815704

Hindawi Publishing CorporationInternational Journal of Digital Multimedia BroadcastingVolume 2010, Article ID 815704, 19 pagesdoi:10.1155/2010/815704

Research Article

Robust MMSE Transceiver Designs for Downlink MIMO Systemswith Multicell Cooperation

Jialing Li, I-Tai Lu, and Enoch Lu

Department of ECE, Polytechnic Institute of NYU, 6 Metrotech Center, Brooklyn, NY 11201, USA

Correspondence should be addressed to Jialing Li, [email protected]

Received 1 July 2009; Accepted 16 December 2009

Academic Editor: Hongxiang Li

Copyright 2010 Jialing Li et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The robust-generalized iterative approach (Robust-GIA), robust-fast iterative approach (Robust-FIA), and robust-decoder covarianceoptimization approach (Robust-DCOA) are proposed for designing MMSE transceivers of downlink multicell multiuser MIMOsystems with per-cell and per-antenna power constraints and possibly imperfect channel state information. The Robust-DCOA isthe most restrictive but is always optimum, the Robust-GIA is the most general, and the Robust-FIA is the most ecient. Whenthe Robust-DCOA is applicable and the decoder covariance matrices are full rank, the three proposed approaches are equivalentand all provide the optimum solution. Numerical results show that the proposed robust approaches outperform their non-robust counterparts in various single-cell and multicell examples with dierent system configurations, channel correlations, powerconstraints, and cooperation scenarios. Moreover, performances of the robust approaches are insensitive to estimation errors ofchannel statistics (correlations and path loss). With cell-cooperation, cell edge interference problems can be remedied withoutreducing the number of data streams by using the proposed robust approaches.

1. Introduction

Joint transceiver designs with criteria such as minimummean square error (MMSE), maximum sum capacity, andminimum bit error rate (BER), and so forth, for multiple-input-multiple-output (MIMO) systems, with both uplinkand downlink configurations, have been studied intensivelyin recent literature (e.g., see [1, 2]). Discussed in this paper isthe robust MMSE transceiver design with respect to channelestimation errors for downlink multicell multiuser MIMOsystems.

Assuming perfect channel state information (CSI),joint MMSE transceiver design has been studied by manyresearchers. A closed form design subject to the total powerconstraint for a single-user MIMO system is derived in [3].Unfortunately though, this closed form design cannot beextended either to the multiuser case or to the per-antennapower constraint. For multiuser uplink MIMO problemssubject to per-user power constraint, numerical solutionsare provided mainly by the transmit covariance optimizationapproach (TCOA) [4, 5] and iterative approaches such as in[4]. We have developed a generalized iterative approach (GIA)for the uplink to deal with arbitrary linear power constraints

(including the more practical per-antenna power constraint)[6]. Recently, we have also extended the TCOA to dealwith arbitrary linear power constraints and have shown thatthe GIA and the TCOA are equivalent and optimum whenthe source covariance matrices are all projection matricesmultiplied by the same constant and the transmit covariancematrices are all full rank [7].

For the downlink configuration, iterative approachessuch as in [8] and a dual uplink approach in [9, 10] areemployed to provide numerical MMSE solutions for mul-tiuser MIMO systems subject to the total power constraint.The extension to deal with the per-antenna and per-cellpower constraints for the downlink scenario is achieved by aniterative approach using a second-order cone programming(SOCP) [11] and by our GIA for the downlink [12]. Recently,we have also developed the decoder covariance optimizationapproach (DCOA) [13] to deal with arbitrary linear powerconstraints (including the per-antenna and per-cell powerconstraints). Furthermore, we have shown that the GIA andthe DCOA are equivalent and optimum when the sourcecovariance matrices are all identity matrices multiplied bythe same constant and the decoder covariance matrices areall full rank [13].

2 International Journal of Digital Multimedia Broadcasting

All of the above mentioned MMSE transceiver designsare based on perfect CSI. However, the CSI is usually esti-mated in practice and is therefore subject to CSI estimationerrors and possibly quantized CSI feedback errors. Hence,in practice, joint transceiver design has to be based onimperfect CSI. One option is to ignore that the CSI isimperfect. This type of approach is herein called non-robust.Unfortunately, the system performances derived from thenon-robust approaches depend strongly on the quality ofthe available CSI (performances get worse quickly if the CSIquality deteriorates). Moreover, an optimum design based onpoor CSI could be worse than suboptimum designs usingthe same CSI. Therefore, a more appealing option is tomodel the CSI error and to incorporate the error modelinto the transceiver design. This type of approach is hereincalled robust. The robust approaches can better mitigate thedegradation of system performances due to imperfect CSIthan the non-robust approaches if the CSI error is modeledcorrectly. Two classes of imperfect CSI models are usuallyemployed: the stochastic model for the CSI estimation errorsand the deterministic model for the CSI feedback errors.If a stochastic model is used, a statistically robust designis usually performed to optimize some system performancefunctions. If a deterministic model is used, a minimax ormaximin design aiming at optimizing the worst-case systemperformance is usually carried out.

To cope with CSI estimation errors, closed form solutionsfor the robust joint MMSE transceiver design subject to thetotal power constraint are developed for single-user MIMOsystems in [1416]. But, similar to the perfect CSI case, noclosed form solution is found when the problem is extendedto deal with either multiuser applications or the per-antennapower constraint. For multiuser uplink MIMO problemssubject to the inequality per-user and sum power constraints,the robust transmit covariance optimization approach (Robust-TCOA) is developed for independent identically distributed(i.i.d.) MIMO channels with CSI estimation errors in [17].For multiuser downlink MIMO problems, when the CSIerrors are bounded, the worst-case design under arbitrarypower constraints is made based on SOCP in [18]; when theCSI errors are statistical errors, the robust design under thetotal power constraint is solved numerically by a dual uplinkapproach in [18].

So far, no statistically robust approach has been shownoptimum in the MMSE sense for the downlink MIMO sys-tems (either single-cell or multicell) under the per-antennapower constraint. Proposed in this paper is the robust MMSEtransceiver design with respect to CSI estimation errors fordownlink multicell MIMO systems subject to arbitrary linearpower constraints. Specifically, the per-antenna and per-cellpower constraints are considered. The work is relevant tofrequency division duplex (FDD) systems where channelestimation is done at each user equipment (UE) and thenfed back to the base station(s) (denoted as evolved NodeB or eNB) via a zero-delay and error free communicationlink. Note that CSI feedback errors are not considered inthis paper. The work may possibly also be extended to timedivision duplex (TDD) systems where channel estimation isdone at the eNBs.

We first extend the statistical model of imperfect CSI in[16] to take into account the path loss eects. This extensionis very crucial for practical multicell systems because thevariances of CSI estimation errors depend on the distancesbetween the UEs and the eNBs. The CSI estimation errorof a UE near the eNB is much smaller than the CSIestimation error of a UE that is far away from the eNB.With the extended imperfect CSI model in hand, we hereinpropose three robust approaches to deal with arbitrary linearequality power constraints. The first, the robust-generalizediterative approach (Robust-GIA), is an extension of the GIA[12, 13] to the imperfect CSI case. The second, the robust-decoder covariance optimization approach (Robust-DCOA), isan extension of the DCOA [13]. The third, the robust-fastiterative approach (Robust-FIA) is completely new. Thoughthe first two are both extensions, their complexities are stillsimilar to those of their predecessors!

The DCOA requires that the numbers of data streams arenot prespecified and that all the source covariance matricesare identity matrices multiplied by the same constant. TheRobust-DCOA is even more restricted since not only itrequires all of the conditions of the DCOA but also thatthe transmit correlation matrix for each user is an identitymatrix. The statistics of the CSI estimation error also needto be the same for all users if the power constraints of theusers are interdependent. The GIA and the Robust-GIA, onthe other hand, do not require any of the above mentionedconditions. The Robust-FIA has the same requirements asthe DCOA, but not the additional restrictions of the Robust-DCOA.

