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Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2012, Article ID 357381, 13 pagesdoi:10.1155/2012/357381

Research Article

SVD-Aided Beamforming and Power Allocation Algorithm forMultiuser Turbo-BLAST System Uplink with Imperfect CSI

Xiaomin Chen,1 Xiangbin Yu,1, 2 and Dazhuan Xu1

1 College of Information Science and Technology, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China2 National Mobile Communications Research Laboratory, Southeast University, Nanjing 210096, China

Correspondence should be addressed to Xiaomin Chen, [email protected]

Received 1 September 2011; Revised 10 December 2011; Accepted 14 January 2012

Academic Editor: Ananda Mohan

Copyright © 2012 Xiaomin Chen 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 SVD-aided joint transmitter and receiver design for the uplink of CDMA-based synchronous multiuser Turbo-BLAST systemsis proposed in the presence of channel state information (CSI) imperfection. At the transmitter, the beamforming and powerallocation schemes are developed to maximize the capacity of the desired user. At the receiver, a suboptimal decorrelating schemeis first proposed to mitigate the multiuser interference (MUI) and decouple the detection of different users with imperfectCSI, and then the iterative detecting algorithm that takes the channel estimation error into account is designed to cancel thecoantenna interference (CAI) and enhance the bit error rate (BER) results further. Simulation results show that the proposed uplinkCDMA-based multiuser Turbo-BLAST model is effective, the detection from every user is completely independent to each otherafter decorrelating, and the system performance can be enhanced by the proposed beamforming and power allocation schemes.Furthermore, BER performance can be enhanced by the modified iterative detection. The effect of CSI imperfection is evaluated,which is proved to be a useful tool to assess the system performance with imperfect CSI.

1. Introduction

Multiple-input multiple-output (MIMO) communicationsystems provide a significant capacity increment over theconventional one through appropriate space-time processing[1, 2]. Bell-Labs Layered Space-Time (BLAST) has shownto be a promising MIMO communication technique thatcan achieve tremendous bandwidth efficiencies [3]. Thecombination of BLAST and turbo-decoding principle iscalled Turbo-BLAST [4, 5], where the coantenna interference(CAI) caused by the major source of channel impairmentcan be removed by a serially concatenated iterative decodingalgorithm.

Although Turbo-BLAST is considered only for single-user application, we show herein that it naturally extends toa multiuser code division multiple access (CDMA) scenario[6], which is called CDMA-based multiuser Turbo-BLASTsystem, where a unique signature code is assigned to eachother and used to identify the user and to spread thespectrum of the user’s data. Since all the users “simulta-neously” occupy the same spectrum, they create multiuser

interference (MUI) to one another because of the nonzerocross correlation of their signatures [7, 8]. The MUI notonly limits the system capacity but also increases the systemerror performance. One of the most promising approachesto meet these challenges is the use of adaptive beamformingand power control [9–13].

In the uplink (UL) of CDMA-based multiuser Turbo-BLAST systems, both the multiple MUI and CAI have tobe mitigated. Adaptive interference suppression techniquesbased on multiuser detection (MUD) and antenna array pro-cessing have recently been considered as powerful methodsfor increasing the quality, capacity, and coverage of thesesystems. However, the method of optimal MUD has a veryhigh computation cost due to its nonlinear nature [14–16].

Moreover, the performance of multiuser Turbo-BLASTsystem is closely related to the channel state information(CSI). Practically, CSI at the receiver is subject to the errorperformance because of the non-real-time data processing,quantization error, and imperfect channel estimations, andso forth [17]. This gives rise to significant challenges tosystem design and analysis.

2 International Journal of Antennas and Propagation

Input datastream ofuth user Encoder

uModulator

uS/P

convertor u

Powerallocator u

Beamformeru

Spreader u

Spreader u

···

···

···

···

···

P(u) V (u)

1

nT

Figure 1: Transmitter structure of the uth mobile station.

Regarding these problems, the SVD-assisted beamform-ing and power allocation schemes are first developed tomaximize the capacity of every active user, and a suboptimaldecorrelating scheme for the uplink of multiuser Turbo-BLAST system is proposed to mitigate the MUI, then thedetection is performed independently for different usersafter decorrelating. Based on this, an improved iterativedetection algorithm is adopted to cancel the CAI. We assumethat the spreading structures of all the users are perfectlyknown to both the transmitter and the receiver at theinstant of transmission and reception, and the channelestimation is imperfect, only the channel estimation matrixand the statistical characteristic of channel estimation errorare known to the system. Besides, in this paper, the commonassumption that true channel matrix and channel estimationerror are complex Gaussian is employed, and a CDMAFrequency Division Duplexing (FDD) system is considered.

Based on the above analysis, in this paper, we derive abeamforming and power allocation algorithm for multiuserTurbo-BLAST system in the presence of imperfect CSI,which avoids that most of the existing schemes are based onperfect CSI and designed for single-user case. Moreover, byusing the obtained beamforming and power allocation, animproved iterative detection scheme is also presented. Withthis detection scheme, the system performance is improvedfurther. Simulation results verify the validity of the presentedschemes.

This paper is organized as follows: after the introductionin Section 1, the basic system is briefly described in Section 2.Section 3 presents an equivalent model for multiuser Turbo-BLAST system, the SVD-aided beamforming and powerallocation scheme, and a suboptimal decorrelating algo-rithm, as well as the iterative detecting method which aredescribed respectively. The simulation results are analyzedin Section 4, and Section 5 makes the conclusion of thepaper.

