Joint Robust Weighted LMMSE Transceiver Design for Dual-Hop AF Multiple-Antenna Relay Systems

0.14 MB·English

by  Chengwen Xing

Joint Robust Weighted LMMSE Transceiver Design
for Dual-Hop AF Multiple-Antenna Relay Systems
Chengwen Xing∗, Shaodan Ma†, Zesong Fei∗, Yik-Chung Wu† and Jingming Kuang∗
∗School of Information and Electronics, Beijing Institute of Technology, Beijing, China
Email: {xingchengwen}@gmail.com {feizesong, jmkuang}@bit.edu.cn
†Department of Electrical and Electronic Engineering, The University of Hong Kong, Hong Kong
Email: {sdma, ycwu}@eee.hku.hk
2
1
Abstract—In this paper, joint transceiver design for dual-hop minimization. In fact, both kinds of criteria can be formu-
0
2 amplify-and-forward (AF) MIMO relay systems with Gaussian latedasweightedMSEminimizationproblems[8].Therefore,
distributedchannelestimationerrorsinbothtwohopsisinvesti-
weightedMSEischosenastheobjectivefunctioninthispaper.
n gated.Duetothefactthatvariouslineartransceiverdesignscan
In this paper, the channel estimation errors are assumed to
a betransformedtoaweightedlinearminimummean-square-error
J (LMMSE) transceiver design with specific weighting matrices, be Gaussian distributed, which is generally true when linear

4 weightedmeansquareerror(MSE)ischosenastheperformance channel estimation algorithms are adopted [9]. The structures
1 metric. Precoder matrix at source, forwarding matrix at relay of the optimalsolutionsfor the robustjointtransceiverdesign
and equalizer matrix at destination are jointly designed with
withminimumweightedMSEastheobjectivearefirstderived.
] channel estimation errors taken care of by Bayesian philosophy.
T Based on these optimal structures, the transceiver design
Several existing algorithms are found to be special cases of the
I proposed solution. The performance advantage of the proposed is then significantly simplified and iterative water-filling is
.
s robust design is demonstrated by the simulation results. adopted to solve the problem. Finally, the performance gain
c of the proposed robust design is verified by simulation.
[ I. INTRODUCTION
The following notations are used throughout this paper.

1 By deploying relays, cooperative communication has great Boldface lowercase letters denote vectors, while boldface
v potential to improve system performance [1]. Among various uppercase letters denote matrices. The notations ZT, Z∗ and
4 relaying strategies, amplify-and-forward (AF) scheme is at- ZH denotethetranspose,conjugateandconjugatetransposeof
8
tractiveforpracticalimplementationduetoitslowcomplexity. thematrixZ,respectivelyandTr(Z)isthetraceofthematrix
9
2 On other hand, it is well-established that employing multiple Z. The symbol IM denotes an M×M identity matrix, while
. antennasisbeneficialtoimprovewirelesssystemperformance 0 denotesanM×N allzeromatrix.ThenotationZ1/2 is
1 M,N
0 due to spatial diversity and multiplexing gains. Consequently, the Hermitian square root of the positive semi-definite matrix
2 inordertoobtainthevirtuesofthesetwotechniques,combina- Z, such that Z1/2Z1/2 = Z and Z1/2 is also a Hermitian
1 tionofAFtransmissionandmulti-inputmulti-output(MIMO) matrix. The symbol E denotes statistical expectation.
: systems attracts more and more researchers’ interest.
v
i Linear transceiver design for AF MIMO relay systems has II. SYSTEMMODEL
X
been widely researched in [2]–[6]. Unfortunately, most of A. Transmitted and Received Signals
r the existing works assume channel state information (CSI) In this paper, a three node dual-hop amplify-and-forward
a
is perfectly known. However channel estimation errors are (AF) cooperativecommunication system is considered. In the
consideredsystem, thereis one sourcewith N antennas,one
inevitable in practical systems. Robust design, which takes S
relay with M receive antennas and N transmit antennas,
channel estimation errors into account, is therefore desirable R R
and one destination with M antennas. Due to long distance
D
inpractice.Jointrobustdesignofrelayforwardingmatrixand and possibly deep fading, the direct link between the source
destinationequalizerunderchannelestimationerrorshasbeen anddestinationisnotconsideredinthispaper.Atthefirsthop,
considered in [7]. However in [7], source precoder design, the source transmits data to the relay. The received signal, x,
at the relay is
which provides further degree of freedom for improving
system performance, is not considered. In this paper, we x=HsrPs+n1 (1)
take a step further to jointly design source precoder matrix,
where H is the MIMO channel matrix between the source
relayforwardingmatrixanddestinationequalizermatrixunder sr
and the relay, andP is the precodermatrix at the source. The
channel estimation errors.