The relationships between the Robust-GIA, the Robust-FIA, and the Robust-DCOA are very interesting. The Robust-GIA is the most general and can provide tradeo betweendiversity and multiplexing gains. The Robust-FIA is themost ecient. Even though the Robust-DCOA is the mostrestricted, it always gives the optimum solution when it isapplicable. But whenever the Robust-DCOA is applicable andall the decoder covariance matrices are full rank, the solu-tions obtained by the three robust approaches are actuallyequivalent (i.e., the Robust-FIA and the Robust-GIA are alsooptimum)! Interestingly, the Robust-GIA and the Robust-DCOA actually become the GIA and the DCOA, respectively,when the CSI is perfectly known, thereby providing a unifiedframework to take care of both perfect and imperfect CSIcases! We also denote the special case of the Robust-FIA whenthe CSI is perfectly known as the fast iterative approach (FIA)for convenience.

MMSE transceiver designs using the proposed robustapproaches are performed for various single-cell and mul-ticell examples with dierent system configurations, powerconstraints, channel correlations, and cooperation scenarios.System performances in terms of MSE and BER of variousnumerical examples are compared. Computational eciencyfor various approaches is studied. Sensitivity studies withrespect to channel statistics (channel correlations and pathloss, estimated independently from channel estimation)are also investigated. The numerical results show that theproposed robust approaches are indeed superior to the non-robust approaches. Moreover, accurate channel correlations

International Journal of Digital Multimedia Broadcasting 3

and path loss are not required in the robust approaches. Withcell cooperation, the cell edge UEs perform as well as thoseUEs without inter-cell interferences.

Notations are as follows. All boldface letters indicate vec-tors (lower case) or matrices (upper case). A

, A, A1, tr(A),

A, and A stand for the transpose, Hermitian, inverse,trace, expectation, and Frobenius norm of A, respectively.Matrix Ia signifies an identity matrix with rank a. diag[. . .]denotes the diagonal matrix with elements [. . .] on the maindiagonal. A B means that A B is positive semidefinite.A B denotes the Schur product of A and B (elementwiseproduct of A and B). CN(, 2) denotes a complex normalrandom variable with mean and variance 2.

2. Formulation

2.1. Downlink Multicell Multiuser MIMO Systems. Considerthe downlink of a multicell multiuser MIMO system with CeNBs and K UEs. Denote the number of transmit antennas atthe cth eNB by tc and the total number of transmit antennasby t, that is, t = Cc=1 tc. Also denote the number of receiveantennas at the ith UE by ri and the number of data streamsintended for the ith UE by mi.

In this system, there may be multiple groups where eachgroup jointly designs its precoders and decoders but does soindependently of the other groups. In the with-cooperationscenario (there is full cooperation among all eNBs), system-wide design is performed and there is only one group. Inthe without-cooperation scenario (there is no cooperationamong eNBs), the eNB and UEs in a cell are one group.Let Da (Db) and Sa (Sb) define one such group in the with-cooperation (without-cooperation) scenario, Da (Db) beingthe set of indices of all eNBs in the group and Sa (Sb) beingthe set of indices of all UEs in the group.

At the ith UE, let yi and ai denote the received signal andnoise, respectively. At eNB c, let si and Fic denote the data andthe precoder for the ith UE, respectively. Also let Hic denotethe channel matrix from eNB c to the ith UE. In the with-cooperation scenario, the data si for the ith UE are jointlytransmitted by all eNBs. Thus, the received signal at the ithUE is:

yi =

cDaHic

jSaF jcs j + ai, (1a)

where Da = {1, . . . ,C} and Sa = {1, . . . ,K}.In the without-cooperation scenario, let the eNB serving

the ith UE be denoted by the index ci where ci {1, 2, . . . ,C}.Thus, the data si for the ith UE are only transmitted by eNBci. Therefore, a system with C eNBs can be decoupled intoC single-cell downlink groups. In the cith group, the receivedsignal vector yi at the ith UE becomes

yi =

cDbHic

jSbF jcs j + bi + ai, bi =

cj /Db

j / SbHic j F jc j s j ,

(1b)

where Db = {ci} consists of the index of eNB ci and Sb = { j |cj = ci} consists of the indexes of all UEs served by eNB ci.

Since there are multiuser precodings at eNB ci, the datatransmitted by eNB ci to the UEs other than the ith UE arenot considered as interference to the ith UE. But, the datatransmitted from the eNBs other than eNB ci result in theinter-cell interference (denoted as bi) to the ith UE.

In order to unify (1a) and (1b), let D and S define a groupin the system, D being the set of indices of all eNBs in thegroup and S being the set of indices of all UEs in the group.For the ith UE, i S, the received signal vector, yi, can thusbe expressed as

yi =

cDHic

jSF jcs j + ni = Hi

jSF js j + ni. (2)

When there is full eNB cooperation,

Hi = [Hi1 HiC], F j =[Fj1 FjC

],

ni = ai, D = Da, S = Sa.(2a)

When there is no eNB cooperation,

Hi = Hici , Fi = Fici ,ni = bi + ai, D = Db, S = Sb.

(2b)

In (2b), Hi denotes the channel matrix from the eNB ci to theith UE, Fi denotes the precoder for the ith UE at eNB ci, andni is the interference plus noise vector at the ith UE. But, in(2a), Hi denotes the composite channel matrix from all eNBsto the ith UE, Fi denotes the composite precoder for the ithuser at all eNBs, and ni is the noise vector. Equation (2) isessentially the same as the formulation in [11].

2.2. Extended Imperfect CSI Model [16]. In order to accountfor path loss and spatial correlation, the channel Hic fromeNB c to the ith UE is modeled as

Hic = dic R1/2R,i HW ,icR1/2T ,c. (3)Here, is a constant, dic denotes the distance between the ithUE and eNB c, and 2 is the path loss exponent. In (3), RR,iand RT ,c are known, normalized (unit diagonal entries), andfull rank. They represent receive and transmit correlationmatrices, respectively. The entries of HW ,ic are i.i.d. CN(0, 1).Here, the subscript W represents spatially white.

In practice, the CSI is estimated, resulting in estimationerror. Thus,

Hic = Hic + Eic, (4)

where Hic is the channel estimate and Eic is the CSIestimation error. By using an orthogonal training methodand MMSE channel estimation, Hic and Eic have been shownin [16] to be independent and

Eic = dic R1/2E,icEW ,icR1/2T ,c, RE,ic =[Iri +

2E,ic R1R,i

]1,

(4a)

2E,ic = 2d2ic tr(

R1T ,c)

2a

Tc, (4b)


where the entries of EW ,ic are i.i.d. CN(0, 2E,ic). Here, 2a is the

noise variance at each of the receive antennas and Tc is thetotal training power transmitted from eNB c. Note that thereis no inter-cell interference when the channel is estimated.Also note that the estimated channels {Hic}iS are fed back bythe UEs to a central processing unit in the with-cooperationscenario and to eNB c in the without-cooperation scenariovia a zero-delay and error free communication link. As to beexpected, Hi, i S, is equal to [Hi1 HiC] in the with-cooperation scenario and to Hic, c D, in the without-cooperation scenario. Also, when perfect CSI is available,Hic = Hic and 2E,ic = 0.

