[22]a mmse vector precoding with bd for mu mimo dl

IEEE TRANSACTIONS ON COMMUNICATIONS, ACCEPTED FOR PUBLICATION 1

A MMSE Vector Precoding With BlockDiagonalization for Multiuser MIMO Downlink

Jungyong Park, Student Member, IEEE, Byungju Lee, Student Member, IEEE,and Byonghyo Shim, Senior Member, IEEE

Abstract— Block diagonalization (BD) algorithm is a general-ization of the channel inversion that converts multiuser multi-input multi-output (MIMO) broadcast channel into single-userMIMO channel without inter-user interference. In this paper,we combine the BD technique with a minimum mean squareerror vector precoding (MMSE-VP) for achieving further gainin performance with minimal computational overhead. Twokey ingredients to make our approach effective are the QRdecomposition based block diagonalization and joint optimizationof transmitter and receiver parameters in the MMSE sense.In fact, by optimizing precoded signal vector and perturbationvector in the transmitter and receiver jointly, we pursue anoptimal balance between the residual interference mitigation andthe noise enhancement suppression. From the sum rate analysisas well as the bit error rate simulations (both uncoded andcoded cases) in realistic multiuser MIMO downlink, we showthat the proposed BD-MVP brings substantial performance gainover existing multiuser MIMO algorithms.

Index Terms— Multiuser MIMO, block diagonalization, QRdecomposition, MMSE vector precoding, Cholesky factorization.

I. INTRODUCTION

IN the multiuser multiple-input multiple-output (MIMO)downlink system, a basestation with multiple antennas

transmits data streams to multiple mobile users, each with oneor more receive antennas. Due to the fact that the co-channelinterference is hard to be managed by the receiver operation,and more importantly, capacity of interference channel canbe made equivalent to capacity without interference, pre-cancellation of the interference via the precoding at thetransmitter has received much attention in recent years. Itis now well-known that the capacity region of the multiuserMIMO downlink can be achieved by dirty paper coding(DPC) [1]–[8]. However, the shortcoming of the DPC is thatsystem implementation leads to the high computational costof successive encodings and decodings. Recently, as a wayto reduce the complexity of DPC while preserving a largefraction of DPC capacity, block diagonalization (BD) hasbeen proposed [9]. The main advantage of the BD technique

Paper approved by D. J. Love, the Editor for MIMO and Adaptive Tech-niques of the IEEE Communications Society. Manuscript received November3, 2010; revised August 4, 2011.

This work is supported by the Basic Science Research Program through theNational Research Foundation (NRF) funded by the Ministry of Education,Science and Technology (MEST) (No. 2011-0012525) and the second BrainKorea 21 project.

The authors are with the School of Information and Communication, KoreaUniversity, Anam-dong, Sungbuk-gu, Seoul, Korea, 136-713 (e-mail: {jypark,bjlee}@isl.korea.ac.kr, [email protected]).

Digital Object Identifier 10.1109/TCOMM.2011.01.xxxxx

is that the transmit precoding matrix assures zero inter-userinterference in the received signal. Extensions of the BD,employing waterfilling or vector perturbation on top of theBD [9] [10], have also been suggested to fill the gap betweenthe DPC and BD.

In this paper, we propose a technique that combines theQR decomposition based BD and minimum mean squareerror vector precoding (MMSE-VP) [11] for pursuing furthergain in performance. By combining the MMSE-VP with theBD, the proposed method, henceforth referred to as the BD-MVP, mitigates noise enhancement effect, thereby achievingconsiderable gain in performance (both capacity and bit er-ror rate). Unlike the conventional VP scheme that controlsparameters at the transmitter only, the MMSE-VP techniquejointly optimizes parameters at the transmit and receiver topursue a balance between the residual interference mitigationand noise enhancement suppression. Moreover, due to the QRdecomposition based block diagonalization, dimension of thesystem being processed is reduced, and hence the proposedBD-MVP brings additional benefit in complexity over theSVD based BD. In a nutshell, the proposed approach improvesthe quality of service (QoS) of mobile users in multiuserMIMO environments with moderate computational cost. Infact, from the sum rate analysis and BER simulation results,we show that the proposed BD-MVP outperforms the BDalgorithm and its variants (BD-VP and BD-WF) [10] as well asTomlinson-Harashima precoding (THP) based schemes (BD-GMD and BD-UCD) [12].

The rest of this paper is organized as follows. In Section II,we describe the system model and the summary of theBD technique and its extensions (BD-WF and BD-VP). InSection III, we present the proposed BD-MVP method. Weprovide the simulation results in Section IV and conclude inSection V.

We briefly summarize notations used in this paper. Weemploy uppercase boldface letters for matrices and lowercaseboldface for vectors. The superscripts (·)H and (·)T denoteconjugate transpose and transpose, respectively. diag(·) is adiagonal matrix where nonzero elements exist only on themain diagonal of the matrix. ‖ · ‖ indicates an L2-norm ofa vector. CN (m, σ2) denotes a complex Gaussian randomvariable with mean m and variance σ2.

II. MULTIUSER MIMO DOWNLINK

In this section, we present the system model for the mul-tiuser MIMO downlink and then review the BD and BD-VPtechniques.