vectorsistheN×1datavectortransmittedbythesourcewith
There are two main kinds of criteria for transceiver de-
sign: capacity maximization and mean-square-error (MSE) the covariance matrix Rs = E{ssH} = IN. Furthermore, n1
is the additive Gaussian noise vector with correlation matrix
R =σ2 I .
ThisresearchworkwassupportedinpartbySino-SwedishIMT-Advanced n1 n1 MR
and Beyond Cooperative Program under Grant No.2008DFA11780 and Pro- At the relay, the received signal x is multiplied by a
gramforChangjiang Scholars andInnovative Research TeaminUniversity. forwarding matrix F. Then the resultant signal is transmitted
tothedestination.Thereceivedsignaly atthedestinationcan where matrices Rx and K2 are defined as
be written as
Rx =E{xxH}=H¯srPPHH¯Hsr+K1
y=HrdFHsrPs+HrdFn1+n2, (2) K1 =Tr(PPHΨsr)Σsr+Rn1
whereHrdistheMIMOchannelmatrixbetweentherelayand K2 =Tr(FRxFHΨrd)Σrd+Rn2. (7)
the destination, and n2 is the additive Gaussian noise vector It is obviousthat Rx is the covariance matrix of the received
at the second hop with correlation matrix R =σ2 I . In signal at the relay. Based on the MSE matrix given by (6), a
n2 n2 MD more generalperformancemetric for transceiver design is the
order to guarantee the transmitted data s can be recovered at
weighted MSE [8]:
the destination, it is assumed that N , M , N , and M are
S R R D
greater than or equal to N [2]. MSEW(G,F,P)=E (Gy s)HW(Gy s)
{ − − }
=Tr[WE (Gy s)(Gy s)H ], (8)
B. Channel Estimation Error Model { − − }
whereW isa N×N positivesemi-definiteweightingmatrix.
Channel state information is usually estimated via using
Under transmit power constraints at the source and relay, the
training sequences. In a dual-hop system, there are two chan-
optimization problem of transceiver design is formulated as
nels to be estimated. For implementation simplicity, training
sequences are transmitted from both source and destination P1: min MSE (G,F,P)
W
to the relay. At the relay, the two channels are separately G,F,P
estimatedandthenbasedonthechannelestimates,transceivers s.t. Tr(PPH) Ps, Tr(FRxFH) Pr. (9)
≤ ≤
are jointly designed. After that, the designed transceiver ma-
trices are forward to the source and destination, respectively.
In general, two hop channels can be written as Notice that based on the definition of Rx in (7), Rx is a
function of P. In order to simplify the analysis, we define a
H =H¯ +Σ1/2H Ψ1/2, H =H¯ +Σ1/2H Ψ1/2, new variable
sr sr sr W,sr sr rd rd rd W,rd rd
F˜ ,FK1/2(K−1/2H¯ PPHH¯HK−1/2+I )1/2, (10)
,∆Hsr ,∆Hrd 1 1 sr sr 1 MR
(3)
| {z } | {z } ,ΠP
where H¯sr and H¯rd are the estimated channels, and ∆Hsr based on which|Tr(FRxFH) ={zTr(F˜F˜H) and}the two con-
and ∆Hrd are the corresponding channel estimation errors straintsinthe optimizationproblem(9) becomesindependent.
