# Power Allocation Optimization: Linear Precoding Adapted to NB-LDPC Coded MIMO Transmission

Authors: Tarek Chehade; Ludovic Collin; Philippe Rostaing; Emanuel Radoi; Oussama Bazzi
Journal: International Journal of Antennas and Propagation
 (2015)

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## Abstract

In multiple-input multiple-output (MIMO) transmission systems, the channel state information (CSI) at the transmitter can be used to add linear precoding to the transmitted signals in order to improve the performance and the reliability of the transmission system. This paper investigates how to properly join precoded closed-loop MIMO systems and nonbinary low density parity check (NB-LDPC). Theq elements in the Galois field, GF(q), are directly mapped to q transmit symbol vectors. This allows NB-LDPC codes to perfectly fit with a MIMO precoding scheme, unlike binary LDPC codes. The new transmission model is detailed and studied for several linear precoders and various designed LDPC codes. We show that NB-LDPC codes are particularly well suited to be jointly used with precoding schemes based on the maximization of the minimum Euclidean distance (max-dmin) criterion. These results are theoretically supported by extrinsic information transfer (EXIT) analysis and are confirmed by numerical simulations.


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## Body

## 1. Introduction

Multiple-input multiple-output (MIMO) schemes have become unavoidable for transmission systems looking for increased throughput and improved reliability. Many methods have lately been developed to exploit the diversity offered by multiantenna systems like antenna distribution or space/polarization isolation [1], space-time block codes (e.g., Alamouti scheme) that do not require any channel state information at the transmitter side (Tx-CSI), or power allocation optimization techniques, also called precoding, that require a full or partial Tx-CSI. In this paper, we focus on NB-LDPC codes that are widely used because of their effectiveness. One advantage of NB-LDPC codes over binary LDPC codes is that nonbinary codes can match very well the underlying modulation, so symbol-to-bit conversion at the receiver can be avoided. The results in [2] confirm the advantages of using NB-LDPC codes to match the underlying high order modulations.To the best of our knowledge, the MIMO linear precoding performances are typically evaluated for uncoded MIMO systems. In this paper, we examine the association of the precoding scheme with channel coding in order to finally determine a power allocation optimization solution that adapts a block of linear precoding to a NB-LDPC coded MIMO transmission, where the Belief Propagation algorithm is used for decoding.We implement this new scheme using linear precoders, having diagonal or nondiagonal structure, and we study the impact of the application of different known precoders on two system configurations ((2, 2) and (2, 4) MIMO). We investigate, by means of an EXIT analysis, how to take advantage of the precoding techniques in a MIMO system that uses NB-LDPC codes and we show why some associations could be harmful and others beneficial.The main contributions of this paper are listed as follows:(i)
a proposed scheme associating NB-LDPC codes to MIMO precoding systems with adequate correspondence between the GF(q) elements and the q received constellation points;(ii)
a comparison between different precoder types and a study of the gain/loss in the different cases;(iii)
a theoretical EXIT analysis in order to predict the performance of different precoders with the NB-LDPC codes;(iv)
a confirmation by numerical simulations, in terms of bit error rate (BER), of the performance of the proposed power allocation optimization scheme.This paper is organized as follows. In Section2 the system model is described with the eigenmode representation. LDPC codes are introduced in Section 3. Linear precoding techniques are presented in Section 4. Section 5 provides a detailed description of the new proposed scheme; Section 6 presents the EXIT analysis, while the simulation results are given in Section 7. Finally, some conclusions are drawn in Section 8.The notations used in this paper are as follows. Matrices and vectors are denoted by symbols in boldface, and(·)T and (·)H represent complex transpose and Hermitian, respectively. Im denotes the m×m identity matrix and diag(x) denotes a diagonal matrix with x on its main diagonal. E[·] denotes statistical expectation. X[i1:i2,j1:j2] denotes a submatrix obtained by extracting rows i1 through i2 and columns j1 through j2 from a matrix X. If no specific range appears at the row or column position in notation, then all rows or columns are considered to constitute the submatrix. Xij denotes the (i,j)th element of a matrix X. x denotes the 2-norm of vector x. XF denotes the Frobenius norm of matrix X and NC(0,σ2) denotes a circular symmetric complex Gaussian distribution with zero mean and variance σ2.


## 2. System Model and Eigenmode Representation

Let us consider a MIMO system withnT transmit antennas and nR receive antennas; the received signal is therefore given by
(1)y=GAFs+Gn,
where y is the b×1 received symbol vector, A is the nR×nT channel matrix, F is the nT×b linear precoding matrix, G is the b×nR linear decoding matrix, s is the b×1 transmitted symbol vector, and n is nR×1 complex additive white Gaussian noise vector with zero mean and covariance matrix N0InR. Let us assume that
(2)EssH=EsIb,EsnH=0,EnnH=N0InR.
The average transmitted power is limited; the precoding matrix is normalized and is therefore subject to the power constraint:
(3)FF2=1.All over the paper, the channel state information is assumed to be perfectly known at both transmitter and receiver sides. We apply a transformation in order to obtain a simplified model of the system. This will result in a diagonalized channel matrix [3]. Let us apply this virtual transformation, which is based on the singular value decomposition (SVD), by means of the following decompositions: A=UAυVH, F=FυFd,   G=Gυ, where Fυ=V(:,1:b) and Gυ=UH(:,1:b).Relation1 can be then rewritten as
(4)y=AυFds+nυ,
where Aυ=GυAFυ is the eigenchannel matrix, nυ=Gυn is the transformed additive noise vector with the covariance matrix Rnυ=EnυnυH=N0Ib, and the unitary matrices Gυ and Fυ are chosen so as to ensure the whitening of the noise, the diagonalization of the channel, and the reduction of the dimension to b. The eigen-channel matrix Aυ can now be expressed as
(5)Aυ=diagσ1,σ2,…,σb,
where the singular values are sorted in descending order.