2. Channel Model and Basic System Description

2.1. Transmitter Model. We consider an uplink (UL) ofsynchronous CDMA-based multiuser Turbo-BLAST systemwith U active users operating in a frequency-nonselectiveRayleigh fading channel, that is, a narrowband multiuserTurbo-BLAST system is considered, where CDMA is mainlyemployed as a multiple access technique to differentiatedifferent users and the corresponding spreading gain isnot considered. At the transmitter, a data stream of eachuser is first encoded, mapped into a symbol stream thatis subsequently demultiplexed into nT substreams, andthen preprocessed by the right singular matrix, of channel

estimation matrix, where each stream is allocated withproper power, finally the preprocessed streams are spread andtransmitted by the MS each of which employs nT transmitantennas. The schematic transmit structure of the uth(u = 1, 2, . . . ,U) mobile station (MS) is shown in Figure 1,where V(u) represents the transmitter’s beamforming matrixformulated for the transmission of the uth MS’s data andP(u) is the power allocation matrix for the uth MS under theconstrain of total transmit power.

2.2. Imperfect Channel Model and Singular Value Decom-position. Consider a multiuser Turbo-BLAST system wherethe base station (BS) employs nR antennas and each useris equipped with nT antennas (nT ≤ nR). We introduce animperfect channel model in this section [17]. For the uthuser, let H(u) ∈ CnR×nT denote the true complex channelmatrix and H(u) ∈ CnR×nT denote the complex channelestimation matrix. Note that the channel estimation matrixand the true channel matrix are different from one anotherfor the uplink. Based on the imperfect channel model, thetrue complex matrix H(u) can be formulated as

H(u) = H(u) + Ξ(u), (1)

where Ξ(u) ∈ CnR×nT is a complex matrix related to theimperfect CSI for the uth user. The true channel matrix H(u)

and channel estimation errors Ξ(u) are complex Gaussian,which leads the estimated channel matrix H(u) to be alsocomplex Gaussian. Hence their statistical distributions are

shown as e(u)i j ∼ CN (0, σ2

e ), h(u)i j ∼ CN (0, 1−σ2

e ), and h(u)i j ∼

CN (0, 1), which are all distributed by complex Gaussian law,and σ2

e indicates the inaccuracy of the CSI.Since the practical system cannot detect the degree of

the imperfection of the CSI, the channel estimation matrixH(u) is treated as the true channel matrix, which can bedecomposed as

H(u) = Φ(u)Λ

(u)[V(u)

]H, (2)

where (·)H denotes a matrix conjugate transpose, Φ(u)

andV(u) are the unitary matrices with left and right singular vec-

tors of H(u) as their columns, Λ(u)

is a nonnegative and diago-

nal matrix, {√λi}nTi=1 are the main diagonal elements of Λ

(u),

and {λ(u)

i }nTi=1 are eigenvalues of H(u)[H(u)]H

, respectively.

2.3. Multiuser Receiver Description. The illustration diagramof the receiver is shown in Figure 2, where the uth user isassumed as the desired user. The decorrelating scheme and

International Journal of Antennas and Propagation 3

1

nR

De-

correlator

Orthogonalconvertor

[ ^ (u)]H

MMSE

detector

P/Sconvertor

S/Pconvertor

DemodulatorSISO

channeldecoder

Transmitsymbol

estimator Outputdata

stream ofthe

expectantuser

···

···

···

···

···

· · ·

Φ

Figure 2: The receiver structure of the multiuser Turbo-BLAST system.

the iterative detection algorithm are derived based on theavailability of the spreading structure of both the desiredand interfering users, as well as the CSI of the desireduser.

At the receiver, the received signal is decorrelated andthen the signal of the desired user can be extracted afterdecorrelating through the use of the orthogonal spreadingcodes. The decoupled signal is orthogonally converted by

Φ(u)

, the left singular matrix of H(u). Finally an iterativedetection strategy based on the “turbo” principles is usedfor the symbol detection after orthogonally converting.For the iterative decoding, the optimal decoding processcan be separated into two stages: soft-input/soft-output(SISO) channel detector and SISO channel decoder, whichmutually exchange the extrinsic information sent from onestage to the other iteratively until the decoding processconverges.

Let nT data streams be transmitted by the uth MS to

the BS hosted by a vector expressed as x(u) = [x(u)1 , . . . ,

x(u)nT ]

T, u = 1, 2, . . . ,U , {s(u)(l)}Ll=1 denotes the spreading

codes of the length L assigned to the desired user, andn(l) = [n1(l), . . . ,nnT (l)]T is a column vector of additiveGaussian noise variables during the lth epoch, where eachcomponent is independent and identically distributed (i.i.d)zero-mean complex Gaussian variables with the variance ofσ2n . Under these assumptions, the baseband discrete-time

signal received by nR receive antennas at the BS can beexpressed as

y(l) =U∑u=1

H(u)V(u)P(u)x(u)s(u)(l) + n(l), (3)

where P(u) = diag(√P(u)

1 , . . . ,√P(u)nT ) is the diagonal transmit

power matrix for the desired user with total power constraint∑nTk=1 P

(u)k = Ptotal = nT .

3. Equivalent System Model and Receiver Design

3.1. Equivalent System Model. In this subsection, an equiva-lent system model is proposed for the uplink of the multiuserTurbo-BLAST system with imperfect CSI.

For a synchronous multiuser Turbo-BLAST system, at asampling instant, the received signal can be formulated as

y(l) =U∑u=1

[H(u) + Ξ(u)

]V(u)P(u)x(u)s(u)(l) + n(l)

=U∑u=1

H(u)V(u)P(u)x(u)s(u)(l)

+U∑u=1

Ξ(u)V(u)P(u)x(u)s(u)(l) + n(l)

=U∑u=1

H(u)V(u)P(u)x(u)s(u)(l) + n(l),

(4)

where

n(l) �U∑u=1

Ξ(u)V(u)P(u)x(u)s(u)(l) + n(l), (5)

n(l) is the equivalent additive noise at the lth instant,which is consisted of the interference caused by the channelestimation errors of all the interfering users and the complexGaussian noise, respectively.

The variance of the equivalent noise n(l) is calculated asfollows.