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2 IEEE TRANSACTIONS ON COMMUNICATIONS, ACCEPTED FOR PUBLICATION

A. System Model

In a multiuser MIMO downlink, a basestation equippedwith NT antennas transmits information to K mobile users.Each mobile user has NR,j receive antennas and thus thetotal number of receive antennas is NR =

∑Kj=1 NR,j . In

the sequel, we use the notation {NR,1, · · · , NR,K} × NT todescribe this multiuser system. For each channel realization,we assume that a block of data symbols with length NB istransmitted.

Under the flat fading channel assumption, the total receivedsignal vector y[n] = [y1[n]T y2[n]T · · · yK [n]T ]T is givenby

y[n] = Hx[n] + w[n], n = 1, 2, · · · , NB, (1)

where H = [HT1 HT

2 · · · HTK ]T is the NR × NT com-

posite channel matrix and x[n] = Fx[n] is the precodedsignal vector being transmitted at the basestation where F =[F1 F2 · · · FK ] and x[n] = [x1[n]T x2[n]T · · · xK [n]T ]T

are the NT × NR precoding matrix and the NR ×1 transmit signal vector, respectively, and w[n] =[w1[n]T w2[n]T · · · wK [n]T ]T is the complex Gaussiannoise vector

(w[n] ∼ CN (0, σ2

wI)). In this setup, the received

signal of the user j can be expressed as

yj [n] = HjFjxj [n] + Hj

K∑l=1,l �=j

Flxl[n] + wj [n]. (2)

Note that the first and second terms in the right side of (2) arethe desired signal and multiuser interferences, respectively.

B. BD Algorithm

The BD algorithm is an extension of the channel inversionfor receivers with multiple antennas [9]. The key feature ofthe BD is to employ the precoding matrix F annihilating allmultiuser interference. The precoding constraint for achievingthis goal is

HiFj = 0 for i = 1, · · · , j − 1, j + 1, · · · , K. (3)

Clearly, Fj should be a null space of Hj = [HT1 · · ·HT

j−1

HTj+1 · · ·HT

K ]T . From the SVD of Hj

Hj = UjΣj [V(1)j V(0)

j ]H (4)

where Uj and Σj are matrices of the left singular vectorsand the ordered singular values of Hj , respectively, and V(1)

j

and V(0)j are right singular matrices corresponding to nonzero

singular values and zero singular values, respectively. It isclear that V(0)

j , right singular matrix corresponding to zerosingular values, forms an orthogonal basis for the null spaceof Hj . Thus, V(0)

j becomes the right choice for the precodingmatrix Fj and hence (2) becomes

yj [n] = HjFjxj [n] + wj [n]. (5)

In order to maximize the achievable sum rate of the BD, thewaterfilling algorithm can be additionally incorporated [13].Since HF has a block diagonal structure, it is possible toperform the SVD for each user’s block matrix HjFj (j =

1, · · · , K). Denoting this effective channel matrix as Heff,j =HjFj = HjV

(0)j , the SVD of Heff,j becomes

Heff,j = Uj [Σj 0][V(1)j V(0)

j ]H . (6)

Since each column of V(1)j corresponds to the singular vector

of nonzero singular value in Σj , by multiplying V(1)j into the

right side of Heff,j , we obtain Heff,jV(1)j = UjΣj . Further,

after identifying and removing Uj1, we can find a power

loading matrix Dj maximizing the information rate of theuser j. In summary, the achievable sum rate of the BD-WF is[9]

RBD-WF = maxD:Tr(D)≤PT

log det(I +

Σ2Dσ2

w

)(7)

where D = diag(D1, · · · ,DK) is determined by the water-filling under the power constraint PT .

C. BD-VP Algorithm

Although the BD-WF is optimal in maximizing the sum rateof the BD algorithm, delivery of precoding matrix index orassignment of dedicated pilot for each user to estimate the ef-fective channel Heff,j might be a bit burdensome. The BD-VP,introduced to avoid this load, employs the vector perturbationtechnique [14] [15] on top of the BD to compensate for thetransmit power suppression caused by the BD. Specifically,the transmitted signal of the user j is

xj [n] =1√γj

H−1eff,j sj [n] =

1√γj

H−1eff,j(sj [n] + τpj [n]) (8)

where sj [n] = sj[n]+τpj [n] where sj [n] and τ are the trans-mit signal vector of the user j and the pre-defined positive realnumber, respectively, and γj = ‖H−1

eff,j(sj [n] + τpj [n])‖2.2

Note that a choice of pj [n] minimizes γj so that

pj [n] = arg minp

′j [n]∈Z

NR,j

∥∥∥H−1eff,j(sj [n] + τp

′j [n])

∥∥∥2

(9)

where Z is the integer set and NR,j is the number of transmitstreams for each user. Since the perturbation vector pj [n]is chosen from the set of NR,j-dimensional lattice points,considerable reduction of γj can be achieved, which helpsto improve the effective SNR of the received signal. In fact,it is shown that the gap of the sum rate between the BD-VPand BD-WF is negligible in high SNR regime [10]. Note thatthe achievable rate of the BD-VP is given by

RBD-VP =K∑

j=1

NR,j∑k=1

log2 (1 + SNRj,k)

=K∑

j=1

NR,j∑k=1

(1 +

ρλ2k

NR,j

)(10)

1Note that Uj can be identified once the precoded effective channel Heff,jis estimated. Heff,j can be estimated from either the dedicated pilot signalso called demodulation reference signal (DM-RS) assigned for each user orcombination of the precoding matrix Fj and the estimated channel Hj .

