal Projection Lemma as in (see the
AppendixA)
ĥj =1√t
[eTj K
−1/2d ⊗ X̃
†][1t
(K−1d ⊗ X̃X̃
†)+ IrTtr
]−1
·(
K−1/2d ⊗ ITtr)
vect(Ỹ), 1 6 j 6 r. (13)
In (13), ej denotes the j-th unit vector ofRr [13], vect(Ỹ)
isthe rTtr-dimensional column vector obtained via the
orderedstacking of theỸ’s columns whileK−1/2d is the positive
squareroot of K−1d [13]. Now, by denoting as² ≡ [²1...²r] , H−
Ĥthe error matrix of the MMSE channel estimates, the
crosscorrelation among its columns may be evaluated as in
E{
²j (²i)†}
= δ(j, i)I t − E{
ĥj(
ĥi)†}
= δ(j, i)I t − 1t
(ej ⊗ I t
)†(K−1/2d ⊗ X̃
)†
·[1t
(K−1d ⊗ X̃X̃
†+ IrTtr
)]−1·(
K−1/2d ⊗ X̃)(
ei ⊗ I t),
1 6 j, i 6 r. (14)
Thus, the resulting total mean square errorσtot ,
||²||2Eequates
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σ2tot ,r∑
j=1
Tra[²j²
†j
]= rt
−1t
r∑
j=1
Tra
[(ej ⊗ I t
)† (K−1/2d ⊗ X̃
)†
·(1
t
(K−1d ⊗ X̃X̃
†)+ IrTtr
)−1 (K−1/2d ⊗ X̃
)(ej ⊗ I t
)].
(15)
A. Condition for the optimal training
Since the total mean square errorσ2tot in (15) depends onthe
employedpilot streamsvia the training matrixX̃ in (9),we are going
to select it for minimizing (15) under the powerconstraint (9.1).
By properly applying the Cauchy inequality[13], we provide the
following condition for the design of theoptimal training matrixX̃
(see the Appendix B).Proposition 1.The training matrixX̃, that
minimizes the totalsquare error in (15) under the power constraint
(9.1), mustmeet the following relationship:
K−1d ⊗ X̃†X̃ = aIrt, (16)
where the positive scalara equates
a , TtrP̃r
Tra[K−1d ]. (16.1)
¨
Therefore, from (16) we deduce that the optimalX̃ depends onthe
spatial coloration property of MAI via the correspondingcovariance
matrixKd. By fact, the practical implication ofthe relationship
(16) is that the pilot streams radiated bytransmit antennas should
beorthogonal after the whiteningfilter performed by the receiver.
In the special case ofKd = I t(e.g., when the MAI is spatially
white), eq.(16) becomesX̃†X̃ = aI t, and the optimalX̃ matrix is
the usual (para)
unitary one [9,10].
B. The MIMO channel MMSE Estimator for the optimaltraining
When X̃ meets the optimality condition in (16), eqs. (13),(14)
assume the following (simpler) forms:
ĥj =1− σ2ε√
t
(eTj K
−1d ⊗ X̃
†)vect(Ỹ), 1 ≤ j ≤ r, (17)
and
E{
²j(²i)†}≡ δ(j, i)I t−E
{ĥj(ĥi)
†}≡ σ2ε I tδ(j, i), 1 ≤ j, i ≤ r,
(18)where
σ2ε , E{||εji||2
}≡ E
{||hji−ĥji||2
}=
(1+
a
t
)−1, 1 ≤ j, i ≤ r,
(19)is the mean square error estimation variance (which is
thesame for alli and j). Furthermore, the estimated path gains
{ĥji} are uncorrelated and identically Gaussian distributed,
sothat the pdfp(Ĥ) of the resulting estimated matrix̂H equates
p(Ĥ) =( 1
π(1− σ2ε))rt
exp
{− 1
(1− σ2ε)Tra[Ĥ
†Ĥ]
}. (20)
Furthermore, all the entries of the resulting MMSE errormatrix ²
= H − Ĥ are mutually independent, identicallydistributed and
Gaussian, with variances given by eq.(19).
Finally, from (17) and (19), we may conclude that
estimatedmatrix Ĥ approaches the actual oneH for σ2ε → 0, while
Ĥvanishes forσ2ε → 1, so that we have the following
limitexpressions:
limσ2ε→0
Ĥ = H (20.1); limσ2ε→1
Ĥ = 0t×r. (20.2)
According to a current taxonomy, we refer to (20.1), (20.2)
asPerfect CSI (PCSI) and No CSI (NCSI) operating conditions,while
Imperfect CSI (ICSI) corresponds to0 < σ2ε < 1.
IV. CONVEYED INFORMATION THROUGHPUT UNDERCHANNEL ESTIMATION
ERRORS AND SPATIALLY COLORED
MAI
The MIMO block fading channel of Sect.II is informationstable,
so that the resulting Shannon Capacity C is the cor-responding
maximum sustainable throughput. By followingquite standard
approaches [14], this capacity may be expressedas in
C = {C(Ĥ)} ≡∫
C(Ĥ)p(Ĥ)dĤ, (nats/payload slot), (21)
wherep(Ĥ) is given by (20), and
C(Ĥ) , sup−→φ :E{
−→φ†−→φ }≤tTpayP
1Tpay
I(−→y ;−→φ |Ĥ
), (22)
is the MIMO channel capacityconditionedon Ĥ. Furthermore,I(−→y
;−→φ |Ĥ
)in (22) is the mutual information conveyed by the
MIMO channel (12) when̂H is the channel estimate availableat Tx
and Rx nodes of Fig.1. Unfortunately, the optimal pdfof input
signals
−→φ achieving the sup in (22) iscurrently
unknown, even in the case of spatially white MAI
[1,2,4,5].Anyway, in [7] it is shown that Gaussian distributed
inputsignals are the capacity-achieving ones for0 ≤ σ2ε ≤ 1 whenthe
payload phase lengthTpay is largely greater than thenumber of
transmit antennas (see [7] about this asymptoticresult). Therefore,
in the sequel we directly consider Gaussiandistributed input
signals. In this case, theTpay components{φ(n) ∈ Ct, TL + Ttr + 1 ≤
n ≤ T} in (11) of theoverall signal vector
−→φ in (12) are modelled as uncorrelated
zero-mean complex Gaussian vectors, with correlation matrixRφ ,
E{φ(n)φ(n)†} constrained as in (11.1).
