Journal of Communications and Information Networks

In the last few decades, multiple-input multiple-output(MIMO)systems have emerged as an important technology amongst the methodologies known to guarantee a high data rate in wireless communication systems. The performance improvement of the MIMO systems in terms of either the link reliability or data throughput depends on the assumption of the availability of channel state information(CSI)at the transmitter(CSIT)and/or that of the state information at the receiver(CSIR). Obtaining the correct CSIT or CSIR in real time is impossible because of the dynamic nature of the channel and the channel estimation errors. However, it is important to outline a system that is sufficient to achieve imperfect CSIT and/or CSIR. MIMO systems can be sub-divided into three fundamental classifications: spatial diversity, spatial multiplexing^[1,2,3], and beamforming^[4,5,6].

In single-user MIMO (SU-MIMO) systems, spatial diversity can be obtained through the utilization of spacetime codes^[7,8]. The transmit beamforming with receive combining^[9,10]was one of the simplest methodologies to enable spatial multiplexing in SU-MIMO systems to accomplish full diversity. Appropriate transmit precoding designs or joint precoder-decoder designs were proposed under a variety of system objectives and different CSI assumptions^[11]. We previously proposed another beamforming method utilizing singular value decomposition(SVD)for closed-loop SUMIMO systems with a convolution encoder and modulation techniques, for example, M-quadrature amplitude modulation (M-QAM)and M-phase shift keying(M-PSK)over Rayleigh fading^[4,5,6].

As design criteria, different performance measures were considered, for example, weighted minimum mean square error(MMSE)^[12], total mean square error(TMSE)^[13], least bit error rate (BER)^[14]. From the point of view of signal processing, TMSE is a critical metric for transceiver design and has been embraced in SU-MIMO systems to minimize the information estimation error from the received signal. A joint transceiver design utilizing an MSE paradigm was also discussed for the SU-MIMO framework^[12].

The above paragraph provides a general introduction and addresses a few optimization criteria such as an extreme data rate, least BER, and MMSE. The design of an optimum linear transceiver for an SU-MIMO channel, possibly with delay spread, utilizing a weighted MMSE paradigm subject to a transmit power constraint was reported^[12]. These studies assumed that the perfect CSI was available on the transmitter side. However, in practical communication systems, the propagation environment may be more challenging, and the receiver and transmitter cannot have perfect knowledge of the CSI. An imperfect CSI may emerge from an assortment of sources, for example, outdated channel estimates, erroneous channel estimation, and quantization of the channel estimate in the feedback channel^[15].

An important problem to investigate with the aim of obtaining a robust communications system is to determine whether it would be possible to design MIMO systems with an imperfect CSI. Optimal precoding strategies in SU-MIMO systems were proposed under the assumption that imperfect CSI is available at the transmitter, and perfect CSI is available at the transmitter^[16]. A robust joint precoder and decoder design to reduce the TMSE with imperfect CSI at both the transmitter and receiver of SU-MIMO systems was proposed in Refs. [17,18].

Novel precoding techniques to enhance the performance of the downlink in MU-MIMO systems were studied with an improper constellation^[19]. The precoder that was designed^[19,20] was more appropriate for a MIMO system with improper signal constellation. The joint precoder and decoder design under the minimum TMSE measure produced exceptional BER performance for proper constellation techniques, e. g. , M-PSK and M-QAM^[21,22]. Then again, when applying the same outline to the improper constellation techniques, e. g. , M-ASK and BPSK, the performance is fundamentally corrupt. A minimum TMSE design for an SU-MIMO system with improper modulation techniques was proposed and found to perform predominantly in terms of BER compared to the traditional design^[23].

A novel optimal strategy for nonlinear precoding in a MIMO system was designed^[24], and simulation results were provided to show that the proposed nonlinear precoding approach clearly outperforms the optimal linear precoding approaches. To avoid the computational complexity in iterativebased linear uplink MU-MIMO systems, a non-iterative joint SVD-based precoder and decoder for uplink MIMO systems with perfect CSI was proposed^[24] and the design was compared with conventional iterative-based linear uplink MIMO systems. Significant performance gains of the non-iterative approach over previous iterative designs in terms of the BER of the system were thoroughly demonstrated with simulation results.