2In principle γj should be delivered to the receiver in the BD-VP technique.However, since the performance difference between γj and Eγj is negligible[15], sporadic use of Eγj is more beneficial in practice.

PARK et al.: A MMSE VECTOR PRECODING WITH BLOCK DIAGONALIZATION FOR MULTIUSER MIMO DOWNLINK 3

Sphere Encoder

dj[n]

pj[n]

OriginalSymbol

sj[n]

τ{-1,0,1}

Aj Fj H

wj[n]

Heff, j

xj[n]

Transmitter Receiver

sj[n]Modulo

dj[n] gj Bj

H

Fig. 1. The transceiver structure of the proposed BD-MVP technique.

where ρ = PT

σ2w

and λk (k = 1, · · · , NR,j) is the singular valueof Heff,j .

In the implementation perspective, the integer least squaresproblem in (9) is processed at the transmitter and solved bya closest lattice point search, referred to as sphere encoder[15]. In the receiver, the received signal vector of the user jis modeled as

yj [n] =1√γj

sj [n] + wj [n]

=1√γj

(sj [n] + τpj [n]) + wj [n]. (11)

In removing τpj [n], we can employ the modulo operation andhence mod(

√γjyj [n], τ ) becomes a sufficient statistic for the

symbol slicing.

III. BD-MVP

In this section, we present the proposed BD-MVP algo-rithm that combines the BD and MMSE-VP. While the mainfeature of the BD-VP is to annihilate inter-stream interferencecompletely, the aim of the proposed BD-MVP is to allow asmall inter-stream interference for optimal balance betweenthe residual interference mitigation and noise enhancementsuppression. We first provide the motivation of our work andthen discuss details of the proposed algorithm.

A. The BD-MVP

Although the BD-VP outperforms the BD, due to the zeroforcing nature of the algorithm, mobile suffers from the noiseenhancement effect. An idea motivated by this observation isto reduce the noise enhancement effect by employing the QRdecomposition based BD together with the MMSE-VP [11].Towards this end, we pursue a tradeoff between the residualinterference mitigation and noise enhancement suppressionin the MMSE sense. In a nutshell, key distinctions of theproposed method over the BD-VP are that 1) elimination ofmultiuser interference by the QR decomposition based BDand 2) per user optimization of perturbation vector using theMMSE criterion.

First, as a computationally efficient alternative of the SVDbased BD, the proposed BD-MVP exploits the QR decomposi-tion of the pseudo inverse of the composite channel matrix H[16]. To be specific, while the original BD employs the SVDof Hj to find out the null space of Hj , the BD-MVP exploitsthe pseudo inverse of H, i.e., H = HH(HHH)−1. DenotingH = [H1 H2 · · · HK ], we perform the QR decompositionfor each submatrix of H (i.e., Hj = QjRj). Then, it is clear

that Qj forms an orthogonal basis for the null space of Hj

(see Appendix A).Second, the MMSE-VP is employed to find the optimal

perturbation vector and the power constraint factor. Denotingthe desired symbol as dj [n] and the received symbol as dj [n](see Fig. 1), the total MSE for whole block can be expressedas

ε(pj [n], xj [n], gj)

=

NBXn=1

E“‖dj [n] − dj [n]‖2

”

=

NBXn=1

E“‖gjB

Hj (Heff,jxj [n] + wj [n]) − dj [n]‖2

”

=

NBXn=1

E((gjBHj (Heff,jxj [n] + wj [n]) − dj [n])H

× (gjBHj (Heff,jxj [n] + wj [n]) − dj [n]))

=

NBXn=1

(g2j x

Hj [n]HH

eff,jHeff,jxj [n] − gjdHj [n]BH

j Heff,j

× xj [n] + g2j tr(Rn) − gjx

Hj [n]HH

eff,jBjdj [n]

+ dHj [n]dj [n]) (12)

where BHj is the receive equalization matrix for the user

j. Note that when BHj = INR,j , this joint optimization

problem turns into the transmitter optimization problem. SinceHeff,j ,dj [n], and tr(Rn) are given, our goal is to find pj [n],xj [n], and gj minimizing the MSE under the power constraint.The corresponding optimization problem can be expressed as

(pj [n],xj [n], gj) = argmin ε(pj [n],xj [n], gj)

subject toNB∑n=1

xHj [n]xj [n] = PT . (13)

This problem can be solved by Lagrangian multiplier [17].One can show that the resulting transmit signal xj [n] is givenby (see Appendix B)

xj [n] = Ajdj [n]

=1gj

HHeff,j(Heff,jHH

eff,j + ξI)−1Bjdj [n]. (14)

