# SVD-Aided Beamforming and Power Allocation Algorithm for Multiuser Turbo-BLAST System Uplink with Imperfect CSI

Authors: Xiaomin Chen; Xiangbin Yu; Dazhuan Xu
Journal: International Journal of Antennas and Propagation
 (2012)

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## Abstract

The SVD-aided joint transmitter and receiver design for the uplink of CDMA-based synchronous multiuser Turbo-BLAST systems is proposed in the presence of channel state information (CSI) imperfection. At the transmitter, the beamforming and power allocation schemes are developed to maximize the capacity of the desired user. At the receiver, a suboptimal decorrelating scheme is first proposed to mitigate the multiuser interference (MUI) and decouple the detection of different users with imperfect CSI, and then the iterative detecting algorithm that takes the channel estimation error into account is designed to cancel the coantenna interference (CAI) and enhance the bit error rate (BER) results further. Simulation results show that the proposed uplink CDMA-based multiuser Turbo-BLAST model is effective, the detection from every user is completely independent to each other after 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.


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## Body

## 1. Introduction

Multiple-input multiple-output (MIMO) communication systems provide a significant capacity increment over the conventional one through appropriate space-time processing [1, 2]. Bell-Labs Layered Space-Time (BLAST) has shown to be a promising MIMO communication technique that can achieve tremendous bandwidth efficiencies [3]. The combination of BLAST and turbo-decoding principle is called Turbo-BLAST [4, 5], where the coantenna interference (CAI) caused by the major source of channel impairment can be removed by a serially concatenated iterative decoding algorithm.Although Turbo-BLAST is considered only for single-user application, we show herein that it naturally extends to a multiuser code division multiple access (CDMA) scenario [6], which is called CDMA-based multiuser Turbo-BLAST system, where a unique signature code is assigned to each other and used to identify the user and to spread the spectrum of the user’s data. Since all the users “simultaneously” occupy the same spectrum, they create multiuser interference (MUI) to one another because of the nonzero cross correlation of their signatures [7, 8]. The MUI not only limits the system capacity but also increases the system error performance. One of the most promising approaches to meet these challenges is the use of adaptive beamforming and power control [9–13].In the uplink (UL) of CDMA-based multiuser Turbo-BLAST systems, both the multiple MUI and CAI have to be mitigated. Adaptive interference suppression techniques based on multiuser detection (MUD) and antenna array processing have recently been considered as powerful methods for increasing the quality, capacity, and coverage of these systems. However, the method of optimal MUD has a very high computation cost due to its nonlinear nature [14–16].Moreover, the performance of multiuser Turbo-BLAST system is closely related to the channel state information (CSI). Practically, CSI at the receiver is subject to the error performance because of the non-real-time data processing, quantization error, and imperfect channel estimations, and so forth [17]. This gives rise to significant challenges to system design and analysis.Regarding these problems, the SVD-assisted beamforming and power allocation schemes are first developed to maximize the capacity of every active user, and a suboptimal decorrelating scheme for the uplink of multiuser Turbo-BLAST system is proposed to mitigate the MUI, then the detection is performed independently for different users after decorrelating. Based on this, an improved iterative detection algorithm is adopted to cancel the CAI. We assume that the spreading structures of all the users are perfectly known to both the transmitter and the receiver at the instant of transmission and reception, and the channel estimation is imperfect, only the channel estimation matrix and the statistical characteristic of channel estimation error are known to the system. Besides, in this paper, the common assumption that true channel matrix and channel estimation error are complex Gaussian is employed, and a CDMA Frequency Division Duplexing (FDD) system is considered.Based on the above analysis, in this paper, we derive a beamforming and power allocation algorithm for multiuser Turbo-BLAST system in the presence of imperfect CSI, which avoids that most of the existing schemes are based on perfect CSI and designed for single-user case. Moreover, by using the obtained beamforming and power allocation, an improved iterative detection scheme is also presented. With this detection scheme, the system performance is improved further. Simulation results verify the validity of the presented schemes.This paper is organized as follows: after the introduction in Section1, 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 power allocation scheme, and a suboptimal decorrelating algorithm, as well as the iterative detecting method which are described respectively. The simulation results are analyzed in Section 4, and Section 5 makes the conclusion of the paper.


## 2. Channel Model and Basic System Description

### 2.1. Transmitter Model

We consider an uplink (UL) of synchronous CDMA-based multiuser Turbo-BLAST system withU active users operating in a frequency-nonselective Rayleigh fading channel, that is, a narrowband multiuser Turbo-BLAST system is considered, where CDMA is mainly employed as a multiple access technique to differentiate different users and the corresponding spreading gain is not considered. At the transmitter, a data stream of each user is first encoded, mapped into a symbol stream that is subsequently demultiplexed into nT substreams, and then preprocessed by the right singular matrix, of channel estimation matrix, where each stream is allocated with proper power, finally the preprocessed streams are spread and transmitted by the MS each of which employs nT transmit antennas. The schematic transmit structure of the uth (u=1,2,…,U) mobile station (MS) is shown in Figure 1, where V̂(u) represents the transmitter’s beamforming matrix formulated for the transmission of the uth MS’s data and P(u) is the power allocation matrix for the uth MS under the constrain of total transmit power.Figure 1
Transmitter structure of theuth mobile station.


### 2.2. Imperfect Channel Model and Singular Value Decomposition

Consider a multiuser Turbo-BLAST system where the base station (BS) employsnR antennas and each user is equipped with nT antennas (nT≤nR). We introduce an imperfect channel model in this section [17]. For the uth user, let H(u)∈ℂnR×nT denote the true complex channel matrix and Ĥ(u)∈ℂnR×nT denote the complex channel estimation matrix. Note that the channel estimation matrix and the true channel matrix are different from one another for the uplink. Based on the imperfect channel model, the true complex matrix H(u) can be formulated as(1)H(u)=Ĥ(u)+Ξ(u),
where Ξ(u)∈ℂnR×nT is a complex matrix related to the imperfect 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 Ĥ(u) to be also complex Gaussian. Hence their statistical distributions are shown as eij(u)~𝒞𝒩(0,σe2), ĥij(u)~𝒞𝒩(0,1-σe2), and hij(u)~𝒞𝒩(0,1), which are all distributed by complex Gaussian law, and σe2 indicates the inaccuracy of the CSI.Since the practical system cannot detect the degree of the imperfection of the CSI, the channel estimation matrixĤ(u) is treated as the true channel matrix, which can be decomposed as(2)Ĥ(u)=Φ̂(u)Λ̂(u)[V̂(u)]H,
where (·)H denotes a matrix conjugate transpose, Φ̂(u) and V̂(u) are the unitary matrices with left and right singular vectors of Ĥ(u) as their columns, Λ̂(u) is a nonnegative and diagonal matrix,  {λ̂i}i=1nT are the main diagonal elements of Λ̂(u), and {λ̂i(u)}i=1nT are eigenvalues of Ĥ(u)[Ĥ(u)]H, respectively.