2.3. Joint MMSE Precoder and Decoder Design Formulation.For a given group and thus a given D and S, the followingis the problem formulation. Define the mean square error(MSE) of the data streams intended for the ith UE, i S, as

i(Fi, Gi) = tr(

Giyi si)(

Giyi si), (5a)

where Gi is the decoder at the ith UE, Fi is the precoderin (2a) or (2b), and yi and si are given in (2). Equation(5a), using the actual channel Hi and actual noise vectorni, is the metric for MSE evaluation for the perfect-CSI,non-robust, and robust approaches. However, in the robustMMSE transceiver design, the following conditional MSE

i(

Fi, Gi | Hi)= tr

(Giyi si

)(Giyi si

),

yi = Hi

jSF js j + ni, ni = ni + Eic

jSF js j

(5b)

is used. Here, when perfect CSI is available, Hi and nirepresent the actual channel and actual noise vector, that is,Hi = Hi and ni = ni, respectively; otherwise, Hi representsthe channel estimate, that is, Hi /=Hi, and ni represents theequivalent interference plus noise vector. We will jointlychoose the decoders and precoders corresponding to all UEsin S and all eNBs in D to minimize the sum MSE :

{Fi, Gi}MMSE = argmin{Fi ,Gi|iS}

{}

, =

jSj(

F j , G j | H j). (6)

Define the positive definite source covariance matrix as si =sisi for the ith UE. The eNB(s) are subject to either the per-antenna or per-cell power constraints. For the per-antennapower constraint, the dth antenna of the cth eNB, c D, haspower

Pad = ed

jSF js jFj

ed. (7a)

For the per-cell power constraint, the cth eNB, c D, haspower

Pbc = tr

Qc

jSF jcs jFjc

Qc

. (7b)

Here, ed in (7a) are an l 1 unit vector with the dth entryequal to one and all other entries equal to zero, Qc in (7b) isan l l matrix whose entries are all equal to zero except forthe diagonal elements corresponding to the antennas of thecth eNB which are equal to one, and l =cD tc.

2.4. Augmented Cost Function. To solve (6) subject to (7a) or(7b), one can use the method of Lagrange multipliers to set upthe augmented cost function:

= + tr

jSF js jFj P

, (8)

where is an unknown diagonal matrix, representing theLagrange multipliers. For the per-antenna power constraintin (7a),

= diag[1, . . . , l], P = diag[Pa1, . . . ,Pal]. (9a)For the per-cell power constraint in (7b), define k = Itk kand Pk = ItkPbk/tk, k=1,. . .,C. Thus,

= c, P = Pc, c D, without cooperation;=diag[1 . . .C], P=diag[P1 . . .PC], with cooperation.

(9b)

2.5. Robust Design. The central processing unit is assumedto have knowledge about the channel estimate(s), H j , j S,and the channel statistics. Define the interference-plus-noisecovariance matrix at the ith UE as ni = nini . Thus,

ni =

ai =aiai

with cooperation;

bi + ai =bibi

+aiai

without cooperation.

(10)

Also define the equivalent interference-plus-noise covariancematrix at the ith UE as ni = nini . After some mathmanipulations, (5b) for the ith UE, i S, becomes

i(

Fi, Gi | Hi)=GiHiFisi siFi Hi Gi

+ si + Gi

Hi

jSF js jFj

Hi + ni

Gi ,

(11)

where ni = i + ni and

i =

cD2c Pbc RE,ic, if RT ,c = Itc , c D,

cD2c

jStr(

F jcs jFjcRT ,c)

RE,ic, if RT ,c /= Itc , c D,

(12)

2c = tr(

R1T ,c)

2a

Tc. (13)


For a given set of precoders {F j} jS, setting the gradient of in (8) with respect to Gi equal to zero, we yield the MMSEdecoder for the ith UE, i S:

Gi = siFi Hi Mi, Mi =

Hi

jSF js jFj

Hi + ni

1

.

(14)

Substituting (11) and (14) into (8), the augmented costfunction in (8) is reduced to

1 =

jStr(s jFj Hj M jH jF js j + s j

)

+ tr

jSF js jFj P

.

(15)

Note that the 1 in (15) no longer depends on {G j} jSexplicitly.

On the other hand, for a given set of decoders {G j} jSand Lagrange multipliers , setting the gradient of in (8)with respect to Fi equal to zero, we have the MMSE precoderfor the ith UE, i S:

Fi = NHi Gi , N =

jSHj G

j G jH j + +

1

, (16)

where

=

0, if RT ,c = Itc , c D,RT , if RT ,c /= Itc , c D,

=

c, c D without cooperation,diag[1,2, . . . ,C] with cooperation,

RT =

RT ,c, c D without cooperation,diag

[RT ,1, RT ,2, . . . , RT ,C

]with cooperation.

(17)

Here, c = 2c

iS tr(Gi GiRE,ic)Itc and 2c is given in (13).Substituting (11) and (16) into (8), the augmented costfunction in (8) is reduced to

2 = tr

jSHj G

j s jG jH j

NP

+

jStr(s j + G jn jGj

),

(18)

where

n j =

n j , if RT ,c = Itc , c D,n j , if RT ,c /= Itc , c D.

(19)

The 2 in (18) no longer depends on {F j} jS explicitly.

3. Robust Iterative Approaches

3.1. Robust-Generalized Iterative Approach (Robust-GIA). Bysetting the gradients of (15) with respect to F j equal to zero,left multiplying the resulting equation with F j , and summingup the resulting equation over j, we obtain

jS

(F js jFj

) = B, (20)

where

B =

jSF j

2s jF

j H

j M jH j

jS

(F js jFj

)

kS

(Hk MkHkFk

2skF

k H

k MkHk

),

(21)

=

0, if RT ,c = Itc , c D,

jSF js jFj

RT

jS j , if RT ,c /= Itc , c D,

(22)

j =

jc, c D without cooperation,diag

[ j1, j2, . . . , jC

]with cooperation

(23)

with jc = 2c tr(RE, jcM jH jF j2siFj Hj M j)Itc . Utilizing(9a) and (9b), we can obtain explicit expressions for theLagrange multipliers as follows. For the per-antenna powerconstraint in (7a), we have

= P1(Il B), (24a)

and for the per-cell power constraint in (7b), we have

c = P1bc tr[Qc(Il B)Qc

]. (24b)

With the explicit expression for the Lagrange multipliers in (24a) or (24b) in hand, a Robust-GIA can be developedusing the MMSE decoder in (14) and MMSE precoder in(16). There are three steps in each iteration of the Robust-GIA.

Step 1. Given {Fi}iS, obtain {Gi}iS by (14).

Step 2. Given {Fi}iS, obtain using (24a) or (24b).

Step 3. Given {Gi}iS and , obtain {Fi}iS by (16).

Note that the Robust-GIA can allow tradeo betweendiversity and multiplexing gains because it can deal withvarious sets of prespecified numbers of data streams intendedfor the UEs.


3.2. Robust-Fast Iterative Approach (Robust-FIA) When si =2Imi . The Robust-FIA can be developed based on theRobust-GIA when the source covariance matrices are allidentity matrices multiplied by the same constant, that is,si = 2Imi , i S. For convenience and without loss ofgenerality, we assume 2 = 1. Define the transmit covariancematrices as

Ui = FiFi , (25)and the decoder covariance matrices as

Vi = Gi Gi. (26)Substituting (14) into (26) and using (25), we obtain

Vi = MiHiUiHi Mi, Mi =

Hi

jSU j

Hi + ni

1

.

(27)

Similarly, substituting (16) into (25) and using (26), weobtain

Ui = NHi ViHiN, N =

jSHj V jH j + +

1

, (28)

where is given in (17). Substituting (26) into c, we canexpress in (17) in terms of {Vi}iS in (26). To remove thedependence of {Vi}iS on {Ui}iS, substitute (28) into (27)to yield

Vi = MiHiNHi ViHiNHi Mi,

Mi =

HiN

jSHj V jH j

NHi + ni

1

,(29)

where N is given in (28). Similarly, using the fact that si =2Imi and substituting (25) and (28) into (21)(23), we canexpress the Lagrange multipliers in (24a) or (24b) in termsof {Vi}iS.