A literature review revealed that a transceiver design of both a nonlinear and non-iterative nature is neither available for SU-MIMO nor for MU-MIMO systems. Our research aimed to address this shortcoming by examining the problem of a nonlinear and non-iterative precoder design for an SU-MIMO system with maximum likelihood (ML) decoding as the main objective of this study. Based on the nonlinear structure of the precoder, three different methods (Method 1, Method 2, and Method 3)are proposed. The approach we followed was to design a less complex and most efficient MIMO transceiver by combining nonlinearity and a non-iterative structure in the MIMO system. The simulation results verified the superiority of all the proposed methods over conventional methods. In the near future, we plan to use the proposed methods in large-scale or massive MIMO^[25,26] to increase the spectral efficiency for next generation wireless systems.

The remainder of the paper is organized as follows. The system model for the proposed nonlinear and non-iterative precoder design for MIMO systems is presented in section Ⅱ. The proposed NCP methods for MIMO systems are presented in section Ⅲ together with a suitable example. The simulation results are presented in section Ⅳ. Section V concludes the paper.

Notation: Throughout this paper, (·)^T denotes matrix transpose, (·)^H represents matrix conjugate transpose, diag[H(1), …, H(n)]is an n×n diagonal matrix with diagonal elements H(i), i=1, …, n, and I_n is an n×n identity matrix.

To enable spatial multiplexing in SU-MIMO systems, the appropriate transmit precoding design or joint precoderdecoder designs were proposed under a variety of system objectives and different CSI assumptions^[27]. Most of the studies assumed that the perfect CSI was available at the transmitter side. However, in practical communication systems, the propagation environment may be more challenging, and the receiver and transmitter cannot have a perfect knowledge of the CSI. A robust communications system can be obtained by designing MIMO systems with imperfect CSI as an important matter to investigate^[28]. The optimum joint linear transceiver is designed for SU-MIMO systems that utilize improper constellation strategies, either under the imperfect or perfect CSI that was proposed^[17]. The computation complexity in an iterative structure was reduced by proposing and designing an SVD-based non-iterative transceiver for MIMO systems^[29]. When the base station (BS) obtains the perfect CSI of all mobile stations, and each of the mobile stations has its own specific perfect CSI, the SVD-assisted method can decouple the multi-user channel into multiple independent SISO subchannels. A novel optimal nonlinear transceiver design for a MIMO system is also proposed to show that the nonlinear precoding-based MIMO system outperforms the equivalent linear system^[24].

In this work, we combined both non-iterative and nonlinear MIMO systems to produce a less complex and more efficient SU-MIMO system. Tab.1 presents a comparison of the various parameters of the different conventional transceiver and proposed transceiver schemes.

In this section, the design of MDS-based nonlinear constel-lation precoding for a MIMO system is considered. The MDS precoder is used to convert information bits $B$ of size 1×n to a code word $\mathcal{S}$ of size 1×m on a fixed constellation with a diversity order of d. A code word is constituted of multi-ple information symbols, where all symbols in a code word are constructed from a q point constellation Q mapper and en-coder $H_{m,d}$. Information bits that are passed to the encoders can be written in a vector form as follows^[30]

The encoder is used to construct q-ary $\left(\mathcal{S},2^n,d\right)$ MDS codes. It chooses the q value as small as possible and a power of 2 for fixed values of $B$, n, and d for the purpose of easy implementation of the algorithm. The encoder is based on binary linear codes as well as the labeling Q, and the encoder performs one-to-one mapping from q symbols to t bits. First, the binary linear code L of[mt, n, d1]is designed with the generated matrix $G$ defined as

where d1 is the minimum Hamming distance of L, ${{\rm I}}_n$ is the identity matrix of size n×n, and $F1$ is an n×t(d−1)binary matrix. The $F1$ can we written as

where $F{1}_{u,v}$ is a submatrix of $F1$, the value of u and v are defined as $1\le u\le m-d+1,and1\le v\le d-1$. The output from the encoder is defined as

The encoders are used to encode the n-length information bits $\left(B\right)$ to a mt-length binary code word $\left(F\right)$. The output from the encoders is given as input to the modulation section, which is designed to have a q value as small as possible. According to a single bound^[31], we have

Further, from Eq. (6), the value of q is

To construct q-ary $\left(\mathcal{S},q^{m-d+1},d\right)$ MDS codes from 2ⁿ code words, the constellation size is chosen as follows

We assume $t\triangleq 2^{\frac{n}{m-d+1}}$ throughout this paper without loss of generality; hence, q=2^t, t bits are mapped to a constellation point of Q. Then the output from mapper $\mathcal{S}$ is expressed as

Then t bits of $F1$ are mapped to the ith symbol ${\mathcal{S}}_i$ of S in accordance with the labeling of Q, 1≤i≤m. The bits are

The resulting code $\mathcal{S}$ is nonlinear on the q-ary field.