### 2.3. Multiuser Receiver Description

The illustration diagram of the receiver is shown in Figure2, where the uth user is assumed as the desired user. The decorrelating scheme and the iterative detection algorithm are derived based on the availability of the spreading structure of both the desired and interfering users, as well as the CSI of the desired user.Figure 2
The receiver structure of the multiuser Turbo-BLAST system.At the receiver, the received signal is decorrelated and then the signal of the desired user can be extracted after decorrelating through the use of the orthogonal spreading codes. The decoupled signal is orthogonally converted byΦ̂(u), the left singular matrix of Ĥ(u). Finally an iterative detection strategy based on the “turbo” principles is used for the symbol detection after orthogonally converting. For the iterative decoding, the optimal decoding process can be separated into two stages: soft-input/soft-output (SISO) channel detector and SISO channel decoder, which mutually exchange the extrinsic information sent from one stage to the other iteratively until the decoding process converges.LetnT data streams be transmitted by the uth MS to the BS hosted by a vector expressed as x(u)=[x1(u),…,xnT(u)]T,u=1,2,…,U, {s(u)(l)}l=1L denotes the spreading codes of the length L assigned to the desired user, and n(l)=[n1(l),…,nnT(l)]T is a column vector of additive Gaussian noise variables during the lth epoch, where each component is independent and identically distributed (i.i.d) zero-mean complex Gaussian variables with the variance of σn2. Under these assumptions, the baseband discrete-time signal received by nR receive antennas at the BS can be expressed as(3)y(l)=∑u=1UH(u)V̂(u)P(u)x(u)s(u)(l)+n(l),
where P(u)=diag(P1(u),…,PnT(u)) is the diagonal transmit power matrix for the desired user with total power constraint ∑k=1nTPk(u)=Ptotal=nT.


## 2.1. Transmitter Model

We consider an uplink (UL) of synchronous CDMA-based multiuser Turbo-BLAST system withU active users operating in a frequency-nonselective Rayleigh fading channel, that is, a narrowband multiuser Turbo-BLAST system is considered, where CDMA is mainly employed as a multiple access technique to differentiate different users and the corresponding spreading gain is not considered. At the transmitter, a data stream of each user is first encoded, mapped into a symbol stream that is subsequently demultiplexed into nT substreams, and then preprocessed by the right singular matrix, of channel estimation matrix, where each stream is allocated with proper power, finally the preprocessed streams are spread and transmitted by the MS each of which employs nT transmit antennas. The schematic transmit structure of the uth (u=1,2,…,U) mobile station (MS) is shown in Figure 1, where V̂(u) represents the transmitter’s beamforming matrix formulated for the transmission of the uth MS’s data and P(u) is the power allocation matrix for the uth MS under the constrain of total transmit power.Figure 1
Transmitter structure of theuth mobile station.


## 2.2. Imperfect Channel Model and Singular Value Decomposition

Consider a multiuser Turbo-BLAST system where the base station (BS) employsnR antennas and each user is equipped with nT antennas (nT≤nR). We introduce an imperfect channel model in this section [17]. For the uth user, let H(u)∈ℂnR×nT denote the true complex channel matrix and Ĥ(u)∈ℂnR×nT denote the complex channel estimation matrix. Note that the channel estimation matrix and the true channel matrix are different from one another for the uplink. Based on the imperfect channel model, the true complex matrix H(u) can be formulated as(1)H(u)=Ĥ(u)+Ξ(u),
where Ξ(u)∈ℂnR×nT is a complex matrix related to the imperfect 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 Ĥ(u) to be also complex Gaussian. Hence their statistical distributions are shown as eij(u)~𝒞𝒩(0,σe2), ĥij(u)~𝒞𝒩(0,1-σe2), and hij(u)~𝒞𝒩(0,1), which are all distributed by complex Gaussian law, and σe2 indicates the inaccuracy of the CSI.Since the practical system cannot detect the degree of the imperfection of the CSI, the channel estimation matrixĤ(u) is treated as the true channel matrix, which can be decomposed as(2)Ĥ(u)=Φ̂(u)Λ̂(u)[V̂(u)]H,
where (·)H denotes a matrix conjugate transpose, Φ̂(u) and V̂(u) are the unitary matrices with left and right singular vectors of Ĥ(u) as their columns, Λ̂(u) is a nonnegative and diagonal matrix,  {λ̂i}i=1nT are the main diagonal elements of Λ̂(u), and {λ̂i(u)}i=1nT are eigenvalues of Ĥ(u)[Ĥ(u)]H, respectively.


## 2.3. Multiuser Receiver Description

The illustration diagram of the receiver is shown in Figure2, where the uth user is assumed as the desired user. The decorrelating scheme and the iterative detection algorithm are derived based on the availability of the spreading structure of both the desired and interfering users, as well as the CSI of the desired user.Figure 2
The receiver structure of the multiuser Turbo-BLAST system.At the receiver, the received signal is decorrelated and then the signal of the desired user can be extracted after decorrelating through the use of the orthogonal spreading codes. The decoupled signal is orthogonally converted byΦ̂(u), the left singular matrix of Ĥ(u). Finally an iterative detection strategy based on the “turbo” principles is used for the symbol detection after orthogonally converting. For the iterative decoding, the optimal decoding process can be separated into two stages: soft-input/soft-output (SISO) channel detector and SISO channel decoder, which mutually exchange the extrinsic information sent from one stage to the other iteratively until the decoding process converges.LetnT data streams be transmitted by the uth MS to the BS hosted by a vector expressed as x(u)=[x1(u),…,xnT(u)]T,u=1,2,…,U, {s(u)(l)}l=1L denotes the spreading codes of the length L assigned to the desired user, and n(l)=[n1(l),…,nnT(l)]T is a column vector of additive Gaussian noise variables during the lth epoch, where each component is independent and identically distributed (i.i.d) zero-mean complex Gaussian variables with the variance of σn2. Under these assumptions, the baseband discrete-time signal received by nR receive antennas at the BS can be expressed as(3)y(l)=∑u=1UH(u)V̂(u)P(u)x(u)s(u)(l)+n(l),
where P(u)=diag(P1(u),…,PnT(u)) is the diagonal transmit power matrix for the desired user with total power constraint ∑k=1nTPk(u)=Ptotal=nT.