Obviously, the MIMO channel information throughput
TG(Ĥ) ,1
Tpaysup
Tra[Rφ
]≤Pt
I(−→y ;−→φ |Ĥ
), (23)
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is upper-bounded byC(Ĥ) in (22), so that, in general, we
haveTG(Ĥ) ≤ C(Ĥ). Anyway, the equality is attained when theabove
mentioned condition ofTpay >> t is met.
About the computation ofI(−→y ;−→φ |Ĥ
)in (23), in general
it resists closed-form evaluation. However, in Appendix C
weprove the following result.Proposition 2.Let us supposẽX to meet
eq.(16). Then, theconditional mutual informationI
(−→y ;−→φ |Ĥ)
in (23) of theMIMO channel (12) equates
I(−→y ;−→φ |Ĥ
)= Tpay
· lg det[(
Ir +1tK−1/2d Ĥ
T · RφĤ∗K−1/2d + σ
2εPK
−1d
)]
− lg det[(
Irt +σ2εTpay
t(K−1d )
∗ ⊗ Rφ)]
, (24)
when (at least) one of following three conditions is verified
:
a) bothTpay and t are large; (24.1)
b) σ2ε vanishes; (24.2)
c) all the SINRsγj , 1 ≤ j ≤ r, in (10.2) vanish. (24.3)¨
Several numerical results confirm that the condition (24.1)
maybe considered virtually met whenTpay ≥ 6t , 7t andt ≥ 4, 5,even
forσ2ε approaching 1 and SINRs of the order of 6dB-7dB.
A. Optimized Power allocation under colored MAI andChannel
Estimation errors
To evaluate the covariance matrixRφ achieving the supin (23),
let us begin with the Singular Value Decomposition(SVD) of the
covariance matrixKd according to
Kd = UdΛdU†d, (25)
where
Λd , diag{µ1, ..., µr}, (25.1)denotes the (r × r) diagonal
matrix ofmagnitude-orderedsingular values ofKd. Furthermore , we
define by
A , Ĥ∗K−1/2d Ud, (26)
the (t × r) matrix which simultaneously accounts for theeffects
of the imperfect channel estimateĤ and MAI spatialcoloration. The
corresponding SVD is
A = UADAV†A, (26.1)
whereUA andVA are unitary matrices, and
DA ,[
diag{k1, ..., ks} 0s×r−s0t−s×s 0t−s×r−s
], (26.2)
is the (t × r) matrix having thes , min{r, t} magnitude-ordered
singular-valuesk1 ≥ k2 ≥ ... ≥ ks > 0 of A alongthe main
diagonal of the sub-matrix starting from elements
(1,1) to (s,s). Finally, let us introduce the following
dummypositions:
αm ,µmk
2m
t(µm + Pσ2ε), 1 ≤ m ≤ s; βl , σ
2εTpaytµl
, 1 ≤ l ≤ r.(27)
Now, the optimized transmit powers{P ?(m), 1 ≤ m ≤ t}achieving
the sup in (23) may be obtained by applying theKuhn-Tucker
conditions [14, eqs.(4.4.10), (4.4.11)]. Theyare detailed by the
followingProposition 3, proved in theAppendix D.
Proposition 3. Let us assume that at least one of the
condi-tions (24.1), (24.2), (24.3) is met. Thus, form = s+1, ...,
t, theoptimal vanish, while form = 1, ..., s they must be
computedaccording to the following two relationships:
P ?(m) = 0, whenk2m ≤(1 +
σ2εP
µm
)( tρ
+ σ2ε Tra[K−1d ]
),
(28)
P ?(m) =1
2βmin
{βmin
[(1− r
Tpay
)ρ− 1
αm
]− 1
+
({βmin
[(1− r
Tpay
)ρ− 1
αm
]− 1
}2
+4βmin
(ρ− 1
αm− rρβmin
αmTpay
))0.5},
whenk2m >(1+
σ2εP
µm
)( tρ
+σ2ε Tra[K−1d ]
), m = 1, ..., s;
(29)where βmin , min{βl, l = 1, .., r}. Furthermore, the
non-negative scalar parameterρ, in (28), (29) must satisfy
thefollowing relationship:
∑
m∈I(ρ)P ?(m) = Pt; (30)
whereI(ρ) ,
{m = 1, ..., s : k2m >
(1 +
σ2εP
µm
)
·( t
ρ+ σ2ε Tra[K
−1d ]
)}, (30.1)
is the (ρ-depending) set of indexes fulfilling the inequali-ty
(29). Finally, the resulting optimized covariance matrixRφ(opt) of
the radiated signals is given by
Rφ(opt) = UA diag{P ?(1), ...P ?(s), 0t−s} U†A, (31)so that the
throughput in (23) may be directly computed as
in
TG(Ĥ) =r∑
m=1
lg(1 +
σ2εP
µm
)
+s∑
m=1
[lg
(1 + αmP ?(m)
)− 1
Tpay
r∑
l=1
lg(1 + βlP ?(m)
)].
¨ (32)
-
7
B. Some explicative remarks
Before proceeding, some explicative comments about themeaning
and practical application of eqs.(28), (29) are in order.
First, the derivation performed in the Appendix D leads tothe
conclusion that theoptimalcovariance matrix in (31)mustbealigned
along the eigenvectors of the matrixA in (26) that,in turn, depend
both on̂H andKd. Therefore,A accountsbothfor the MAI spatial
coloration and errors possibly present inthe channel estimateŝH
availableat the receiver. Thus, matrixA plays the key-role of
”effective” MIMO channel viewed bythe receiver.
Second, since for smallx we have that√
1 + x u 1+0.5x,for vanishingσ2ε we may rewrite (according to
Taylor seriesapproximation) eqs.(28), (29) as follows:
limσ2ε→0P?(m) = max
{0, ρ− t
k2m
}, m = 1, .., s. (33)
Thus, from (33), it follows that the proposed power
allocationalgorithm reduces to the standard water filling one for
vanish-ing σ2ε .