whose elements are zero mean Gaussian random variables. Furthermore, with (10) the objective function becomes
Furthermore,the M ×N matrix ∆H can be decomposed
as∆H =Σ1/2HR ΨS1/2,wheretheserlementsoftheM × MSEW(G,F˜,P)
NS mastrrix HWsr,sr aWre,srindesrpendent and identically distribuRted =Tr[WG(H¯rdF˜F˜HH¯Hrd+K2)GH]
(i.i.d.) Gaussian random variables with zero mean and unit Tr(WGH¯ F˜Π−1/2K−1/2H¯ P)
variance. When MMSE channel estimator is used, the row − rd P 1 sr
and column covariance matrices of ∆H can be derived as Tr(WPHH¯HK−1/2Π−1/2F˜HH¯HGH)+Tr(W). (11)
sr − sr 1 P rd
[9]
As in the optimization problem (9) there is no constraint on
Σsr =IMR, ΨTsr =(σh−s2rINS + σ12 D1∗DT1)−1 (4) Gfun,ctthieonoopftimPalansdolFu˜tio[7n],foanrdGiscgainvetnhebryefore be written as a
n1
where σh2sr is channel variance in the first hop and D1 is the G=(H¯rdF˜Π−P1/2K−11/2H¯srP)H(H¯rdF˜F˜HH¯Hrd+K2)−1. (12)
training sequenceused in the first hop. Similarly, the row and Substituting(12)intotheMSEformulation(11),theweighted
column covariance matrices of ∆Hrd, can also be derived to MSE can be rewritten as
be
MSE (F˜,P)
Σrd =(σh−r2dIMD + σ1n22D2DH2)−1, ΨTrd =INR (5) =TrW(W)−Tr[(H¯rdF˜Π−P1/2K−11/2H¯srPW1/2)H
whereσ2 denotesthechannelvarianceofthesecondhopand ×(H¯rdF˜F˜HH¯Hrd+K2)−1(H¯rdF˜Π−P1/2K−11/2H¯srPW1/2)].
hrd (13)
D2 represents the training sequence used in the second hop.
Notice that the directionsof training sequencestransmitted in Finally,itcanbeeasilyprovedthattheoptimalPandF˜ must
occur on the boundary [7]
the two hops are opposite. Therefore, the results given by (4)
and (5) are different. Tr(PPH)=P , Tr(F˜F˜H)=P . (14)
s r
Clearly, it means that for the minimum weighted MSE, both
III. PROBLEM FORMULATION
relay and source should transmit at the maximum power. As
Atthe destination,a linearequalizerGis adoptedto detect a result, the optimization problem for joint transceiver design
the data vector s. The mean-square-error (MSE) matrix is is formulated as
E{(Gy−s)(Gy−s)H} , where theexpectationis taken with
respect to random data, channel estimation errors, and noise. P2: min MSEW(F˜,P)
F˜,P
In [7], it is shown that
s.t. Tr(F˜F˜H)=P , Tr(PPH)=P . (15)
r s
E (Gy s)(Gy s)H
{ − − } Inthefollowing,thestructuresoftheoptimalsolutionsarefirst
=G(H¯rdFRxFHH¯Hrd+K2)GH+IN −(PHH¯HsrFHH¯HrdGH) derived and then the transceiver design problem is simplified
(GH¯rdFH¯srP), (6) and solved in Section VI.
−
IV. THE STRUCTURE OF OPTIMALF˜ where Λ˜A is the N ×N principal submatrix of ΛA and is
a diagonalmatrix with diagonalelements in decreasing order.
With
Using (24), the weighted MSE given by (17) is rewritten as
A,(Π−1/2K−1/2H¯ PW1/2)
P 1 sr MSE (F˜,P)=Tr(W) Tr(AHMA)
the weightMed,MF˜SHEH¯cHradn(H¯berdfF˜urF˜thHeH¯rHrrdef+orKm2u)l−at1eH¯dradsF˜, (16) W =Tr(W)−−Tr(VAΛ˜MVAHAHA). (25)
It is shown in Appendix A that for the minimum MSE the
MSEW(F˜,P)=Tr(W) Tr(AHMA). (17) following identity holds
−
U(F˜sHinH¯gHKth−e1H¯maF˜trix+ Ii)n−v1e,rsiothne lewmeimgha,tedMMSE= furIthe−r VA =UW, (26)
rd 2 rd
becomes where the unitary matrix UW is obtained from the eigen-
MSEW(F˜,P)=Tr(W) Tr(AHA) decomposition of W = UWΛWUHW in which ΛW is a
− diagonalmatrixwithdecreasingdiagonalelements.Theabove
+Tr[AH(F˜HH¯HK−1H¯ F˜+I )−1A]. (18)
rd 2 rd MR equation will also be useful in the following derivation.