## 3. Equivalent System Model and Receiver Design

### 3.1. Equivalent System Model

In this subsection, an equivalent system model is proposed for the uplink of the multiuser Turbo-BLAST system with imperfect CSI.For a synchronous multiuser Turbo-BLAST system, at a sampling instant, the received signal can be formulated as(4)y(l)=∑u=1U[Ĥ(u)+Ξ(u)]V̂(u)P(u)x(u)s(u)(l)+n(l)=∑u=1UĤ(u)V̂(u)P(u)x(u)s(u)(l)+∑u=1UΞ(u)V̂(u)P(u)x(u)s(u)(l)+n(l)=∑u=1UĤ(u)V̂(u)P(u)x(u)s(u)(l)+n̂(l),
where(5)n̂(l)≜∑u=1UΞ(u)V̂(u)P(u)x(u)s(u)(l)+n(l),n̂(l) is the equivalent additive noise at the lth instant, which is consisted of the interference caused by the channel estimation errors of all the interfering users and the complex Gaussian noise, respectively.The variance of the equivalent noisen̂(l) is calculated as follows.First, letG(u)=Ξ(u)V̂(u); the mean and variance of G(u) can be evaluated as(6)ε{G(u)}=ε{Ξ(u)V̂(u)}=ε[Ξ(u)]V̂(u)=0,(7)ε{G(u)[G(u)]H}=ε{Ξ(u)V̂(u)[V̂(u)]H[Ξ(u)]H}=ε{Ξ(u)[Ξ(u)]H}=σe2InR.
Clearly, the components of G(u) are i.i.d zero-mean complex Gaussian variables with the variance of σe2, thus statistical distributions can be expressed as gij(u)~𝒞𝒩(0,σe2) and ε{gki(u)[gki′(u′)]}=σe2δ(i-i′)δ(u-u′).Then, letf(l)=∑u=1UG(u)P(u)x(u)s(u)(l); the mean and variance of f(l) can be evaluated as(8)ε[f(l)]=ε[∑u=1UG(u)P(u)x(u)s(u)(l)]=∑u=1Uε[G(u)]P(u)x(u)s(u)(l)=0,(9)f(l)=∑u=1UG(u)P(u)x(u)s(u)(l)=∑u=1U{[g11(u)⋯g1k(u)⋯g1nT(u)⋮⋱⋮⋱⋮gk1(u)⋯gkk(u)⋯gknT(u)⋮⋱⋮⋱⋮gnR1(u)⋯gnRk(u)⋯gnRnT(u)]×[P1(u)⋯0⋯0⋮⋱⋮⋱⋮0⋯Pk(u)⋯0⋮⋱⋮⋱⋮0⋯0⋯PnT(u)]×[x1(u)s(u)(l)⋮xk(u)s(u)(l)⋮xnT(u)s(u)(l)]}=[∑u=1U∑i=1nTg1i(u)Pi(u)xi(u)s(u)(l)⋮∑u=1U∑i=1nTgki(u)Pi(u)xi(u)s(u)(l)⋮∑u=1U∑i=1nTgnRi(u)Pi(u)xi(u)s(u)(l)]=[f1(l)⋮fk(l)⋮fnR(l)].
Clearly, f(l) is a zero-mean complex Gaussian column vector with its component fk(l)=∑u=1U∑i=1nTgki(u)Pi(u)xi(u)s(u)(l),k=1,…,nR,l=1,…,LS, whose variance can be evaluated as(10)σfk2(l)=ε{[fk(l)][fk(l)]*}=ε{[∑u=1U∑i=1nTgki(u)Pi(u)xi(u)s(u)(l)]×[∑u′=1U∑i′=1nTgki′(u′)Pi′(u′)xi′(u′)s(u′)(l)]*}.
Since the normalized spreading codes are used, that is, [su(l)]2=1/LS and ε{gki(u)[gki′(u′)]}=σe2δ(i-i′)δ(u-u′), (10) can be written as(11)σfk2(l)=∑u=1U∑i=1nTε{[gki(u)Pi(u)xi(u)][gki(u)Pi(u)xi(u)]*}×{su(l)[su(l)]*}=1LS∑u=1U∑i=1nTPi(u)ε{gki(u)[gki(u)]*}ε{xi(u)[xi(u)]*}=1LS∑u=1U∑i=1nTPi(u)ε{|gki(u)|2}ε{|xki(u)|2}=σe2LS∑u=1U∑i=1nTPi(u)=UnTσe2LS.
Because n̂(l)=f(l)+n(l), the component of n̂(l) is given as n̂k(l)=fk(l)+nk(l),k=1,…,nR by the aforementioned definition with the variance calculated as(12)σn̂k2(l)=ε[n̂k(l)n̂k*(l)]=ε{[fk(l)+nk(l)][fk(l)+nk(l)]*}=ε{|fk(l)|2}+ε{|nk(l)|2}=UnTσe2LS+σn2.
Each component of n̂(l) is i.i.d zero-mean complex Gaussian variables with the variance of UnTσe2/LS+σn2, so the variance of n̂(l) is σn̂2(l)=(UnTσe2/LS+σn2)InR. When σe2=0, that is, the channel matrix is estimated perfectly, the variance of the equivalent noise is σn̂2=σn2, which is equal to the variance of the scenario where the channel is estimated perfectly.


### 3.2. Decorrelating Scheme

Optimal coherent and noncoherent detection based on maximum likelihood (ML) processing can be readily derived from (4) [14, 15]. These optimal detectors perform joint detection for all users and incur an exponential complexity in both the number of active users and that of transmit antennas. The enormous complexity of the optimum receivers renders them mainly of theoretical interest.In this part, a suboptimal decorrelating receiver is presented which decouples the detection of different users with a linear complexity in the number of active users [16].In order to express it compactly, we define(13)Y≜[y(1),…,y(l),…,y(LS)],N̂≜[n̂(1),…,n̂(l),…,n̂(LS)],s(u)≜[s(u)(1),…,s(u)(l),…,s(u)(LS)],S≜{[s(1)]T,…,[s(u)]T,…,[s(U)]T}T,Â(u)≜Ĥ(u)V̂(u)P(u)x(u),Â≜[Â(1),…,Â(u),…,Â(U)].
Then, (4) is expressed as(14)Y=ÂS+N̂.
The variance of any component of N̂ is σn̂2(l)=(UnTσe2/LS+σn2)InR, so the variance of N̂ is σN̂2(l)=ε[N̂N̂H]=(UnTσe2/LS+σn2)InR too.Since the information about all the transmitted symbols is contained inÂ, the key to the differential and coherent receivers introduced in this paper is to obtain an initial estimate of Â, which is used for the subsequent detection.When we assume thatSSH has a full rank, the ML estimate of Â, which is conditioned on {Ĥ(u)}u=1U, {V̂(u)}u=1U, P(u) and {S(u)}u=1U, is given by(15)R=YSH(SSH)-1.
The equation of (15) is suitable for all the spreading structure, but if the nonorthogonal spreading structures, such as GOLD code, are used, it is very difficult to get the variance of the equivalent noise. We use normalized orthogonal WALSH code in this paper, which satisfies SSH=IU.The ML estimateR is effectively the output of a decorrelator with the input being the received matrix Y. The decorrelator mitigates the MUI and decouples the detection of different users.Then we study the impact of the decorrelation on the variance of the equivalent noise, via (14), (15) which can be rewritten as(16)R=(ÂS+N̂)SH(SSH)-1=ÂSSH(SSH)-1+N̂SH(SSH)-1=Â+N̂SH=Â+N̂̂,
where N̂̂=N̂SH is the equivalent noise after decorrelating and the variance of N̂̂ is calculated as follows:(17)σN̂̂2=ε{N̂̂N̂̂H}=ε{[N̂SH][N̂SH]H}=ε{N̂SHSN̂H}=ε{N̂N̂H}=(UnTσe2LS+σn2)InR,
that is, the variance of equivalent noise is unchanged after decorrelating.Due to the block structure ofÂ, the ML estimate of the desired user denoted by r(u) can be easily obtained from R, which is the uth column of R(18)r(u)=R(:,u)=Ĥ(u)V̂(u)P(u)x(u)+N̂̂(:,u)=Λ̂(u)P(u)x(u)+n̂̂(u),
where n̂̂(u)=N̂̂(:,u) is the uth column of N̂̂, whose variance is σn̂̂(u)2=(UnTσe2/LS+σn2)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 the transmitter, and the decoupled signal is first orthogonally converted byΦ̂(u), which is denoted as(19)r̃(u)=[Φ̂(u)]Hr(u)=[Φ̂(u)]HĤ(u)V̂(u)P(u)x(u)+[Φ̂(u)]Hn̂̂(u)=Λ̂(u)P(u)x(u)+n̂̂̂(u),
where n̂̂̂(u)=[Φ̂(u)]Hn̂̂(u) the variance of the equivalent noise n̂̂̂(u) is calculated as follows:(20)ε{n̂̂̂(u)(l)}=ε{[Φ̂(u)]Hn̂̂(u)}=[Φ̂(u)]Hε{n̂̂(u)}=0nR,(21)ε{[n̂̂̂(u)(l)][n̂̂̂(u)(l)]H}=ε{[n̂̂̂(u)(l)]H{[Φ̂(u)]Hn̂̂(u)}×{[Φ̂(u)]Hn̂̂(u)}H}=ε{[Φ̂(u)]Hn̂̂(u)[n̂̂(u)]HΦ̂(u)}=[Φ̂(u)]Hε{n̂̂(u)[n̂̂(u)]H}Φ̂(u)=(UnTσe2LS+σn2)InR.
Clearly, the components of n̂̂̂(u) are also i.i.d zero-mean Gaussian variables with the variance of (UnTσe2/LS+σn2).


