# SVD-Aided Power Allocation and Iterative Detection Scheme for Turbo-BLAST System with Imperfect Channel State Information

Authors: Xiaomin Chen; Xiangbin Yu; Dazhuan Xu
Journal: International Journal of Antennas and Propagation
 (2011)

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## Abstract

A new technique that combines adaptive power allocation and iterative detection based on singular value decomposition (SVD) is introduced for the modified Turbo-BLAST system with imperfect channel state information (I-CSI). At the transmitter, in order to maximize the capacity performance, the MIMO channel is decomposed into several parallel eigen subchannels by SVD, and then proper power based on the water-filling principle is allocated to every subchannel subject to the total transmit power constraint. At the receiver, the modified MMSE detector taking the CSI imperfection into account is used to remove the coantenna interference, and then the turbo idea is employed for iterative detection to lower the system BER. As a result, the BER performance is effectively enhanced. Numerical results show that the introduced SVD-aided adaptive power allocation method is valid to improve not only the capacity but also the BER performance in the presence of channel state information imperfection, while the iterative detector can further lower the BER results.


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## Body

## 1. Introduction

Multiple-input multiple-output (MIMO) systems can provide a significant capacity increment over the conventional one through appropriate space-time processing [1, 2]. Bell labs layered space-time (BLAST) structure is one of the promising architectures thanks to the advantages of low detection complexity and high data rates [3]. The new system which combined BLAST structure with turbo idea is called Turbo-BLAST that can offer a reliable and practical solution to high data-rate transmission for wireless communications [4].One of the attractive features of MIMO system is a multiplexing gain and consequently a higher capacity performance over single-input single-output system. When channel state information (CSI) is perfect, the water-filling power allocation strategy [5–7] is optimized to improve the capacity performance for MIMO systems. Practically, CSI at the receiver is subject to the error due to nonreal time data processing, and so forth [8]. For the modified Turbo-BLAST system, the task of designing the effective link adaptive technique becomes more challenging in the presence of CSI imperfection. Recently, various contributions have also considered the MIMO transceiver design problem for systems with either partial or imperfect CSI at the transmitter [9–12]. In these references, the effect of I-CSI on the capacity performance of MIMO system is investigated in [9, 10]. Based on the perfect and imperfect CSI, the performance of adaptive modulation schemes in MIMO systems is analyzed in [11]. The transmit power allocation for a two-input multiple-output system is presented in [12] by minimizing the BER, but the scheme is only suitable for the systems with two-transmit antennas. The above references, however, consider the uncoded MIMO system only, and thus the superiority of channel coding is not fully utilized. Especially, the Turbo-BLAST structure with superior performance is not considered. As a result, the performance improvement is limited.Motivated by the reason above, we will study the performance of Turbo-BLAST system in this paper and develop a new SVD-aided adaptive power allocation strategy and iterative detection scheme for modified Turbo-BLAST system in the presence of CSI imperfection. At the transmitter, aiming at maximizing the capacity performance of the overall system, MIMO channel is decomposed into several parallel eigen subchannels by singular value decomposition (SVD) [13–16], and then appropriate power based on water-filling principle is allocated to each subchannel subject to the total transmit power constraint. At the receiver, the iterative turbo strategy is employed for signal detection to enhance the BER performance. The theoretical analysis and numerical results show that the proposed SVD-assisted transmit power allocation (TPA) approach is an effective method to improve both the capacity and the BER performance. Furthermore, the corresponding BER results of the modified Turbo-BLAST system can be further enhanced through the iterative detection.This paper is organized as follows: after the introduction in Section1, the channel model and system description are briefly introduced in Section 2. Section 3 is concerned with SVD-aided power allocation scheme in the presence of imperfect CSI. Section 4 presents an iterative detection algorithm for the modified Turbo-BLAST system with adaptive power allocation. The numerical results are given in Section 5. Finally, Section 6 summarizes the conclusion of the paper.