First, let G(u) = Ξ(u)V(u); the mean and variance of G(u)

can be evaluated as

ε{

G(u)}= ε

{Ξ(u)V(u)

}= ε

[Ξ(u)

]V(u) = 0, (6)

ε{

G(u)[

G(u)]H} = ε

{Ξ(u)V(u)

[V(u)

]H[Ξ(u)

]H}

= ε{Ξ(u)

[Ξ(u)

]H}= σ2

e InR .

(7)

Clearly, the components of G(u) are i.i.d zero-mean complexGaussian variables with the variance of σ2

e , thus statistical


distributions can be expressed as g(u)i j ∼ CN (0, σ2

e ) and

ε{g(u)ki [g(u′)

ki′ ]} = σ2e δ(i− i′)δ(u− u′).

Then, let f(l) = ∑Uu=1 G(u)P(u)x(u)s(u)(l); the mean and

variance of f(l) can be evaluated as

ε[f(l)] = ε

⎡⎣ U∑u=1

G(u)P(u)x(u)s(u)(l)

⎤⎦

=U∑u=1

ε[

G(u)]

P(u)x(u)s(u)(l) = 0,

(8)

f(l) =U∑u=1

G(u)P(u)x(u)s(u)(l)

=U∑u=1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

g(u)11 · · · g(u)

1k · · · g(u)1nT

.... . .

.... . .

...

g(u)k1 · · · g(u)

kk · · · g(u)knT

.... . .

.... . .

...

g(u)nR1 · · · g(u)

nRk· · · g(u)

nRnT

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

×

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

√P(u)

1 · · · 0 · · · 0

.... . .

.... . .

...

0 · · ·√P(u)k · · · 0

.... . .

.... . .

...

0 · · · 0 · · ·√P(u)nT

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

×

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

x(u)1 s(u)(l)

...

x(u)k s(u)(l)

...

x(u)nT s(u)(l)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

U∑u=1

nT∑i=1

g(u)1i

√P(u)i x(u)

i s(u)(l)

...

U∑u=1

nT∑i=1

g(u)ki

√P(u)i x(u)

i s(u)(l)

...

U∑u=1

nT∑i=1

g(u)nRi

√P(u)i x(u)

i s(u)(l)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

f1(l)

...

fk(l)

...

fnR(l)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.

(9)

Clearly, f(l) is a zero-mean complex Gaussiancolumn vector with its component fk(l) =∑U

u=1

∑nTi=1 g

(u)ki

√P(u)i x(u)

i s(u)(l), k = 1, . . . ,nR, l = 1, . . . ,LS,whose variance can be evaluated as

σ2fk

(l) = ε{[

fk(l)][fk(l)

]∗}

= ε

⎧⎨⎩⎡⎣ U∑u=1

nT∑i=1

g(u)ki

√P(u)i x(u)

i s(u)(l)

⎤⎦

×⎡⎣ U∑u′=1

nT∑i′=1

g(u′)ki′

√P(u′)i′ x(u′)

i′ s(u′)(l)

⎤⎦∗⎫⎬⎭.(10)

Since the normalized spreading codes are used, that is,

[su(l)]2 = 1/LS and ε{g(u)ki [g(u′)

ki′ ]} = σ2e δ(i− i′)δ(u− u′), (10)

can be written as

σ2fk

(l) =U∑u=1

nT∑i=1

ε

{[g(u)ki

√P(u)i x(u)

i

][g(u)ki

√P(u)i x(u)

i

]∗}

×{su(l)[su(l)]∗

}

= 1LS

U∑u=1

nT∑i=1

P(u)i ε

{g(u)ki

[g(u)ki

]∗}ε{x(u)i

[x(u)i

]∗}

= 1LS

U∑u=1

nT∑i=1

P(u)i ε

{∣∣∣g(u)ki

∣∣∣2}ε{∣∣∣x(u)

ki

∣∣∣2}

= σ2e

LS

U∑u=1

nT∑i=1

P(u)i = UnTσ2

e

LS.

(11)

Because n(l) = f(l) + n(l), the component of n(l) is given asnk(l) = fk(l) + nk(l), k = 1, . . . ,nR by the aforementioneddefinition with the variance calculated as

σ2nk

(l) = ε[nk(l)n∗k (l)

]= ε

{[fk(l) + nk(l)

][fk(l) + nk(l)

]∗}= ε

{∣∣ fk(l)∣∣2}

+ ε{|nk(l)|2

}= UnTσ2

e

LS+ σ2

n .

(12)

Each component of n(l) is i.i.d zero-mean complex Gaussianvariables with the variance of UnTσ2

e /LS + σ2n , so the variance

of n(l) is σ2n(l) = (UnTσ2

e /LS + σ2n)InR . When σ2

e = 0, that is,the channel matrix is estimated perfectly, the variance of theequivalent noise is σ2

n = σ2n , which is equal to the variance of

the scenario where the channel is estimated perfectly.

3.2. Decorrelating Scheme. Optimal coherent and noncoher-ent detection based on maximum likelihood (ML) process-ing can be readily derived from (4) [14, 15]. These optimaldetectors perform joint detection for all users and incur anexponential complexity in both the number of active usersand that of transmit antennas. The enormous complexity ofthe optimum receivers renders them mainly of theoreticalinterest.


In this part, a suboptimal decorrelating receiver is pre-sented which decouples the detection of different users witha linear complexity in the number of active users [16].

In order to express it compactly, we define

Y �[

y(1), . . . , y(l), . . . , y(LS)],

N � [n(1), . . . , n(l), . . . , n(LS)],

s(u) �[s(u)(1), . . . , s(u)(l), . . . , s(u)(LS)

],

S �{[

s(1)]T

, . . . ,[

s(u)]T

, . . . ,[

s(U)]T}T

,

A(u) � H(u)V(u)P(u)x(u),

A �[

A(1), . . . , A(u), . . . , A(U)].