Further, using the matrix inversion lemma, xj [n] = 1gj

(HHeff,j

Heff,j + ξI)−1HHeff,jBjdj [n] and hence the power constraint

factor gj is given by

gj =

√√√√ 1PT

NB∑n=1

dHj [n]BH

j Heff,j(H†eff,j)−2HH

eff,jBjdj [n] (15)


where H†eff,j = Heff,jHH

eff,j + ξI, ξ = NBtr(Rn)/PT . Oncexj [n] and gj are available, what remains is to find the optimalperturbation vector pj [n]. By plugging xj [n] and gj into (12),one can show that (see Appendix C)

pj [n] = arg minp

′j [n]

‖LjBj(sj [n] + τp′j [n])‖2 (16)

where we employ the Cholesky factorization (Heff,jHHeff,j +

ξI)−1 = LHj Lj (Lj is the lower triangular matrix). Similar to

the BD-VP, the perturbation vector pj [n] can be found by thesphere encoder.

In the receiver, the signal vector yj [n] is

yj [n] = Heff,jxj [n] + wj [n]

=1gj

Heff,jHHeff,j(H

†eff,j)

−1Bj(sj [n] + τpj [n])

+ wj [n]

≈ 1gj

Bj(sj [n] + τpj [n]) + w′j [n]. (17)

Note that w′j contains the residual interference introduced by

the regularization factor ξI as well as the Gaussian noisewj . Since τ is known to the receiver, the effect of τpj [n]can be removed using a simple matrix multiplication (gjBH

j )followed by the modulo operation. The proposed algorithm issummarized in Appendix D.

B. BD-MVP With Geometric Mean Decomposition

In this subsection, we consider the optimization problem toobtain unitary receive equalization matrix BH

j . The geometricmean decomposition (GMD) [18] [19], developed for thispurpose, partitions the channel matrix Heff,j such that

Heff,j = QjRjPHj (18)

where Qj and Pj are unitary matrices and Rj is a real uppertriangular matrix with identical diagonal elements. We canchoose the unitary matrix Bj = Qj to equalize the effectivechannel of the user j. Then, the transmit signal and powerconstraint factor are given by

xj [n] =1gj

PjRHj

(RjRH

j + ξI)−1

dj [n], (19)

gj =

√√√√ 1PT

NB∑n=1

‖RHj (RjRH

j + ξI)−1dj [n]‖2, (20)

and the total MSE becomes

ε = ξ

NB∑n=1

dHj [n]

(RjRH

j + ξI)−1

dj [n]. (21)

Using the Cholesky factorization(RjRH

j + ξI)−1 = TH

j Tj ,(21) can be simplified to

ε = ξ

NB∑n=1

‖Tj (sj[n] + τpj [n]) ‖2. (22)

In order to minimize the total MSE, each independent termshould be minimized and hence pj [n] is chosen as

pj [n] = arg minp

′j [n]

‖Tj(sj [n] + τp′j [n])‖2. (23)

C. The Sum Rate of the BD-MVP

The sum rate of the proposed method is equivalent to thesum of single user rate by virtue of the fact that the precodingmatrix Fj annihilates multiuser interference. Recalling thatthe transmit and received signal vectors of the user j arexj [n] = 1√

gjHH

eff,j(Heff,jHHeff,j + ξI)−1Bjdj [n] and yj [n] =

Heff,jxj [n]+wj [n], respectively, and also using the eigenvaluedecomposition Heff,jHH

eff,j = QΛQH , Heff,jxj [n] becomes

Heff,jxj [n] =1√gj

QΦQBjdj [n] (24)

where Q is the unitary matrix and Φ is the diagonal matrixwhose jth element is λj

λj+ξ (λj is the jth diagonal element ofΛ).

In (24), the kth entry of the user j is

[Heff,jxj [n]]k =1√gj

»qk,1

λ1

λ1 + ξ· · · qk,NR,j

λNR,j

λNR,j + ξ

–

×

2664

qH1,1 · · · qH

NR,j ,1

... · · ·...

qH1,NR,j

· · · qHNR,j ,NR,j

3775

264

cj,1[n]...

cj,NR,j [n]

375 (25)

where qm,n is the (m, n)th entry of Q and cj,i[n]is the ith entry of Bjdj [n]. In (25), we see that

1√gj

(∑NR,j

l=1λl

λl+ξ |qk,l|2)cj,k[n] is the desired term and all

of the rest (containing cj,l[n], l �= k) are interferences. In thisperspective, kth received signal for the user j can be expressedas

yj,k[n] =1√gj

⎛⎝NR,j∑

l=1

λl

λl + ξ|qk,l|2

⎞⎠ cj,k[n] + zj,k[n] (26)

where zj,k[n] = 1√gj

∑NR,j

m=1,m �=k

(∑NR,j

l=1λl

λl+ξ qk,lqHm,l

)× cj,m[n]+wj,k[n] contains the inter-stream interference andadditive noise wj,k. Therefore, the SINR of each stream ofthe user j becomes

SINRj,k =NB∑n=1

‖scj,k[n]‖2

‖tcj,m[n]‖2 + ‖wj,k[n]‖2(27)

where s = 1√gj

(∑NR,j

l=1λl

λl+ξ |qk,l|2)

and t = 1√gj∑NR,j

m=1,m �=k

(∑NR,j

l=1λl

λl+ξ qk,lqHm,l

)and the corresponding

sum rate becomes

RBD-MVP =K∑

j=1

NR,j∑k=1

log2 (1 + SINRj,k) . (28)