## 2. Channel Model and System Description

### 2.1. Channel Model

Consider a Turbo-BLAST system withnT transmit antennas and nR receive antennas (nT≤nR) as depicted in Figure 1. Based on CSI obtained from the feedback channel, the transmitter optimally varies its power allocation parameters as will be detailed in Section 3. We adopt a CSI imperfection model from [8]. The CSI is estimated at the receiver and sent to the transmitter by feedback channel. To assist the receiver in performing channel estimation and symbol detection, known pilot symbols are periodically inserted at the transmitter—a technique that is known as pilot symbol assisted modulation (PSAM) [17]. At the receiver, the samples corresponding to the known pilots are extracted, based on which CSI is interpolated using optimal Wiener filtering [17].Figure 1
An illustrative diagram of the modified Turbo-BLAST system.The imperfect CSI is represented by the complex channel matrixĤ, which is related to the true channel matrix H as (1)H=Ĥ+Ξ,
where Ξ is a complex matrix related to the imperfect CSI. Note that the same CSI is known to both the transmitter and receiver. The elements of both Ĥ and Ξ can be modeled as independent complex Gaussian variables [8], which subsequently lead to the entries of H as complex Gaussian variables, and furthermore, Ĥ and Ξ are independent to each other [8, 11]. Thus we can express their statistical distributions as eij~C𝒩(0,σe2), ĥij~C𝒩(0,1-σe2), and hij~C𝒩(0,1), which are all distributed by complex Gaussian law with σe2 indicating the accuracy of the CSI.In order to maximize the capacity of the Turbo-BLAST system, the MIMO channel is decomposed into several parallel eigen subchannels by SVD. In practice, true CSI cannot be obtained, soĤ is treated as the true channel matrix which can be decomposed as(2)Ĥ=ÛΛ̂V̂H,
where (·)H denotes a matrix conjugate transpose, Û and V̂ are the unitary matrices with left and right singular vector of Ĥ as their columns, Λ̂ is a nonnegative and diagonal matrix, {λ̂i}i=1nT are the main diagonal elements of Λ̂, and {λ̂i}i=1nT are eigenvalues of ĤĤH, respectively. The left singular vector V̂ is used for the input signal shaping at the transmitter and the conjugate transpose of right singular vector Û is used for orthogonal transform at the receiver.


### 2.2. System Description

In this part we propose a system model for a modified Turbo-BLAST system using a novel SVD-assisted power allocation strategy with imperfect CSI.Figure1 shows an illustrative diagram of a modified Turbo-BLAST system. At the transmitter, the input data stream is firstly encoded and then mapped into a symbol stream that is subsequently demultiplexed into nT sub-streams, and the transmit power at each eigen subchannel is varied according to the instantaneous values and/or statistics of the subchannel gains under an total transmit power constraint.Letx=[x1,…,xnT]T and r=[r1,…,rnR]T denote input and output column vectors at a sampling instant, respectively. Similarly, n=[n1,…,nnR]T is a column vector, where each component is independent and identically distributed (i.i.d) zero-mean complex Gaussian variables with variance σn2.The transmitted signals are received by thenR antennas at a sampling instant which can be expressed as (3)r=ÛH(HV̂Px+n)=ÛHHV̂Px+ÛHn,
where P=diag(P1,…,PnT) is the diagonal transmit power matrix with total power constraint ∑k=1nTPk=nT.At the receiver, an iterative detection strategy based on the “turbo” principle is used for the symbol detection. The encoding and demultiplexing for the data stream can be equivalently viewed as a serially concatenated encoding process as depicted in Figure1. The concatenated code can be decoded using a low-complexity iterative decoder which is similar to the iterative decoder for a serially concatenated turbo code. For the iterative decoding, the optimal decoding process can be separated into two stages, that is, soft-input/soft-output (SISO) detector and SISO channel decoder, respectively, which mutually exchange the extrinsic information sent from one stage to another iteratively until the decoding process converges.