It should be highlighted that the two constraints in (15) are From (26) and the definition of A in (16) and using the
independentandforanygivenPtheoptimizationproblemfor matrix inversionlemma again,the weighted MSE (25) can be
F˜ is equivalent to further rewritten as
min Tr[AH(F˜HH¯HK−1H¯ F˜+I )−1A] MSEW(F˜,P)
rd 2 rd MD
sF˜.t. Tr(F˜F˜H)=P (19) =Tr(W)−Tr[(Π−P1/2K−11/2H¯srP)W1/2UWΛ˜MUHWW1/2
r (Π−1/2K−1/2H¯ P)H]
wglheecrteed.thWeitchonthsteanfatctpathrtast Ψindep=enIdefnrotmof(5F˜), whaevheavbeeen ne- =T×r(WP)+Tr[1UWΛs1Wr/2Λ˜MΛ1W/2UHW(PHH¯HsrK−11H¯srP+I)−1]
K2 =Tr(F˜F˜HΨrd)Σrd+σn2rd2IMD =PrΣrd+σn22IMD (20) No−ticTer(tWha1t/2bUasWedΛ˜oMnUthHWeWde1fi/2n)i.tion of M in (16), Λ˜M i(s27a)
which is a constant matrix. Furthermore, based on singular function of F˜ only and independent of P. Only the second
value decomposition
termin(27)isafunctionofP.Then,theoptimizationproblem
A=UAΛAVAH, K−21/2H¯rd =UrdΛrdVrHd. (21) for P becomes
wherethediagonalelementsofΛA andΛrd areindecreasing mpin Tr[UWΛ1W/2Λ˜MΛ1W/2UHW(PHH¯HsrK−11H¯srP+IN)−1]
order. Using Majorization theory, the objective function of
s.t. Tr(PPH)=P . (28)
(19) can be transformed into a Schur-concave function of s
the diagonal elements of (UHAF˜HH¯HrdK−21H¯rdF˜UA+IMR)−1. Forsourceprecoderdesign,themaindifferencefromforward-
Therefore, the optimal solution of (19) has the following ing matrix F˜ design is that K1 is not constant. As mentioned
structure [7]
previously Σsr =I (4), then K1 equals to
F˜ =Vrd,NΛF˜UHA,N (22) K1 =[Tr(PPHΨsr)+σn21]IMR,ηpIMR. (29)
mewlehamterrieexnΛtΛ˜sF˜rodfisΛisaF˜NtΛh˜e2r×dNΛNF˜×daiNaregoipnnraidnlecmcipraeatarlisxsinusgbumcohradttrehirxa.tTothhfeeΛddriidaa.ggooTnnhaaell With the powηper=cTonr(sPtrPaiHntΨTsrr)(P+PσHn21)=Ps, we have
matrices Vrd,N and UA,N are the first N columns of Vrd =Tr(PPHΨsr)+σn21Tr(PPH)/Ps
and UA, respectively. As A is a function of P, the value =1
of UA,N will be given in the next section, after giving the =Tr(PPH(PsΨsr+σn2|1INS{)z)/Ps.} (30)
structure of optimal P.