## 3. Low Density Parity Check Codes

Low density parity check (LDPC) codes were proposed by Gallager in 1962 [4]. They are defined by a sparse parity check matrix over a Galois field GF(q), where q denotes the order of the Galois field. LDPC codes are binary when q=2 and nonbinary when q>2. Binary LDPC codes have been shown to approach Shannon limit performance [4–7] for very long code lengths. NB-LDPC codes are usually preferred to their binary counterparts when the block length is small to moderate [8] or when the order of the symbols sent through the channel is not binary [9], which is the case for high order modulations or for multiple-antennas channels [10, 11]. Davey and MacKay studied in 1998 NB-LDPC codes [12] and showed that they may achieve superior performance than the binary codes when constructed over higher order Galois fields at the expense of increased decoding complexity [12, 13].A sparse parity check matrixH describes the LDPC code and can be efficiently represented by a bipartite graph called Tanner graph. Iterative decoding of LDPC codes has been addressed efficiently in [12, 14] using the Belief Propagation (BP) algorithm.The BP algorithm is a suboptimal decoding algorithm proposed by Gallager in [4] for binary LDPC codes decoding and then generalized in [12] for the NB-LDPC codes. The iterative decoding is done by repeating the steps of the algorithm until a valid codeword has been obtained or a fixed number of iterations have been completed. Therefore, the rapidity of convergence of the algorithm appears to be an important issue.One advantage of NB-LDPC coding is that one can match the field order with the constellation size. This way, one element in GF(q) is mapped to one point in the signal constellation. This is because the likelihood probabilities (or LLRV) for each coded symbol over GF(q) are independent of other coded symbols.The LDPC codes we chose to use have the following properties.(i)
Ultrasparse NB-LDPC codes with column weightdv=2 at the variable-node side that gives large girths compared to classical binary graphs [8, 15].(ii)
These codes are well designed for both good waterfall and error floor properties [8, 15], compared to random choice of the Tanner graph structure and the nonzero values assignment.(iii)
At the check mode side, several row weights are chosen (dc=4,8, and 12) to have precoding comparison at several code rates.(iv)
These codes are targeted for LTE-A with small packet lengths (small to moderate codeword lengths) [15].(v)
Small packet lengths are typically well suited for closed-loop MIMO systems.


## 4. Linear Precoding

The full knowledge of the channel state information at the transmitter side (Tx-CSI) permits the application of linear precoding operation in MIMO transmissions. It consists in a matrix multiplication operation that is applied to the transmitted signal which combines the symbols on the different antennas. This antenna allocation operation is determined and designed in such a way that allows the optimization of a well-defined criterion. Therefore, different linear precoders are defined because of the different optimization criteria. These precoders are mainly divided into two categories: diagonal precoders that allow transforming the MIMO transmission system into parallel independent SISO systems and nondiagonal precoders.The precoders are detailed hereafter for the sake of presenting them on the same formalism, which facilitates the presentation of our proposed scheme and allows the comparison in terms of both EXIT analysis (Section6) and numerical simulations (Section 7).


### 4.1. Diagonal Precoders

For diagonal precoders, the precoding matrix has the following expression:(6)Fd=diagp1,…,pb,
where pi(i=1,…,b) represents the power allocation of the ith subchannel.Knowing that the transformation of the channel model gives a diagonal channelAυ the use of a diagonal precoder will result in completely independent subchannels and the transmission system is therefore equivalent to distributing the symbols onto parallel independent subsystems.The power constraint in this case is expressed as(7)∑i=1bpi=1.The main optimization criteria that lead to diagonal precoders are the maximization of the postprocessing SNR [16], the maximization of the channel capacity [17], the minimization of the BER [18], the minimization of the mean square error (MSE) [19], the maintaining of fixed ratios between the postprocessing SNR values of the different subchannels [19].


#### 4.1.1. Max-SNR Precoder

Maximizing the postprocessing SNR leads to a transmission over one subchannel only, corresponding to the highest subchannel singular valueσ1. If we apply the power constraint, we obtain p1=1 and pi=0 for i=2,…,b. The max-SNR solution simplifies the signal model equation that becomes
(8)y=σ1s+n.


#### 4.1.2. Water-Filling Precoder

The optimization criterion that maximizes the channel capacity leads to the Water-Filling (WF) precoder. The channel capacity can be expressed as(9)C=∑i=1blog⁡2⁡1+EsN0×piσi2
with Es=Es2 and N0=En2.We set(10)σ~i=σiEsN0
and then the maximization of C gives the following power allocation strategy:
(11)pi=Ψ-1σ~i2ifΨ>1σ~i20otherwisefori=1,…,b,
where the threshold Ψ is defined by
(12)Ψ=1+γΨbΨ;γΨ=∑i=1bΨ1σ~i2
and bΨ=i the greatest value in {1,…,b} such that Ψ>1/σ~i2.The power allocation strategy of the WF precoder depends on the SNR values of the subchannels (i.e.,σ~i2 for i=1,2,…,b). The precoder allows the cancellation of the most disadvantaged subchannels. The total average power is then distributed among the strongest bΨ subchannels.In a (2, 2) MIMO system, this results in either transmitting over 2 subchannels or using only the most advantageous one. We will show later the switch of this precoder in function of the SNR values.