## 2.1. Channel Model

Consider a Turbo-BLAST system withnT transmit antennas and nR receive antennas (nT≤nR) as depicted in Figure 1. Based on CSI obtained from the feedback channel, the transmitter optimally varies its power allocation parameters as will be detailed in Section 3. We adopt a CSI imperfection model from [8]. The CSI is estimated at the receiver and sent to the transmitter by feedback channel. To assist the receiver in performing channel estimation and symbol detection, known pilot symbols are periodically inserted at the transmitter—a technique that is known as pilot symbol assisted modulation (PSAM) [17]. At the receiver, the samples corresponding to the known pilots are extracted, based on which CSI is interpolated using optimal Wiener filtering [17].Figure 1
An illustrative diagram of the modified Turbo-BLAST system.The imperfect CSI is represented by the complex channel matrixĤ, which is related to the true channel matrix H as (1)H=Ĥ+Ξ,
where Ξ is a complex matrix related to the imperfect CSI. Note that the same CSI is known to both the transmitter and receiver. The elements of both Ĥ and Ξ can be modeled as independent complex Gaussian variables [8], which subsequently lead to the entries of H as complex Gaussian variables, and furthermore, Ĥ and Ξ are independent to each other [8, 11]. Thus we can express their statistical distributions as eij~C𝒩(0,σe2), ĥij~C𝒩(0,1-σe2), and hij~C𝒩(0,1), which are all distributed by complex Gaussian law with σe2 indicating the accuracy of the CSI.In order to maximize the capacity of the Turbo-BLAST system, the MIMO channel is decomposed into several parallel eigen subchannels by SVD. In practice, true CSI cannot be obtained, soĤ is treated as the true channel matrix which can be decomposed as(2)Ĥ=ÛΛ̂V̂H,
where (·)H denotes a matrix conjugate transpose, Û and V̂ are the unitary matrices with left and right singular vector of Ĥ as their columns, Λ̂ is a nonnegative and diagonal matrix, {λ̂i}i=1nT are the main diagonal elements of Λ̂, and {λ̂i}i=1nT are eigenvalues of ĤĤH, respectively. The left singular vector V̂ is used for the input signal shaping at the transmitter and the conjugate transpose of right singular vector Û is used for orthogonal transform at the receiver.


## 2.2. System Description

In this part we propose a system model for a modified Turbo-BLAST system using a novel SVD-assisted power allocation strategy with imperfect CSI.Figure1 shows an illustrative diagram of a modified Turbo-BLAST system. At the transmitter, the input data stream is firstly encoded and then mapped into a symbol stream that is subsequently demultiplexed into nT sub-streams, and the transmit power at each eigen subchannel is varied according to the instantaneous values and/or statistics of the subchannel gains under an total transmit power constraint.Letx=[x1,…,xnT]T and r=[r1,…,rnR]T denote input and output column vectors at a sampling instant, respectively. Similarly, n=[n1,…,nnR]T is a column vector, where each component is independent and identically distributed (i.i.d) zero-mean complex Gaussian variables with variance σn2.The transmitted signals are received by thenR antennas at a sampling instant which can be expressed as (3)r=ÛH(HV̂Px+n)=ÛHHV̂Px+ÛHn,
where P=diag(P1,…,PnT) is the diagonal transmit power matrix with total power constraint ∑k=1nTPk=nT.At the receiver, an iterative detection strategy based on the “turbo” principle is used for the symbol detection. The encoding and demultiplexing for the data stream can be equivalently viewed as a serially concatenated encoding process as depicted in Figure1. The concatenated code can be decoded using a low-complexity iterative decoder which is similar to the iterative decoder for a serially concatenated turbo code. For the iterative decoding, the optimal decoding process can be separated into two stages, that is, soft-input/soft-output (SISO) detector and SISO channel decoder, respectively, which mutually exchange the extrinsic information sent from one stage to another iteratively until the decoding process converges.