The non-iterative transceiver design for a MIMO system with perfect CSI is discussed in this section. As shown in Fig.1, let N_t be the number of antennas used at the transmitter and N_r be the number of antennas used at the receiver. The nonlinear encoded information symbols to be sent are denoted by an m×1 vector $\mathcal{S}={\left[S_1,S_2,\cdots ,S_m\right]}^T$, where m is the number of data streams. The symbol vector $\mathcal{S}$ is precoded using an N_t×m precoding matrix $F2$ to produce an N_t×1 precoded vector $x=F2\ast \mathcal{S}$, which is then transmitted simultaneously over N_t antennas. The data symbols are assumed to be uncorrelated and have zero mean and unit energy, i. e. , $E\left[\mathcal{S}{\mathcal{S}}^H\right]={{\rm I}}_m$.

Here we use the SVD method to develop a non-iterative precoder. Using SVD channel decomposition, the MIMO channel, $\widehat{H}$, can be decomposed as follows.

where $U$ and $V$ are two unitary matrices of size N_r×N_r and N_t×N_t, respectively, and (·)^H denotes the conjugate transpose and $\Lambda$ is the N_t×N_t diagonal matrix with non-negative real numbers on the diagonal i. e. , $\Lambda =diag\left\{{\Lambda }_1,{\Lambda }_2,\cdots ,{\Lambda }_{N_t}\right\}$. Then, the precoder can be defined as

where $\beta =\sqrt{P_T/Tr\left(V{\Lambda }^{-1}{\Lambda }^{-H}{V}^H\right)}$ where P_T is the total transmit power at the transmitter end. In the MIMO channel, the transmitter simultaneously transmits m symbols(i. e. , ) S₁, S₂, …, S_m from a finite constellation point of Q. At the receiver end, the received signal vector is defined as follows

where the vector w represents N_r-dimensional noise with $\zeta \left(0;{\sigma }_w^2\right)$. Upon substituting Eqs. (12) and (13) to Eq. (14), the vector $y$ of the received signal can be expressed as

The received signal $y$ is fed to the decoder $G$, which is an n×N_r matrix. Then the estimation of the resultant vector at the receiver is:

In practice, the CSI is usually imperfect and partially known for many reasons such as poor channel estimation, erroneous or outdated feedback, and time delays or frequency offsets between the reciprocal channels. Therefore, MIMO systems design under perfect CSI is no longer suitable for MIMO systems operating with estimated channel information. To improve the robustness of communication systems, the imperfectness of CSI with transmit and receive correlations has to be taken into consideration. In addition, the channel estimation is performed based on an orthogonal training method^[23].

The MIMO channel with transmit and receive correlation information is denoted as $H=R_r^{1/2}{H}_w{R}_t^{1/2}$ where $H_w$ is a spatially white matrix whose entries are independent and identically distributed(i. i. d. ) $N_c\left(0,1\right)$. The matrices $R_t$ and $R_r$ represent the normalized transmit and receive correlations, respectively.

In general, the channel is estimated at the receiver, the estimated information is fed back to the transmitter but that feedback information is not perfect due to feedback delays and errors. The transmitter can only obtain an erroneous estimate $\widehat{H}$ of the true channel $H$. The MIMO channel with imperfect CSI and both the transmit and receive correlation information can be written as

where $\widehat{H}=R_r^{1/2}{\widehat{H}}_w{R}_t^{1/2}$ and ${\rm E}=R_r^{1/2}{\widehat{{\rm E}}}_w{R}_t^{1/2}$. The entries of $H_w$ and Ew are independent. By substituting the value of $\widehat{H}$ and E in Eq. (18), we obtain

where $R_{e,r}={\left[{{\rm I}}_{N_r}+{\sigma }_{ce}^2{R}_r^{-1}\right]}^{-1}$ and ${\sigma }_{ce}^2=Tr\left(R_t^{-1}\right){\sigma }_n^2/P_{tr}$ and P_tr is the training power and ${{\rm E}}_w$ is independent with the data and noise vector. As in perfect CSI, the precoder corresponding to the imperfect CSI is derived by decomposing ${\widehat{H}}_w$ using the SVD channel decomposition method.