### 3.3. SVD-Aided Beamforming and Power Allocation Scheme

Given the proposed equivalent multiuser Turbo-BLAST system model, we give an SVD-aided beamforming and transmit power allocation scheme in this part, which aims at maximizing the capacity performance of the desired user in the presence of imperfect CSI.When CSI is imperfect, the right singular matrix ofĤ(u) is used for beamforming and the channel capacity of the desired user with SVD-aided beamforming and power allocation can be expressed as [1, 18](22)C(u)[λ̂̅(u)]=∑k=1nTlog2[1+Pk(u)λ̂k(u)(UnTσe2)/LS+σn2],u=1,…,U,
where λ̂̅(u)=diag[λ̂1(u),…,λ̂nT(u)] is the diagonal matrix with the eigenvalues of Ĥ(u)[Ĥ(u)]H and Pk(u) is the transmit power allocated for the kth subchannel of the desired user, which satisfies the total transmit power constrain. The capacity subject to an optimization problem is given as below(23)[P1(u),…,PnT(u)]opt=argmax⁡⁡{∑k=1nTlog2[1+Pk(u)λ̂k(u)(UnTσe2)/LS+σn2]}subjectto∑k=1nTPk(u)=PTotal.
The Lagrange multiplier method is employed to find the optimum power allocation matrix that can maximize the capacity with total power constraint. After solving the equations in (23), we find that the optimal power allocated for the kth subchannel is(24)Pk(u)={μ-(UnTσe2LS+σn2)[λ̂k(u)]-1}+,
where (x)+≜max{x,0} and μ is the Lagrange multiplier restricted by ∑k=1nT{μ-(UnTσe2/LS+σn2)[λ̂k(u)]-1}+=PTotal.In a frequency-nonselective, Rayleigh fading MIMO channel, the channel capacity is the statistical mean ofλ̂̅(u) approximately in the presence of CSI imperfection.(25)C(u)=ελ̂̅[C(u)(λ̂̅(u))]=ε{∑k=1nTlog2[1+Pk(u)λ̂k(u)(UnTσe2)/LS+σn2]}=ε{∑k=1nT[log2(μλ̂k(u))]+}.
Because {λ̂k(u)}k=1nT are i.i.d variables and all of them have the same probability density p[λ̂k(u)], formula (25) can be expressed as(26)C(u)=nT∫0+∞[log2(μλ̂k(u))]p(λ̂k(u))dλ̂k(u),u=1,…,U.
Since singular values are random variables, it is hard to achieve the function of probability density straightly. The channel capacity can be calculated by averaging (26) over a large number of channel realizations, which is expressed as(27)C(u)≈ε{∑k=1nTlog2[1+Pk(u)λ̂k(u)(UnTσe2)/LS+σn2]}≈1N∑i=1N{∑k=1nTlog2[1+Pk,i(u)λ̂k,i(u)(UnTσe2)/LS+σn,i2]},u=1,…,U,
where N is the number of channel realizations, which is used for simulation, {Pk,i(u)}k=1nT are the transmit power sequence for ith channel matrix Ĥi(u), {λ̂k,i(u)}k=1nT are eigenvalues of Ĥi(u)[Ĥi(u)]H, and σe,i2 and σn,i2 are the variances of channel estimation error and AWGN, respectively.


### 3.4. Iterative Detection Algorithm for the Uplink of Multiuser Turbo-BLAST Systems

In this section, an iterative detection strategy is employed to enhance the error performance [4] in the presence of imperfect CSI. Because the orthonormal spreading codes are used, the MUI is removed absolutely after decorrelating, so the iterative detection of every user is completely independent from each other.Letxk(u) be the kth transmitted signal of desired user at a sampling epoch, by (19); the received symbol vector r̃(u) which is corrupted by the channel noise and interferences, can be written as the sum of desired response, the CAI, and the equivalent noise:(28)r̃(u)=λ̂k(u)Pk(u)xk(u)+Λ̂k̅(u)Pk̅(u)xk̅(u)+n̂̂̂(u).
In (27), λ̂k(u) is the kth column of the matrix Λ̂(u), Pk(u) is the transmit power allocated to the kth subchannel of the desired user, and Λ̂k̅(u),xk̅(u),Pk̅(u) is defined as follows:(29)Λ̂k̅(u)=[λ̂1(u),…,λ̂k-1(u),λ̂k+1(u),…,λ̂nT(u)],xk̅(u)=[x1(u),…,xk-1(u),xk+1(u),…,xnT(u)]T,Pk̅(u)=diag⁡(P1(u),…,Pk-1(u),Pk+1(u),…,PnT(u)).
By (29), the decision statistic of kth substream of the uth user using a linear filter ŵk(u) can be expressed as(30)yk(u)=[ŵk(u)]Hλ̂k(u)Pk(u)xk(u)︸q̂k(u)+[ŵk(u)]HΛ̂k̅(u)Pk̅(u)xk̅(u)︸d̂k(u)+[ŵk(u)]Hn̂̂̂(u)︸ẑk(u),
where q̂k(u), d̂k(u), and ẑk(u) are the desired response obtained by the linear filter, the coantenna interference, and the phase-rotated equivalent noise, respectively.The CAI can be removed fromyk(u) by the proposed iterative detector and soft interference cancellation based on mean square error (MMSE) principle. The improved estimation of the transmitted symbol x̃k(u) can be formulated as(31)x̃k(u)=[ŵk(u)]Hr̃(u)-d̂k(u),
where x̃k(u) is the estimate of the symbol xk(u) and d̂k(u) is the linear combination of the interfering substreams. The estimation error is defined as Δxk(u)=x̃k(u)-xk(u). The weighted vector ŵk(u) and the interference combination d̂k(u) are optimized by minimizing the mean-square estimation error Δxk(u) between each substream and the related estimation, by the following cost function:(32)[w̅̂k(u),d̂k(u)]=argminε[‖x̃k(u)-xk(u)‖2],
where the expectation is taken over the equivalent noise n̂̂̂ and the statistics of the data sequence x(u).We use standard minimization techniques to solve the optimization problem formulated in (32). In order to arrive at the solution, we write the cost function as(33)Cost(u)=ε[‖x̃k(u)-xk(u)‖2]=ε[‖(ŵk(u))Hr̃(u)-d̂k(u)-xk(u)‖2]=[ŵk(u)]Hε{r̃(u)[r̃(u)]H}ŵk(u)-[ŵk(u)]Hε{r̃(u)[d̂k(u)+xk(u)]*}-ε{r̃(u)[d̂k(u)+xk(u)]*}Hŵk(u)+ε{[d̂k(u)+xk(u)][d̂k(u)+xk(u)]*},
where(34)ε{r̃(u)[r̃(u)]H}=ε{[λ̂k(u)n̂̂̂(u)]H[+Λ̂k̅(u)Pk̅(u)xk̅(u)+n̂̂̂(u)λ̂k(u)Pk(u)xk(u)+Λ̂k̅(u)Pk̅(u)xk̅(u)+n̂̂̂(u)Pk(u)xk(u)]×[+n̂̂̂(u)λ̂k(u)Pk(u)xk(u)+Λ̂k̅(u)Pk̅(u)xk̅(u)+n̂̂̂(u)Pk(u)xk(u)]H}=[λ̂k(u)Pk(u)][λ̂k(u)Pk(u)]H+[Λ̂k̅(u)Pk̅(u)]ε{xk̅(u)[xk̅(u)]H}×[Λ̂k̅(u)Pk̅(u)]H+(UnTσe2LS+σn2)InRε{r̃(u)[d̂k(u)+xk(u)]*}=ε{[n̂̂̂(u)λ̂k(u)Pk(u)xk(u)+Λ̂k̅(u)Pk̅(u)xk̅(u)+n̂̂̂(u)Pk(u)xk(u)]×[d̂k(u)+xk(u)]*[λ̂k(u)Pk(u)]H}=λ̂k(u)Pk(u)+Λ̂k̅(u)Pk̅(u)ε[xk̅(u)][d̂k̅(u)]*.
Via (34), the cost function in (33) can be further evaluated as(35)Cost(u)=[ŵk(u)]H{[λ̂k(u)Pk(u)][λ̂k(u)Pk(u)]H+[Λ̂k̅(u)Pk̅(u)]ε{xk̅(u)[xk̅(u)]H}[Λ̂k̅(u)Pk̅(u)]H+(UnTσe2LS+σn2)InR}ŵk(u)-[ŵk(u)]H{λ̂k(u)Pk(u)+Λ̂k̅(u)Pk̅(u)ε[xk(u)][d̂k(u)]*}-{λ̂k(u)Pk(u)+Λ̂k̅(u)Pk̅(u)ε{xk(u)}[d̂k(u)]*}Hŵk(u)+ε{d̂k(u)[d̂k(u)]*+xk(u)[d̂k(u)]*+d̂k(u)[xk(u)]*+xk(u)[xk(u)]*}.
The linear combination of interfering substreams d̂k(u) and the weighted vector ŵk(u) is obtained in (36) by letting ∂Cost(u)/∂[d̂k(u)]*=0 and ∂Cost(u)/∂[ŵk(u)]*=0,respectively,(36)d̂k(u)=[ŵk(u)]H[λ̂k(u)Pk(u)][λ̂k(u)Pk(u)]Hε[xk̅(u)],ŵk(u)=[Q(u)+S(u)+(UnTσe2LS+σn2)InR]-1[λ̂k(u)Pk(u)],
where(37)Q(u)=[λ̂k(u)Pk(u)][λ̂k(u)Pk(u)]H,S(u)=[Λ̂k̅(u)Pk̅(u)]{I(nT-1)-diag⁡{ε[xk̅(u)]ε[xk̅(u)]H}}×[Λ̂k̅(u)Pk̅(u)]H.
Therefore, for the desired user, the weighted vector ŵk(u) is used for the iterative detection in the presence of imperfect CSI. Thus (31) can be rewritten as(38)x̃k(u)=[ŵk(u)]Hr̃(u)-d̂k(u)=[λ̂k(u)Pk(u)]H×{[Q(u)+S(u)+(UnTσe2LS+σn2)InR]-1}H×{r(u)-[Λ̂k̅(u)Pk̅(u)]ε[xk̅(u)]}.
For the first iteration, we assume ε[xk̅(u)]=0 and thus the linear MMSE detection for the kth substream of the desired user becomes(39)x̃k(u)=[λ̂k(u)Pk(u)]H×{[Q(u)+S(u)+(UnTσe2LS+σn2)InR]-1}Hr̃(u).
Next, we assume that ε[xk̅(u)]→xk̅(u) with the increasing number of iterations, the CAI interference canceller for the kth substream can be reduced to the linear MMSE receiver expressed as (40), where the SVD-assisted beamforming and power allocation scheme are used(40)x̃k(u)=[λ̂k(u)Pk(u)]H×{[Q(u)+S(u)+(UnTσe2LS+σn2)InR]-1}H×{r̃(u)-[Λ̂k̅(u)Pk̅(u)]xk̅(u)}.The performance of iterative scheme depends on the veracity of channel estimation matrix and the power allocation scheme