From (30), the constraint of the optimization problem (28)
V. THESTRUCTURE OFOPTIMALP becomes as
Substituting the structure of optimal F˜ (22) into the defini- Tr(PPH)=Tr[PPH(P Ψ +σ2 I )]/η =P , (31)
tionof M in (16), andtogetherwith the factthatthediagonal s sr n1 NS p s
elements of ΛF˜Λ˜2rdΛF˜ are in decreasing order, we have btoased on which the optimization problem (28) is equivalent
M=UA,N[IN −(ΛF˜Λ˜2rdΛF˜ +IN)−1]UHA,N (23) mPin Tr[UWΛ1W/2Λ˜MΛ1W/2UHW(PH1/ηpH¯HsrH¯srP+IN)−1]
,Λ˜M s.t. Tr[PPH(P Ψ +σ2 I )]/η =P . (32)
s sr n1 NS p s
whereΛ˜M isaN×N| diagonalm{zatrixwithdia}gonalelements
in decreasing order. It is also straightforward that (23) is the
Defining a new variable,
eigen-decomposition of M. Based on (23), and the singular
vsuablusetitduetcioonmwpoesihtiaovneothfeAfoglilvoewniinng(2id1e)n,taifttyerastraightforward P˜ =1/√ηp(PsΨsr+σn21INS)1/2P (33)
the optimization problem (32) becomes as (34) as shown at
AHMA=VAΛ˜AΛ˜MΛ˜AVAH =VAΛ˜MVAHAHA (24) the top of the nextpage. This formulationis exactly the same
min Tr[UWΛ1W/2Λ˜MΛ1W/2UHW(P˜H(PsΨsr+σn21INS)−1/2H¯HsrH¯sr(PsΨsr+σn21INS)−1/2P˜+IN)−1]
P˜
s.t. Tr(P˜P˜H)=P (34)
s
as the optimization problem (19) for F˜. Following the same where Λ˜ = diag{λ } means that the ith diagonal element
a a,i
argument for F˜ and defining unitary matrices Usr and Vsr ofthediagonalmatrixΛ˜aisdenotedasλa,i.Thisoptimization
based on the following singular value decomposition is non-convex, and thus generally speaking it is difficult to
solve. However, notice that when p ’s are fixed, f ’s can be
H¯sr(PsΨsr+σn21INS)−1/2 =UsrΛsrVsHr, (35) computed as i i
with the diagonal elements of the diagonal matrix Λ in
sr + 1/2
decreasing order, the optimal P˜ has the following structure w p2λ2 1
f = i i sr,i , (43)
P˜ =Vsr,NΛP˜UHW (36) i  rµfλ2rd,is1+p2iλ2sr,i − λ2rd,i! 
 
where ΛP˜ is a N2×N diagonalmatrix such that the diagonal whereµf istheLagrangemultiplierwhichmakesPfi2 =Pr.
elements of ΛP˜Λ˜srΛP˜ are in decreasing order. The diagonal On the other hand, when fi’s are fixed, pi’s can be computed
matrixΛ˜ is theN×N principalsubmatrixofΛ . Further- as
sr sr
more, based on the definition of P˜ given by (33), the optimal + 1/2
P has the following structure p = wi fi2λ2rd,i 1 , (44)
P=√ηp(PsΨsr+σn21INS)−1/2Vsr,NΛP˜UHW. (37) i  rµpλ2sr,is1+fi2λ2rd,i − λ2sr,i! 
 
Substituting (37) into the definition of A in (16), we have where µ is the Lagrange multiplier which makes Pp2 =
p i
UA,N = Usr,N. Therefore, the optimal F˜ has the following Ps hold. Notice that this iterative water-filling algorithm is
structure guaranteed to converge, as discussed in [10].
F˜ =Vrd,NΛF˜UHsr,N. (38) Special cases: Several existing algorithms can be considered
as special cases of our proposed solution.
Remark: Given (37) and (38), the remaining problem is how
• When CSI is perfectly known, W = I and P = I, the
to determine two diagonal matrices ΛF˜ and ΛP˜. proposed solution for F reduces to that in [2].
Notice that after ΛP˜ is computed, the remaining unknown • When CSI is perfectly known and W = I, the proposed
parameter in (37) is only η . In order to solve η , substitute
p p
theformulationofP(37)intothedefinitionofη in(29),and solution for P and F reduces to that given in [5].
p
then we get • When the second hop channel is an identity matrix and
η =Tr(PPHΨ )+σ2 noiseless, the proposed solution for source precoder design
p sr n1 reduces to that given in [11].