## 3. Power Allocation Scheme with Imperfect CSI

### 3.1. Equivalent System Model

In this subsection, we propose an equivalent system model under the conditions of imperfect CSI for the modified Turbo-BLAST system.With the channel model and the SVD technique introduced aforementioned, when the CSI is imperfect, the received signal at a sampling instant can be formulated as(4)r=ÛH(Ĥ+Ξ)V̂Px+ÛHn=ÛHĤV̂Px+ÛHΞV̂Px+ÛHn=ÛHÛΛ̂V̂HV̂Px+ÛHΞV̂Px+ÛHn=Λ̂Px+ÛHΞV̂Px+n′=Λ̂Px+n̂,
where n′=ÛHn, Û is the unitary matrix with the right singular vectors of Ĥ as its columns, thus n′=ÛHn is also a column vector of additive Gaussian noise variables, where each component is i.i.d zero-mean complex Gaussian variables with the variance of σn2, n̂=ÛHΞV̂Px+n′ is an equivalent additive noise consisting of the interference caused by the channel estimation errors and the complex Gaussian noise, respectively. Here we make an approximation that assumes that the entries of x are i.i.d. Gaussian variables, that is, xi~C𝒩(0,1) [11]. Under this assumption, the variance of the equivalent noise n̂ can be calculated as follows.Firstly, letG=ÛHΞV̂, the mean and variance of G can be evaluated as(5)ε{G}=ε{ÛHΞV̂}=ÛHε{Ξ}V̂=0nR×nT,ε{GGH}=ε{ÛHΞV̂V̂HΞHÛ}=ε{ÛHΞΞHÛ}=ÛHε{ΞΞH}Û=σe2InR.
Clearly, the components of G are i.i.d zero-mean complex Gaussian variables with the variance of σe2: thus we can express the statistical distributions as(6)gij~CN(0,σe2),ε{gqkgqi*}=σe2δ(k-i).
Then let f be denoted as follows:(7)f=GPx=[g11g12⋯g1nTg21g22⋯g2nT⋮⋮⋱⋮gnR1gnR2⋯gnRnT][P10⋯00P2⋯0⋮⋮⋱⋮00⋯PnT]×[x1x2⋮xnT]=[∑k=1nTg1kPkxk∑k=1nTg2kPkxk⋮∑k=1nTgnRkPkxk]=[f1f2⋮fnR].
Clearly, f is a zero-mean complex Gaussian column vector with its components fq=∑k=1nTgqkPkxk,(q=1,…,nR), whose variance can be evaluated as          (8)σq2=ε(fqfq*)=ε{[∑k=1nTgqkPkxk][∑i=1nTgqiPixi]*}=∑k=1nTPkε[(gqkgqk*)(xkxk*)]=∑k=1nTPkε{|gqk|2}ε{|xk|2}=σe2∑k=1nTPk=nTσe2.
Thus the component of n̂ is given as n̂q=fq+nq′,(q=1,…,nR) by the aforementioned definition with the variance calculated as(9)σn̂q2=ε{n̂qn̂q*}=ε{(fq+nq′)(fq+nq′)*}=ε{|fq|2}+ε{|nq′|2}=nTσe2+σn2.Given the proposed equivalent system model, a SVD-assisted transmit power allocation scheme aiming at maximizing the capacity of the whole system in the presence of CSI imperfection will be described in the next part.


### 3.2. SVD-Aided Power Allocation Strategy

In the presence of channel state information imperfection, the channel capacity for Turbo-BLAST system with SVD-assisted power allocation scheme can be expressed as [1](10)C(λ̂̅)=∑k=1nTlog2[1+Pkλ̂knTσe2+σn2],
where λ̂̅=diag[λ̂1,…,λ̂nT] and Pk is the transmit power allocated for the k-th subchannel. The capacity subject to an optimization problem is given as
(11)[P1,…,PnT]opt=argmax⁡{∑k=1nTlog2[1+Pkλ̂knTσe2+σn2]}subjectto∑k=1nTPk=nT.
The Lagrange multiplier method is employed to find the optimized power allocation matrix that can maximize the overall capacity with total power constraint. By solving the equation in (11), we can find that the transmit power allocated for the k-th subchannel is
(12)Pk=[μ-(nTσe2+σn2)λ̂k-1]+,
where (x)+≜max{x,0} and μ is the Lagrange multiplier restricted by(13)∑k=1nT[μ-(nTσe2+σn2)λ̂k-1]+=nT.In a flat fading MIMO channel, the channel capacity is the statistical mean ofλ̂̅ approximately in the presence of CSI imperfection
(14)C=ελ̂̅{C(λ̂̅)}=ε{∑k=1nTlog2(1+Pkλ̂kσn̂2)}=ε{∑k=1nTlog2(μλ̂k)}.
Because {λ̂i}i=1nT is i.i.d and has the same probability density function p(λ̂k), formula (14) can be expressed as(15)C=nT∫0+∞[log2(μλ̂k)]+p(λ̂k)dλ̂k.
Since singular values are random variables, it is hard to achieve the function of its probability density straightly. We use the algebraic mean to achieve the capacity results in the simulation which can be expressed as(16)C≈ε{∑k=1nTlog2[1+Pkλ̂knTσe2+σn2]}≈1N∑i=1N{∑k=1nTlog2[1+Pk,iλ̂k,inTσe,i2+σn,i2]},
where N is the number of channel estimation matrix which is used for simulation, {Pk,i}k=1nT is the transmit power sequence for i-th channel matrix Ĥi, {λ̂k,i}k=1nT are eigenvalues of ĤiĤiH,σe,i2, and σn,i2 are the variances of channel estimation and AWGN, respectively.