(13)

Then, (4) is expressed as

Y = AS + N. (14)

The variance of any component of N is σ2n(l) = (UnTσ2

e /LS +σ2n)InR , so the variance of N is σ2

N(l) = ε[NNH] =

(UnTσ2e /LS + σ2

n)InR too.Since the information about all the transmitted symbols

is contained in A, the key to the differential and coherentreceivers introduced in this paper is to obtain an initialestimate of A, which is used for the subsequent detection.

When we assume that SSH has a full rank, the ML

estimate of A, which is conditioned on {H(u)}Uu=1, {V(u)}Uu=1,

P(u) and {S(u)}Uu=1, is given by

R = YSH(

SSH)−1

. (15)

The equation of (15) is suitable for all the spreadingstructure, but if the nonorthogonal spreading structures,such as GOLD code, are used, it is very difficult to getthe variance of the equivalent noise. We use normalizedorthogonal WALSH code in this paper, which satisfies SSH =IU .

The ML estimate R is effectively the output of adecorrelator with the input being the received matrix Y. Thedecorrelator mitigates the MUI and decouples the detectionof different users.

Then we study the impact of the decorrelation on thevariance of the equivalent noise, via (14), (15) which can berewritten as

R =(

AS + N)

SH(

SSH)−1 = ASSH

(SSH

)−1+ NSH

(SSH

)−1

= A + NSH = A + N,(16)

where N = NSH is the equivalent noise after decorrelating

and the variance of N is calculated as follows:

σ2N = ε

{N NH}= ε

{[NSH

][NSH

]H} = ε{

NSHSNH}

= ε{

NNH}=(UnTσ2

e

LS+ σ2

n

)InR ,

(17)

that is, the variance of equivalent noise is unchanged afterdecorrelating.

Due to the block structure of A, the ML estimate of thedesired user denoted by r(u) can be easily obtained from R,which is the uth column of R

r(u) = R(:,u) = H(u)V(u)P(u)x(u) + N(:,u)

= Λ(u)

P(u)x(u) + n(u),

(18)

where n(u) = N(:,u) is the uth column of N, whose varianceis σ2n(u) = (UnTσ2

e /LS+σ2n)InR , that is, the variance is increased

with the number of active users.The received signal of the desired user is achieved after

decorrelating, corresponding to the preprocessing at thetransmitter, and the decoupled signal is first orthogonally

converted by Φ(u)

, which is denoted as

r(u) =[Φ

(u)]H

r(u) =[Φ

(u)]H

H(u)V(u)P(u)x(u)

+[Φ

(u)]H n(u) = Λ

(u)P(u)x(u) +

n

(u)

,

(19)

wheren

(u)

= [Φ(u)

]H n(u)

the variance of the equivalent noisen

(u)

is calculated as follows:

ε

{n

(u)

(l)

}= ε

{[Φ

(u)]H n(u)

}

=[Φ

(u)]H

ε{n(u)

}= 0nR ,

(20)

ε

⎧⎨⎩[

n(u)

(l)

][n

(u)

(l)

]H⎫⎬⎭ = ε

⎧⎨⎩{[

Φ(u)]H n(u)

}

×{[

Φ(u)]H n(u)

}H⎫⎬⎭

= ε

{[Φ(u)

]H n(u)[n(u)

]HΦ

(u)}

=[Φ

(u)]H

ε

{n(u)[n(u)

]H}Φ

(u)

=(UnTσ2

e

LS+ σ2

n

)InR .

(21)


Clearly, the components ofn

(u)

are also i.i.d zero-meanGaussian variables with the variance of (UnTσ2

e /LS + σ2n).

3.3. SVD-Aided Beamforming and Power Allocation Scheme.Given the proposed equivalent multiuser Turbo-BLASTsystem model, we give an SVD-aided beamforming andtransmit power allocation scheme in this part, which aimsat maximizing the capacity performance of the desired userin the presence of imperfect CSI.

When CSI is imperfect, the right singular matrix ofH(u) is used for beamforming and the channel capacity ofthe desired user with SVD-aided beamforming and powerallocation can be expressed as [1, 18]

C(u)

[λ

(u)]=

nT∑k=1

log2

⎡⎣1 +P(u)k λ(u)

k(UnT σ2

e

)/LS + σ2

n

⎤⎦,

u = 1, . . . ,U ,

(22)

where λ(u)

= diag [λ(u)1 , . . . , λ(u)

nT ] is the diagonal matrix

with the eigenvalues of H(u)[H(u)]H

and P(u)k is the transmit

power allocated for the kth subchannel of the desired user,which satisfies the total transmit power constrain. Thecapacity subject to an optimization problem is given asbelow

[P(u)

1 , . . . ,P(u)nT

]opt

= arg max

⎧⎨⎩nT∑k=1

log2

⎡⎣1 +P(u)k λ(u)

k(UnT σ2

e

)/LS + σ2

n

⎤⎦⎫⎬⎭subject to

nT∑k=1

P(u)k = PTotal.

(23)

The Lagrange multiplier method is employed to find theoptimum power allocation matrix that can maximize thecapacity with total power constraint. After solving theequations in (23), we find that the optimal power allocatedfor the kth subchannel is

P(u)k =

{μ−

(UnTσ2

e

LS+ σ2

n

)[λ

(u)

k

]−1}+

, (24)

where (x)+ � max{x, 0} and μ is the Lagrange multiplier

restricted by∑nT

k=1 {μ− (UnTσ2e /LS + σ2

n)[λ(u)k ]

−1}+

= PTotal.

In a frequency-nonselective, Rayleigh fading MIMO

channel, the channel capacity is the statistical mean of λ(u)

approximately in the presence of CSI imperfection.

C(u) = ελ

[C(u)

(λ

(u))]

= ε

⎧⎪⎨⎪⎩nT∑k=1

log2

⎡⎣1 +P(u)k λ

(u)

k(UnT σ2

e

)/LS + σ2

n

⎤⎦⎫⎪⎬⎪⎭

= ε

⎧⎨⎩nT∑k=1

[log2

(μλ(u)

k

)]+

⎫⎬⎭.(25)

Because {λ(u)k }

nT

k=1 are i.i.d variables and all of them have

the same probability density p[λ(u)k ], formula (25) can be

expressed as

C(u) = nT

∫ +∞

0

[log2

(μλ(u)

k

)]p(λ

(u)

k

)dλ(u)

k , u = 1, . . . ,U.