=η Tr[VH (P Ψ +σ2 I)−1/2Ψ (P Ψ +σ2 I)−1/2
p sr,N s sr n1 sr s sr n1
×Vsr,NΛ2P˜]+σn21. (39) VII. SIMULATION RESULTSAND DISCUSSIONS
This is a simple linear function of η , and η can be easily In this section, simulation results are presented to demon-
p p
solved to be strate the performance of the proposed algorithm. For the
η =σ2 / 1 Tr[VH (P Ψ +σ2 I)−1/2Ψ purpose of comparison, the algorithm based on the estimated
p (Pn1Ψ{ −+σ2 I)s−r,1N/2Vs srΛ2]n.1 sr (40) channel only (without taking the channel errors into account)
× s sr n1 sr,N P˜ } [5] and the robust algorithm without source precoder design
VI. PROPOSED SOLUTIONFORΛF˜ ANDΛP˜ in [7] arealso simulated.Inthe following,we consideran AF
In this section, the optimal ΛF˜ and ΛP˜ will be derived. MIMO relay system where the source, relay and destination
NoticethatΛ˜ andΛ˜ aretheN×N principalsubmatrices are equipped with the same number of antennas, i.e., N =
sr rd S
of Λsr and Λrd, respectively. Based on (31), substituting the MR =NR =MD =4.TheelementsofchannelmatricesHsr
optimalstructures(37) and (38) into the originaloptimization
and H are randomlygeneratedas i.i.d. Gaussian distributed
problem (15), and denoting rd
random variables.
Λ˜sr =diag{λsr,i} Λ˜rd =diag{λrd,i} ΛF˜ =diag{fi} The widely used exponential correlation matrix Rα =
the oΛpPt˜im=izdaitaigo{nppi}robΛleWm=(1d5i)agb{ewcoi}mes (41) {DαD|i−Hj|∝}ijRisαu.sTedhetno mΨosdrel=thΣe rcdorr=ela(tIio4n+mSaNtriRxEoSfTDR,αi).−e.1,
where SNREST is the signal-to-noise ratio (SNR) in channel
N w (f2λ2 +p2λ2 +1) estimation process. In the simulation, the weighting matrix
min i i rd,i i sr,i
fi,pi Xi=N1 (p2iλ2sr,i+1N)(fi2λ2rd,i+1) cisovsaerlieacntceedmasatrWices=aredRiang1{[0=.3σ012.I34 0an.2d 0R.2n]2}.=Thσe22In4o.iIsne
s.t. f2 =P p2 =P , (42) data transmission stage the SNRs at relay and destination are
i r i s
Xi=1 Xi=1 defined as Ps/σ12 and Pr/σ22, respectively.
100 based on which, the weighted MSE (18) can be rewritten as
MSE (F˜,P)=Tr(W) Tr(WN)
W
−
Tr(W) λ (W)λ (N), (48)
SE ≥ − i i i
M10−1
Weighted wNheuermeaλnni(Zin)eqdueanloittyes[1t2h]e, fiothr tlhaergmeXsitniemiguemnvwaleuieghotfedZM. USEsinNg
The algorithm based on estimated CSI only andW havethesameeigen-vectors.Inotherwords,giventhe
The robust design without source precoder design in [7] eigen-decompositionof W as
10−2 The proposed robust design
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 W=UWΛWUHW (49)
1/SNR
EST where the diagonal elements of the diagonal matrix ΛW are
indecreasingorder,fortheminimumMSE,Ncouldbeeigen-
Fig.1. WeightedMSEsofthedetected datafordifferent algorithms, when
α=0.3andPs/σ12=Pr/σ22=30dB. decomposed as
N=UWΛNUHW. (50)
For the source node, four independent data streams are where ΛN is a diagonal matrix whose diagonal elements are
in decreasingorder.Based on (49) and (50), for the minimum
transmitted and in each data stream, NData = 10000 inde- MSE it holds that
pendent QPSK symbols are transmitted. Each point in the
figureisanaverageof10000independentchannelrealizations. AHMA=W1/2NW1/2 =UWΛWΛNUHW. (51)
Fig.1showstheweightedMSEsatthedestinationfordifferent =Λ2
algorithms when α = 0.3 and Ps/σ12 = Pr/σ22 = 30dB. It As the diagonal elements of the diagon|al{mzat}rices ΛW and
can be seen that the performance of the proposed algorithm ΛN are positive and both in decreasing order, the diagonal
is always better than that based on the estimated CSI only. elements of Λ2 are also in decreasing order. Clearly, the
second equation of (51) also denotes an eigen-decomposition
It can also be observed that the proposed robust algorithm ofAHMA.Comparing(45)and(51),itcanbeconcludedthat
with source precoderdesign performsbetter than thatwithout forthe minimumMSE thereexistsan eigen-decompositionof
source precoder design in [7], illustrating the importance of AHMA with eigenvalues in decreasing order such that
joint transceiver design involving source precoder.
VA =UW. (52)
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