## 3.1. Equivalent System Model

In this subsection, we propose an equivalent system model under the conditions of imperfect CSI for the modified Turbo-BLAST system.With the channel model and the SVD technique introduced aforementioned, when the CSI is imperfect, the received signal at a sampling instant can be formulated as(4)r=ÛH(Ĥ+Ξ)V̂Px+ÛHn=ÛHĤV̂Px+ÛHΞV̂Px+ÛHn=ÛHÛΛ̂V̂HV̂Px+ÛHΞV̂Px+ÛHn=Λ̂Px+ÛHΞV̂Px+n′=Λ̂Px+n̂,
where n′=ÛHn, Û is the unitary matrix with the right singular vectors of Ĥ as its columns, thus n′=ÛHn is also a column vector of additive Gaussian noise variables, where each component is i.i.d zero-mean complex Gaussian variables with the variance of σn2, n̂=ÛHΞV̂Px+n′ is an equivalent additive noise consisting of the interference caused by the channel estimation errors and the complex Gaussian noise, respectively. Here we make an approximation that assumes that the entries of x are i.i.d. Gaussian variables, that is, xi~C𝒩(0,1) [11]. Under this assumption, the variance of the equivalent noise n̂ can be calculated as follows.Firstly, letG=ÛHΞV̂, the mean and variance of G can be evaluated as(5)ε{G}=ε{ÛHΞV̂}=ÛHε{Ξ}V̂=0nR×nT,ε{GGH}=ε{ÛHΞV̂V̂HΞHÛ}=ε{ÛHΞΞHÛ}=ÛHε{ΞΞH}Û=σe2InR.
Clearly, the components of G are i.i.d zero-mean complex Gaussian variables with the variance of σe2: thus we can express the statistical distributions as(6)gij~CN(0,σe2),ε{gqkgqi*}=σe2δ(k-i).
Then let f be denoted as follows:(7)f=GPx=[g11g12⋯g1nTg21g22⋯g2nT⋮⋮⋱⋮gnR1gnR2⋯gnRnT][P10⋯00P2⋯0⋮⋮⋱⋮00⋯PnT]×[x1x2⋮xnT]=[∑k=1nTg1kPkxk∑k=1nTg2kPkxk⋮∑k=1nTgnRkPkxk]=[f1f2⋮fnR].
Clearly, f is a zero-mean complex Gaussian column vector with its components fq=∑k=1nTgqkPkxk,(q=1,…,nR), whose variance can be evaluated as          (8)σq2=ε(fqfq*)=ε{[∑k=1nTgqkPkxk][∑i=1nTgqiPixi]*}=∑k=1nTPkε[(gqkgqk*)(xkxk*)]=∑k=1nTPkε{|gqk|2}ε{|xk|2}=σe2∑k=1nTPk=nTσe2.
Thus the component of n̂ is given as n̂q=fq+nq′,(q=1,…,nR) by the aforementioned definition with the variance calculated as(9)σn̂q2=ε{n̂qn̂q*}=ε{(fq+nq′)(fq+nq′)*}=ε{|fq|2}+ε{|nq′|2}=nTσe2+σn2.Given the proposed equivalent system model, a SVD-assisted transmit power allocation scheme aiming at maximizing the capacity of the whole system in the presence of CSI imperfection will be described in the next part.