#### 4.1.3. MMSE Precoder

The optimization criterion of the MMSE precoder is the minimization of the mean square error that is expressed as(13)MSE=Ey-s2=∑i=1bEσipi-1si+ni2
and then the minimization of MSE gives the following power allocation strategy:
(14)pi=1σ~iΨMMSE-1σ~iifΨMMSE>1σ~i0otherwisefori=1,…,b,
where the threshold ΨMMSE is defined by
(15)ΨMMSE=1+γΨMMSE∑i=1bΨMMSE1/σ~i;γΨMMSE=∑i=1bΨMMSE1σ~i2.The power allocation strategy of the MMSE precoder depends on the SNR values of the subchannels also and may lead to subchannel cancellation; that is,bΨMMSE subchannels are used (in the case of b=2, only one subchannel is used).


#### 4.1.4. Quality of Service and Equal Error Precoders

The power allocation strategy of the Quality of Service (QoS) precoder aims to maintain fixed ratios between the postprocessing SNR values of the different subchannels [19], which are expressed as
(16)γi=piσ~i2=wi∑k=1bwk/σ~k2fori=1,…,b,
where w1=1>w2>⋯>wb are the fixed SNR ratios between subchannel i≠1 and subchannel 1 (i.e., wi=γi/γ1).The Equal Error (EE) precoder is the particular case of the QoS precoder where all subchannels have the same postprocessing SNR valueγi by setting w1=w2=⋯=wb=1.


### 4.2. Nondiagonal Precoders

Unlike diagonal precoders, the precoding matrix of nondiagonal precoders is a nondiagonal matrix. In our study, we focus on two of the best performing nondiagonal precoders which are based on the minimal Euclidean distance criterion: the max-dmin⁡ precoder [3] and the max-dmin⁡-DFT precoder (Discrete Fourier Transform) [20].


#### 4.2.1. Max-dmin⁡

The minimum Euclidean distance between signal points on the receiver’s constellation affects the system performances, especially with the ML detector [21]. The max-dmin⁡ precoder is designed with the criterion of maximizing this minimal Euclidean distance dmin⁡ which is expressed as
(17)dmin⁡=min⁡ϵ∈E⁡AυFdϵ,
where ϵ=sk-sl is the error vector between vector symbols sk and sl for k≠l, E being the set of all error vectors. The max-dmin⁡ precoder can then be obtained by solving
(18)Fddmin⁡=argmaxFd⁡dmin⁡(Fd)
under the power constraint FdF2=1.Collin et al. give in [3] a solution for 17 for the case of b=2 and a 4-QAM modulation. For two data streams, the precoding matrix is written as
(19)Fd=cos⁡⁡ψ00sin⁡ψcos⁡⁡θsin⁡θ-sin⁡θcos⁡⁡θ100ejϕ
and the channel matrix is written as
(20)Aυ=ρcos⁡⁡γ00sin⁡γ,
where ρ2=σ12+σ22 and tan⁡γ=σ2/σ1.For 4-QAM modulation, the values of the tripletψ,θ,ϕ that maximizes dmin⁡ depend on the parameter γ, and two cases are possible:(i)
ifγ<γ0, we have
(21)ψ=0,θ=atan⁡26+2,ϕ=15°;(ii)
ifγ≥γ0, we have
(22)ψ=atan⁡2-1tan⁡γ,θ=45°,ϕ=45°withγ0≈17.28°.


#### 4.2.2. Max-dmin⁡-DFT

The general form of this precoder can be written as(23)Fd=diagp1,…,pbWb,
where Wb is the DFT matrix defined as
(24)Wb=wklk,l=0,…,b-1
with w=ej2π/b.Forb=2, the DFT matrix is written as
(25)W2=111-1.The power allocation strategy aims to optimize the minimal Euclidean distancedmin⁡, which leads to a switch to the max-SNR solution at γDFT=atan⁡(1/7)=20.70°.Forγ<γDFT, the max-SNR precoder is used, while for γ>γDFT, considering the form expressed in 19, we obtain the following triplet values:
(26)ψ=atan⁡13tan⁡γ,θ=45°ϕ=0.


## 4.1. Diagonal Precoders

For diagonal precoders, the precoding matrix has the following expression:(6)Fd=diagp1,…,pb,
where pi(i=1,…,b) represents the power allocation of the ith subchannel.Knowing that the transformation of the channel model gives a diagonal channelAυ the use of a diagonal precoder will result in completely independent subchannels and the transmission system is therefore equivalent to distributing the symbols onto parallel independent subsystems.The power constraint in this case is expressed as(7)∑i=1bpi=1.The main optimization criteria that lead to diagonal precoders are the maximization of the postprocessing SNR [16], the maximization of the channel capacity [17], the minimization of the BER [18], the minimization of the mean square error (MSE) [19], the maintaining of fixed ratios between the postprocessing SNR values of the different subchannels [19].


### 4.1.1. Max-SNR Precoder

Maximizing the postprocessing SNR leads to a transmission over one subchannel only, corresponding to the highest subchannel singular valueσ1. If we apply the power constraint, we obtain p1=1 and pi=0 for i=2,…,b. The max-SNR solution simplifies the signal model equation that becomes
(8)y=σ1s+n.