At the receiver end, we use ML as a decoding method to estimate the transmitted symbol, $\widehat{\mathcal{S}}$. The optimal ML receiver tries to minimize the probability of error(i. e. , $P_e=P\left(\mathcal{S}\ne \widehat{\mathcal{S}}\right)$). In other words, it can also be defined as maximizing the probability of correct estimation $\mathcal{S}\left(i.e.,P\left(\mathcal{S}=\widehat{\mathcal{S}}|r,U\Lambda V^H\right)\right)$. Most commonly, maximizing the probability of correct estimation can be defined as follows

where $f_{r|S,H}$ is the conditional probability density function (CPDF) of r given $\left(\mathcal{S},H\right)$ and $f_{r|H}$ is the CPDF of the r given $H$. Further, from Eq. (21), we can see that both the $P\left(\mathcal{S}=\widehat{\mathcal{S}}\right)$ and $f_{r|H}\left(r|H\right)$ are independent on $\widehat{\mathcal{S}}$ and the criterion of $\left(P\left(\mathcal{S}=\widehat{\mathcal{S}}|r,U\Lambda V^H\right)\right)$ is maximized by the $\widehat{\mathcal{S}}$, which maximizes $f_{r|H}\left(r|\mathcal{S}=\widehat{\mathcal{S}},H\right)$. The ML detector is defined as the following equation

Eq. (21)can be further simplified by applying the model of Eq. (17)as given by Ref. [32]. The ML detector tries to find $\widehat{\mathcal{S}}$, which minimizes the following equation.

By substituting Eq. (12)and(17)in Eq. (22), we obtain

Thus, the ML detector tries to find $\widehat{\mathcal{S}}$, which produces the smallest distance between $GU\beta S+G\omega$ and $U\Lambda V^H\widehat{\mathcal{S}}$.

The 3-Nonlinear constellation and Non-iterative precoding for MIMO systems(3-NCNP-MIMO)is a design example for novel MDS-based NCP in MIMO systems. In this example, we are constructing a 64-ary(3, 2¹², 2)MDS code $S_3$ by using the following parameters m=3, n=12, d=2, q=64. The input vector $B$ to the NCP encoders can be defined as

The binary generator matrix $G$ of size 12×18 as in Eq. (2) is

where $F_{1,1}$, $F_{2,1}$ are 6×6 matrices and 0 denotes an all-zero matrix. According to Theorem 1^[31], $F_{1,1}$, $F_{2,1}$ are matrices of the size 6×6. In order to achieve the diversity order of 2 as per our proposed example, $F_{1,1}$ and $F_{2,1}$ are necessarily required to be full rank matrices. Therefore, we choose the following condition $F_{1,1}=F_{2,1}={{\rm I}}_6$. The output code words from the NCP encoder are represented as

Here we required a 64-QAM constellation mapper from Eq. (8), and the output from the mappers are represented as follows

In this work, three different nonlinear precoding methods for a MIMO system based on the design of the generator matrix $G_{3,2}$ are proposed. These are

1. Method 1—It follows the generator matrix $G_{3,2}$, which is defined in Eq. (2)and Theorem 1^[31]. The coding gain for the MIMO system using Method 1 with 64-QAM is defined as ${\zeta }_{Method1}={\left(d_{\min }^{64}\right)}^2$. Because of the nature of the poor coding gain in Method 1, it may not suitable for an application that needs to achieve additional coding gain. To resolve this issue, we propose Method 2, which is coding gain efficient.

2. Method 2—To achieve a coding gain(ζ)as large as possible, we propose Method 2 based on Theorem 3^[31] to design an $F$. Accordingly the pair matrix $F_{1,1}=F_{2,1}$ is designed as follows

The coding gain for the MIMO system using Method 2 with 64-QAM is defined as ${\zeta }_{Method2}={\left(d_{\min }^{64}\right)}^2$. The use of Method 2 enables us to achieve very good coding gain with little added complexity. We propose another method named Method 3 to reduce the complexity in Method 3 but achieve the same coding gain.

3. Method 3—the pair matrix $F_{1,1}=F_{2,1}$ is designed by using gray labeling and violating Theorem 3^[31], as follows

The coding gain for the MIMO system using Method 3 with 64-QAM is the same as that for Method 2 (i. e. , ) ${\zeta }_{Method3}={\left(d_{\min }^{64}\right)}^2$.

This section provides numerical results to illustrate the performance improvement of the proposed 3-NCNP-MIMObased Method 1, Method 2, and Method 3 in MIMO systems, in terms of the bit error rate(BER)vs. the signal-to-noise ratio (SNR). In particular, the following comparisons are made:

1. The performance of the proposed 3-NCNP-MIMO-based Method 1, Method 2, and Method 3 is compared with the linear and non-iterative precoding(LNIP)^[29]. This comparison is to show the benefit of the proposed nonlinear and non-iterative(NCNP)precoding technique in MIMO compare to linear and non-iterative precoding(LNIP).