## 3.2. SVD-Aided Power Allocation Strategy

In the presence of channel state information imperfection, the channel capacity for Turbo-BLAST system with SVD-assisted power allocation scheme can be expressed as [1](10)C(λ̂̅)=∑k=1nTlog2[1+Pkλ̂knTσe2+σn2],
where λ̂̅=diag[λ̂1,…,λ̂nT] and Pk is the transmit power allocated for the k-th subchannel. The capacity subject to an optimization problem is given as
(11)[P1,…,PnT]opt=argmax⁡{∑k=1nTlog2[1+Pkλ̂knTσe2+σn2]}subjectto∑k=1nTPk=nT.
The Lagrange multiplier method is employed to find the optimized power allocation matrix that can maximize the overall capacity with total power constraint. By solving the equation in (11), we can find that the transmit power allocated for the k-th subchannel is
(12)Pk=[μ-(nTσe2+σn2)λ̂k-1]+,
where (x)+≜max{x,0} and μ is the Lagrange multiplier restricted by(13)∑k=1nT[μ-(nTσe2+σn2)λ̂k-1]+=nT.In a flat fading MIMO channel, the channel capacity is the statistical mean ofλ̂̅ approximately in the presence of CSI imperfection
(14)C=ελ̂̅{C(λ̂̅)}=ε{∑k=1nTlog2(1+Pkλ̂kσn̂2)}=ε{∑k=1nTlog2(μλ̂k)}.
Because {λ̂i}i=1nT is i.i.d and has the same probability density function p(λ̂k), formula (14) can be expressed as(15)C=nT∫0+∞[log2(μλ̂k)]+p(λ̂k)dλ̂k.
Since singular values are random variables, it is hard to achieve the function of its probability density straightly. We use the algebraic mean to achieve the capacity results in the simulation which can be expressed as(16)C≈ε{∑k=1nTlog2[1+Pkλ̂knTσe2+σn2]}≈1N∑i=1N{∑k=1nTlog2[1+Pk,iλ̂k,inTσe,i2+σn,i2]},
where N is the number of channel estimation matrix which is used for simulation, {Pk,i}k=1nT is the transmit power sequence for i-th channel matrix Ĥi, {λ̂k,i}k=1nT are eigenvalues of ĤiĤiH,σe,i2, and σn,i2 are the variances of channel estimation and AWGN, respectively.