D. MSE Performance Between the BD-VP and BD-MVP

In this subsection, we provide the MSE analysis of the BD-VP and BD-MVP algorithms. Since the optimization problemof the conventional VP can be interpreted as a zero forcing(ξ = 0) MMSE-VP problem [11], we can compare theMSE of two schemes. Moreover, due to the fact that theminimum MSE of the BD-VP and BD-MVP equals γj andgj , respectively, the MSE of an instantaneous block for both


TABLE I

COMPLEXITY COMPARISON BETWEEN THE BD-VP AND BD-MVP

Operations BD-VP BD-MVPAnnihilation of multiuser interference BD is done by the SVD BD is achieved by the QR decomposition

CSVD = K`4N2

T NR + 13(NR − NR,j)3

´CQR = 11

3N3

T + 53N2

T + KN2R,j(NT − 1

3NR,j)

Computation of perturbation vector SE employing zero forcing criterion SE exploiting MMSE criterionwith complexity CVP-SE with complexity CMVP-SE (≈ CVP-SE)

Receiver equalizer operation - BHj multiplication is required per each user

Modulo operation Remove τpj to map the received signal Same as BD-VPinto the original vector

γj and gj can be compared without loss of generality. First,γj of the BD-VP algorithm is expressed as

γj � ‖HHeff,j(Heff,jHH

eff,j)−1(sj + τpj)‖2

= (sj + τpj)H(Heff,jHHeff,j)

−1(sj + τpj)

= (sj + τpj)HQΛ−1QH(sj + τpj)

=NR,j∑l=1

1λj

|qHl (sj + τpj)|2. (29)

In a similar way, gj of the BD-MVP scheme is

gj � ‖Lj(sj + τpj)‖2

= (sj + τpj)H(Heff,jHHeff,j + ξI)−1(sj + τpj)

= (sj + τpj)HQ (Λ + ξI)−1 QH(sj + τpj)

=NR,j∑l=1

1λj + ξ

|qHl (sj + τpj)|2. (30)

Due to the inclusion of ξ(> 0) in the denominator, it is clearthat

γj − gj > 0. (31)

Note that since the total MSE of the BD-VP is the sum ofeach individual block MSE (MSEBD-VP =

∑Kj=1 γj) and the

same is true for the BD-MVP (MSEBD-MVP =∑K

j=1 gj), weconclude that the BD-MVP outperforms the BD-VP in theMSE sense. Note also that we only consider the case whereBH

j = INR,j in gj computation so that further improvementin the MSE can be achieved using the GMD based receiveequalization.

E. Comments on Complexity

In this subsection, we discuss the complexity of the BD-MVP and BD-VP. Since the receiver operation of the BD-VPand BD-MVP (the modulo operation and receive equalization)is very simple to analyze and in fact trivial, we focus onlyon the computations associated with transmitter precoding. Inour analysis, we measure the complexity of the algorithm bythe number of floating point operations (flops). The majoroperations of the proposed method and BD-VP include 1)annihilation of multiuser interference using Fj and 2) per useroptimization of perturbation vector pj .

The computational complexity of the SVD for the BD-VPCSVD and QR decomposition for the BD-MVP CQR are [20]

• CSVD : K × (4N2

T NR + 13(NR − NR,j)3).

• CQR : 113 N3

T + 53N2

T for channel inversion and K×N2R,j×

(NT − 13NR,j) for orthogonalization.

0 5 10 15 20 25 300

5

10

15

20

25

30

35

40

SNR(dB)

Sum

Rat

e (b

its/s

ec/H

z)

DPCBDBD−WFBD−VP [10]BD−GMD [21]BD−UCD [21]BD−MVP

Fig. 2. Achievable sum rates of DPC, conventional BD algorithms (BD,BD-WF and BD-VP), THP based schemes (BD-GMD and BD-UCD), andthe proposed BD-MVP.

Comparing to the SVD based BD, the QR based BD hassmaller computational complexity since the QR based orthog-onalization of NT ×NR,j matrix Hj is simpler than the SVDoperation on (NR −NR,j)×NT matrix Hj . For example, therequired flops of the BD-VP and BD-MVP for the {4, 4}× 8system are 2688 and 2137, respectively. In fact, computationalbenefit of the BD-MVP over the BD-VP grows with NT andK mainly because the SVD is operated in a higher dimension.

Next, we investigate the complexity of perturbation vectorcomputation. Due to the fact that the vector precoding isimplemented using the sphere encoding, the computationalcomplexity associated with this operation is nondeterministic[21]. The lower bound on the complexity of the sphereencoding CSE is given by [22]

CSE ≥ βνNs − 1√β − 1

(32)

where β is the modulation order, Ns is the number of searchdimension, and ν is the complexity component given by ν =12

(1 + 4(β−1)

3ω2 SNR)

where ω is a constant. Noting that thedominant factor of CSE is Ns and both schemes have the sameNs, we expect that the complexity of the BD-VP and BD-MVPwould be more or less similar. We summarize the complexityanalysis of this subsection in Table I.