2. The performance of the proposed 3-NCNP-MIMO-based Method 1, Method 2, and Method 3 is compared with the linear and iterative precoding(LIP)in Ref. [20]. The purpose of this comparison is to show the benefit of the nonlinear and non-iterative precoding technique in MIMO compared to linear and iterative precoding methods.

3. Method 1, Method 2, and Method 3 are compared to show the performance difference between the proposed methods.

The simulation results are averaged over at least 15 000 channel realizations. In all the simulation results reported in this section, the number of parallel date streams are set as B=12 and the number of transmit and receiver antennas are fixed as N_t =N_r=3. The transmit correlation metric is defined as $R_t\left(i,j\right)={\rho }_t^{\left|i-j\right|}$ for i, j=1, 2, …, N_t, where the receive correlation metric is defined as $R_r\left(i,j\right)={\rho }_r^{\left|i-j\right|}$ for i, j=1, 2, …, N_r. The SNR for all the simulation results in this paper is defined as $SNR=P_T/{\sigma }_n^2$ and the training phase SNR is defined as $SN{R}_{tr}=P_{tr}/{\sigma }_n^2=26.016dB$.

First, Fig.2 shows the simulation results of the proposed 3-NCNP-MIMO-based Method 1 for the ML decoding algorithm under perfect CSI and uncorrelated Rayleigh channel (N_t =N_r=3. The values of ${\sigma }_{ce}^2$ are 0 for ρ_t =0.0 and ρ_t =0.0). It compares the performance of the proposed 3-NCNPMIMO-based Method 1 with LNIP-based MIMO systems^[29]. It also compares the performance of the proposed 3-NCNPMIMO-based Method 1 with LIP-based MIMO systems^[20]. We considered MMSE as a decoding algorithm for both of the conventional methods considered in this study for comparison. It is clear from the simulation results that a performance improvement is achieved by 3-NCNP-MIMO-based Method 1 compared to both the conventional methods. These same results are presented in Tab.2 for clarity and comparison purpose.

Fig.3 compares the performance of the proposed 3-NCNPMIMO-based Method 2 for the ML decoding algorithm under perfect CSI and for an uncorrelated Rayleigh channel. In addition, the 3-NCNP-MIMO-based Method 2 designs are com pared with both LNIP^[29]and LIP in Ref. [20]. It can be seen from Fig.3 that the 3-NCNP-MIMO-based MIMO system based on Method 2 leads to a performance improvement compared to the conventional methods. Again, the performance improvement by the proposed 3-NCNP-MIMO-based Method 3 over the conventional design is clearly observed in Fig.4. A significant improvement in performance by the proposed 3NCNP-MIMO compared to the conventional linear-based precoding methods is shown.

A performance comparison of all three proposed methods along with the conventional LNIP and LIP is also conducted and the results are shown in Fig.5. This comparison shows that the performance of proposed Method 2 and Method 3 is almost identical, and both outperform Method 1. All three of the proposed methods are found to outperform the conventional methods under perfect CSI and for an uncorrelated Rayleigh channel.

Fig.6 shows the same performance as in Fig.5, but for the case of imperfect CSI with transmit and receive corre

lation. Note that, with $SN{R}_{tr}=P_{tr}/{\sigma }_n^2=26.016dB$ and ρ_t=ρ_r=0.5, one has ${\sigma }_{ce}^2=0.015$. Again, the performance improvement by all three of our different proposed designs compared to the conventional designs is clearly observed in Fig.6. These same results are presented in Tab.3 for clarity and comparison purpose.

Fig.7 examines the effect of channel correlations on the proposed MIMO system BER performance under imperfect CSI for proposed Method 1, Method 2, and Method 3, respectively. The two different sets of transmit/receive correlations that were considered are{ρ_t=0.5, ρ_r=0.5}; {ρ_t=0.0, ρ_r=0.0}. In general, Fig.7 shows that higher values of the transmit and receive correlations lead to larger performance losses.

This paper proposes a low-complexity high-performance precoder design for MIMO systems with both nonlinear and non-iterative processing strategies under both perfect and imperfect CSI. This nonlinear precoding method provides the flexibility of supporting any number of diversity channels and desired diversity order. The computational complexity of an iterative-based algorithm is reduced by using SVD-based noniterative algorithms to design a MIMO precoder and decoder. Significant performance gains of the proposed designs over previous designs in terms of the BER of the system are thoroughly demonstrated with simulation results. Finally, it is pointed out that the proposed nonlinear and non-iterative precoder design for the MIMO can be extended to the case of a massive MIMO system under imperfect CSI.