(26)

Since singular values are random variables, it is hard toachieve the function of probability density straightly. Thechannel capacity can be calculated by averaging (26) over alarge number of channel realizations, which is expressed as

C(u) ≈ ε

⎧⎪⎨⎪⎩nT∑k=1

log2

⎡⎣1 +P(u)k λ

(u)

k(UnTσ2

e

)/LS + σ2

n

⎤⎦⎫⎪⎬⎪⎭

≈ 1N

N∑i=1

⎧⎨⎩nT∑k=1

log2

⎡⎣1 +P(u)k,i λ

(u)k,i(

UnTσ2e

)/LS + σ2

n,i

⎤⎦⎫⎬⎭,

u = 1, . . . ,U ,

(27)

where N is the number of channel realizations, which is usedfor simulation, {P(u)

k,i }nT

k=1are the transmit power sequence

for ith channel matrix H(u)i , {λ(u)

k,i }nT

k=1are eigenvalues of

H(u)i [H(u)

i ]H

, and σ2e,i and σ2

n,i are the variances of channelestimation error and AWGN, respectively.

3.4. Iterative Detection Algorithm for the Uplink of MultiuserTurbo-BLAST Systems. In this section, an iterative detectionstrategy is employed to enhance the error performance [4]in the presence of imperfect CSI. Because the orthonormalspreading codes are used, the MUI is removed absolutelyafter decorrelating, so the iterative detection of every user iscompletely independent from each other.

Let x(u)k be the kth transmitted signal of desired user at

a sampling epoch, by (19); the received symbol vector r(u)

which is corrupted by the channel noise and interferences,can be written as the sum of desired response, the CAI, andthe equivalent noise:

r(u) = λ(u)

k

√P(u)k x(u)

k + Λ(u)k P(u)

kx(u)k

+n

(u)

. (28)


In (27), λ(u)

k is the kth column of the matrix Λ(u)

, P(u)k is the

transmit power allocated to the kth subchannel of the desired

user, and Λ(u)k , x(u)

k, P(u)

kis defined as follows:

Λ(u)k =

[λ

(u)

1 , . . . , λ(u)

k−1, λ(u)

k+1, . . . , λ(u)

nT

],

x(u)k=[x(u)

1 , . . . , x(u)k−1, x(u)

k+1, . . . , x(u)nT

]T,

P(u)k= diag

(√P(u)

1 , . . . ,√P(u)k−1,

√P(u)k+1, . . . ,

√P(u)nT

).

(29)

By (29), the decision statistic of kth substream of the uth user

using a linear filter w(u)k can be expressed as

y(u)k =

[w(u)k

]Hλ

(u)

k

√P(u)k x(u)

k︸︷︷︸q(u)k

+[

w(u)k

]HΛ

(u)k P(u)

kx(u)k︸︷︷︸

d(u)k

+[

w(u)k

]H n

(u)

︸︷︷︸z(u)k

,

(30)

where q(u)k , d(u)

k , and z(u)k are the desired response obtained by

the linear filter, the coantenna interference, and the phase-rotated equivalent noise, respectively.

The CAI can be removed from y(u)k by the proposed

iterative detector and soft interference cancellation basedon mean square error (MMSE) principle. The improvedestimation of the transmitted symbol x(u)

k can be formulatedas

x(u)k =

[w(u)k

]Hr(u) − d(u)

k , (31)

where x(u)k is the estimate of the symbol x(u)

k and d(u)k is

the linear combination of the interfering substreams. The

estimation error is defined as Δx(u)k = x(u)

k − x(u)k . The

weighted vector w(u)k and the interference combination d(u)

kare optimized by minimizing the mean-square estimation

error Δx(u)k between each substream and the related estima-

tion, by the following cost function:

[w

(u)k , d(u)

k

]= argmin ε

[∥∥∥x(u)k − x(u)

k

∥∥∥2]

, (32)

where the expectation is taken over the equivalent noisen

and the statistics of the data sequence x(u).

We use standard minimization techniques to solve theoptimization problem formulated in (32). In order to arriveat the solution, we write the cost function as

Cost(u) = ε[∥∥∥x(u)

k − x(u)k

∥∥∥2]

= ε

[∥∥∥∥(w(u)k

)Hr(u) − d(u)

k − x(u)k

∥∥∥∥2]

=[

w(u)k

]Hε{

r(u)[

r(u)]H}

w(u)k

−[

w(u)k

]Hε{

r(u)[d(u)k + x(u)

k

]∗}

− ε{

r(u)[d(u)k + x(u)

k

]∗}H

w(u)k

+ ε{[

d(u)k + x(u)

k

][d(u)k + x(u)

k

]∗},

(33)

where

ε{

r(u)[

r(u)]H} = ε

⎧⎨⎩[λ

(u)

k

√P(u)k x(u)

k

+Λ(u)k P(u)

kx(u)k

+n

(u)]

×[λ

(u)

k

√P(u)k x(u)

k

+Λ(u)k P(u)

kx(u)k

+n

(u)]H

⎫⎬⎭=[λ

(u)

k

√P(u)k

][λ

(u)

k

√P(u)k

]H+[Λ

(u)k P(u)

k

]ε{

x(u)k

[x(u)k

]H}

×[Λ

(u)k P(u)

k

]H

+

(UnTσ2

e

LS+ σ2

n

)InR

ε{

r(u)[d(u)k + x(u)

k

]∗} = ε

{[λ

(u)

k

√P(u)k x(u)

k

+Λ(u)k P(u)

kx(u)k

+n

(u)]

×[d(u)k + x(u)

k

]∗}

= λ(u)

k

√P(u)k

+ Λ(u)k P(u)

kε[

x(u)k

][d(u)k

]∗.