Third, in the case of NCSI (e.g, whenσ2ε = 1), the
channelestimateĤ equates0t×r (see(20.2)). As a consequence,
theresulting throughputTG(Ĥ) in (23) becomes
limσ2ε→1
TG(Ĥ) , TG(0) =r∑
m=1
lg
(1 + Pµm
)
(1 + PTpayµm
)1/Tpay
,
(nats/payload slot). (34)
Since this relationship is valid for large t andTpay
regardlessof employed power level P, the relationship (34) supports
theconjecture in [7] that forlarge Tpay the channel capacity
isattained by employing input signals with Gaussian pdf,evenwhen H
is fully unknown at Rx. Thus, we conclude that,for vanishingσ2ε
and/or small SINRs, the throughputTG(Ĥ)approaches the MIMO channel
capacityC(Ĥ) regardlessofTpay andt values. Several numerical
trials confirmed that, for0 < σ2ε ≤ 1, TG(Ĥ) in (32) is close
to the capacityC(Ĥ)when t ≥ 4 andTpay ≥ 6t.
C. A Numerical Algorithm implementing the proposed
PowerAllocation
The first step for computing (28), (29) is to properly set
theparameterρ in order to meet the power constraint (30). Forthis
purpose, we note that the size of the set (30.1) vanishes atρ = 0
and grows for increasing values ofρ. As consequence,for evaluating
theρ value meeting the relationship (30), wemay adopt the (very)
simple iterative procedure which startsby settingρ = 0 and then
increasesρ by using a properlychosen step-size8 of Table I.
8Several numerical trials confirmed that∆ = 0.1Pt is adequate
for thispurpose. The iterative procedure of Table I is stopped when
the summationin (30) attains the power constraint.
1. Compute and order the eigenvalues of the MAI covariance
matrixKd;2. Compute the SVD of matrixA in (26.1) and order its
singular values;3. SetP ?(m) = 0, 1 ≤ m ≤ t;4. Setρ = 0 andI(ρ) =
∅;5. Set the step size∆;
6. While
�Pm∈I(ρ) P
?(m) < Pt
�do
7. Updateρ = ρ + ∆;8. Update the setI(ρ) via eq. (30.1);9.
Compute the powers{P ?(m), m ∈ I(ρ)} via eq.(29);10. End;11.
Compute the optimized powers{P ?(m), 1 ≤ m ≤ s} via eqs. (28),
(29);12. Compute the optimized shaping matrixR�(opt.) ;13. Compute
the conveyed throughputTG(Ĥ) via eq.(32).
TABLE I
A PSEUDO-CODE FOR THE NUMERICAL IMPLEMENTATION OF THE
PROPOSED OPTIMIZED POWER ALLOCATION ALGORITHM.
V. A TOPOLOGY-BASED MAI MODEL FORMULTI -ANTENNA ” AD-HOC” N
ETWORKS
To test the proposed power allocation algorithm, we con-sider
the application scenario of Fig.3 that captures the key-features of
Multi-Antenna ”ad-hoc” networks impaired byspatial MAI
[15,18,20].
l0
lN
l2
l1
Tx1
Tx2
TxN
Rx1
Rx2
RxN
Tx0 Rx0
(1)dθ
(2)dθ
( )Ndθ
(1)aθ
(2)aθ ( )N
aθ
... ...
Fig. 3. A general scheme for an ”ad-hoc” network composed of
(N+1)point-to-point links active over the same hot-spot area.
Shortly, we assume that the network of Fig.3 is composedof (N+1)
no cooperative, mutually interfering, point-to-pointlinks Txf → Rxf
, 0 ≤ f ≤ N . The signal received by thereference node Rx0 is the
combined effect of that transmittedby Tx0 and those radiated by the
other interfering transmitters(Txf ,1 ≤ f ≤ N ). The transmit node
Txf and the receivenode Rxf are equipped withtf andrf antennas,
respectively.Thus, after indicating aslf the Txf → Rx0 distance,
then thed(n) disturbance vector in (11) may be modelled as
d(n) =N∑
f=1
√( l0lf
)4 1√tf
χf HTf φ(f)(n) + w(n). (35)
The vector w(n) in (35) accounts for the thermal noise(see
(11)); theφ(f)(n) term represents thetf -dimensional(Gaussian)
signal radiated by the Txf interfering transmitter;
-
8
χf accounts for the shadowing effects9; the matrixHf modelsthe
Ricean-distributed fast-fading affecting the interfering linkTxf →
Rx0. Furthermore, according to the faded spatialinterference model
recently proposed in [1,2], the channelmatrix Hf in (35) may be
modeled as in
Hf ≡√
kf1 + kf
H(sp)f +
√1
1 + kfH(sc)f , 1 ≤ f ≤ N, (36)
wherekf ∈ [0, +∞) is thef -th Ricean-factor and all the (tf×r0)
terms of the matrixH
(sc)f are mutually independent, zero-
mean, unit-variance Gaussian distributed r.v.s, that account
forthe scattering phenomena impairing thef -th interfering linkTxf
→ Rx0. The (tf × r0) matrix H(sp)f in (36) captures forthe specular
components of the interfering signals and may bemodelled as in
[1,2]
H(sp)f =(
a(f)b(f)T)T
, 1 ≤ f ≤ N, (36.1)where a(f) and b(f) are (r0 × 1) and (tf × 1)
column
vectors. They are used to model the specular array responsesat
the receive node Rx0 and transmit node Txf , respectively[1,2].
When isotropic regularly-spaced linear arrays are em-ployed at the
Txf and Rx0 nodes, the above vectors may beevaluated as in
[1,2,15]
a(f) =[1, exp
(j2πν cos(θ(f)a )
),
... exp(j2πν(r0 − 1) cos(θ(f)a )
)]T, (36.2)
b(f) =[1, exp
(j2πν cos(θ(f)d )
),
... exp(j2πν(tf − 1) cos(θ(f)d )
)]T, (36.3)
whereθ(f)a , θ(f)d are the arrival and departure angles of
the
radiated signals (see Fig.3), whileν is the antenna spacing
inmultiple of RF wavelengths10.
