1

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.

2

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].

3

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.

4

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

5

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)

6

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.