9 12 15 18 21 24 27

10−5

10−4

10−3

10−2

10−1

ρ (dB)

Unc

oded

BER

BD−VP [10]BD−GMD [21]BD−UCD [21]BD−MVPBD−MVP w/ GMD

Fig. 3. Uncoded BER performance for {4, 4}×8 multiuser MIMO systems.

IV. SIMULATION RESULTS AND DISCUSSIONS

In this section, we compare the performance of the proposedBD-MVP method with the BD, BD-VP, BD-WF, as wellas THP based schemes (BD-GMD and BD-UCD). Whilethe BD-GMD generates subchannels with identical SNRsusing zero forcing based the GMD, the BD-UCD employsMMSE extension of uniform channel decomposition (UCD)for obtaining further gain in performance (readers are referredto [12] for more details).

The simulation setup is based on the 16-QAM transmission.We assume that elements of each user’s channel matrix arei.i.d. zero mean complex Gaussian random variables with unitvariance, which almost surely ensures that each user’s channelmatrix has full rank (rank(Hj) = min(NR,j , NT )). In order toobserve the comprehensive picture of theoretical and practicalperformance, we measure the sum rate and bit error rate (BER)(both uncoded and coded BER), respectively.

In Fig. 2, we plot the sum rate for the {2, 2}× 4 system asa function of SNR. Note that the achievable rate of the DPCis [3]–[5]

RDPC = supD∈A

log |I + HHDH| (33)

where A is the set of nonnegative diagonal matrices Dsatisfying tr(D) ≤ PT . Generally, we observe that the sumrate of the BD-UCD, BD-GMD, BD-WF, and BD-MVP is notmuch different from the sum rate of the DPC. In particular,the sum rate of the BD-MVP becomes close to the sum rateof the DPC as the SNR increases. However, the BD algorithmleaves nonnegligible gap to the sum rate of the DPC.

In Fig. 3, we plot the uncoded BER for the {4, 4} × 8systems. We observe that the BD-MVP outperforms the BD-VP and BD-UCD resulting in about 0.8 dB and 1.5 dB gain atBER = 10−3, respectively. We also observe that the BD-MVPwith receive equalization provides additional gain (around 0.3dB at BER = 10−3) over the BD without equalization.

In order to assess the computational complexity of al-gorithms under test, we examine the running time of eachscheme using same set of symbols under same environments.The program is coded with the Matlab program and the

6 7 8 9 10 11 12 13 14 1510

−6

10−5

10−4

10−3

10−2

10−1

100

ρ (SNR)

Cod

ed B

ER

BD−UCD [21]BD−VP [10]BD−MVP

Fig. 4. Coded BER performance for {4, 4} × 8 systems with 16-QAMmodulation (Turbo encoding with rate r = 1/2).

results are obtained as an average time (in seconds) for 104

symbols running under a duo-core processor with Windows7 environment. As summarized in Table II, the BD-GMDachieves the lowest complexity among all candidates by virtueof the fact that small number of computations is needed forthe THP modulo operation. Due to the considerable numberof computations associated with the sphere encoding, we seethat the running time of the BD-MVP and BD-VP is 3.3 and3.9 times higher than that of the BD-GMD. Since number ofiterations for performing waterfilling is substantial, BD-UCDachieves the highest computational complexity.

Finally, in order to observe if this gain can be transferred tothe gain after decoding, we check the coded BER performance.Fig. 4 illustrates the coded BER performance of the proposedmethod with half-rate (r = 1/2) turbo encoded data. Thestandard Gray mapping is used to map the information bitsto 16-QAM symbols. As an encoder, we use a universalmobile telecommunications systems (UMTS) standard withfeedforward polynomial 1+D+D3 and feedback polynomial1+D2+D3 [23]. The size of codebook is set to 4000. Similarto [15], y′

j,re[n] = yj,re[n] − (τ/gj)pj,re[n] is employed tocompute the likelihood function

p(y′j,re[n]|sj,re) =

∞Xm=−∞

1√2πσ2

× exp {−(y′j,re[n] − (sj,re − mτ )/gj)

2/2σ2}y′

j,re[n] ∈»− τ

2gj,

τ

2gj

«. (34)

The imaginary part is handled in the same way.In our simulation, we replace the infinite sum in (34) with

a sum of a few terms on both sides of m = 0 since theperformance loss over the setup using infinite sum is negligi-ble. From the Fig. 4, we observe that the proposed methodoutperforms the BD-VP and BD-UCD. In particular, gainof the BD-MVP over the BD-VP and BD-UCD at waterfallregion is around 2 dB and 4 dB, respectively. Since theperformance requirement of coded system typically lies on thewaterfall regime, the proposed method can bring a substantial


TABLE II

TIME COMPLEXITY FOR {4, 4} × 8 SYSTEM (RUNNING TIME FOR 104

SYMBOLS)

Algorithm BD-MVP BD-VP BD-GMD BD-UCDRunning time (sec) 4.0531 4.8308 1.2125 19.2539

reduction on the transmitter power over the BD-UCD andBD-VP. Clearly, we conclude that the proposed scheme iscompetitive in both practical and theoretical perspectives.