A. The resulting model for the MAI Covariance Matrix
Therefore, after assuming the spatial covariance matrixR(f)
φ, E{φ(f)(n)φ(f)(n)†}, 1 ≤ f ≤ N , of signals
radiated by thef−th transmit node Txf power-limited as
(seeeq.(11.1))
Tra[R(f)φ
] = tfP (f), (37)
then the covariance matrixKd of the MAI vector in
(35)equates
Kd , E{
d(n)d(n)†}
=
{N0 +
N∑
f=1
( l0lf
)4 E{χ2f}1 + kf
P (f)
}Ir0
9Without loss of generality, we may assumeχf to fall in the
interval[0, 1].When χf = 1 (worst case), MAI impairing effects
arising from transmitinterfering node Txf are the largest.
10Several tests show that rays impinging receive antennas may
beconsidered virtually uncorrelated whenν is of the order of 1/2
[15].
+
{N∑
f=1
( l0lf
)4 kf1 + kf
E{χ2f}tf
a(f)bT (f)R(f)φ
b∗(f)a†(f)
}.
(38)This relationship captures the MAI effects due to the
topolog-ical and propagation features of the considered
multi-antennaad-hoc network. Specifically, eq.(38) points out that
MAIinterference may be considered spatially white when all
theinterfering links’ Ricean factors may be neglected. On
thecontrary, for high Ricean factors the MAI spatial colorationis
not negligible, as confirmed by the numerical results of thenext
Sect.VI.
B. A Worst-Case Application Scenario
Let us consider the hexagonal network of Fig.4. All transmitand
receive nodes have the same number of antennas (e.g.,t0 = t1 = t2 =
t and r0 = r1 = r2 = r) and all transmitnodes radiate the same
power level (e.g.,P0 = P1 = P2 = P ).We assume that the array
elements are one-half wavelengthapart (e.g.,ν=1/2), and all Ricean
factors are equal (e.g.,k1 = k2 = k). Furthermore, let us consider
aworst-operatingscenario with all shadowing coefficients equal to
unity (e.g.,χ1 = χ2 = 1) and the correlation matricesR
(1)
φ, R(2)
φof the
signals radiated by the interfering transmit nodes Tx1,
Tx2equatingP I t [1,2]. Therefore, in this case eq.(38) becomes
Kd ={
N0+29
P
1 + k
}Ir+{ k1 + k
P
9
2∑
f=1
a(f)bT (f)b∗(f)a†(f)},(39)
where
a(1) =
[1, exp(jπ
√3
2), ..., exp(jπ(r − 1)
√3
2)
]T, (39.1)
b(2) =
[1, exp(jπ
√3
2), ..., exp(jπ(t− 1)
√3
2)
]T. (39.2)
while b(1), a(2) are column vectors composed byt andr
unitentries respectively.
VI. N UMERICAL RESULTS AND PERFORMANCECOMPARISONS
Although the MIMO channel pdf in (20) is in closed form,the
corresponding throughput expectation
TG , E{TG(Ĥ)
}, (40)
resists closed-form evaluations, even in the case of
spatiallywhite MAI with vanishingσ2ε [4,5,17 and references
therein].Thus, as in [1,2,4], we evaluate the expected
throughputTGin (40) by resorting to a Monte-Carlo approach based on
thegeneration of 10,000 independent samples ofTG(Ĥ). All
thereported numerical plots refer to the hexagonal network ofFig.4
with unit noise levelN0.
-
9
Rx1
Rx2
Rx0
Tx1
Tx2
Tx0
Fig. 4. A hexagonal network with two interfering links.
A. Effect of the channel estimation errors
The first plots’ set of Fig.5 shows the sensitivity of
thethroughputTG of reference link Tx0→ Rx0 on MIMO
channelestimation errors. All nodes are equipped withr = t =
8antennas, all the Ricean factors in (39) are set to 10 andTpay
=40. Fig.5 shows that throughput loss is at most1% for σ2εvalues
below0.01.
2=0.001εσ
2=0.01εσ
2=0.1εσ
2=0εσ
2=1 (eq.(34))εσ
(nat
s/sl
ot)
G�
Fig. 5. Sensitivity of the throughputTG conveyed by the
reference link Tx0→ Rx0 of Fig.4 on the squared error levelσ2ε
affecting the available channelestimates (Tpay = 40, k=10,
r=t=8).
B. Effect of the number of transmit/receive antennas
The numerical plots drawn in Fig.6 allow us to evaluate
theeffect on the throughput of the numberr = t of antennasequipping
each node of the network of Fig.4. Specifically,Fig.6 shows the
average throughput (40) of the reference link
Tx0→ Rx0 of Fig.4 when the Ricean factor in (39) equates10, σ2ε
= 0.01 andTpay = 80.
(nat
s/sl
ot)
G�
Fig. 6. Sensitivity of the throughput conveyed by the reference
link Tx0→Rx0 of Fig.4 on the number t=r of antennas (Tpay = 80,
k=10,σ2ε = 0.01 ).
An examination of these plots leads to the conclusion that,by
increasing the number of antennas, we are able to quicklygain in
terms of channel throughput.
C. Effect of Errors in the Estimation of the MAI
covariancematrix
As anticipated in Sect.II.A, the estimation accuracy ofK̂din (6)
is mainly limited by the learning phase lengthTL, so itcan be of
interest to test the sensitivity of the proposed powerallocation
algorithm on errors possibly affecting the estimatedK̂d. For this
purpose, we perturbed the actualKd by usinga randomly generated (r
× r) matrix N, composed by zero-mean unit-variance independent
Gaussian entries. Hence, the(analytical) expression for the
resulting perturbedK̂d is
K̂d = Kd +
√||Kd||2E
r2
√δN, (41)
whereδ , E{||Kd− K̂d||
}2E
/||Kd||2E in (41) is a determin-istic parameter which may be
tuned so to obtain the desiredsquare estimation error. Thus, after
replacing theKd matrix bythe corresponding perturbed̂Kd version, we
have implementedthe proposed power allocation algorithm as dictated
by therelationships (28), (29). Finally, we evaluated the new
valueof TG(Ĥ) according to eq.(32) and that we computed{αm}and
then{βl} according to (27) on the basis of theactualMAI matrix Kd.