(34)


Via (34), the cost function in (33) can be further evaluated as

Cost(u) =[

w(u)k

]H{[λ

(u)

k

√P(u)k

][λ

(u)

k

√P(u)k

]H

+[Λ

(u)k P(u)

k

]ε{

x(u)k

[x(u)k

]H}[Λ

(u)k P(u)

k

]H

+

(UnTσ2

e

LS+ σ2

n

)InR

}w(u)k

−[

w(u)k

]H{λ

(u)

k

√P(u)k + Λ

(u)k P(u)

kε[

x(u)k

][d(u)k

]∗}

−{λ

(u)

k

√P(u)k + Λ

(u)k P(u)

kε{

x(u)k

}[d(u)k

]∗}H

w(u)k

+ ε{d(u)k

[d(u)k

]∗+ x(u)

k

[d(u)k

]∗+d(u)

k

[x(u)k

]∗+ x(u)

k

[x(u)k

]∗}.

(35)

The linear combination of interfering substreams d(u)k and

the weighted vector w(u)k is obtained in (36) by letting

∂Cost(u)/∂[d(u)k ]

∗ = 0 and ∂Cost(u)/∂[w(u)k ]

∗ = 0, respec-tively,

d(u)k =

[w(u)k

]H[λ

(u)

k

√P(u)k

][λ

(u)

k

√P(u)k

]Hε[

x(u)k

],

w(u)k =

[Q(u) + S(u) +

(UnTσ2

e

LS+ σ2

n

)InR

]−1[λ

(u)

k

√P(u)k

],

(36)

where

Q(u) =[λ

(u)

k

√P(u)k

][λ

(u)

k

√P(u)k

]H,

S(u) =[Λ

(u)k P(u)

k

]{I(nT−1) − diag

{ε[

x(u)k

]ε[

x(u)k

]H}}

×[Λ

(u)k P(u)

k

]H.

(37)

Therefore, for the desired user, the weighted vector w(u)k is

used for the iterative detection in the presence of imperfectCSI. Thus (31) can be rewritten as

x(u)k =

[w(u)k

]Hr(u) − d(u)

k

=[λ

(u)

k

√P(u)k

]H

×⎧⎨⎩[

Q(u) + S(u) +

(UnTσ2

e

LS+ σ2

n

)InR

]−1⎫⎬⎭H

×{

r(u) −[Λ

(u)k P(u)

k

]ε[

x(u)k

]}.

(38)

For the first iteration, we assume ε[x(u)k

] = 0 and thus thelinear MMSE detection for the kth substream of the desireduser becomes

x(u)k =

[λ

(u)

k

√P(u)k

]H

×⎧⎨⎩[

Q(u) + S(u) +

(UnTσ2

e

LS+ σ2

n

)InR

]−1⎫⎬⎭H

r(u).

(39)

Next, we assume that ε[x(u)k

] → x(u)k

with the increasingnumber of iterations, the CAI interference canceller for thekth substream can be reduced to the linear MMSE receiverexpressed as (40), where the SVD-assisted beamforming andpower allocation scheme are used

x(u)k =

[λ

(u)

k

√P(u)k

]H

×⎧⎨⎩[

Q(u) + S(u) +

(UnTσ2

e

LS+ σ2

n

)InR

]−1⎫⎬⎭H

×{

r(u) −[Λ

(u)k P(u)

k

]x(u)k

}.

(40)

The performance of iterative scheme depends on theveracity of channel estimation matrix and the power alloca-tion scheme

4. Simulation Results

In this section, we compare the capacity and BER perfor-mance of the traditional and modified multiuser Turbo-BLAST systems, where the traditional multiuser Turbo-BLAST systems adopt the equal power allocated strategy andwithout the SVD-aided preprocessing and postprocessing,denoted by “EPA”, and the modified multiuser Turbo-BLASTsystems employ the proposed SVD-aided beamforming andtransmit power allocation scheme, denoted by “BF-TPA”,respectively.

At the transmitter, the data stream is first encoded by arate-1/2 convolutional code with generator (7,5), modulatedby 8-PSK modulation scheme, and spread by WALSH codes,whose capacity performance over a large number of channelrealizations is exhibited in Figures 3–5, extra capacity, whichis defined as the capacity gap between the modified multiuserTurbo-BLAST system and the conventional one, is givenin Figure 6, and finally, BER performance is presented inFigures 7-8 with different inaccuracy of channel estimationand different number of active users. 4 active users areconsidered in Figure 3 and Figures 5–7.

Figure 3 shows capacity performance of traditional andmodified multiuser Turbo-BLAST systems with four trans-mit and four receive antennas under the conditions ofdifferent channel estimation errors. Figure 3 indicates thatthe proposed SVD-aided beamforming and transmit powerallocation strategy is an effective means to improve capacityperformance even with the imperfect CSI, and the larger


0 2 4 6 8 10 12 14 16

5

10

15

20

25

Cap

acit

y (b

it/s

/Hz)

Eb/N0 (dB)

BF-TPA,EPA,BF-TPA,EPA,

BF-TPA,EPA,BF-TPA,EPA,

σ2e = 0

σ2e = 0σ2e = 0.1

σ2e = 0.1

σ2e = 0.4

σ2e = 0.4σ2e = 0.6

σ2e = 0.6

Figure 3: Capacity performance of traditional and modified multiuser Turbo-BLAST systems (U = 4).

0 5 10 15 20 25 305

10

15

20

25

30

35

40

Cap

acit

y (b

it/s

/Hz)

Eb/N0 (dB)

U = 1U = 16

U = 32U = 64

Figure 4: Capacity performance of the modified multiuser Turbo-BLAST systems with different number of active users in the presence ofimperfect CSI (σ2

e = 0.01).

capacity gains can be achieved for lower Eb/N0 and/or higher

σ2e . For example, at Eb/N0 = 10 dB, there are 0.4

(bit/s/Hz)

and 0.8 (bit/s/Hz) capacity gains for σ2e = 0.1 and σ2

e = 0.4,respectively.