### 4.1.2. Water-Filling Precoder

The optimization criterion that maximizes the channel capacity leads to the Water-Filling (WF) precoder. The channel capacity can be expressed as(9)C=∑i=1blog⁡2⁡1+EsN0×piσi2
with Es=Es2 and N0=En2.We set(10)σ~i=σiEsN0
and then the maximization of C gives the following power allocation strategy:
(11)pi=Ψ-1σ~i2ifΨ>1σ~i20otherwisefori=1,…,b,
where the threshold Ψ is defined by
(12)Ψ=1+γΨbΨ;γΨ=∑i=1bΨ1σ~i2
and bΨ=i the greatest value in {1,…,b} such that Ψ>1/σ~i2.The power allocation strategy of the WF precoder depends on the SNR values of the subchannels (i.e.,σ~i2 for i=1,2,…,b). The precoder allows the cancellation of the most disadvantaged subchannels. The total average power is then distributed among the strongest bΨ subchannels.In a (2, 2) MIMO system, this results in either transmitting over 2 subchannels or using only the most advantageous one. We will show later the switch of this precoder in function of the SNR values.


### 4.1.3. MMSE Precoder

The optimization criterion of the MMSE precoder is the minimization of the mean square error that is expressed as(13)MSE=Ey-s2=∑i=1bEσipi-1si+ni2
and then the minimization of MSE gives the following power allocation strategy:
(14)pi=1σ~iΨMMSE-1σ~iifΨMMSE>1σ~i0otherwisefori=1,…,b,
where the threshold ΨMMSE is defined by
(15)ΨMMSE=1+γΨMMSE∑i=1bΨMMSE1/σ~i;γΨMMSE=∑i=1bΨMMSE1σ~i2.The power allocation strategy of the MMSE precoder depends on the SNR values of the subchannels also and may lead to subchannel cancellation; that is,bΨMMSE subchannels are used (in the case of b=2, only one subchannel is used).


### 4.1.4. Quality of Service and Equal Error Precoders

The power allocation strategy of the Quality of Service (QoS) precoder aims to maintain fixed ratios between the postprocessing SNR values of the different subchannels [19], which are expressed as
(16)γi=piσ~i2=wi∑k=1bwk/σ~k2fori=1,…,b,
where w1=1>w2>⋯>wb are the fixed SNR ratios between subchannel i≠1 and subchannel 1 (i.e., wi=γi/γ1).The Equal Error (EE) precoder is the particular case of the QoS precoder where all subchannels have the same postprocessing SNR valueγi by setting w1=w2=⋯=wb=1.


## 4.1.1. Max-SNR Precoder

Maximizing the postprocessing SNR leads to a transmission over one subchannel only, corresponding to the highest subchannel singular valueσ1. If we apply the power constraint, we obtain p1=1 and pi=0 for i=2,…,b. The max-SNR solution simplifies the signal model equation that becomes
(8)y=σ1s+n.


## 4.1.2. Water-Filling Precoder

The optimization criterion that maximizes the channel capacity leads to the Water-Filling (WF) precoder. The channel capacity can be expressed as(9)C=∑i=1blog⁡2⁡1+EsN0×piσi2
with Es=Es2 and N0=En2.We set(10)σ~i=σiEsN0
and then the maximization of C gives the following power allocation strategy:
(11)pi=Ψ-1σ~i2ifΨ>1σ~i20otherwisefori=1,…,b,
where the threshold Ψ is defined by
(12)Ψ=1+γΨbΨ;γΨ=∑i=1bΨ1σ~i2
and bΨ=i the greatest value in {1,…,b} such that Ψ>1/σ~i2.The power allocation strategy of the WF precoder depends on the SNR values of the subchannels (i.e.,σ~i2 for i=1,2,…,b). The precoder allows the cancellation of the most disadvantaged subchannels. The total average power is then distributed among the strongest bΨ subchannels.In a (2, 2) MIMO system, this results in either transmitting over 2 subchannels or using only the most advantageous one. We will show later the switch of this precoder in function of the SNR values.


## 4.1.3. MMSE Precoder

The optimization criterion of the MMSE precoder is the minimization of the mean square error that is expressed as(13)MSE=Ey-s2=∑i=1bEσipi-1si+ni2
and then the minimization of MSE gives the following power allocation strategy:
(14)pi=1σ~iΨMMSE-1σ~iifΨMMSE>1σ~i0otherwisefori=1,…,b,
where the threshold ΨMMSE is defined by
(15)ΨMMSE=1+γΨMMSE∑i=1bΨMMSE1/σ~i;γΨMMSE=∑i=1bΨMMSE1σ~i2.The power allocation strategy of the MMSE precoder depends on the SNR values of the subchannels also and may lead to subchannel cancellation; that is,bΨMMSE subchannels are used (in the case of b=2, only one subchannel is used).


## 4.1.4. Quality of Service and Equal Error Precoders

The power allocation strategy of the Quality of Service (QoS) precoder aims to maintain fixed ratios between the postprocessing SNR values of the different subchannels [19], which are expressed as
(16)γi=piσ~i2=wi∑k=1bwk/σ~k2fori=1,…,b,
where w1=1>w2>⋯>wb are the fixed SNR ratios between subchannel i≠1 and subchannel 1 (i.e., wi=γi/γ1).The Equal Error (EE) precoder is the particular case of the QoS precoder where all subchannels have the same postprocessing SNR valueγi by setting w1=w2=⋯=wb=1.


## 4.2. Nondiagonal Precoders

Unlike diagonal precoders, the precoding matrix of nondiagonal precoders is a nondiagonal matrix. In our study, we focus on two of the best performing nondiagonal precoders which are based on the minimal Euclidean distance criterion: the max-dmin⁡ precoder [3] and the max-dmin⁡-DFT precoder (Discrete Fourier Transform) [20].