The resulting average throughput is plottedin Fig.7 for the
reference link Tx0→ Rx0 of Fig.4 (Tpay = 40,r = t = 8, σ2ε = 0.015,
k = 10). From these plots we mayconclude that throughput loss due
to errors in the estimated ofK̂d may be neglected when the
parameterδ is at most0.01.
-
10
Fig. 7. Sensitivity of the throughputTG conveyed by the
reference linkTx0 → Rx0 of Fig.4 on the estimation errors affecting
the available MAIcovariance matrix (Tpay = 40, r=t=8, k=10 ,σ2ε =
0.015 ).
D. Coordinated versus Uncoordinated Medium Access Strate-gies:
some MAC considerations
Although in these last years the MAI-mitigation capabilityof
multi-antenna systems has been often claimed [8,15,18]. Totest
there claims, it may be of interest we want to comparethe
information throughputTG of the proposed power alloca-tion
algorithm with that of orthogonal MAI-free TDMA (orFDMA)-based
access techniques. Till now, it appears that noneof them
definitively perform the best. In particular, this is truewhen
application scenarios as those of Fig.3 are considered,where SINRs
are usually low, so that multiuser detectionstrategies based on
iterative cancellation of the MAIdo noteffectively work [22].
Therefore, on the basis of the aboveconsiderations, we have
computed the average informationthroughputTG , E
{TG(Ĥ)
}(nats/ payload slot) conveyed
by the reference link Tx0→ Rx0 of Fig.4 when MAI-freeTDMA-based
access is used11. The numerical plots of Fig.8for the network in
Fig.4 have been obtained by settingTpay =80, σ2ε = 0.1, k = 1000
and then by varying the number oftransmit/receive antennas from 4
to 12.
Although TG has been evaluated in the worst MAI case(see (39)
and related remarks), the plots of Fig.8 show howmuch greaterTG is
thanTTDMA, specially when low powerlevels P are used and the
transceivers are equipped with alarge number of transmit/receive
antennas. This conclusion is
11According to [22, Sect.VI.C], the conditional information
throughputTTDMA(Ĥ) has been evaluated by fixing the estimation
matrixĤ and byrunning the algorithm of Table I under the following
operating conditions:i) all shadowing factors in (39) have been
zeroed;ii) the power level P in (39) has been replaced by 3P;iii)
the resulting throughputTG(Ĥ) in (32) has been scaled by 1/3.The
condition i) is for modelling the MAI-free condition of the
TDMAtechnique , while the conditions ii) and iii) are due to the
fact that the referencelink Tx0→ Rx0 of Fig.4 is in TDMA mode, and
then it is active only over1/3 of the overall transmission
time.
Fig. 8. Throughput comparisons for the reference link Tx0→ Rx0
of Fig.4for Tpay = 80, k=1000,σ2ε = 0.1.
confirmed by the plots of Fig.9, which refer to
Rayleigh-fadedapplication scenarios.
Fig. 9. Throughput comparisons for the reference link Tx0→ Rx0
of Fig.4(Tpay = 80, k=0, σ2ε = 0.1).
Therefore, from the outset we may conclude that when thenumber
of antennas increases, by using the spatial-shapingalgorithm of
Table I we are able to achieve channel throughputlarger than those
attained by conventional orthogonal accessmethods.
VII. C ONCLUSIONS
The main contribution of this paper is the development ofan
optimized spatial signal-shaping for multi-antenna systemsimpaired
by spatially colored MAI and channel estimation
-
11
errors. From our analysis we may draw three main con-clusions.
First, throughput loss induced by estimation errorsis not very
critical, especially when the system operates atmedium/low SINRs.
Second, the throughput comparisons ofSect.VI confirm the
MAI-suppressing capability of multi-antenna transceivers, even in
”ad-hoc” operating scenarios.Third, the plots of Figs.8,9 show the
throughput improvementattained by uncoordinated spatial-based
multiple access tech-niques respect to coordinated orthogonal ones
(as, for example,TDMA). Currently, we are going to test the
validity of theseconclusions in the mesh-like operating scenarios
consideredby WOMEN project [27].
APPENDIX A - THE MIMO CHANNEL MMSE ESTIMATOR
By using the following property [13]:vect(AB) =[I ⊗ A] vect(B),
we may rewrite (9) as
vect(Ỹ) =1√t
[Ir ⊗ X̃
]vect(H) + vect(D̃). (A.1)
Therefore, since E{vect(D̃)(vect(D̃))†} = Kd ⊗ ITtr ,
andE{vect(Ỹ)(vect(Ỹ))†} = 1t (Ir ⊗ X̃X̃
†) + (Kd ⊗ ITtr), via
an application of the Orthogonal Projection Lemma we obtaineqs.
(13), (14).