## 4. Iterative Detection Algorithm for the Modified Turbo-BLAST System

In this section, the iterative detection strategy is employed for the modified Turbo-BLAST system with imperfect CSI to enhance the BER performance. The iterative decoding structure of serially concatenated turbo codes provides the principal model for the iterative decoding algorithm [4].In practice, true CSI cannot be obtained, so the receiver treatsĤ as the true channel matrix and n̂ as the equivalent noise. The iterative detection scheme for the modified Turbo-BLAST system will be described next.Letxk be the k-th transmitted signal at a sampling epoch, with (4), the received symbol vector r corrupted by the channel noise and interferences is given as(17)r=λ̂kPkxk+Λ̂k̅Pk̅xk̅+n̂,
where λ̂k is the k-th column of the matrix Λ̂ and(18)Λ̂k̅=[λ̂1,…,λ̂k-1,λ̂k+1,…,λ̂nT],xk̅=[x1,…,xk-1,xk+1,…,xnT]T,Pk̅=diag⁡(P1,…,Pk-1,Pk+1,…,PnT).
With (18), the decision statistic of k-th substream using a linear filter ŵk can be expressed as (19)yk=ŵkHλ̂kPkxk︸q̂k+̂ŵkHΛ̂k̅Pk̅xk̅︸d̂k+ŵkHn̂︸ẑk,
where q̂k, d̂k, and ẑk are the desired response obtained by the linear filter, the coantenna interference, and the phase-rotated noise, respectively.The coantenna interference can be removed fromyk by the proposed multisubstream detector and soft interference cancellation based on MMSE principle. The improved estimation of the transmitted symbol xk can be formulated as(20)x̃k=ŵkHr-d̂k,
where x̃k is the estimate of the symbol xk, and d̂k is the linear combination of the interfering substreams. The estimation error is defined as Δxk=x̃k-xk. The weighted vector ŵk and the interference estimation d̂k are optimized by minimizing the mean-square estimation error of Δxk between each substream and the related estimate by the following cost function(21)(w̅̂k,d̅̂k)=argmin⁡(w̅̂k,d̅̂k)⁡ε[‖x̃k-xk‖22],
where the expectation is taken over the equivalent noise n̂ and the statistics of the data sequence x.We use standard minimization techniques to solve the optimization problem formulated in (21). In order to arrive at the solution, we write the cost function as (22)Cost=ε[‖x̃k-xk‖22]=ε[‖ŵkHr-d̂k-xk‖22]=ŵkHε(rrH)ŵk-ŵkHε[r(d̂k+xk)*]-ε[r(d̂k+xk)*]Hŵk+ε[(d̂k+xk)(d̂k+xk)*],
where(23)ε[rrH]=ε[(λ̂kPkxk+Λ̂k̅Pk̅xk̅+n̂)Hλ̂kPkxk+Λ̂k̅Pk̅xk̅+n̂×(λ̂kPkxk+Λ̂k̅Pk̅xk̅+n̂)H]=Pkλ̂kλ̂kH+Λ̂k̅Pk̅ε(xk̅xk̅H)(Λ̂k̅Pk̅)H+(nTσe2+σn2)InR,ε[r(d̂k+xk)*]=ε[(λ̂kPkxk+Λ̂k̅Pk̅xk̅+n̂)(d̂k+xk)*]=λ̂kPk+Λ̂k̅Pk̅ε(xk̅)d̂k*.
By (23), the cost function in (22) can be further evaluated as(24)Cost=ŵkH{Pkλ̂kλ̂kH+(Λ̂k̅Pk̅)ε[xk̅xk̅H](Λ̂k̅Pk̅)H+(nTσe2+σn2)InRPkλ̂kλ̂kH+(Λ̂k̅Pk̅)ε[xk̅xk̅H](Λ̂k̅Pk̅)H}ŵk-ŵkH[λ̂kPk+Λ̂k̅Pk̅ε(xk̅)d̂k*]-[Pkλ̂kH+d̂kε(xk̅)H(Λ̂k̅Pk̅)H]ŵk+ε[d̂kd̂k*+d̂kxk*+xkd̂k*+xkxk*].
The linear combination of interfering substreams d̂k and the weighted vector ŵk is obtained from (24) by letting ∂Cost/∂d̂k*=0 and ∂Cost/∂ŵk*=0, respectively,(25)d̂k=ŵkH(Λ̂k̅Pk̅)ε(xk̅),ŵk=[Q+S+(nTσe2+σn2)InR]-1(λ̂kPk),
where(26)Q=(λ̂kPk)(λ̂kPk)H,S=(Λ̂k̅Pk̅){Ι(nT-1)-diag⁡[ε(xk̅)ε(xk̅)H]}(Λ̂k̅Pk̅)H.
Therefore the weighted vector ŵk is used for the modified Turbo-BLAST system in the presence of imperfect CSI. Thus (20) can be rewritten as (27)x̃k=ŵkHr-d̂k={[Q+S+(nTσe2+σn2)InR]-1(λ̂kPk)}H×[r-(Λ̂k̅Pk̅)ε(xk̅)].For the first iteration, we assume thatε(xk̅)=0, and thus the linear MMSE detection for the k-th substream becomes(28)x̃k={[(λ̂kPk)(λ̂kPk)H+(Λ̂k̅Pk̅)(Λ̂k̅Pk̅)H+(nTσe2+σn2)InR(λ̂kPk)(λ̂kPk)H+(Λ̂k̅Pk̅)(Λ̂k̅Pk̅)H]-1(λ̂kPk)(λ̂kPk)(λ̂kPk)H}Hr=(λ̂kPk)H[(Λ̂P)(Λ̂P)H+(nTσe2+σn2)InR]-1r.
Next, we assume that ε(xk̅)→xk̅ with the increasing number of iterations, where the MMSE interference canceller for the modified Turbo-BLAST system using SVD-based power allocation with imperfect CSI is (29)x̃k=(λ̂kPk)H[(λ̂kPk)H(λ̂kPk)+(nTσe2+σn2)]-1×[r-(Λ̂k̅Pk̅)xk̅].The MMSE detector gets the estimation of all the transmit signal using (29) and generates soft estimates of the coded bits conditioned on the received signal. The iterative decoder depicts message passing between the MMSE detector and SISO channel decoder. During the first iteration, the initial intrinsic probabilities of all symbol bits are assumed to be 1/2 (i.e., equally likely). The posteriori information is computed and fed back to the SISO channel decoder as the intrinsic information of its all coded bits. Then the SISO modules, in turn, process the soft information and compute refined estimates of soft information on both coded and information bits, based on the trellis structure of the channel codes, which is the channel code constraint. The SISO channel decoders are implemented by using the generalized BCJR algorithm [18].The soft information is sent to MMSE detector for interference cancellation [4]. Steps above are repeated until the decoding algorithm converges.