### 4.2.1. Max-dmin⁡

The minimum Euclidean distance between signal points on the receiver’s constellation affects the system performances, especially with the ML detector [21]. The max-dmin⁡ precoder is designed with the criterion of maximizing this minimal Euclidean distance dmin⁡ which is expressed as
(17)dmin⁡=min⁡ϵ∈E⁡AυFdϵ,
where ϵ=sk-sl is the error vector between vector symbols sk and sl for k≠l, E being the set of all error vectors. The max-dmin⁡ precoder can then be obtained by solving
(18)Fddmin⁡=argmaxFd⁡dmin⁡(Fd)
under the power constraint FdF2=1.Collin et al. give in [3] a solution for 17 for the case of b=2 and a 4-QAM modulation. For two data streams, the precoding matrix is written as
(19)Fd=cos⁡⁡ψ00sin⁡ψcos⁡⁡θsin⁡θ-sin⁡θcos⁡⁡θ100ejϕ
and the channel matrix is written as
(20)Aυ=ρcos⁡⁡γ00sin⁡γ,
where ρ2=σ12+σ22 and tan⁡γ=σ2/σ1.For 4-QAM modulation, the values of the tripletψ,θ,ϕ that maximizes dmin⁡ depend on the parameter γ, and two cases are possible:(i)
ifγ<γ0, we have
(21)ψ=0,θ=atan⁡26+2,ϕ=15°;(ii)
ifγ≥γ0, we have
(22)ψ=atan⁡2-1tan⁡γ,θ=45°,ϕ=45°withγ0≈17.28°.


### 4.2.2. Max-dmin⁡-DFT

The general form of this precoder can be written as(23)Fd=diagp1,…,pbWb,
where Wb is the DFT matrix defined as
(24)Wb=wklk,l=0,…,b-1
with w=ej2π/b.Forb=2, the DFT matrix is written as
(25)W2=111-1.The power allocation strategy aims to optimize the minimal Euclidean distancedmin⁡, which leads to a switch to the max-SNR solution at γDFT=atan⁡(1/7)=20.70°.Forγ<γDFT, the max-SNR precoder is used, while for γ>γDFT, considering the form expressed in 19, we obtain the following triplet values:
(26)ψ=atan⁡13tan⁡γ,θ=45°ϕ=0.


## 4.2.1. Max-dmin⁡

The minimum Euclidean distance between signal points on the receiver’s constellation affects the system performances, especially with the ML detector [21]. The max-dmin⁡ precoder is designed with the criterion of maximizing this minimal Euclidean distance dmin⁡ which is expressed as
(17)dmin⁡=min⁡ϵ∈E⁡AυFdϵ,
where ϵ=sk-sl is the error vector between vector symbols sk and sl for k≠l, E being the set of all error vectors. The max-dmin⁡ precoder can then be obtained by solving
(18)Fddmin⁡=argmaxFd⁡dmin⁡(Fd)
under the power constraint FdF2=1.Collin et al. give in [3] a solution for 17 for the case of b=2 and a 4-QAM modulation. For two data streams, the precoding matrix is written as
(19)Fd=cos⁡⁡ψ00sin⁡ψcos⁡⁡θsin⁡θ-sin⁡θcos⁡⁡θ100ejϕ
and the channel matrix is written as
(20)Aυ=ρcos⁡⁡γ00sin⁡γ,
where ρ2=σ12+σ22 and tan⁡γ=σ2/σ1.For 4-QAM modulation, the values of the tripletψ,θ,ϕ that maximizes dmin⁡ depend on the parameter γ, and two cases are possible:(i)
ifγ<γ0, we have
(21)ψ=0,θ=atan⁡26+2,ϕ=15°;(ii)
ifγ≥γ0, we have
(22)ψ=atan⁡2-1tan⁡γ,θ=45°,ϕ=45°withγ0≈17.28°.


## 4.2.2. Max-dmin⁡-DFT

The general form of this precoder can be written as(23)Fd=diagp1,…,pbWb,
where Wb is the DFT matrix defined as
(24)Wb=wklk,l=0,…,b-1
with w=ej2π/b.Forb=2, the DFT matrix is written as
(25)W2=111-1.The power allocation strategy aims to optimize the minimal Euclidean distancedmin⁡, which leads to a switch to the max-SNR solution at γDFT=atan⁡(1/7)=20.70°.Forγ<γDFT, the max-SNR precoder is used, while for γ>γDFT, considering the form expressed in 19, we obtain the following triplet values:
(26)ψ=atan⁡13tan⁡γ,θ=45°ϕ=0.


## 5. Proposed Scheme

The block diagram of the proposed scheme is detailed in Figure1. A sparse parity check matrix H of dimension (M×N) whose elements are in the Galois field GF(q) defines the LDPC code, with M=N-K, where K and N are, respectively, the source block length and the transmitted code block length. Data stream is encoded through the LDPC encoder into the codeword v and then mapped into b×2m-QAM symbol vectors. Note that the code alphabet matches the b-dimensional constellation alphabet (i.e., q=2mb). The precoding matrix Fd is applied to the transmitted symbol vectors before transmission through the channel AυAυ=GυAFυ where the transformed AWGN noise vector is added. The received symbol vector y is used to compute the log likelihood ratio vector (LLRV) required for log-BP initialization.Figure 1
Proposed block diagram scheme.The LLRV corresponding to the variable-nodev, according to the channel model, is given by
(27)Lv=Lvv=1Lvv=α⋯Lvv=αq-2,
where {0,1,α,…,αq-2} are the elements of GF(q), α represents the primitive element, and
(28)Lvv=αk=log⁡⁡Py∣v=αkPy∣v=0fork=0,…,q-2.The channel LLR value computation is actually done using the following expression:(29)Lvv=αk=-1N0y-AυFdsk+12-y-AυFds02,
where sk represent all possible elements of vectors s for k=0,1,…,q-2.The LLRV values of28, calculated for all nonbinary GF symbols of a codeword, are the intrinsic messages given as input to the Belief Propagation algorithm. They serve to initialize extrinsic messages, which are then updated through the iterated exchange of messages between variable and check nodes in the Tanner graph representing the code. At the end of every iteration, a posteriori messages are calculated for a decoding attempt. The procedure stops when all the parity check equations are satisfied (null syndrome) or when the maximum number of iterations is reached (decoding failure).