APPENDIX B - OPTIMIZATION OF THE TRAINING MATRIX
Since [13,p.64](
1t
(K−1d ⊗ X̃X̃
†)+ I rTtr
)−1
= IrTtr −1t
(K−1/2d ⊗ X̃
) [I rt +
1t
(K−1d ⊗ X̃
†X̃
)]−1
·(
K−1/2d ⊗ X̃)†
, (B.1)
the right-hand-side (r.h.s) of eq.(15) may be recast in
thefollowing form:
σ2tot = rt−1t
r∑
j=1
Tra
[(ej ⊗ I t
)†(K−1d ⊗ X̃
†X̃
)(ej ⊗ I t
)]
+1t2
r∑
j=1
Tra[Λj(X̃)
], (B.2)
where
Λj(X̃) ,(
ej⊗ I t)†(
K−1d ⊗X̃†X̃
) [I rt +
1t(K−1d ⊗ X̃
†X̃)
]−1
·(
K−1d ⊗ X̃†X̃
)(ej ⊗ I t
), 1 ≤ j ≤ r, (B.3)
is semidefinite positive and Hermitian. Now, traces present
inthe first summation of (B.2) may be developed as
Tra[(
ej ⊗ I t)†(
K−1d ⊗ X̃†X̃
)(ej ⊗ I t
)]
(a)= Tra
[(eTj K
−1d ej
)⊗ X†X̃
]
(b)= Tra
[eTj K
−1d ej
]Tra
[X†X̃
]≡ tP̃TtrTra
[eTj K
−1d ej
], (B.4)
where (a) follows from an application of the proper-ty Tra [AB]
= Tra [BA], (b) stems from the propertyTra [A ⊗ B] = Tra [A]Tra
[A], while (c) arises from thepower constraint in (9.1). Hence,
after inserting (B.4) into(B.2), this last may be equivalently
rewritten as
σ2tot = rt−TtrP̃(
r∑
j=1
eTj K−1d ej
)+
1t2
r∑
j=1
Tra[Λj(X̃)
]. (B.5)
Now, our next task is to find the minimum value of thetraces in
the summation (B.5). For accomplishing this task,we resort to a
suitable application of the Cauchy inequality.Specifically, after
indicating by{λj(i), i = 1, .., t} theΛj(X̃)matrix eigenvalues, we
have that
t2 =
(t∑
i=1
√λj(i)
1√λj(i)
)2(a)
≤(
t∑
i=1
λj(i)
)
t∑
j=1
(λj(i))−1
≡t∑
j=1
(λj(i)
)−1Tra
[Λj(X̃)
], (B.6)
where (a) from an application of the Cauchy inequality[13, p.42]
to the sequences{√λj(i), i = 1, .., t} and{(λj(i))−1/2, i = 1, ..,
t}. Obviously, eq. (B.6) may berewritten as
Tra[Λj(X̃)
] ≥ t2/( t∑
j=1
λj(i)−1), (B.7)
that gives arise to a lower bound onTra[Λj(X̃)
]. Further-
more, the Cauchy inequality also allows us to conclude thatthe
lower bound (B.3) is attained whenΛj(X̃) is equal to thefollowing
diagonal matrix (see (B.3)):
Λj(X̃) =a2t
t + aI t, 1 ≤ j ≤ r. (B.8)
As a direct consequence, the condition (16) arises for
theoptimal X̃.
APPENDIX C - DERIVATION OF THROUGHPUT FORMULA IN(24)
The whitening filter−→B of the (non singular) MAI covariance
matrix Kd is defined as−→B , (I⊗Kd)−1/2 = ITpay⊗K−1/2d .
(C.1)
It is a (rTpay × rTpay) (non singular) block matrix, so thatthe
resulting transformed observations12 −→ω , −→B−→y
constitutesufficient statistics for the detection of the
transmitted messageM of Fig.1. On the basis of the above property,
we maydirectly write the following equality:
I(−→y ;−→φ |Ĥ) ≡ I(−→ω ;−→φ |Ĥ) , h(−→ω |Ĥ)− h(−→ω |−→φ ,
Ĥ), (C.2)12By applying the linear transformation (C.1) to the
disturbance vector−→d
in (12) we arrive at the following relationship E{−→B−→d (−→B−→d
)†} = IrTpay .So, according to our current taxonomy, we denote as
”spatial whitening filter”the matrix
−→B in (C.1).
-
12
whereh(·) denotes the differential entropy operator.
Further-more, from the channel model in (12) and the linear
transfor-mation performed by the whitening filter in (C.1), it
followsthat the conditional r.v.−→ω |−→φ , Ĥ is Gaussian
distributed andits covariance matrix is given by
Cov(−→ω |−→φ , Ĥ) = IrTpay +σ2εt
{( [φ(TL + Ttr + 1)...φ(T )
]T
· [φ?(TL + Ttr + 1)...φ?(T )] )⊗ K−1d
}, (C.3)
whereσ2ε in (C.3) arises from the fact that̂H = H − ²and the
elements{εji} of the MMSE estimation error matrix² are uncorrelated
zero-mean Gaussian r.v.s whose variancesE{‖εji‖2} equateσ2ε for any
(j,i) indexes. Thus, being theconditional r.v.−→ω |−→φ , Ĥ proper,
complex and Gaussian dis-tributed, its differential entropy in
(C.2) may be directlycomputed as in [29, Th.2]
h(−→ω |−→φ , Ĥ) = lg[(πe)rTpay det
[Cov(−→ω |−→φ , Ĥ)
]], (C.4)
that due to (C.3), may be developed as
h(−→ω |−→φ , Ĥ) = rTpay lg(πe)+E{
lg det[Irt +
σ2εt
((K−1d )
?
⊗( T∑
n=TL+Ttr+1
φ(n)φ(n)†))]}
, (C.5)
where the expectation in (C.5) follows from definition
ofconditional differential entropy [12].Now, although the pdf of
signal vector
−→φ is assumed to
be Gaussian distributed too, forσ2ε > 0 the
correspondingexpectation in (C.5) cannot be put in closed-form,
even in thesimplest case of spatially white MAI (see [6] and
referencetherein). Anyway, by resorting to the Law of Large
Numbers[26, eqs.(8.95), (8.96)], we may conclude that for largeTpay
=T − TL − Ttr the summation in (C.5) converges (in the meansquare
sense) to the expectationTpayRφ, so that the followinglimit holds
for largeTpay:
h(−→ω |−→φ , Ĥ) = rTpay lg(πe) + lg det[I rt +
σ2εTpayt
((K−1d )
?
⊗Rφ)]≡ rTpay lg(πe)+
r∑
l=1
t∑m=1
lg(1+
σ2εTpayt
P (m)µl
), (C.6)
where{P (m), 1 ≤ n ≤ t} in (C.7) are the eigenvalues of
thesignal correlation matrixRφ, while {µl, l = 1, ..., r} are
theeigenvalues of the MAI covariance matrixKd. Furthermore,since
the disturbance in (12) is Gaussian distributed, therelationships
(C.5), (C.6) are still valid,regardlessof Tpay,when σ2ε vanishes
and/or the SINRs{P (m)/µl, 1 ≤ m ≤t, 1 ≤ l ≤ r} in (C.6) approach
zero.