## 5. Simulation Results

In this section we compare the capacity and the BER performance of the traditional and modified Turbo-BLAST systems, where the proposed adaptive SVD-aided power allocation strategy is adopted in the latter case. At the transmitter, the data stream is first encoded by a rate-1/2 convolutional code with generator (7, 5), then modulated by 4-QAM scheme, whose capacity performance over a large number of channel realizations is exhibited in Figures2 and 3, extra capacity, which is defined as the capacity gap between the modified Turbo-BLAST system and the conventional one, is given in Figure 4, and finally BER performance is presented in Figure 5, respectively, with the traditional Turbo-BLAST system adopting the equal power allocation strategy, denoted by “EPA”, and the modified Turbo-BLAST system employing the proposed transmit power allocation, denoted by “TPA”, respectively.Figure 2
Capacity performance of traditional and modified Turbo-BLAST systems with four transmit and four receive antennas under the conditions of different inaccuracy of channel estimation.Figure 3
Impacts of CSI on the capacity performance of modified Turbo-BLAST systems with four transmit and four receive antennas in various values ofEb/N0.Figure 4
Extra capacity results between modified and traditional Turbo-BLAST systems with eight receive antennas and various transmit antennas configuration under the conditions of imperfect CSI (σe2=0.01).Figure 5
BER performance of traditional and modified Turbo-BLAST systems with four transmit and four receive antennas under the conditions of different inaccuracy of channel estimation.Figure2 shows that the capacity performance of the modified Turbo-BLAST system outperforms the traditional one in all cases under the same conditions. This indicates that the proposed SVD-aided power allocation strategy is an effective means to improve capacity performance even with the imperfect CSI. For example, at Eb/N0=10 dB, there are 0.8 (bit/s/Hz) and 1 (bit/s/Hz) capacity gains for σe2=0.04 and σe2=0.1, respectively.Figure3 shows that the capacity degradation is relatively insensitive to the CSI imperfection when CSI is more accurate. Only when σe2 increases to nearly 10-2, the capacity starts to fall. The over SNR degrades as the noise contribution due to the use of imperfect CSI increases which determine the detection performance. Figure 3 also shows that the impact of CSI imperfection on the capacity results is higher for larger Eb/N0.It is clear that from Figure4, we can see that the extra capacity is higher for low SNR and/or large antenna systems in the presence of CSI imperfection. For instance, at Eb/N0=4 dB, there are 3.2 (bit/s/Hz) and 1.3 (bit/s/Hz) capacity gains for the Turbo-BLAST system with eight and seven transmit antennas, respectively.It can be observed from Figure5 that as the BER results concerned, the modified Turbo-BLAST system outperforms the traditional system under the same conditions, regardless of the status of CSI. This implies that the proposed power allocation strategy is also valid to improve BER performance even with the imperfect CSI. For example, at a BER of 10-4,we can see that there are 5 dB gains for the modified system over the traditional one in the 1st detection iteration under the condition of perfect CSI, that is, σe2=0. Moreover, the BER results are quickly lowered with the increasing detection iterations. For instant, at a BER of 10-4,there are 3 dB extra gains attained for the modified Turbo-BLAST system using TPA scheme through 3rd iterative detection under the condition of perfect CSI.


## 6. Conclusion

In this paper, we propose a new strategy that combines SVD technique, transmit power allocation, and iterative detection techniques to improve both the capacity results and the BER performance for the modified Turbo-BLAST system.Based on the equivalent system model used in this paper and the derived variance of the equivalent noise, the SVD-assisted transmit power allocation strategy is theoretically obtained. The Lagrange multiplier method is then employed to find the optimized power allocation matrix which can maximize the capacity for the whole system subject to the total transmit power constraint. Finally, the iterative detection technique is adopted for the modified Turbo-BLAST system to enhance the BER performance. Simulation results show that the newly introduced method is effective to enhance not only the capacity results but also the BER performance of Turbo-BLAST system in both cases of perfect and imperfect CSI.



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*Source: 505606-2011-07-26.xml*