## 6. EXIT Analysis

EXIT charts predict the behavior of the BP decoder on the basis of only simulated behavior of the individual components decoders (variable-node decoder (VND) and check-node decoder (CND)). Using this approach it is only necessary to simulate the behavior of each component decoder once. Such charts that track the mutual information at each iteration give an excellent visual representation to analyze the decoding convergence.In [9], Bennatan and Burshtein extend the development of EXIT charts for binary LDPC codes to NB-LDPC codes. In the following, the main results are presented.The mutual informationIC;L between the code symbol C and its corresponding LLR vector L (a priori or extrinsic information from VND and CND at each iteration step) can be expressed as
(30)IC;L=1-Elog⁡q⁡1+∑i=1q-1e-Li∣C=0.
The conditioning on C=0 results in the classical all-zero codeword assumption where the decoder performance is independent of the transmitted codeword.Using a Gaussian distribution approximation ofL (mean m and covariance matrix Σ) and symmetry and permutation invariance assumptions as defined in [9], the number of parameters is reduced from q-1 to one parameter denoted by σ such that
(31)m=σ22σ22⋮σ22;Σ=σ2σ22σ2⋯σ22σ2.
That is, mi=σ2/2,i=1,…,q-1, and Σi,j=σ2 if i=j and σ2/2 otherwise.The computation of30 can be numerically evaluated by using the Gaussian approximation 31 and results in a monotonically increasing function Jqσ=IC;L with values from 0 to 1. We denote by Jq-1(I) the corresponding inverse function. These functions can be obtained by using fitting polynomial approximation [9].These assumptions result in the expression of the classical mutual information relations for the CND and VND regular LDPC codes, similarly to the binary case:(32)IE,VNDIA;dv,I0≈Jqdv-1Jq-1IA2+Jq-1I02,(33)IE,CNDIA;dc≈1-Jqdc-1Jq-11-IA,
where IA and I0 denote the mutual information of the incoming and the initial message, respectively. I0 equals the capacity of the channel and is given by
(34)I0=1-1q∑i=1qEnυlog⁡q∑j=1qe-di,j
with di,j=(1/N0)(AυFdsi-sj+nυ2-nυ2) and q=2mb.Unlike33, 32 depends on the channel information (SNR value and MIMO precoding). Based on 32 the computation of IE,VND highly depends on the mutual information I0. Higher I0 implies a higher IE,VND curve (Jq(σ) and Jq-1(I) monotonically increasing functions), which means that the convergence point with the IE,CND curve will be closer to one. Therefore, by examining the I0 curves for the different precoders, we should be able to predict the performance of the association with the LDPC code.In Figure2, the mutual information I0 is plotted for every precoder for b=2 data streams in function of γ (cf. 20 and 34). This plot is shown for a given received normalized SNR equal to 8.0 dB and defined as
(35)SNRR=AF2nRnTEsN0.Figure 2
Mutual informationI0 in function of γ for different precoders.Since the channel energyAF2=ρ2 is included in SNRR, I0 does not depend on ρ2. Thus I0 shows only the influence of the parameter γ for each precoder (cf. 20 and 34). Note that the values of I0 are between 0 and 1 since logarithm log⁡q uses base q=2mb. In terms of bit per symbol time, the channel capacity is equal to bmI0, with b=2, and m=2 (4-QAM); values are between 0 and bm=4.In order to maintain a constant transmission throughput and knowing that some precoders may cancel the weak subchannel and transmit only on the best one, the modulation of the data stream was set to switch to 16-QAM whenever the precoder cancels one subchannel. Therefore the precoders will be denoted in the simulation results as presented in Table1.Table 1
Precoder notations.
Precoder
Notation
Max-SNR (1 × 16QAM)
MaxSNR
Water-Filling (2 × 4QAM/1 × 16QAM)
WF
MMSE (2 × 4QAM/1 × 16QAM)
MMSE
Equal Error (2 × 4QAM)
EE
Quality of Service(2 × 4QAM-WdB with W=10log10⁡w2)
QoS
Max-dmin⁡ (2 × 4QAM)
MaxDmin
Max-dmin⁡-DFT-2 × 4QAM/1 × 16QAM
MaxdminDFTIn order to analyze the performance of the precoders in a given Rayleigh MIMO system, it is important to examine the distribution (pdf) of theγ angle. Therefore, Figure 3 plots the probability density function of γ for Rayleigh channel model (Aij~NC0,1) for both MIMO (2, 2) and MIMO (2, 4) systems [22].Figure 3
Probability density functions of theγ channel angle for Rayleigh MIMO (2, 2) and MIMO (2, 4) systems.Figure3 shows that, in the case of MIMO (2, 2) system, the pdf is larger for low γ values than for high γ values, while, in the case of MIMO (2, 4) system, the pdf is considerably larger for high γ values than for low γ values. This will be helpful in order to explain the BER performance with regard to the I0 values in function of γ. The I0 values for low γ values will affect the MIMO (2, 2) system and the I0 values for high γ values will affect the MIMO (2, 4) system performance.Now thatγ distribution is known for both MIMO systems, the impact of γ on I0 for every precoder can be analyzed, and the performance of each precoder can be predicted for both MIMO systems.In Figure2, the curve corresponding to the Equal Error precoder has the lowest values for small γ values and does not increase much for large γ values, compared to other precoders. This will affect the performance of this precoder that should be among the worse compared to other precoders especially in the case of MIMO (2, 2) system.The curve of the QoS precoder is slightly higher than the EE one, which will result in a slight performance improvement for both MIMO systems.The WF precoder is a diagonal precoder that is capable of using two subchannels or cancelling the second when the threshold stated in11 is exceeded. This appears in its curve where high I0 values are recorded for small γ values, and a switch is clearly seen above a γ limit, where this precoder becomes even worse than the EE. This deterioration will be retrieved in the performance level when γ values increase (MIMO (2, 4)).The MMSE precoder applies a similar cancellation strategy as the WF with a smaller switch angle. Its curve shows lowerI0 values than the WF for small γ and greater I0 values for large γ. This will be reflected in better performance of MMSE compared to WF in the case of MIMO (2, 4) system.These curves show that all diagonal precoders have lowI0 values, which predicts poor performance in terms of BER.The max-SNR precoder, which can be considered as a special precoder because of its use of one subchannel only, reaches highI0 values for small γ values. For high γ values, the precoder strategy is to always use one subchannel only, thus wasting the second subchannel that became more advantageous when γ values are high. The max-SNR I0 curve decreases with increasing γ and becomes low. The max-SNR precoder will have performance as good as nondiagonal precoders for MIMO (2, 2) system and will see its performance deteriorated for MIMO (2, 4) system.Finally, the nondiagonal precoders based on the minimum Euclidean distance criterion (max-dmin⁡ and max-dmin⁡-DFT) have both a constant I0 value larger than all the others, whatever the γ values are. This will result in a better performance of both MIMO systems.As it is shown in34, the minimum Euclidean distance appears in the computation of I0, and the curves of Figure 2 confirm that this criterion is well-suited with the LDPC code association.In addition, the high values ofI0 for the nondiagonal precoders predict not only better performance in terms of BER, but also a higher convergence rate due to the smaller number of iterations needed in order to reach convergence point on the EXIT charts.This analysis of theI0 curves for different precoders will be verified through simulations and the predicted results will be confirmed hereafter (Figures 4 and 5).Figure 4
BER performance of the proposed scheme for a (2, 2) MIMO system.Figure 5
BER performance of the proposed scheme for a (2, 4) MIMO system.