V. CONCLUSION

In this paper, we investigated an approach based on blockdiagonalization (BD) achieving robustness of multiuser MIMOdownlink. Motivated by the fact that mobile users suffer fromnoise enhancement effect due to the zero forcing nature ofthe BD algorithm, we combined the BD with the MMSE-VP to achieve the balance between residual interferencemitigation and noise enhancement suppression. We couldobserve from the sum rate analysis as well as practical BERperformance (both coded and uncoded) that the proposedBD-MVP approach uniformly outperforms the BD algorithmand its variants. Future study needs to be directed towardsthe investigation of the performance when the channel stateinformation at the transmitter (CSIT) is imperfect.

APPENDIX IQR DECOMPOSITION BASED BD

The pseudo inverse of NR,j×NT composite channel matrixH =

[HT

1 HT2 · · · HT

K

]Tis H = HH

(HHH

)−1 =[H1 H2 · · · HK

]where Hj is NT × NR,j matrix. Then,

one can show that

HH =

⎡⎢⎣

H1

...HK

⎤⎥⎦ [

H1 · · · HK

]

(35)

=

⎡⎢⎣

H1H1 · · · H1HK

...HKH1 · · · HKHK

⎤⎥⎦

=

⎡⎢⎣

INR,1 0. . .

0 INR,K

⎤⎥⎦ = INR .

(36)

Clearly, HjHk = 0 when j �= k. By defining Hj =[HT

1 · · · HTj−1 HT

j+1 · · · HTK

]T, the zero inter-user inter-

ference constraint is satisfied as HjHj = 0. Now considerthe QR decomposition of Hj

Hj = QjRj for j = 1, · · · , K (37)

where Rj is NR,j × NR,j upper triangular matrix and Qj isNT ×NR,j matrix whose columns form an orthonormal basisfor Hj . From the zero inter-user interference constraint, wehave HjQjRj = 0. Since Rj is invertible, it follows thatHjQj = 0.

APPENDIX IIPROOF OF JOINT OPTIMIZATION PROBLEM

The Lagrangian function to solve the joint optimizationproblem is

f(pj [n],xj [n], gj , λ)

= ε(pj [n],xj [n], gj) + λ(NB∑n=1

xHj [n]xj [n] − PT ). (38)

By equating the derivatives of f(·) with respect to xj [n], gjand λ to zero, we can obtain the following equation

∂f(·)∂xj [n]

= g2j x

Hj [n]HH

eff,jHeff,j − gjdHj [n]BH

j Heff,j

+ λxHj [n] = 0, (39)

∂f(·)∂gj

=

NBXn=1

{2gjxHj [n]HH

eff,jHeff,jxj [n] − dHj [n]BH

j

× Heff,jxj [n] + 2gj tr(Rn) − xHj [n]HH

eff,jBj

× dj [n]} = 0, (40)

∂f(·)∂λ

=

NBXn=1

xHj [n]xj [n] − PT = 0. (41)

Using the matrix inversion lemma, (39) becomes

xj [n] =1gj

(HHeff,jHeff,j +

λ

g2j

I)−1HHeff,jBjdj [n]

=1gj

HHeff,j(Heff,jHH

eff,j +λ

g2j

I)−1Bjdj [n]. (42)

By multiplying xj [n]gj

at the right side of (39), we have

dHj [n]BH

j Heff,jxj [n]

= gjxHj [n]HH

eff,jHeff,jxj [n] +λ

gjxH

j [n]xj [n]. (43)

Plugging (43) into (40), we have

NB∑n=1

( − 2λ

gjxH

j [n]xj [n] + 2gjtr(Rn))

= 0. (44)

After some manipulation, we have

λ

g2j

=tr(Rn)

1NB

∑NB

n=1 xHj [n]xj [n]

=NBtr(Rn)

PT. (45)

Let ξ = NBtr(Rn)PT

, then (42) can be rewritten as

xj [n] =1gj

(HHeff,jHeff,j + ξI)−1HH

eff,jBjdj [n]

=1gj

HHeff,j(Heff,jHH

eff,j + ξI)−1Bjdj [n]. (46)


APPENDIX IIIPROOF OF (16)

Plugging xj [n] in (14) and gj in (15) into (12) and aftersome manipulations, we have

ε(pj [n]˛xj [n], gj)

=

NBXn=1

(g2j x

Hj [n]HH

eff,jHeff,jxj [n] − gjdHj [n]BH

j Heff,jxj [n]

+ g2j tr(Rn) − gjx

Hj [n]HH

eff,jBjdj [n] + dHj [n]dj [n])

=

NBXn=1

(dHj [n]BH

j (Heff,jHHeff,j + ξI)−1Heff,jH

Heff,jHeff,jH

Heff,j

× (Heff,jHHeff,j + ξI)−1Bjdj [n] + dH

j [n]dj [n]

− dHj [n]BH

j Heff,jHHeff,j(Heff,jH

Heff,j + ξI)−1Bjdj [n]

− dHj [n]BH


Heff,jBjdj [n]

+ ξdHj [n]BH


Heff,j(Heff,j

× HHeff,j + ξI)−1Bjdj [n])