## 3.1. Equivalent System Model

In this subsection, an equivalent system model is proposed for the uplink of the multiuser Turbo-BLAST system with imperfect CSI.For a synchronous multiuser Turbo-BLAST system, at a sampling instant, the received signal can be formulated as(4)y(l)=∑u=1U[Ĥ(u)+Ξ(u)]V̂(u)P(u)x(u)s(u)(l)+n(l)=∑u=1UĤ(u)V̂(u)P(u)x(u)s(u)(l)+∑u=1UΞ(u)V̂(u)P(u)x(u)s(u)(l)+n(l)=∑u=1UĤ(u)V̂(u)P(u)x(u)s(u)(l)+n̂(l),
where(5)n̂(l)≜∑u=1UΞ(u)V̂(u)P(u)x(u)s(u)(l)+n(l),n̂(l) is the equivalent additive noise at the lth instant, which is consisted of the interference caused by the channel estimation errors of all the interfering users and the complex Gaussian noise, respectively.The variance of the equivalent noisen̂(l) is calculated as follows.First, letG(u)=Ξ(u)V̂(u); the mean and variance of G(u) can be evaluated as(6)ε{G(u)}=ε{Ξ(u)V̂(u)}=ε[Ξ(u)]V̂(u)=0,(7)ε{G(u)[G(u)]H}=ε{Ξ(u)V̂(u)[V̂(u)]H[Ξ(u)]H}=ε{Ξ(u)[Ξ(u)]H}=σe2InR.
Clearly, the components of G(u) are i.i.d zero-mean complex Gaussian variables with the variance of σe2, thus statistical distributions can be expressed as gij(u)~𝒞𝒩(0,σe2) and ε{gki(u)[gki′(u′)]}=σe2δ(i-i′)δ(u-u′).Then, letf(l)=∑u=1UG(u)P(u)x(u)s(u)(l); the mean and variance of f(l) can be evaluated as(8)ε[f(l)]=ε[∑u=1UG(u)P(u)x(u)s(u)(l)]=∑u=1Uε[G(u)]P(u)x(u)s(u)(l)=0,(9)f(l)=∑u=1UG(u)P(u)x(u)s(u)(l)=∑u=1U{[g11(u)⋯g1k(u)⋯g1nT(u)⋮⋱⋮⋱⋮gk1(u)⋯gkk(u)⋯gknT(u)⋮⋱⋮⋱⋮gnR1(u)⋯gnRk(u)⋯gnRnT(u)]×[P1(u)⋯0⋯0⋮⋱⋮⋱⋮0⋯Pk(u)⋯0⋮⋱⋮⋱⋮0⋯0⋯PnT(u)]×[x1(u)s(u)(l)⋮xk(u)s(u)(l)⋮xnT(u)s(u)(l)]}=[∑u=1U∑i=1nTg1i(u)Pi(u)xi(u)s(u)(l)⋮∑u=1U∑i=1nTgki(u)Pi(u)xi(u)s(u)(l)⋮∑u=1U∑i=1nTgnRi(u)Pi(u)xi(u)s(u)(l)]=[f1(l)⋮fk(l)⋮fnR(l)].
Clearly, f(l) is a zero-mean complex Gaussian column vector with its component fk(l)=∑u=1U∑i=1nTgki(u)Pi(u)xi(u)s(u)(l),k=1,…,nR,l=1,…,LS, whose variance can be evaluated as(10)σfk2(l)=ε{[fk(l)][fk(l)]*}=ε{[∑u=1U∑i=1nTgki(u)Pi(u)xi(u)s(u)(l)]×[∑u′=1U∑i′=1nTgki′(u′)Pi′(u′)xi′(u′)s(u′)(l)]*}.
Since the normalized spreading codes are used, that is, [su(l)]2=1/LS and ε{gki(u)[gki′(u′)]}=σe2δ(i-i′)δ(u-u′), (10) can be written as(11)σfk2(l)=∑u=1U∑i=1nTε{[gki(u)Pi(u)xi(u)][gki(u)Pi(u)xi(u)]*}×{su(l)[su(l)]*}=1LS∑u=1U∑i=1nTPi(u)ε{gki(u)[gki(u)]*}ε{xi(u)[xi(u)]*}=1LS∑u=1U∑i=1nTPi(u)ε{|gki(u)|2}ε{|xki(u)|2}=σe2LS∑u=1U∑i=1nTPi(u)=UnTσe2LS.
Because n̂(l)=f(l)+n(l), the component of n̂(l) is given as n̂k(l)=fk(l)+nk(l),k=1,…,nR by the aforementioned definition with the variance calculated as(12)σn̂k2(l)=ε[n̂k(l)n̂k*(l)]=ε{[fk(l)+nk(l)][fk(l)+nk(l)]*}=ε{|fk(l)|2}+ε{|nk(l)|2}=UnTσe2LS+σn2.
Each component of n̂(l) is i.i.d zero-mean complex Gaussian variables with the variance of UnTσe2/LS+σn2, so the variance of n̂(l) is σn̂2(l)=(UnTσe2/LS+σn2)InR. When σe2=0, that is, the channel matrix is estimated perfectly, the variance of the equivalent noise is σn̂2=σn2, which is equal to the variance of the scenario where the channel is estimated perfectly.