## 7. Simulation Results

The performance of the proposed power allocation scheme has been validated through a series of simulations that assess the achieved improvement in terms of both BER and convergence rate.The precoders under evaluation are all the above cited diagonal and nondiagonal precoders. LDPC codes are constructed over GF(16) and different parity check matrices are used, having different rates, all derived from matrices designed in the framework of DAVINCI Project [8, 23]. Simulations have been run on two different MIMO configurations: a symmetric (2, 2) system and an asymmetric (2, 4) system. The number of subchannels is set to b=2. The modulation is set to 4-QAM. We consider a flat-fading channel with a Rayleigh distribution model. A new channel realization is considered at each transmitted codeword with Aij~NC0,1. The SNR per bit is defined in Table 2.Table 2
SNR per bit definition.
System with precoding
E
b
/
N
0
=
(
E
s
/
N
0
)
(
1
/
4
R)
System without precoding(spatial multiplexing)
E
b
/
N
0
=
(
n
T
E
s
/
N
0
)
(
1
/
4
R)Figure4 illustrates the BER performances versus SNR for a (2, 2) MIMO transmission system with LDPC code designed from the parity check matrix having the following parameters:(i)
matrix size:N=384, M=64, the numbers of nonzero elements in columns and rows being, respectively, dv=2 and dc=12,code rate:R=5/6=0.83,girth: 8.The curves on this figure assess the performance of the proposed scheme predicted above by EXIT analysis and confirm the results for the MIMO (2, 2) system.Figure4 shows that applying diagonal precoders on MIMO systems with LDPC codes deteriorates the performance of the transmission. All diagonal precoders present higher BER than the system with no precoding that was plotted as reference. The only precoders that permit a gain of about 4 dB at high Eb/N0 are the one based on the minimal Euclidean distance criteria, along with the max-SNR precoder.Figure5 illustrates another simulation done with the same configuration over an asymmetric (2, 4) MIMO system. The results show that the diagonal precoders are still damaging the performance and that the max-dmin⁡ and the max-dmin⁡-DFT still allow a gain of up to 1 dB. The max-SNR precoder shows a weaker performance than that of the symmetric (2, 2) MIMO system.These simulations confirm the performance predicted from theI0 values in Figure 2.These results suggest that the power allocation optimization of LDPC coded MIMO systems should be done using minimal Euclidean distance based precoders.The use of SVD allows transforming the MIMO channel matrix intob parallel independent eigensubchannels. The diagonal precoder (Fd is a diagonal matrix) preserves the diagonal structure, which allows transmitting each symbol over each eigensubchannel. The diagonal matrix Fd allows only obtaining a power allocation strategy among all the subchannels used. In this particular case, the diagonal structure leads to diversity loss. The diversity order of diagonal precoders using b substreams is (nT-b+1)(nR-b+1). Hence, increasing the throughput by sending multiple symbols at a time loses the full diversity order over flat-fading channel [24, 25]. We obtain performance degradation in terms of BER. Only the special case for max-SNR precoder (b=1) can reach full diversity order. The proposed NB-LDPC codes are not well suited for diagonal closed-loop systems (b>1) due to the bad LLRV initialization of the log-BP algorithm. This bad initialization was clearly examined in the VND curves of the diagonal precoders (Figure 2). The max-dmin⁡ precoder that presents full diversity order [22] is well suited with the proposed NB-LDPC. The I0 values of Figure 2 have already predicted this good match.Therefore, other simulations have been conducted on the max-dmin⁡ precoder (in which performances appear to be equivalent to the max-dmin⁡-DFT) in order to validate these results, and four different systems have been compared. The first one is a (2, 2) MIMO transmission without any error correcting coding or precoding. This system is equivalent to “spatial multiplexing (SM).” The second system consists in the same MIMO scheme with the LDPC coding block, which is referred to as “SM-LDPC.” The third system consists in a MIMO scheme where the max-dmin⁡ precoder is applied and is referred to as “max-dmin⁡.” The last system is the MIMO scheme with LDPC coding and where the power allocation optimization is done by applying the max-dmin⁡ precoder and is referred to as “max-dmin⁡-LDPC.” Figures 6, 7, and 8 show BER performance of these four systems for LDPC codes rates of 0.5 (dc=4), 0.75 (dc=8), and 0.83 (dc=12), respectively.Figure 6