About the differential entropyh(−→ω |Ĥ) in (C.2), forσ2ε >
0it cannot be expressed in closed-form, even in the simplestcase
ofr = t = 1 with white MAI [6]. However, since
H = Ĥ + ², the r.v. −→ω , −→B−→y equates (see channelmodel in
(12))
−→ω = 1√t
[ITpay ⊗ K−1/2d Ĥ
T]−→
φ
+1√t
[ITpay ⊗ K−1/2d ²T
]−→φ +−→w , (C.7)
where the zero-mean Gaussian r.v.−→w , −→B−→d is the”whitened”
version of the colored MAI
−→d (see note 12). Thus,the conditional r.v.−→ω |Ĥ is zero-mean
and the correspondingcovariance matrixCov
(−→ω |Ĥ)
may be developed as
Cov(−→ω |Ĥ
)(a)= 1
t
[ITpay ⊗ K−1/2d Ĥ
T] (
ITpay ⊗ Rφ)
·[ITpay ⊗ Ĥ
∗K−1/2d
]+
1tE{(
ITpay ⊗ K−1/2d ²T)−→
φ−→φ†
·(
ITpay ⊗ ²∗K−1/2d)| Ĥ
}+ IrTpay
(b)= I rTpay +
1t
[ITpay ⊗
(K−1/2d Ĥ
T)
Rφ
(K−1/2d Ĥ
T)†]
+1tE{(
ITpay ⊗ K−1/2d ²T Rφ²∗K−1/2d
)}
(c)= IrTpay +
1t
[ITpay ⊗
(K−1/2d Ĥ
T)
Rφ
(K−1/2d Ĥ
T)†]
+1t
(ITpay ⊗ σ2εtPK−1d
)
(d)= ITpay ⊗
{Ir +
1t
(K−1/2d Ĥ
T)
Rφ
(K−1/2d Ĥ
T)†
+σ2εtPK−1d
}, (C.9)
where(a) follows from E{−→
φ−→φ†}
= ITpay ⊗ Rφ, (b) arisesfor the mutual independence of the
r.v.s
−→φ ,², Ĥ, (c) stems
from the relationship E{²T Rφ²
}= σ2εtP I r, and, finally,(d)
exploits the propertyIrTpay = ITpay ⊗ I r. Therefore,
althoughthe differential entropy of−→ω |Ĥ is upper bounded as [29,
th.2]
h(−→ω |Ĥ) ≤ lg{
(πe)rTpay det[Cov(−→ω |Ĥ)
] }, (C.10)
nevertheless, the Limit Central Theorem guarantees that,
forlarge numbert of transmit antennas, the r.v.−→ω |Ĥ
becomesGaussian, so that the upper bound in (C.10) may be
assumedattained for large t. Furthermore, since forσ2ε → 0
and/orvanishing SINRs,̂H converges toH and the r.v.−→ω |Ĥ
becomesGaussian distributed, then the upper bound in (C.10) can
beattained regardlessof t value. Hence, after inserting (C.5)and
(C.10) into (C.2), we directly obtain eq.(24).
APPENDIX D - DERIVATION OF THE POWER ALLOCATIONFORMULAS IN (28),
(29)
-
13
Since the eigenvalues of the Kronecker matrix productA ⊗Bare
given by the products of the eigenvalues of the matrixAby those ofB
(see [13], Corollary 1, p.412], we may directlyexpress the second
determinant in (24) as
lg det[(
Irt +σ2εTpay
t
(K−1d
)∗ ⊗ Rφ)]
=r∑
l=1
t∑m=1
lg(
1 +σ2εTpay
t
P (m)µl
), (D.1)
where{µl} are the eigenvalues ofKd and{P (m)} are thoseof Rφ
(e.g.,P (m) is the power allocated to the m-th modeof the
considered MIMO channel). Now, after introducing theSVD in (25) of
Kd into the first determinant in (24), we mayrewrite this last
equation in the following equivalent form
I(−→y ;−→φ |Ĥ
)= lg det
[Ir + σ2εP (Λ)
−1d +
1tA†RφA
]
−r∑
l=1
t∑m=1
lg(
1 +σ2εTpay
t
P (m)µl
), (D.2)
with A given by eq.(26). Therefore, after introducing in(D.2)
the SVD in (26.1) ofA, an application of Hadamardinequality [12]
allows us to develop the constrained supremumin (23) as
TG(Ĥ) =r∑
m=1
lg(
1 +σ2εP
µm
)
+ supPtm=1 P
?(m)≤Pt
{f(P ?(1), ..., P ?(s))
− 1Tpay
t∑m=s+1
r∑
l=1
lg(
1 +σ2εTpay
t
P ?(m)µl
) }, (D.3)
where
f(P ?(1), ..., P ?(s)) ,s∑
m=1
fm(P ?(m))
=s∑
m=1
[lg(1 + αmP ?(m))− (1/Tpay)
·r∑
l=1
lg(1 + βlP ?(m)
)], (D.4)
is an additive objective function which just depends on
thepowers radiated by firsts transmit antennas. Since the
lasttwo-fold summation into brackets in (D.3) vanishes only whenP
?(s + 1) = ... = P ?(t) = 0, we may directly rewrite (D.3)as
TG(Ĥ) =r∑
m=1
lg(1 +
σ2εP
µm
)
+ supPsm=1 P
?(m)≤Pt
{f(P ?(1), ..., P ?(s)
)}. (D.5)
Now, after denoting byβmax , max{βl, l = 1, ..., r},we see that
the second derivatives of logarithmic functions
{fm(P ?), 1 ≤ m ≤ s}, in (D.4) are not positive over regionD of
Rs given by
D ,{
(P ?(1), ..., P ?(s)) : P ?(m)
≥ max{
0,βmax
√r − αm
√Tpay
αmβmax[√
Tpay −√
r]}
,m = 1, ..., s
}. (D.6)
Then, we conclude that the sum-functionf(P ?(1), ..., P ?(s)) in
(D.2) is
⋂−convex (at least) overD.This region approaches the overall
positive orthantRs+ of Rswhen σ2ε is vanishing and/or largeTpay is
considered (seeeq. (27))13. Thus, after applying the Kuhn-Tucker
conditions[14, eqs.(4.4.10), (4.4.11)] for carrying out the
constrainedmaximization of objective function (D.4) we arrive at
thefollowing relationships:
(P ?(m) + α−1m )−1 − 1
Tpay
r∑
l=1
(P ?(m) + β−1l )−1 ≤ 1/ρ,
for all m such that:P ?(m) = 0, (D.7)
(P ?(m) + α−1m )−1 − 1
Tpay
r∑
l=1
(P ?(m) + β−1l )−1 = 1/ρ,
for all m such that:P ?(m) > 0. (D.8)
Now, while (D.7) directly gives arise to eq.(28), to
proveeq.(29) we need to rewrite (D.8) in the following form:
Tpay
{r∏
l=1
(P ?(m) + β−1l )
}[ρ−
(P ?(m) + α−1m
)]
−ρ(P ?(m)+α−1m
){ r∑
l=1
[ r∏
j=1,j 6=l(P ?(m)+β−1j )
]}= 0. (D.9)
Eq.(D.9) is an (r+1)-th order algebraic equation whichcannot
generally be expressed in closed form as a functionof optimal power
levelP ?(m). Anyway, when
β1 = β2 = ... = βr = βmin, (D.10)
then (D.9) reduces to the following 2nd-order
algebraicequation:
βminP?(m)2+
{1−βmin
[ρ(1− r
Tpay
)− α−1m
] }P ?(m)
−ρα−1m(αm−ρ−1− rβmin
Tpay
)= 0, (D.11)
whose positive roots are given by (29). Thus, directly fromthe
relationship (27), it follows that the above condition (D.10)
13In practice, some sufficient conditions for theT−convexity og
the
objective function are:k2m ≥ (σ2ε
prTpay(µm + Pσ2ε))/(µmµmin), 1 ≤ m ≤ s,
whereµmin denotes the minimum eigenvalue ofKd.