## 3.2. Decorrelating Scheme

Optimal coherent and noncoherent detection based on maximum likelihood (ML) processing can be readily derived from (4) [14, 15]. These optimal detectors perform joint detection for all users and incur an exponential complexity in both the number of active users and that of transmit antennas. The enormous complexity of the optimum receivers renders them mainly of theoretical interest.In this part, a suboptimal decorrelating receiver is presented which decouples the detection of different users with a linear complexity in the number of active users [16].In order to express it compactly, we define(13)Y≜[y(1),…,y(l),…,y(LS)],N̂≜[n̂(1),…,n̂(l),…,n̂(LS)],s(u)≜[s(u)(1),…,s(u)(l),…,s(u)(LS)],S≜{[s(1)]T,…,[s(u)]T,…,[s(U)]T}T,Â(u)≜Ĥ(u)V̂(u)P(u)x(u),Â≜[Â(1),…,Â(u),…,Â(U)].
Then, (4) is expressed as(14)Y=ÂS+N̂.
The variance of any component of N̂ is σn̂2(l)=(UnTσe2/LS+σn2)InR, so the variance of N̂ is σN̂2(l)=ε[N̂N̂H]=(UnTσe2/LS+σn2)InR too.Since the information about all the transmitted symbols is contained inÂ, the key to the differential and coherent receivers introduced in this paper is to obtain an initial estimate of Â, which is used for the subsequent detection.When we assume thatSSH has a full rank, the ML estimate of Â, which is conditioned on {Ĥ(u)}u=1U, {V̂(u)}u=1U, P(u) and {S(u)}u=1U, is given by(15)R=YSH(SSH)-1.
The equation of (15) is suitable for all the spreading structure, but if the nonorthogonal spreading structures, such as GOLD code, are used, it is very difficult to get the variance of the equivalent noise. We use normalized orthogonal WALSH code in this paper, which satisfies SSH=IU.The ML estimateR is effectively the output of a decorrelator with the input being the received matrix Y. The decorrelator mitigates the MUI and decouples the detection of different users.Then we study the impact of the decorrelation on the variance of the equivalent noise, via (14), (15) which can be rewritten as(16)R=(ÂS+N̂)SH(SSH)-1=ÂSSH(SSH)-1+N̂SH(SSH)-1=Â+N̂SH=Â+N̂̂,
where N̂̂=N̂SH is the equivalent noise after decorrelating and the variance of N̂̂ is calculated as follows:(17)σN̂̂2=ε{N̂̂N̂̂H}=ε{[N̂SH][N̂SH]H}=ε{N̂SHSN̂H}=ε{N̂N̂H}=(UnTσe2LS+σn2)InR,
that is, the variance of equivalent noise is unchanged after decorrelating.Due to the block structure ofÂ, the ML estimate of the desired user denoted by r(u) can be easily obtained from R, which is the uth column of R(18)r(u)=R(:,u)=Ĥ(u)V̂(u)P(u)x(u)+N̂̂(:,u)=Λ̂(u)P(u)x(u)+n̂̂(u),
where n̂̂(u)=N̂̂(:,u) is the uth column of N̂̂, whose variance is σn̂̂(u)2=(UnTσe2/LS+σn2)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 the transmitter, and the decoupled signal is first orthogonally converted byΦ̂(u), which is denoted as(19)r̃(u)=[Φ̂(u)]Hr(u)=[Φ̂(u)]HĤ(u)V̂(u)P(u)x(u)+[Φ̂(u)]Hn̂̂(u)=Λ̂(u)P(u)x(u)+n̂̂̂(u),
where n̂̂̂(u)=[Φ̂(u)]Hn̂̂(u) the variance of the equivalent noise n̂̂̂(u) is calculated as follows:(20)ε{n̂̂̂(u)(l)}=ε{[Φ̂(u)]Hn̂̂(u)}=[Φ̂(u)]Hε{n̂̂(u)}=0nR,(21)ε{[n̂̂̂(u)(l)][n̂̂̂(u)(l)]H}=ε{[n̂̂̂(u)(l)]H{[Φ̂(u)]Hn̂̂(u)}×{[Φ̂(u)]Hn̂̂(u)}H}=ε{[Φ̂(u)]Hn̂̂(u)[n̂̂(u)]HΦ̂(u)}=[Φ̂(u)]Hε{n̂̂(u)[n̂̂(u)]H}Φ̂(u)=(UnTσe2LS+σn2)InR.
Clearly, the components of n̂̂̂(u) are also i.i.d zero-mean Gaussian variables with the variance of (UnTσe2/LS+σn2).


## 3.3. SVD-Aided Beamforming and Power Allocation Scheme

Given the proposed equivalent multiuser Turbo-BLAST system model, we give an SVD-aided beamforming and transmit power allocation scheme in this part, which aims at maximizing the capacity performance of the desired user in the presence of imperfect CSI.When CSI is imperfect, the right singular matrix ofĤ(u) is used for beamforming and the channel capacity of the desired user with SVD-aided beamforming and power allocation can be expressed as [1, 18](22)C(u)[λ̂̅(u)]=∑k=1nTlog2[1+Pk(u)λ̂k(u)(UnTσe2)/LS+σn2],u=1,…,U,
where λ̂̅(u)=diag[λ̂1(u),…,λ̂nT(u)] is the diagonal matrix with the eigenvalues of Ĥ(u)[Ĥ(u)]H and Pk(u) is the transmit power allocated for the kth subchannel of the desired user, which satisfies the total transmit power constrain. The capacity subject to an optimization problem is given as below(23)[P1(u),…,PnT(u)]opt=argmax⁡⁡{∑k=1nTlog2[1+Pk(u)λ̂k(u)(UnTσe2)/LS+σn2]}subjectto∑k=1nTPk(u)=PTotal.
The Lagrange multiplier method is employed to find the optimum power allocation matrix that can maximize the capacity with total power constraint. After solving the equations in (23), we find that the optimal power allocated for the kth subchannel is(24)Pk(u)={μ-(UnTσe2LS+σn2)[λ̂k(u)]-1}+,
where (x)+≜max{x,0} and μ is the Lagrange multiplier restricted by ∑k=1nT{μ-(UnTσe2/LS+σn2)[λ̂k(u)]-1}+=PTotal.In a frequency-nonselective, Rayleigh fading MIMO channel, the channel capacity is the statistical mean ofλ̂̅(u) approximately in the presence of CSI imperfection.(25)C(u)=ελ̂̅[C(u)(λ̂̅(u))]=ε{∑k=1nTlog2[1+Pk(u)λ̂k(u)(UnTσe2)/LS+σn2]}=ε{∑k=1nT[log2(μλ̂k(u))]+}.
Because {λ̂k(u)}k=1nT are i.i.d variables and all of them have the same probability density p[λ̂k(u)], formula (25) can be expressed as(26)C(u)=nT∫0+∞[log2(μλ̂k(u))]p(λ̂k(u))dλ̂k(u),u=1,…,U.
Since singular values are random variables, it is hard to achieve the function of probability density straightly. The channel capacity can be calculated by averaging (26) over a large number of channel realizations, which is expressed as(27)C(u)≈ε{∑k=1nTlog2[1+Pk(u)λ̂k(u)(UnTσe2)/LS+σn2]}≈1N∑i=1N{∑k=1nTlog2[1+Pk,i(u)λ̂k,i(u)(UnTσe2)/LS+σn,i2]},u=1,…,U,
where N is the number of channel realizations, which is used for simulation, {Pk,i(u)}k=1nT are the transmit power sequence for ith channel matrix Ĥi(u), {λ̂k,i(u)}k=1nT are eigenvalues of Ĥi(u)[Ĥi(u)]H, and σe,i2 and σn,i2 are the variances of channel estimation error and AWGN, respectively.