With (24a), (24b), and (29) being available, the Robust-FIA can be readily developed. There are two steps in eachiteration of the Robust-FIA.

Step 1. Given {Vi}iS, obtain using (24a) or (24b).

Step 2. Given , use (29) to obtain {Vi}iS for the nextiteration.

Note that the number of data streams intended for theUEs {mi}iS, has to be equal to the ranks of {Vi}iS thatthe Robust-FIA returns and thus cannot be prespecified whenusing the Robust-FIA. When the Robust-FIA converges, thedecoders {Gi}iS can be obtained by the decompositiondefined in (26) and the precoders {Fi}iS can then beobtained from (16). Note that the decomposition in (26) isnot unique: Vi = Gi Gi = Gi AiAi Gi where Ai is an arbitraryunitary matrix. One can easily show that when si = 2Imi ,i S, if (Fi, Gi) is a pair of joint MMSE precoder and

decoder, so is (FiAi, Ai Gi). Both (Fi,Gi) and (FiAi, Ai Gi)

give the same MMSE . However, dierent choices of {Ai}matrices may lead to dierent BER results.

Note that when the CSI is perfectly known, the Robust-FIA is reduced to the FIA by replacing {Hi}iS by {Hi}iS in(24a), (24b), and (29), setting ni = 0 in Mi in (29) andsetting = 0 in N in (28).

4. Robust-Decoder Covariance OptimizationApproach (Robust-DCOA)

When the source covariance matrices are all identity matricesmultiplied by the same constant, that is, si = 2Imi ,i S, and when the transmit correlation matrices are allidentity matrices, that is, RT ,c = Itc , c D, a robust-decoder covariance optimization approach (Robust-DCOA)can be used for jointly designing the MMSE transceivers.For convenience, we assume 2 = 1. The augmented costfunction in (18) becomes

2 = tr

iSHi ViHi +

1

+

iStr(

Vini) tr[P] +

iSmi l.

(30)

The robust MMSE transceiver design problem becomes

min{Vi}iS

max

2,

subject to Vi 0, rank(Vi) = mi, i S, 0.

(31a)

The problem in (31a) is not convex because of the implicitrank constraints dealing with the numbers of data streams,that is, rank(Vi) = mi. Allowing {mi}iSs to be unspecifiedand noting that l is a known constant, we obtain the rank-relaxed decoder covariance optimization problem:

minVi0,iS

max0

2,rel,

2,rel = tr

iSHi ViHi +

1

P

+

iStr(

Vini).

(31b)

The cost function 2,rel in (31b) is convex with respectto {Vi}iS and concave with respect to . Definemin{Vi0,iS} max{0} 2,rel as the primal problem andmax{0} min{Vi0,iS} 2,rel as the dual problem. Since boththe primal problem and the dual problem are convex andstrictly feasible, strong duality holds; that is, the optimumvalues of {Vi}iS, , and 2,rel obtained from the primalproblem are the same as those obtained from the dualproblem.


4.1. Primal-Dual Algorithm. We propose a novel primal-dual algorithm to solve the rank-relaxed decoder covarianceoptimization problem in (31b). Denote the feasible set ofvalues for {Vi}iS as the primal domain and the feasible setof values for as the dual domain. In short, the approachconsists of iterating between a primal domain step (lookingin the primal domain for the best {Vi}iS for a given ) and adual domain step (looking in the dual domain for the best for a given {Vi}iS). The iterative procedure stops when the2,rels corresponding to the primal domain step and the dualdomain step converge to the same value and when {Vi}iSconverge and converge. The two steps of the ( j + 1)thiteration are as follows.

Step 1 (Primal domain step). Given = ( j), find the{Vi}iS which solves (32). Denote them as {V( j+1)i }iS:

min{Vi}iS

tr

iSHi ViHi +

1

+

iStr(

Vini)

,

subject to Vi 0, i S.

(32)

Step 2 (Dual domain step). Given {Vi}iS = {V( j+1)i }iS, findthe which solves (33). Denote it as ( j+1):

max

tr

iSHi ViHi +

1

P

,

subject to 0.

(33)

Both subproblems, defined in (32) and (33), are convexbecause their cost functions are convex and concave, respec-tively, and their constraints are all linear matrix inequalities.Therefore the resulting solution of each subproblem isoptimum. Furthermore, the convexity of the rank-relaxeddecoder covariance optimization problem guarantees thatthe resulting solution provided by the primal-dual algorithmis global optimum. Once the optimal and {Vi}iS areobtained, the optimum numbers of data streams {mi}iS, theoptimum decoders {Gi}iS, and the corresponding optimumprecoders {Fi}iS are obtained in the same way as in theRobust-FIA. And, in all this, the power constraints have beenaccounted for by the Lagrange multipliers.

In practice, the Robust-DCOA given by solving (32) and(33) is considered to have converged at the ( j + 1)th iteration

when {V( j+1)i V( j)i }iS, ( j+1) ( j), and the dualitygap of the values of 2,rel derived from the two steps

gap( j+1)=2,rel({

V( j+1)i

}

iS,( j+1)

) 2,rel

({V

( j+1)i

}

iS,( j))

,

(34)

is less than some prespecified thresholds.

4.2. Semidefinite Programming (SDP) Procedure. Similar tothe TCOA [5, 7] in uplink MIMO systems, (32) and (33)can be reformulated as SDP formulations, which can besolved numerically using existing codes such as SeDuMi [19]

and Yalmip [20]. Equation (32) can be reformulated as SDPformulation:

minWp ,{Vi}iS

tr[Wp

]+

iStr(

Vini)

,

subject to Vi 0, i S,

Wp I

I

iSHi ViHi +

0.

(35)

Since

iSHi ViHi +

1

= Il

iSHi ViHi +

1

iSHi ViHi

,

(36)

(33) can also be reformulated as SDP formulation:

minWd ,

tr

Wd

iSHi ViHi

+ tr(P),

subject to 0,

Wd I

I

iSHi ViHi +

0.

(37)

4.3. Numerically Ecient Procedure. We observe poor con-vergence behavior of the SDP procedure for the Robust-DCOA at high SNRs due to numerical errors introducedby SDP solvers. We therefore use the explicit closed formexpression of given in (24a) and (24b) in the Robust-FIAfor the dual domain step in (33). The SDP procedure in (35)is still employed for the primal domain step in (32). Thisimproves the convergence of the Robust-DCOA greatly.

4.4. Equivalence of the Robust-DCOA, the Robust-FIA, and theRobust-GIA. When the Robust-DCOA is applicable and thedecoder covariance matrices {Vi}iS are full rank, we claimthat the Robust-DCOA, the Robust-FIA, and the Robust-GIA are equivalent. Thus, the solution of the Robust-FIAor the Robust-GIA is optimum under the above mentionedconditions because the solution given by the Robust-DCOAis always optimal (due to convexity).

Note that the Robust-FIA is equivalent to the Robust-GIAbecause the Robust-FIA is a special case of the Robust-GIAwhen si = 2Imi , i S. To prove the equivalence betweenthe Robust-DCOA and the Robust-FIA, it suces to show thatthe KKT conditions of the two approaches are equivalent.The KKT conditions common to both approaches are (16)and the power constraint, (7a) or (7b).