BER performance for max-dmin⁡ with LDPC code rate R=0.5(dc=4).Figure 7
BER performance for max-dmin⁡ with LDPC code rate R=0.75dc=8.Figure 8
BER performance for max-dmin⁡ with LDPC code rate R=0.83(dc=12).In Figure6 we can see that the SM-LDPC system reaches the BER value of 10-5 for Eb/N0 equal to 12.5 dB while the max-dmin⁡-LDPC system reaches the same BER value for only 11 dB. Consequently, the max-dmin⁡-LDPC system has a gain of 1.5 dB over the SM-LDPC system for this same BER value.In Figure7, it can be noted that, for a higher code rate, the gain of the max-dmin⁡-LDPC system over the SM-LDPC is increased and attains 4.2 dB for a BER value of 10−5.In Figure8, the increase of the gain between the max-dmin⁡-LDPC and the SM-LDPC systems reaches 5 dB for a BER value of 10-5 and confirms that the improvement brought by the max-dmin⁡ precoder to an LDPC coded MIMO system is more significant for LDPC codes with higher rates.The presence of the max-dmin⁡ precoder allows an increase of the spectral efficiency compared to the SM-LDPC system. For example, we can see that the BER performance of the SM-LDPC for code rate R=0.5 (Figure 6) and the BER performance of the max-dmin⁡-LDPC for code rate R=0.83 (Figure 8) are very similar. The transmission rate is 4×0.5=2 for SM-LDPC and 4×0.83=3.33 for max-dmin⁡-LDPC. This is equivalent to a 67% increase in the transmission rate for the same BER performance, due to the use of the max-dmin⁡ precoder.The convergence rate of the LDPC decoder, when combined with linear precoding, has been appraised through the simulations and the results point out that a slight improvement is obtained compared to the SM-LDPC transmissions. This result has been already predicted and explained via the EXIT charts analysis above.We denote by “Nb-iter” the required number of iterations for the BP algorithm to converge (i.e., null syndrome). Figure9 plots the probability of Nb-iter to be less than or equal to the numbers on the x-axis (abscissa), P (Nb-iter ≤ X) for X=1,2,…,20 and for a fixed SNR per bit at 6 dB. For X=1 the BP algorithm converges at the iteration number 1 in 23% of the cases for the SM-LDPC system, while it converges at the first iteration in 49% of the cases for the max-dmin⁡-LDPC. For larger values of X, the gain becomes smaller, but still noticeable.Figure 9
Probability of number of iterations till convergence for SM-LDPC and max-dmin⁡-LDPC transmissions.


## 8. Conclusion

We have presented in this paper a new power allocation optimization scheme that deals with MIMO transmission scheme using ultrasparse NB-LDPC codes adapted for closed-loop MIMO systems and targeted for LTE-A with small to moderate codeword lengths. The proposed scheme makes use of a nondiagonal linear precoder based on the minimal Euclidean distance criteria in order to improve performance in terms of BER. Our study has shown that it is possible to join precoding techniques to such MIMO systems with NB-LDPC codes, but our simulation results also reveal that no improvement is brought by diagonal linear precoders when NB-LDPC codes are used. However, the special case of the max-SNR is competitive fornT=nR=2 only. The results of our simulation are predicted and explained by a theoretical EXIT chart analysis that shows the impact of using different precoders on the initial mutual information of the decoder and subsequently its influence on the performance of the proposed association. We have established that using a full diversity order precoder (max-dmin⁡ or max-dmin⁡-DFT) along with NB-LDPC results in a considerable improvement of the transmission performance and leads to a slight improvement of the convergence rate for the log Belief Propagation algorithm of LDPC decoder. Consequently, thanks to the proposed scheme, a smaller SNR is required to achieve a given BER value. The spectral efficiency is also increased due to the use of LDPC codes of higher rate along with the corresponding precoding. This result implies less energy consumption in the system and promotes green communication applications.



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*Source: 975139-2015-07-27.xml*