The capacity performance of the modified multiuserTurbo-BLAST systems with different number of active usersin the presence of imperfect CSI are given in Figure 4,where U denotes the number of active users. It is clearthat from Figure 4, in the presence of imperfect CSI, thecapacity is lower for larger number of active users, and thatthe impact of the number of active users on the capacity

performance is higher for higher Eb/N0. For instance, atEb/N0 = 0 dB, the capacity of desired user is almost the samefor different number of active users. As Eb/N0 becomes high,the capacity degradation becomes large with the numberof active users; for example, at Eb/N0 = 30 dB, thereare 17 (bit/s/Hz) capacity degradation for the desired userwhen system with 64 active users is compared to that withonly one active user. This is due to the existence of thechannel estimation error, which will make the equivalentnoise of the system large, especially for large U (whichcan be seen from (17) and (21)). This large noise willdegrade the system performance greatly because it can affect


10

15

20

25

30

35

40

45

Cap

acit

y (b

it/s

/Hz)

σ2e

Eb/N0 = 30 dBEb/N0 = 25 dBEb/N0 = 20 dB

Eb/N0 = 15 dBEb/N0 = 10 dB

10010−110−210−310−4

Figure 5: Capacity performance of the modified multiuser Turbo-BLAST system in different values of Eb/N0 with imperfect CSI (U = 4).

Eb/N0 = 20 dBEb/N0 = 15 dB

Eb/N0 = 10 dB

Eb/N0 = 0 dBEb/N0 = 5 dB

4 6 8 10 12 14 16

0

2

4

6

8

10

12

14

16

18

Ext

ra C

apac

ity

(bi

t/s/

Hz)

nT

Figure 6: Extra capacity results between modified and traditional multiuser Turbo-BLAST systems with sixteen receive antennas and varioustransmit antennas configuration under the conditions of imperfect CSI (σ2

e = 0.01, U = 4).

the following iterative detection and the resultant capacitycalculation.

Figure 5 shows the capacity performance of the modifiedmultiuser Turbo-BLAST systems in different values of Eb/N0

with imperfect CSI. The simulation results are obtained forfive different Eb/N0 values of 10 dB, 15 dB, 20 dB, 25 dB,and 30 dB, and under various degrees of CSI imperfection.From Figure 5, we see that for CSI imperfection with σ2

e <10−3, the actual capacity is almost constant, as the CSIimperfection gets more severe; in particular, σ2

e > 10−2;the capacity becomes very sensitive to the CSI imperfection.

And we see that the CSI imperfection on the capacityperformance is higher for higher Eb/N0 because for higherEb/N0, that is, σ2

n → 0, the variance of the equivalent noisemainly depends on the error of channel estimation, andtherefore, the capacity performance becomes sensitive to theCSI imperfection.

In Figure 6, we give the extra capacity results, which aredefined as the capacity gap between modified and traditionalmultiuser Turbo-BLAST systems with sixteen receive anten-nas and various transmit antennas configuration under theconditions of imperfect CSI (σ2

e = 0.01) for five different


100

10−1

10−2

10−3

10−4

10−5

10−6

0 5 10 15 20

BE

R

BF-TPA, I=1BF-TPA, I=3

EPA, I=1EPA, I=3

Eb/N0 (dB)

σ2e = 0.1

σ2e = 0.01

Figure 7: BER performance of traditional and modified Turbo-BLAST systems under the conditions of different inaccuracy of channelestimation (U = 4).

100

10−1

10−2

10−3

10−4

10−5

10−6

BE

R

0 5 10 15 20

Eb/N0 (dB)

I = 1I = 1I = 1I = 1

I = 3I = 3I = 3I = 3

K = 64,K = 32,K = 16,K = 1,

K = 64,K = 32,K = 16,K = 1,

Figure 8: BER performance of modified Turbo-BLAST systems with different number of active users under the conditions of imperfect CSI(σ2

e = 0.01).

Eb/N0 values of 0 dB, 5 dB, 10 dB, 15 dB, and 20 dB. It is clearthat from Figure 6, we can see the extra capacity is higherfor lower Eb/N0 and/or large number of transmit antennas inthe presence of CSI imperfection. For instance, at Eb/N0 =0 dB, there are 16 (bit/s/Hz) capacity gains for the multiuserTurbo-BLAST system with sixteen transmit antennas, whileonly 3 (bit/s/Hz) capacity gains can be achieved for twelvetransmit antennas.

BER performance of traditional and modified multiuserTurbo-BLAST systems under the condition of differentinaccuracy of channel estimation is shown in Figure 7, where

I denotes the number of iterative detection. It is observedfrom Figure 7 that as the BER results are concerned, themodified multiuser Turbo-BLAST system outperforms thetraditional one under the same conditions, regardless ofthe status of CSI. This implies that the proposed SVD-aided beamforming and power allocation strategy are alsovalid to improve BER performance even with the imperfectCSI. For example, at a BER of 10−3, we can see that thereare 4 dB gains for the modified system over the traditionalone in the 1st detection iteration under the condition ofimperfect CSI, that is, σ2

e = 0.01. Moreover, the BER results


are quickly lowered with the increasing detection iterations.For instance, at a BER of 10−3, there are 2.5 dB extra gainsattained for the modified system, where BF-TPA scheme isadopted through 3rd iterative detection under the conditionof σ2

e = 0.01, which indicates that the proposed iterativedetection scheme is an effective means to enhance the BERperformance further even with imperfect CSI.