1

0.5

0

0.5

1

1 0.5 0 0.5 1

eNB3

eNB1

UE1

UE3

eNB2

UE2

(a) Example 3

1

0.5

0

0.5

1

1 0.5 0 0.5 1

eNB3

eNB1

UE1

UE2

UE5

UE6

eNB2

UE3

UE4

(b) Example 4

Figure 1: System configurations of two multicell examples: (a) 3 eNBs and 3 UEs. (b) 3 eNBs and 6 UEs. The coordinates of relevant eNBsand UEs shown here are employed for simulations. Note that UE1 is right on the edge of three cells in both systems. In example 3, UE3 isnear the cell edge between cells 1 and 3. In example 4, UE2 and UE2 are near the cell edge between cells 1 and 3, and UE4 and UE6 are nearthe cell edge between cells 2 and 3. (Legends: Each UE is associated with the eNB in the same color.)

For the Robust-DCOA, we set up the following aug-mented cost function from (31b) to include the nonnegativeconstraint on {Vi}iS:

= tr

iSHi ViHi +

1

P

+

iStr(

Vini iVi)

,

(38)

where {i}iS are the Lagrange Multipliers satisfying

tr(iVi) = 0, i 0, i S. (39)

Setting the gradient of (38) with respect to Vi equal to zero,we have

HiNNHi = ni i, i S. (40a)

When {Vi}iS are full rank, the Lagrangian variable {i}iSis zero matrices and (40a) becomes

HiNNHi = ni, i S. (40b)

The task of showing the equivalence of the KKT condi-tions of the two approaches which boils down to showingthe above KKT condition of the Robust-DCOA (40b), canbe derived from (and can be used to derived to) the KKTcondition unique to the Robust-FIA, which are (14), (24a),and (24b). Substitute (16) and (28) into (14) to obtain

GiHiNHi

HiN

jSHj V jH j

NHi + ni

1

= Gi. (41a)

Then left multiply (41a) by V1i Gi to get

HiNHi

HiN

jSHj V jH j

NHi + ni

1

= Iri . (41b)

Summing up (41b) over i S and using some matrixmanipulations, we can show that the resulting equation and(40b) are equivalent. To get (24a) and (24b) from (40b),note that (24a) and (24b) can be obtained by using (20)and the power constraints. In turn, (20) can be obtained bysubstituting (14) and (26) into (16). Since (14) and (40b) canbe derived from each other, this proof is complete. The aboveproof is done assuming si = Imi , i S. It is also applicablewhen si = 2Imi , i S, with 2 /= 1.

5. Numerical Results

Without loss of generality, let = 1 and = 3.5 (i.e., 2= 7) in the path loss model. Each cell is a hexagon withsides normalized to be 1 in length. The noise and sourcecovariance matrices ai and si are all identity matrices ofdimension ri and mi, respectively. Let the per-antenna powerconstraint Pad for antenna d in cell c (d = 1, 2, . . . tc) be equalto P (see (7a)) and let the per-cell power constraint Pbc forcell c, c = 1, 2, . . . ,C, be equal to tcP (see (7b)). Therefore,the total transmission power from eNB c is Pbc = tcP underboth per-antenna and per-cell power constraints.

Four examples (two single-cell and two 3-cell examples)will be considered. Their system parameters are shown inTable 1. The configurations of the two single-cell exampleswill be detailed later on while the configurations of the two3-cell examples are shown here in Figure 1. In example 3


0.5

0

1

1.5

2

2.5

MSE

11 15 20 25

10 log10 tP

Per-antenna power

(a)

0.5

0

1

1.5

2

2.5

MSE

11 15 20 25

10 log10 tP

Per-cell power

(b)

106

104

102

101

BE

R

11 15 20 25

10 log10 tP

(c)

106

104

102

101

BE

R

11 15 20 25

10 log10 tP

(d)

Figure 2: MSE and BER as functions of 10log10tP under per-antenna and per-cell power constraints. System parameters are shown inexamples 1 and 2 in Table 1. (Legends: Solid square lines: the DCOA with (m1,m2) = (2, 2); Dotted diamond lines: the GIA (FIA) with(m1,m2) = (2, 2); Dash + lines: the GIA with (m1,m2) = (1, 1).)

Table 1: System parameters.

Example 1 2 3 4

No. of cells (eNBs): C 1 1 3 3

No. of Tx Antennas per eNB: tc, c = 1, . . . ,C 4 4 2 4Total No. of Tx Antennas of the System: t 4 4 6 12

Number of UEs associated per eNB 2 2 1 2

Total No. of UEs of the System: K 2 2 3 6

No. of Rx Antennas per UE: rj ( j = 1, . . . ,K) 2 2 2 2No. of data streams per UE: mj ( j = 1, . . . ,K) 2 1 2 2

(shown in Figure 1(a)), only one UE is associated with eacheNB, and therefore, there are three UEs in total (K = 3).In example 4 (shown in Figure 1(b)), two UEs are associatedwith each eNB and there are 6 UEs in total (K = 6).

In the simulation, no CSI feedback error is assumed. Theonly CSI error is the CSI estimation error. 2a = 1 and Tc, thetotal transmission power of the cth eNBs training signal usedfor channel estimation, is the same as the total transmissionpower of the data signal, tcP. Three types of designs (perfect-CSI, robust, and non-robust) will be performed. Take the

family of generalized iterative approaches as an example. Forthe perfect-CSI design (denoted as the GIA), there is no CSIestimation error and the perfect CSI is employed for the jointMMSE design of precoders and decoders. On the other hand,there are CSI estimation errors for the non-robust design,Non-robust-GIA, and the robust design, Robust-GIA; only anestimated CSI is available to them. The dierence betweenthe non-robust and robust designs is simple; the non-robustdesign is unaware that the CSI it has is estimated and thustreats it as if it were perfect while the robust design is awareand thus incorporates the statistics of the CSI estimationerror and the CSI into its design.

5.1. Equivalence of the Various Proposed Approaches. Withoutloss of generality, we will numerically show the equivalenceof the Robust-GIA, Robust-FIA, and Robust-DCOA when theCSI is perfect (recall that the Robust-GIA, Robust-FIA, andRobust-DCOA are actually the GIA, FIA, and DCOA, resp.,when the CSI is perfect!). To this end, consider two single-cellexamples: examples 1 and 2 of Table 1. Also, for convenience,consider dic = 1, RT ,c = Itc , and RR,i = Iri for i = 1, 2 andc = 1.


102

100

dG

5 10 15 20

Number of iteration

10 log10 P = 10 dB

(a)

102

100

dG

5 10 15 20

Number of iteration

10 log10 P = 20 dB

(b)

105

100

dP

100 102 104

Number of iteration

Per-cellPer-antenna

(c)

105

100

dP

100 102 104

Number of iteration

Per-cellPer-antenna

(d)

Figure 3: Convergence of the GIA (FIA): the dierences dG and dP, defined in (42), as functions of the number of iterations under theper-antenna and per-cell power constraints. System parameters are given in example 1 of Table 1.

Figure 2 shows the MSEs and BERs as functions of thesum power of the system, that is, 10 log10tP where t =4. The results are obtained by averaging over 20 channelrealizations. When two data streams are transmitted for bothusers (i.e., example 1 in Table 1 where (m1,m2) = (2, 2) and{Vi} are full rank), both the GIA (or the FIA) and the DCOAcan be employed to find the globally optimum precoders anddecoders. Comparing the two 2-data-stream curves in eachsubplot of Figure 2, we observe that the GIA (or the FIA)indeed has the same globally optimum performance as theDCOA. It is remarkable that the performances for the per-cell and per-antenna power constraints are similar to eachother.

For the 1-data-stream scenario (i.e., example 2 in Table 1where (m1,m2) = (1, 1) and {Vi} are not full rank), onlythe GIA can be employed because both the DCOA and theFIA result in (m1,m2) = (2, 2) and thus are not applicablehere. Comparing the 1-data-stream curve against the 2-data-stream curves in each subplot of Figure 2, the MSE and BERperformances, as predicted, are improved by transmitting

fewer data streams than transmit antennas. But the increasein diversity gain is accompanied by a reduction in themultiplexing gain. For the 1-data-stream scenario, only thelocal optimality of the GIA can be guaranteed.