## 3.4. Iterative Detection Algorithm for the Uplink of Multiuser Turbo-BLAST Systems

In this section, an iterative detection strategy is employed to enhance the error performance [4] in the presence of imperfect CSI. Because the orthonormal spreading codes are used, the MUI is removed absolutely after decorrelating, so the iterative detection of every user is completely independent from each other.Letxk(u) be the kth transmitted signal of desired user at a sampling epoch, by (19); the received symbol vector r̃(u) which is corrupted by the channel noise and interferences, can be written as the sum of desired response, the CAI, and the equivalent noise:(28)r̃(u)=λ̂k(u)Pk(u)xk(u)+Λ̂k̅(u)Pk̅(u)xk̅(u)+n̂̂̂(u).
In (27), λ̂k(u) is the kth column of the matrix Λ̂(u), Pk(u) is the transmit power allocated to the kth subchannel of the desired user, and Λ̂k̅(u),xk̅(u),Pk̅(u) is defined as follows:(29)Λ̂k̅(u)=[λ̂1(u),…,λ̂k-1(u),λ̂k+1(u),…,λ̂nT(u)],xk̅(u)=[x1(u),…,xk-1(u),xk+1(u),…,xnT(u)]T,Pk̅(u)=diag⁡(P1(u),…,Pk-1(u),Pk+1(u),…,PnT(u)).
By (29), the decision statistic of kth substream of the uth user using a linear filter ŵk(u) can be expressed as(30)yk(u)=[ŵk(u)]Hλ̂k(u)Pk(u)xk(u)︸q̂k(u)+[ŵk(u)]HΛ̂k̅(u)Pk̅(u)xk̅(u)︸d̂k(u)+[ŵk(u)]Hn̂̂̂(u)︸ẑk(u),
where q̂k(u), d̂k(u), and ẑk(u) are the desired response obtained by the linear filter, the coantenna interference, and the phase-rotated equivalent noise, respectively.The CAI can be removed fromyk(u) by the proposed iterative detector and soft interference cancellation based on mean square error (MMSE) principle. The improved estimation of the transmitted symbol x̃k(u) can be formulated as(31)x̃k(u)=[ŵk(u)]Hr̃(u)-d̂k(u),
where x̃k(u) is the estimate of the symbol xk(u) and d̂k(u) is the linear combination of the interfering substreams. The estimation error is defined as Δxk(u)=x̃k(u)-xk(u). The weighted vector ŵk(u) and the interference combination d̂k(u) are optimized by minimizing the mean-square estimation error Δxk(u) between each substream and the related estimation, by the following cost function:(32)[w̅̂k(u),d̂k(u)]=argminε[‖x̃k(u)-xk(u)‖2],
where the expectation is taken over the equivalent noise n̂̂̂ and the statistics of the data sequence x(u).We use standard minimization techniques to solve the optimization problem formulated in (32). In order to arrive at the solution, we write the cost function as(33)Cost(u)=ε[‖x̃k(u)-xk(u)‖2]=ε[‖(ŵk(u))Hr̃(u)-d̂k(u)-xk(u)‖2]=[ŵk(u)]Hε{r̃(u)[r̃(u)]H}ŵk(u)-[ŵk(u)]Hε{r̃(u)[d̂k(u)+xk(u)]*}-ε{r̃(u)[d̂k(u)+xk(u)]*}Hŵk(u)+ε{[d̂k(u)+xk(u)][d̂k(u)+xk(u)]*},
where(34)ε{r̃(u)[r̃(u)]H}=ε{[λ̂k(u)n̂̂̂(u)]H[+Λ̂k̅(u)Pk̅(u)xk̅(u)+n̂̂̂(u)λ̂k(u)Pk(u)xk(u)+Λ̂k̅(u)Pk̅(u)xk̅(u)+n̂̂̂(u)Pk(u)xk(u)]×[+n̂̂̂(u)λ̂k(u)Pk(u)xk(u)+Λ̂k̅(u)Pk̅(u)xk̅(u)+n̂̂̂(u)Pk(u)xk(u)]H}=[λ̂k(u)Pk(u)][λ̂k(u)Pk(u)]H+[Λ̂k̅(u)Pk̅(u)]ε{xk̅(u)[xk̅(u)]H}×[Λ̂k̅(u)Pk̅(u)]H+(UnTσe2LS+σn2)InRε{r̃(u)[d̂k(u)+xk(u)]*}=ε{[n̂̂̂(u)λ̂k(u)Pk(u)xk(u)+Λ̂k̅(u)Pk̅(u)xk̅(u)+n̂̂̂(u)Pk(u)xk(u)]×[d̂k(u)+xk(u)]*[λ̂k(u)Pk(u)]H}=λ̂k(u)Pk(u)+Λ̂k̅(u)Pk̅(u)ε[xk̅(u)][d̂k̅(u)]*.
Via (34), the cost function in (33) can be further evaluated as(35)Cost(u)=[ŵk(u)]H{[λ̂k(u)Pk(u)][λ̂k(u)Pk(u)]H+[Λ̂k̅(u)Pk̅(u)]ε{xk̅(u)[xk̅(u)]H}[Λ̂k̅(u)Pk̅(u)]H+(UnTσe2LS+σn2)InR}ŵk(u)-[ŵk(u)]H{λ̂k(u)Pk(u)+Λ̂k̅(u)Pk̅(u)ε[xk(u)][d̂k(u)]*}-{λ̂k(u)Pk(u)+Λ̂k̅(u)Pk̅(u)ε{xk(u)}[d̂k(u)]*}Hŵk(u)+ε{d̂k(u)[d̂k(u)]*+xk(u)[d̂k(u)]*+d̂k(u)[xk(u)]*+xk(u)[xk(u)]*}.
The linear combination of interfering substreams d̂k(u) and the weighted vector ŵk(u) is obtained in (36) by letting ∂Cost(u)/∂[d̂k(u)]*=0 and ∂Cost(u)/∂[ŵk(u)]*=0,respectively,(36)d̂k(u)=[ŵk(u)]H[λ̂k(u)Pk(u)][λ̂k(u)Pk(u)]Hε[xk̅(u)],ŵk(u)=[Q(u)+S(u)+(UnTσe2LS+σn2)InR]-1[λ̂k(u)Pk(u)],
where(37)Q(u)=[λ̂k(u)Pk(u)][λ̂k(u)Pk(u)]H,S(u)=[Λ̂k̅(u)Pk̅(u)]{I(nT-1)-diag⁡{ε[xk̅(u)]ε[xk̅(u)]H}}×[Λ̂k̅(u)Pk̅(u)]H.
Therefore, for the desired user, the weighted vector ŵk(u) is used for the iterative detection in the presence of imperfect CSI. Thus (31) can be rewritten as(38)x̃k(u)=[ŵk(u)]Hr̃(u)-d̂k(u)=[λ̂k(u)Pk(u)]H×{[Q(u)+S(u)+(UnTσe2LS+σn2)InR]-1}H×{r(u)-[Λ̂k̅(u)Pk̅(u)]ε[xk̅(u)]}.
For the first iteration, we assume ε[xk̅(u)]=0 and thus the linear MMSE detection for the kth substream of the desired user becomes(39)x̃k(u)=[λ̂k(u)Pk(u)]H×{[Q(u)+S(u)+(UnTσe2LS+σn2)InR]-1}Hr̃(u).
Next, we assume that ε[xk̅(u)]→xk̅(u) with the increasing number of iterations, the CAI interference canceller for the kth substream can be reduced to the linear MMSE receiver expressed as (40), where the SVD-assisted beamforming and power allocation scheme are used(40)x̃k(u)=[λ̂k(u)Pk(u)]H×{[Q(u)+S(u)+(UnTσe2LS+σn2)InR]-1}H×{r̃(u)-[Λ̂k̅(u)Pk̅(u)]xk̅(u)}.The performance of iterative scheme depends on the veracity of channel estimation matrix and the power allocation scheme