=

NBXn=1

dHj [n]QH

j ((Heff,jHHeff,j + ξI)−1Heff,jH

Heff,jHeff,j

× HHeff,j(Heff,jH

Heff,j + ξI)−1 + I − Heff,jH

Heff,j

× (Heff,jHHeff,j + ξI)−1 − (Heff,jH

Heff,j + ξI)−1Heff,j

× HHeff,j + ξ(Heff,jH

Heff,j + ξI)−1Heff,jH

Heff,j(Heff,j

× HHeff,j + ξI)−1)Bjdj [n]

=

NBXn=1

dHj [n]BH


Heff,jHeff,jH

Heff,j

× (Heff,jHHeff,j + ξI)−1 − I + I −Heff,jH

Heff,j

× (Heff,jHHeff,j + ξI)−1 + I− (Heff,jH

Heff,j + ξI)−1Heff,j

× HHeff,j + ξ(Heff,jH

Heff,j + ξI)−1Heff,jH

Heff,j(Heff,jH

Heff,j

+ ξI)−1)Bjdj [n]

=

NBXn=1

dHj [n]BH


Heff,jHeff,jH

Heff,j

× (Heff,jHHeff,j + ξI)−1 − I + 2ξ(Heff,jH

Heff,j + ξI)−1

+ ξ(Heff,jHHeff,j + ξI)−1Heff,jH

Heff,j(Heff,jH

Heff,j

+ ξI)−1)Bjdj [n]

=

NBXn=1

dHj [n]BH

j (ξ(Heff,jHHeff,j + ξI)−1)Bjdj [n], (47)

where I− (Heff,jHHeff,j + ξI)−1Heff,jHH

eff,j = ξ(Heff,jHHeff,j +

ξI)−1 follows from the matrix inversion lemma. Employingthe Cholesky factorization (Heff,jHH

eff,j + ξI)−1 = LHj Lj and

also noting that dj [n] = sj [n] + τpj [n], (47) becomes

ε(pj [n]∣∣xj [n], gj) = ξ

NB∑n=1

dHj [n]BH

j LHj LjBjdj [n]

= ξ

NB∑n=1

‖LjBj(sj [n] + τpj [n])‖2. (48)

In summary, the optimization problem to find the perturbationvector pj is expressed as

pj [n] = arg minp

′j [n]

‖LjBj(sj [n] + τp′j [n])‖2. (49)

APPENDIX IVSUMMARY OF THE BD-MVP ALGORITHM

• Input : H, s[n]• Output : p[n]• Variable

1) j denotes the jth user being examined2) n denotes the nth block being examined

Step 1 : (QR based BD step)In order to compute the nullspace of Hj , we define

the pseudo-inverse of the channel matrix H as H =HH(HHH)−1 = [H1 · · · Hj · · · HK ]

for j = 1 : K

Hj = QjRj

Fj = Qj

Heff,j = HjFj

endStep 2 : (MMSE-VP step) for j = 1 : K

factorize (Heff,jHHeff,j + ξInj )−1 = LH

j Lj

for n = 1, . . . , NB

pj [n] = argminp′j[n] ‖LjBj(sj [n] + τp

′j [n])‖2

xj [n] = HHeff,j(Heff,jHH

eff,j + ξInj )−1Bjdj [n]= HH

eff,j(Heff,jHHeff,j + ξInj )−1Bj(sj [n] + τpj [n])

endgj =

√1

PT

∑NB

n=1 dHj [n]BH

j Heff,j(H†eff,j)−2HH

eff,jBjdj [n]for n = 1, . . . , NB

xj [n] = 1gj

xj [n]end

end

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Jungyong Park (S’09) received the B.S. degree ininformation and communication engineering fromDongguk University, Seoul, Korea in 2008, and theM.S. degree from the School of Information andCommunication, Korea University, Seoul, Korea, in2010, where he is currently working toward thePh.D. degree. His research interests include wirelesscommunications and information theory.

Byungju Lee (S’09) received the B.S. and M.S.degrees in the School of Information and Commu-nication, Korea University, Seoul, Korea, in 2008and 2010, where he is currently working toward thePh.D. degree. His research interests include wirelesscommunications and information theory.

Byonghyo Shim (SM’09) received the B.S. andM.S. degrees in control and instrumentation engi-neering (currently electrical engineering) from SeoulNational University, Korea, in 1995 and 1997, re-spectively, and the M.S. degree in mathematics andthe Ph.D. degree in electrical and computer engi-neering from the University of Illinois at Urbana-Champaign, in 2004 and 2005, respectively.

From 1997 and 2000, he was with the Departmentof Electronics Engineering at the Korean Air ForceAcademy as an Officer (First Lieutenant) and an

Academic Full-time Instructor. From 2005 to 2007, he was with QualcommInc., San Diego, CA, working on CDMA systems with the emphasis on next-generation UMTS receiver design. Since September 2007, he has been withthe School of Information and Communication, Korea University, where he isnow an Associate Professor. His research interests include signal processingfor wireless communications, compressive sensing, applied linear algebra, andinformation theory.

Dr. Shim was the recipient of the 2005 M. E. Van Valkenburg researchaward from the ECE department of the University of Illinois and the 2010Haedong award from IEEK.

[22]a mmse vector precoding with bd for mu mimo dl

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