5.2. Computational Eciency: The GIA (FIA) versus theDCOA. Without loss of generality, we will compare thecomputational eciency of the various proposed approacheswith perfect CSI. Consider example 1 in Table 1. The numberof data streams is two for each of the two UEs so that theDCOA can be applicable. For convenience, we also choosedic = 1, RT ,c = Itc , RR,i = Iri for i = 1, 2 and c = 1.

Note that the GIA and the FIA have the same convergenceproperty because the FIA is a special case of the GIA whenthe source covariance matrices are all identity matricesmultiplied by the same constant. The FIA is slightly moreecient than the GIA because it combines, into one step,two of the three steps in each iteration of the GIA. Theconvergence property (expressed as dG and dP) of theGIA (or the FIA) for both per-antenna and per-cell power


1010

105

100D

ual

ity

gap

1 10 20 30 40

Number of iterations

SDP procedure

(a)

1010

105

100

Du

alit

yga

p

1 10 20 30 40


Numerically ecient procedure

(b)

103

102

1 10 20 30 40


(c)

103

102

1 10 20 30 40


(d)

12

14

16

17

Per-

ante

nn

apo

wer

(dB

)

1 10 20 30 40


(e)

12

14

16

17

Per-

ante

nn

apo

wer

(dB

)

1 10 20 30 40


(f)

Figure 4: Convergence of the DCOA: duality gap, four Lagrange multipliers, and four per-antenna transmission power, as functions of thenumber of iterations under the per-antenna power constraint with 10 log10P = 15 dB. (Legend: the 4 dierent colors in the two middle subplotsand two lower subplots correspond to the 4 transmit antennas.)

constraints is shown in Figure 3. The dierence in thedecoders between the jth iteration and the ( j + 1)th iterationand the distance from the power constraints at the jthiteration are defined as, respectively,

dG( j) =2

i=1

G

( j+1)i G( j)i

,

dP( j) = 1tP

tr

2

i=1F

( j)i

(F

( j)i

) P

.

(42)

In Figure 3, the convergence rates for both power con-straints are similar. It is remarkable that the GIA (or theFIA) converges much slower in higher power. This is dueto the fact that, when P increases, the Lagrange multipliers1 decrease quickly. For large Ps, the Lagrange multipliersare very small. For example, when 10 log10P = 30 dB, theLagrange multipliers can be as small as 1010. Under sucha situation, the equality power constraints in (7a) and (7b)are dicult to be met because the usage of (24a) or (24b)merely enforces the corresponding complementary slackness


3

3.5

4M

SE

7 10 15 20 23

10 log10 tP

Without cooperation

(a)

0

1

2

3

4

MSE

7 10 15 20 23

10 log10 tP

With cooperation

(b)

0.1

0.15

0.2

BE

R

7 10 15 20 23

10 log10 tP

(c)

104

103

102

101

BE

R

7 10 15 20 23

10 log10 tP

(d)

Figure 5: MSE and BER as functions of 10 log10tP under per-antenna and per-cell power constraints. System parameters and configurationsare shown in example 3 of Table 1 and Figure 1(a). (Legends: the blue solid lines, red dashed lines, and black dotted lines represent, resp., theGIA, the Non-robust-GIA, and the Robust-GIA results under the per-cell power constraint. And the blue plus markers, red circle markers, andblack square markers represent, resp., the GIA, the Non-robust-GIA, and the Robust-GIA results under the per-antenna power constraint.)

conditions:

It

2

j=1F js jFj

P

= 0, (43a)

1

tr

2

j=1F js jFj

Pb1

= 0. (43b)

Thus, the number of iteration increases drastically as Pincreases if equality in the power constraints in (7a) or(7b) is insisted. However, if the equality constraints arerelaxed and only inequality constraints (the per-antennaor per-cell transmission powers are allowed to be lessthan the corresponding power constraints) are required, theconvergence rate at high power will be improved greatly.

Using the same single-cell example, the convergentproperties of the SDP Procedure and the NumericallyEcient Procedure of the DCOA are shown in Figure 4.Here, 10 log10P = 15 dB and, for convenience, only theper-antenna power constraint is considered. Observing the

convergence rates of the duality gap in (34), the Lagrangemultipliers in (9a), and the per-antenna transmission powerfrom Figure 4, we conclude that the Numerical EcientProcedure converges faster than the SDP procedure.

Comparing the DCOA with the GIA (or the FIA), theGIA (or the FIA) is numerically more ecient than theDCOA. This is because, for the GIA (or the FIA), closedform expressions are available for the precoders, decodersand Lagrange multipliers; but for the DCOA, a numericaloptimization procedure has to be carried out to find thedecoder covariance matrices in the primal step. Note that,just like the GIA (or the FIA), the number of iterationsbetween the primal and dual steps of the DCOA increasesdrastically as P increases. This is because the convergenceproblem due to very small Lagrange multipliers at highpower exists for both the DCOA and the GIA (or the FIA).In fact, the DCOA does not even converge at times dueto the lack of numerical precision of the numerical solversused. Thus, both the DCOA and the GIA (or the FIA) havediculty in convergence at high power. Fortunately, withinthe practical power range, both the DCOA and the GIA


0

1

2

3

4

MSE

7 10 15 20 23

10 log10 tP

Example 3

(a)

0

2

4

6

8

MSE

10 15 20 26

10 log10 tP

Example 4

(b)

104

103

102

101

BE

R

7 10 15 20 23

10 log10 tP

(c)

105

104

103

102

101

BE

R

10 15 20 26

10 log10 tP

(d)

Figure 6: MSE and BER as functions of 10 log10tP under per-antenna and per-cell power constraints for examples 3 and 4 in Table 1 andFigure 1. (Legends are the same as in Figure 5.)

(or the FIA) worked fine as long as some attention was paidto the selection of the initial values of the iteration processat high power. Note that the robust approaches have only asmall increase in complexity compared to their perfect-CSIcounterparts and the conclusion made here for complexityanalysis is also applicable to the robust approaches.

In the following sections, we will consider the situationwhere {si} are all identity matrices and {Vi} are full rank.Under such a situation, the DCOA, the FIA, and the GIAare equivalent. Moreover, the Non-robust-DCOA, the Non-robust-FIA, and the Non-robust-GIA are equivalent, and theRobust-FIA, and the Robust-GIA are also equivalent. If theRobust-DCOA is applicable, the Robust-DCOA, the Robust-FIA, and the Robust-GIA are equivalent. Thus, only theGIA, the Robust-GIA, and the Non-robust-GIA results arepresented for convenience.

5.3. Multicell: With Cooperation versus without Cooperation.Using the 3-cell configuration in Figure 1(a) and the systemparameters of example 3 in Table 1, two dierent coopera-tion scenarios will be simulated. In the first scenario, thereis no cooperation among the eNBs. In the second scenario,

there is full cooperation among the three eNBs. Note that theinterference-plus-noise covariance matrix in (10) needs to beestimated in the without-cooperation scenario. With somederivations, we can show that (10) can be approximated as

ni

c /Db2d

2ic PbcRR,i + ai (44)

for the without-cooperation scenario. For convenience, wechoose RT ,c = Itc and RR,i = Iri for i = 1, 2, 3 and c = 1, 2, 3.Channel matrices are estimated and Hic /=Hic and 2E,ic /= 0.Figure 5 shows the MSE and BER results derived with andwithout eNB cooperation. All the MSE and BER results areobtained by averaging over 30 channel realizations.