## 4. Simulation Results

In this section, we compare the capacity and BER performance of the traditional and modified multiuser Turbo-BLAST systems, where the traditional multiuser Turbo-BLAST systems adopt the equal power allocated strategy and without the SVD-aided preprocessing and postprocessing, denoted by “EPA”, and the modified multiuser Turbo-BLAST systems employ the proposed SVD-aided beamforming and transmit power allocation scheme, denoted by “BF-TPA”, respectively.At the transmitter, the data stream is first encoded by a rate-1/2 convolutional code with generator (7,5), modulated by 8-PSK modulation scheme, and spread by WALSH codes, whose capacity performance over a large number of channel realizations is exhibited in Figures3–5, extra capacity, which is defined as the capacity gap between the modified multiuser Turbo-BLAST system and the conventional one, is given in Figure 6, and finally, BER performance is presented in Figures 7-8 with different inaccuracy of channel estimation and different number of active users. 4 active users are considered in Figure 3 and Figures 5–7.Figure 3
Capacity performance of traditional and modified multiuser Turbo-BLAST systems (U=4).Figure 4
Capacity performance of the modified multiuser Turbo-BLAST systems with different number of active users in the presence of imperfect CSI (σe2=0.01).Figure 5
Capacity performance of the modified multiuser Turbo-BLAST system in different values ofEb/N0 with imperfect CSI (U=4).Figure 6
Extra capacity results between modified and traditional multiuser Turbo-BLAST systems with sixteen receive antennas and various transmit antennas configuration under the conditions of imperfect CSI (σe2=0.01, U=4).Figure 7
BER performance of traditional and modified Turbo-BLAST systems under the conditions of different inaccuracy of channel estimation (U=4).Figure 8
BER performance of modified Turbo-BLAST systems with different number of active users under the conditions of imperfect CSI (σe2=0.01).Figure3 shows capacity performance of traditional and modified multiuser Turbo-BLAST systems with four transmit and four receive antennas under the conditions of different channel estimation errors. Figure 3 indicates that the proposed SVD-aided beamforming and transmit power allocation strategy is an effective means to improve capacity performance even with the imperfect CSI, and the larger capacity gains can be achieved for lower Eb/N0 and/or higher σe2. For example, at Eb/N0=10 dB, there are 0.4(bit/s/Hz) and 0.8 (bit/s/Hz) capacity gains for σe2=0.1 and σe2=0.4, respectively.The capacity performance of the modified multiuser Turbo-BLAST systems with different number of active users in the presence of imperfect CSI are given in Figure4, where U denotes the number of active users. It is clear that from Figure 4, in the presence of imperfect CSI, the capacity is lower for larger number of active users, and that the impact of the number of active users on the capacity performance is higher for higher Eb/N0. For instance, at Eb/N0=0 dB, the capacity of desired user is almost the same for different number of active users. As Eb/N0 becomes high, the capacity degradation becomes large with the number of active users; for example, at Eb/N0=30 dB, there are 17 (bit/s/Hz) capacity degradation for the desired user  when system with 64 active users is compared to that with only one active user. This is due to the existence of the channel estimation error, which will make the equivalent noise of the system large, especially for large U (which can be seen from (17) and (21)). This large noise will degrade the system performance greatly because it can affect the following iterative detection and the resultant capacity calculation.Figure5 shows the capacity performance of the modified multiuser Turbo-BLAST systems in different values of Eb/N0 with imperfect CSI. The simulation results are obtained for five 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 σe2<10-3, the actual capacity is almost constant, as the CSI imperfection gets more severe; in particular, σe2>10-2; the capacity becomes very sensitive to the CSI imperfection. And we see that the CSI imperfection on the capacity performance is higher for higher Eb/N0 because for higher Eb/N0, that is, σn2→0, the variance of the equivalent noise mainly depends on the error of channel estimation, and therefore, the capacity performance becomes sensitive to the CSI imperfection.In Figure6, we give the extra capacity results, which are defined as the capacity gap between modified and traditional multiuser Turbo-BLAST systems with sixteen receive antennas and various transmit antennas configuration under the conditions of imperfect CSI (σe2=0.01) for five different Eb/N0 values of 0 dB, 5 dB, 10 dB, 15 dB, and 20 dB. It is clear that from Figure 6, we can see the extra capacity is higher for lower Eb/N0 and/or large number of transmit antennas in the presence of CSI imperfection. For instance, at Eb/N0=0 dB, there are 16 (bit/s/Hz) capacity gains for the multiuser Turbo-BLAST system with sixteen transmit antennas, while only 3 (bit/s/Hz) capacity gains can be achieved for twelve transmit antennas.BER performance of traditional and modified multiuser Turbo-BLAST systems under the condition of different inaccuracy of channel estimation is shown in Figure7, where I denotes the number of iterative detection. It is observed from Figure 7 that as the BER results are concerned, the modified multiuser Turbo-BLAST system outperforms the traditional one under the same conditions, regardless of the status of CSI. This implies that the proposed SVD-aided beamforming and power allocation strategy are also valid to improve BER performance even with the imperfect CSI. For example, at a BER of 10-3,we can see that there are 4 dB gains for the modified system over the traditional one in the 1st detection iteration under the condition of imperfect CSI, that is, σe2=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 gains attained for the modified system, where BF-TPA scheme is adopted through 3rd iterative detection under the condition of σe2=0.01, which indicates that the proposed iterative detection scheme is an effective means to enhance the BER performance further even with imperfect CSI.Figure8 gives the BER performance of the modified Turbo-BLAST system with different number of active users under the condition of imperfect CSI (σe2=0.01), where U denotes the number of active users and I denotes the number of iterative detection. Figure 8 shows that at the same inaccuracy of channel estimation and for the identical number of iterative detection, the BER performance gets worse for large number of active users. For example, at a BER of 10-4, comparing the system that has sixty-four active users with the one having only one active user, there is 4.5 dB degradation of Eb/N0. This is because the variance of each components of the equivalent noise is UnTσe2/LS+σn2; when the multiuser system was with more active users; the variance is approximately linear to the number of active users, so the BER results get worse with the increase of the number of active users.


## 5. Conclusions

In this paper, we first introduce a CDMA-based multiuser Turbo-BLAST system model, based on the imperfect channel model, the equivalent system model, and the variance of the equivalent noise. In order to maximize the capacity performance of the desired user, the SVD-assisted beamforming and power allocation strategy are theoretically obtained and the Lagrange multiplier method is then employed to find the optimized power allocation matrix, which is subject to the total transmit power constraint. And then, to circumvent the complexity of multiuser detection, the suboptimal decorrelating scheme is proposed to decouple the detection of different users. Finally, the iterative detection technique is adopted after postprocessing to enhance the BER results further. The complexity of the suboptimal receivers is exponential only in the transmission rate, similarly to single-user Turbo-BLAST receivers. Numerical results show that the newly introduced method is effective to enhance the capacity results and the BER performance of multiuser Turbo-BLAST systems in the presence of imperfect CSI.



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*Source: 357381-2012-04-22.xml*