-
14
is satisfied for vanishingσ2ε and/or largeTpay and/or diagonalKd
and/or low SINRs. Furthermore, when all above conditionsfall short,
the worst-case application scenario is obtained whenthe MAI
covariance matrixKd is equal to the diagonal oneµmaxIr, whereµmax
is the maximum eigenvalue ofKd. Inthis case, the optimal power
levelP ?(m) is still given bythe positive root (29) of the
algebraic equation (D.11). Thus,we may conclude that,in any case,
(29) represents themin-max solutionof the constrained maximization
of the objectivefunction in (D.4).
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Enzo Baccarelli Enzo Baccarelli received the Lau-rea Degree
summa cum laude in electronic engineer-ing, the Ph.D. degree in
communication theory andsystems, and the Post-doctorate Degree in
informa-tion theory and application from the University ofRome
”Sapienza” Rome, Italy in 1989, 1992, and1995, respectively. He is
currently with the Univer-sity of Rome ”Sapienza”, where he was
ResearcherScientist from 1996 to 1998 and Associate Professorin
signal processing and radio communication from1998 to 2003. Since
2003 he is Full Professor in data
communication and coding. He is also Dean of the
Telecommunication Board,and member of the Educational Board, both
within the Faculty of Engineering.From 1990 to 1995, he was Project
Manager with SELTI ELETTRONICACorporation, where he worked on the
design of high-speed modems for data-transmission. From 1996 to
1998 he attended the international project AC-104 Mobile
Communication Service for High-Speed Trains (MONSTRAIN),where he
worked on equalization and coding for fast-time varying
radio-mobile links. He is currently the Coordinator of the national
Project Wireless802.16 Multi-antenna mEsh Networks (WOMEN). He is
author of more than100 international IEEE publications and coauthor
of two international patentson adaptive equalization and
turbo-decoding for high-speed wireless andwired data-transmission
systems licensed by international corporations. Dr.Baccarelli is
Associate Editor of the IEEE COMMUNICATION LETTERS,and his
biography isis listed inWho’s WhoandContemporary Who’s Who.
Mauro Biagi Mauro Biagi was born in Rome in1974. He received his
”Laurea degree” in Telecom-munication Engineering in 2001 from ”La
Sapienza”University of Rome. He obtained the Ph.D. oninformation
and communication theory in January2005, at INFO-COM Dept. of the
”La Sapienza”University of Rome and actually he covers the
posi-tion of Assistant Professor in the same department.His
teaching activity deals with coding and statisticalsignal
processing. His research is focused on Wire-less Communications
(Multiple Antenna systems
and Ultra Wide Band transmission technology) mainly dealing with
space-time coding techniques and power allocation/ interference
suppression inMIMO-ad-hoc networks with special attention to game
theory applications.Concerning UWB his interests are focused on
transceiver design for UWB-MIMO applications. His research is
focused also on Wireline Communicationsand in particular bit
loading algorithms and channel equalization for xDSLsystems and
Power Line Communication and he is member of IEEE PLC com-mittee
and he joined several International Conferences as Technical
ProgrammCommittee member. Actually he is involved in the Italian
National ProjectWireless 8O2.16 Multi-antenna mEsh Networks (WOMEN)
in research andproject managing activities.
-
15
Cristian Pelizzoni Cristian Pelizzoni was born inRome, Italy, in
1977. He received the Laurea De-gree in Telecommunication
Engineering from theUniversity of Rome ”Sapienza” in 2003. From
2003to 2006 he was Ph.D student of Information andCommunication
Engineering at Faculty of Engineer-ing in University ”Sapienza”.
Waiting for discussingthe final Ph.D thesis, related to
optimization ofwireless transceivers for Multiple-Input
Multiple-Output Ultra Wide Band (MIMO-UWB) systems,he currently
works as contractor researcher at the
INFOCOM dept. of the Faculty of Engineering (University
”Sapienza”).He participates in the technical committee of the
Italian Project ’Wireless802.16 Multi-antenna mEsh Networks”
(WOMEN). His research areas includeProject and Optimization of very
high speed Wireless transceivers for theemerging 4GWLANs, based on
MIMO-UWB technology; Space-Time codingfor wireless (UWB-like)
channels, affected by dense multipath, Space-Timecoding and game
theory approach for power optimal allocation of wirelessad-hoc
networks; novel Physical and MAC layer solutions for Wireless
MeshNetworks.
Nicola Cordeschi Nicola Cordeschi was born inRome, Italy, in
1978. He received the Laurea Degree(summa cum laude) in
Telecommunication Engi-neering in 2004 from University of Rome
”Sapien-za”. He is pursuing the Ph.D. at the INFOCOMDepartment of
the Engineering Faculty of ”Sapien-za”. His research activity
focuses on wireless com-munications, in particular dealing with the
designand optimization of high performance transmissionsystems for
wireless multimedia applications, bothin centralized and
decentralized multiple antenna
scenarios.