It is not surprising to see that the BER and the MSE ofthe without-cooperation scenario are much larger (worse)than the BER and the MSE of the with-cooperation scenario,respectively. Even with perfect CSI, the without-cooperationBER is larger than 10% even at high power. It is obvious thatsome kinds of time/frequency scheduling or code spreadingare needed in order to reduce the cell edge interferences if nocooperation among eNBs is available. On the other hand, inthe with-cooperation scenario, the BER of the GIA is below


0

1

2

2.5

MSE

7 10 15 20 23

10 log10 tP

T = 0, R = 0

(a)

0

1

2

2.5

MSE

7 10 15 20 23

10 log10 tP

T = 0, R = 0.8

(b)

0

1

2

3

MSE

7 10 15 20 23

10 log10 tP

T = 0.8, R = 0

(c)

0

1

2

3

MSE

7 10 15 20 23

10 log10 tP

T = 0.8, R = 0.8

(d)

Figure 7: MSEs as functions of 10 log10tP under per-antenna and per-cell power constraints for dierent values of T and R. Systemparameters are shown in example 1 of Table 1. (Legends are the same as in Figure 5.)

1% at the low transmission power (10 log10tP = 10 log106 +5 dB) when the perfect CSI is available. When the perfectCSI is not available, the Robust-GIA result is decent. Eventhe Non-robust-GIA result in the with-cooperation scenariois better than the GIA result in the without-cooperationscenario. The Robust-GIA result loses around 8 dB in SNRwith respect to the GIA result and has around a 3 dBgain in SNR with respect to the Non-robust-GIA. Note thatresults obtained from both per-antenna and per-cell powerconstraints are similar.

5.4. Multicell: Example 3 versus Example 4. We now comparethe results of example 3 with the results of example 4 inTable 1. The system configurations of examples 3 and 4 areshown in Figures 1(a) and 1(b), respectively. Note that thereis one UE per cell in example 3 but there are two UEs percell in example 4. For convenience, we choose RT ,c = Itcand RR,i = Iri for i = 1, 2, . . . ,K and c = 1, 2, 3. Channelmatrices are estimated and Hic /=Hic and 2E,ic /= 0. Note thatthe coordinates of the eNBs and UEs are shown in Figures1(a) and 1(b); UE1 is right on the 3-cell edge and each of theother UEs is near at least one of the 2-cell edges.

Figure 6 shows the MSE and BER results of examples 3and 4 with full cooperation among 3 eNBs. All the MSEand BER results are obtained by averaging over 25 channel

realizations. Note that the average per-antenna power P inexamples 3 and 4 is the same. But the total power of example4 is twice of (3 dB larger than) the total power of example3 since the tc in example 4 is twice of the tc in example 3.Therefore, there is a 3 dB dierence in the scales of the x-axesof examples 3 and 4 in Figure 6.

We make four main observations. First, the results forthe per-cell and per-antenna power constraints are moreor less the same for all of the approaches (the GIA, theRobust-GIA, and the Non-robust-GIA) in both examples 3and 4. This is remarkable because the per-antenna powerconstraint, though more practical, is much stricter than theper-cell power constraint.

Secondly, as expected, the Robust-GIA yields better MSEand BER performances than the Non-robust-GIA. In thepower ranges shown in Figure 6, the performance gain ofthe Robust-GIA over the Non-robust-GIA for the MSE resultsis around 5 dB for example 4 and 3 dB for example 3. Theperformance gain of the Robust-GIA over the Non-robust-GIA for the BER results is around 25 dB for example 4 and03 dB for example 3. The performance gain for the MSEresults decreases as power P increases. This is due to the factthat CSI estimation errors decrease as P increases (Tc = tcP).However, the performance gain for the BER results increasesas P increases. This is because the BER is expressed in logscale. We conclude that the robust approach is more crucial


0.2

0.5

1

1.5

2

2.5

MSE

0 0.3 0.5 0.7 0.9

R

T = 0

(a)

0.2

0.5

1

1.5

2

2.5

MSE

0 0.3 0.5 0.7 0.9

R

T = 0.5

(b)

0.2

0.5

1

1.5

2

2.5

MSE

0 0.3 0.5 0.7 0.9

T

R = 0

(c)

0.2

0.5

1

1.5

2

2.5

MSE

0 0.3 0.5 0.7 0.9

T

R = 0.5

(d)

Figure 8: MSE as functions of T (with fixed R) and functions of R (with fixed T) under per-antenna and per-cell power constraints. Thetotal transmission power is 16 dB. System parameters are shown in example 1 of Table 1. (Legends are the same as in Figure 5.)

to larger MIMO systems such as example 4 than smallerMIMO systems such as example 3.

Thirdly, also as expected, the Robust-GIA yields larger(worse) MSE and BER than the GIA. In the power rangesshown in Figure 6, the performance degradation of theRobust-GIA with respect to the GIA for the MSE resultsis around 910 dB for example 4 and around 7-8 dB forexample 3. The performance degradation of the Robust-GIAwith respect to the GIA for the BER results is around 912 dB for example 4 and 9 dB for example 3. For the samereasons stated before, the performance degradation of MSEresults decreases as power P increases, but the performancedegradation of BER results increases as power P increases.We conclude that larger MIMO systems such as example 4are more sensitive to the CSI estimation errors than smallerMIMO systems such as example 3.

Lastly, compared to the results in example 3, the MSEresults for all the approaches are noticeably higher inexample 4, but the degradation of BER results in example 4compared to example 3 is not significant if the per-antennapower P is the same in both examples. We conclude thatcooperation among the eNBs is very eective in mitigatinginter-cell interferences at cell edges. And, increasing the

antenna numbers is an eective way to increase the systemcapacity even at cell edges as long as full eNB cooperationis allowed for the joint design of robust precoders anddecoders.

5.5. Spatial Channel Correlations. Using the example 1 inTable 1, system performances of various approaches underdierent antenna correlation conditions are studied. Thechannel correlation matrices are defined as

RT ,c =

1 T

T 1

, RR,i =

1 R

R 1

, i = 1, 2, c = 1.

(45)

We choose d1c = 1 and d2c = 0.78. Channel matrices areestimated and Hic /=Hic and 2E,ic /= 0.

Figures 7 and 8 show the MSE results for various valuesof T and R. In Figure 7, the MSE is plotted against the sumpower; in Figure 8, the MSE is plotted against either T or R.The MSE results are obtained by averaging over 20 channelrealizations. Again, we observe that the results for the per-cell and per-antenna power constraints are more or less thesame for all of the approaches (the GIA, the Robust-GIA,


1.9

2.10.5

1

2

2.5

MSE

0 0.3 0.5 0.7 0.9

Estimated T

R = 0, actual T = 0.5

(a)

1.61

1.6450.5

1

2

2.5

MSE

0 0.3 0.5 0.7 0.9

Estimated R

T = 0, actual R = 0.5

(b)

103

102

101

BE

R

0 0.3 0.5 0.7 0.9

Estimated T

(c)

103

102

101

BE

R

0 0.3 0.5 0.7 0.9

Estimated R

(d)

Figure 9: MSE and BER as functions of estimated T or R under per-antenna and per-cell power constraints (10 log10tP = 10 log103+10 dB,averaging over 20 channel realizations). (Legends are the same as in Figure 5.)

and the Non-robust-GIA). As T and/or R increase, the MSEincreases, the performance gain of the Robust-GIA over theNon-robust-GIA decreases, and the performance degradationof the Robust-GIA with respect to the GIA increases. Theeect due to increasing T is more profound than the eectdue to increasing R. We conclude that the robust approacheswork satisfactorily in wireless channels with high channelcorrelations.