Figure 8 gives the BER performance of the modifiedTurbo-BLAST system with different number of active usersunder the condition of imperfect CSI (σ2

e = 0.01), where Udenotes the number of active users and I denotes the numberof iterative detection. Figure 8 shows that at the same inac-curacy of channel estimation and for the identical number ofiterative detection, the BER performance gets worse for largenumber of active users. For example, at a BER of 10−4, com-paring the system that has sixty-four active users with theone having only one active user, there is 4.5 dB degradation ofEb/N0. This is because the variance of each components of theequivalent noise is UnTσ2

e /LS+σ2n ; when the multiuser system

was with more active users; the variance is approximatelylinear to the number of active users, so the BER results getworse with the increase of the number of active users.

5. Conclusions

In this paper, we first introduce a CDMA-based multiuserTurbo-BLAST system model, based on the imperfect channelmodel, the equivalent system model, and the variance of theequivalent noise. In order to maximize the capacity perfor-mance of the desired user, the SVD-assisted beamformingand power allocation strategy are theoretically obtainedand the Lagrange multiplier method is then employedto find the optimized power allocation matrix, which issubject to the total transmit power constraint. And then,to circumvent the complexity of multiuser detection, thesuboptimal decorrelating scheme is proposed to decouple thedetection of different users. Finally, the iterative detectiontechnique is adopted after postprocessing to enhance the BERresults further. The complexity of the suboptimal receivers isexponential only in the transmission rate, similarly to single-user Turbo-BLAST receivers. Numerical results show that thenewly introduced method is effective to enhance the capacityresults and the BER performance of multiuser Turbo-BLASTsystems in the presence of imperfect CSI.

Acknowledgments

This work is supported by National Natural Science Foun-dation of China (61172077), Doctoral Fund of Ministry ofEducation of China (20093218120021), and FundamentalResearch Funds for the Central Universities (NS2012075,NP2011036). The authors would like to thank the anony-mous reviewers for their valuable comments.

References

[1] G. J. Foschini and M. J. Gans, “On limits of wireless com-munications in a fading environment when using multiple

antennas,” Wireless Personal Communications, vol. 6, no. 3, pp.315–335, 1998.

[2] A. A. Abouda and S. G. Haggman, “Effect of mutual couplingon capacity of mimo wireless channels in high SNR scenario,”Progress in Electromagnetics Research, vol. 65, pp. 27–40, 2006.

[3] P. W. Wolniansky, G. J. Foschini, G. D. Golden, and R.A. Valenzuela, “V-BLAST: an architecture for realizing veryhigh data rates over the rich-scattering wireless channel,” inProceedings of the URSI International Symposium on Signals,Systems, and Electronics (ISSSE ’98), pp. 295–300, Pisa, Italy,October 1998.

[4] M. Sellathurai and S. Haykin, “Turbo-BLAST for wirelesscommunications: theory and experiments,” IEEE Transactionson Signal Processing, vol. 50, no. 10, pp. 2538–2546, 2002.

[5] X. Chen, X. Yu, and D. Xu, “SVD-aided power allocationand iterative detection scheme for turbo-BLAST system withimperfect channel state information,” International Journal ofAntennas and Propagation, vol. 2011, Article ID 505606, pp.1–8, 2011.

[6] R. L. Pickholtz, D. L. Schilling, and L. B. Milstein, “Theory ofspread-spectrum communications—a tutorial,” IEEE Transac-tions on Communications, vol. 30, no. 5, pp. 855–884, 1982.

[7] R. Lupas and S. Verdu, “Linear multiuser detectors forsynchronous code-division multiple-access channels,” IEEETransactions on Information Theory, vol. 35, pp. 123–136,1989.

[8] D. Tse and O. Zeitouni, “Performance of linear multiuserreceivers in random environments,” IEEE Transactions onInformation Theory, vol. 46, no. 1, pp. 171–188, 2000.

[9] T. J. Willink, “Efficient adaptive SVD algorithm for MIMOapplications,” IEEE Transactions on Signal Processing, vol. 56,no. 2, pp. 615–622, 2008.

[10] W. Liu, L. L. Yang, and L. Hanzo, “SVD aided joint transmitterand receiver design for the uplink of multiuser detectionassisted MIMO systems,” in Proceedings of the IEEE Interna-tional Conference on Communications (ICC ’08), pp. 4095–4099, Beijing, China, May 2008.

[11] D. P. Palomar and J. R. Fonollosa, “Practical algorithms for afamily of waterfilling solutions,” IEEE Transactions on SignalProcessing, vol. 53, no. 2, pp. 686–695, 2005.

[12] A. Yoshimoto and T. Hattori, “Area coverage of a multi-linkMIMO system with water filling power allocation strategy,” inProceedings of the 66th IEEE Vehicular Technology Conference(VTC ’07) Fall, pp. 1137–1141, Baltimore, Md, USA, October2007.

[13] F. Wang and Z. Liu, “Adaptive water-filling power control forwireless communications networks,” IEEE CommunicationsLetters, vol. 12, no. 10, pp. 737–739, 2008.

[14] M. Brehler and M. K. Varanasi, “Coherent multiuser space-time communications: optimum receivers and signal design,”in Proceedings of the 35th Annual Conference InformationScience System, pp. 372–378, Baltimore, Md, USA, 2001.

[15] M. Brehler and M. K. Varanasi, “Noncoherent multiuserspace-time communications: optimum receivers and signaldesign,” in Proceedings of the 35th Annual Conference Informa-tion Science System, pp. 379–383, Baltimore, Md, USA, 2001.

[16] H. B. Li and J. Li, “Differential and coherent decorrelatingmultiuser receivers for space-time-coded CDMA systems,”IEEE Transactions on Signal Processing, vol. 50, no. 10, pp.2529–2537, 2002.

[17] S. L. Zhou and G. B. Giannakis, “How accurate chan-nel prediction needs to be for transmit-beamforming withadaptive modulation over Rayleigh MIMO channels?” IEEE


Transactions on Wireless Communications, vol. 3, no. 4, pp.1285–1294, 2004.

[18] I. E. Telatar, “Capacity of multi-antenna Gaussian channels,”European Transactions on Telecommunications, vol. 10, no. 6,pp. 585–595, 1999.

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