5.6. Sensitivity with Respect to Estimation Errors of T or R.Using the 3-cell configuration in Figure 1(a) and the systemparameters of example 3 in Table 1, the sensitivity of MSEand BER performances with respect to estimation errors ofT or R is studied. Channel matrices are estimated andHic /=Hic and 2E,ic /= 0. Full cell cooperation is assumed. InFigure 9, MSE and BER are plotted against the estimatedT for a fixed R or against the estimated R for a fixedT . The enlarged MSE results of the Robust-GIA are shownin the middle two subplots. First of all, the GIA results areindependent of estimation errors of T and R because theperfect CSI is employed in the design. Similarly, the Non-robust-GIA results are also independent of estimation errors

of T and R because channel correlation statistics are notneeded in estimating the instantaneous channel matrices inpractice. Secondly, the Robust-GIA outperforms the Non-robust-GIA in terms of both MSE and BER regardless ofthe estimation error in T or R. Thirdly, the performancedegradation due to the estimation error in R (for a fixedT) is less profound than that due to the estimation errorin T (for a fixed R). This is because the variance 2E,ic (see(4b)) of EW ,ic depends only on T and the accuracy of RE,icin (4a) is not significantly aected by the estimation error inR if SNR is suciently large. Lastly, the Robust-GIA is lesssensitive to underestimates of T or R than overestimatesof T or R. The same observations as above are made fromsensitivity studies for various nonzero values of actual T orR. We conclude that eects of estimation errors (especiallyunderestimates) of channel correlations T or R on thesystem performances of the Robust-GIA are very small.

5.7. Sensitivity with Respect to Estimation Errors of PathLoss. Using the 3-cell configuration in Figure 1(a) and thesystem parameters of example 3 in Table 1, the sensitivityof MSE and BER performances with respect to estimation


0.5

1

1.5

2

2.3

MSE

0.25 0.75 1.25 1.75 2.25

EAPLR

(a)

1.517

1.525

MSE

0.25 0.75 1.25 1.75 2.25

EAPLR

(b)

103

102

101

BE

R

0.25 0.75 1.25 1.75 2.25

EAPLR

(c)

1.498

1.511

MSE

0.25 0.75 1.25 1.75 2.25

EAPLR

(d)

Figure 10: MSE and BER as functions of the estimated-to-actual-path loss ratio (EAPLR) under the per-antenna and per-cell powerconstraints (10 log10tP = 10 log103 + 10 dB, averaging over 20 channel realizations). (Legends are the same as in Figure 5.)

errors of path loss (PL = d2ic in (4a)) is studied.Full cell cooperation is assumed. For convenience, wechoose RT ,c = Itc and RR,i = Iri for i = 1, 2, 3 andc = 1, 2, 3. Channel matrices are estimated and Hic /=Hicand 2E,ic /= 0. Define the estimated-to-actual-path-loss ratio(EAPLR) as PLestimated/PLactual. In Figure 10, MSE and BERare plotted against the EAPLR ranging from 0.25 to 2.25.The enlarged MSE results of the Robust-GIA under the per-antenna and per-cell power constraints are shown in thetwo right subplots, respectively. First of all, the GIA resultsare independent of estimation errors of path loss becausethe perfect CSI is employed in the design. Similarly, theNon-robust-GIA results are also independent of estimationerrors of path loss because channel statistics are not usedin estimating the instantaneous channel matrices. Secondly,the Robust-GIA outperforms the Non-robust-GIA in termsof both MSE and BER regardless of estimation errors ofpath loss. Thirdly, the degradation of MSE due to estimationerrors of path loss is negligible. This is due to the fact that 2cin (13) is independent of path loss, and the accuracy of RE,icin (4a) is not significantly aected by path loss errors if SNRis suciently large. We conclude that the eects of estimation

errors of path loss on the system performances of the Robust-GIA are negligible.

6. Conclusion

Three robust approaches, the Robust-GIA, the Robust-FIA,and the Robust-DCOA, are proposed for designing MMSEtransceivers in the downlink of multicell multiuser MIMOSystems under general linear equality power constraints andwith CSI estimation errors. The GIA, the FIA, and theDCOA (the perfect CSI approaches) are special cases ofthe Robust-GIA, the Robust-FIA, and the Robust-DCOA thusgiving a general framework to deal with both perfect andimperfect CSI! Note that the robust approaches have only asmall increase in complexity compared to their perfect-CSIcounterparts.

The Robust-DCOA always gives optimum solutions but isonly applicable when the rank constraints on the precodersare relaxed, the transmit correlation matrix of each user isan identity matrix, and the source covariance matrices areall identity matrices multiplied by the same constant. Thestatistics of the CSI estimation error also need to be the


same for all users if the power constraints of the users areinterdependent. The Robust-GIA, on the other hand, hasno such restrictions and is the most general among thethree proposed robust approaches. It allows tradeo betweendiversity and multiplexing gains, which is not possible in theRobust-DCOA or the Robust-FIA. The multiplexing gains ofthe Robust-DCOA or the Robust-FIA are determined by theranks of the decoder covariance matrices. The Robust-FIA isa special case of the Robust-GIA. It, requiring that the sourcecovariance matrices are identity matrices multiplied by thesame constant, is a bit less flexible than the Robust-GIA.But, it is much more flexible than the Robust-DCOA sinceit does not require all of the transmit correlation matrices tobe identity matrices. Both the Robust-GIA and the Robust-FIA are numerically more ecient than the Robust-DCOA.The Robust-FIA is slightly more ecient than the Robust-GIA because it combines two of the three steps in eachiteration of the Robust-GIA into one step. All approachesshow diculties in convergence when the transmit poweris very high. Relaxing the equality power constraints willimprove the numerical eciency greatly. Both the Robust-GIA and the Robust-FIA can only guarantee local optimality.But, whenever the Robust-DCOA is applicable and all thedecoder covariance matrices are full rank, the three robustapproaches are actually equivalent (i.e., the Robust-GIA andthe Robust-FIA are also optimum).

MMSE transceiver designs using the three proposedapproaches are performed for various single-cell and mul-ticell examples with dierent system configurations, powerconstraints, channel spatial correlations, and cooperationscenarios. System performances in terms of MSE and BERare investigated. Important concluding remarks made fromthese numerical examples are list below. First of all, therobust approaches outperform their non-robust counter-parts in most of the numerical simulations (even whenthe channel is highly correlated, when the CSI estimationerrors are large, and when there exist estimation errors instatistics of channel parameters). Secondly, the performanceof the with-cooperation scenario is much better than thatof the without-cooperation scenario. With cell cooperation,the cell edge UEs perform as well as those UEs withoutinter-cell interferences and therefore the cell edge dicultiescan be remedied. Thus, with full cell cooperation, thesystem throughput can increase linearly with the numbersof antennas for both transmission and reception. Thirdly,the robust approaches are insensitive to the estimationerrors of the channel statistics (e.g., to channel correla-tions and path loss). This important feature makes robustapproaches practical. Fourthly, the system performancesderived under the more practical per-antenna power con-straint are very similar to those with the per-cell powerconstraint. Thus, the practical per-antenna power constraintinflicts little performance losses compared to the optimumper-cell power constraint. Fifthly, the performance gainof the robust approaches over the non-robust approachesis more profound in larger MIMO systems. Sixthly, theperformance gain of the robust approaches over the non-robust approaches is reduced if the channel correlationsincrease.

In short, we have herein proposed, for joint MMSEtransceiver designs, three novel robust approaches: theRobust-GIA (the most general), Robust-FIA (the mostecient), and the Robust-DCOA (which guarantees theglobal optimality). The proposed approaches are indeedrobust with respect to dierent system configurations,CSI estimation errors, channel correlations, and channelmodeling errors. When cell cooperation is available, therobust approaches provide a remedy for solving the cell edgeproblem without reducing the number of data streams.

Acknowledgment

The authors would like to thank InterDigital Communica-tions Corporation for its financial support.

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