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Robust beamforming and spatial precoding for quasi-OSTBC massive MIMO communications

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Abstract

This paper considers the problem of massive multiple-input-multiple-output (MIMO) wireless communication systems with quasi-orthogonal space-time block code (QOSTBC) transmission in the presence of spatial correlation effect (SCE) and mutual coupling effect (MCE). Conventional MIMO channels with QOSTBC transmission suffer from several drawbacks. The number of transmit antennas is restricted to a few in order to achieve full transmission rate. Therefore, it is difficult to make a quasi-orthogonal space time block coded massive MIMO system to achieve full transmission rate. Moreover, MIMO channel with QOSTBC transmission is usually considered for the case where the number of user equipments (UEs) is one. We present a joint beamforming and spatial precoding method to deal with the above drawbacks. The proposed method incorporates a beamforming scheme and spatial precoding to formulate an appropriate optimization process to effectively alleviate the considered problems. The resulting optimization problem will be solved by using a cooperative coevolutionary particle swarm optimization algorithm under two proposed fitness functions. During the optimization process, the proposed method finds the optimal beamforming coefficients of the precoding matrix, the optimal normalized positions of transmit and receive antenna elements for the case of using linear antenna arrays, and the optimal angle differences of transmit and receive antenna elements for the case of using circular arrays. Based on the proposed method, we are able to cure the performance degradation of a quasi-orthogonal space time block coded massive MIMO system due to the SCE and MCE. Moreover, the proposed method makes QOSTBC MIMO communications with full transmission rate for any number of transmit antennas achievable. Several simulation examples are presented to show the superior bit error rate (BER) performances of QOSTBC wireless MIMO scenarios with linear as well as circular antenna arrays by using the proposed method as compared to the existing methods.

Introduction

Achieving higher transmission rates and increasing number of users have become the essential requirements of modern wireless MIMO communications. However, the performance and capacity of MIMO wireless communication systems are degraded due to the interference problem at the mobile users. Nowadays, it is well known that employing array beamforming techniques, we can make the transmitted information stream weighted with appropriate phases and amplitudes at each antenna sensor to create a directed beam form. As a result, the interference problem due to multiple UEs can be effectively mitigated. However, downlink beamformnig for alleviating the interference problem at the mobile users is not an easy task for the frequency-division duplex (FDD) mode of transmission in MIMO systems. This is due to only limited knowledge of the downlink channel available for the base station (BS). In the literature, the existing beamforming and precoding methods, such as [13], employ a set of pre-defined fixed beamforming coefficients to incorporate the necessary precoding design. The key idea of using the beamforming-based spatial precoding (BBSP) scheme of [3] is to gather all the antennas deployed at BS into several beam groups and employ a set of pre-defined fixed beamforming coefficients to design the necessary precoding matrix. As a result, each antenna in one beam group can transmit the downlink training signal to UEs simultaneously, and hence, the user equipments (UEs) only need to feed the selected beam indices back to the BS; the feedback overhead is drastically reduced. It has been shown [3] that the BBSP scheme preserves an acceptable downlink transmission performance with reduced downlink training overhead and has a significant reduction of the total feedback bits from each UE. Nevertheless, these methods cause severe interference problem between UEs. Consequently, the bit error rate (BER) performance of the whole system becomes poor. Recently, a joint beamforming and spatial precoding technique was presented by [4] to optimally design the beamforming coefficients to effectively cure the interference problem between UEs. It has been shown in [4] that this technique not only can cure the problem of the BBSP scheme, but also preserves the advantages of requiring lower overheads of the downlink training and the channel state information (CSI) feedback for massive wireless MIMO systems.

For developing the next generation wireless mobile communications, it has been realized that employing wireless MIMO systems with massive antenna arrays deployed at transmitter and receiver is the key manner of achieving the necessary requirements of channel capacity, energy efficiency, and spectral efficiency [5, 6]. Unfortunately, massive MIMO wireless systems suffer from the degradation of diversity gain due to the spatial correlation effect (SCE) of the fading signals between the array elements with limited spacing. Moreover, a phenomenon called mutual coupling effect (MCE) in addition to the SCE inevitably occurs in a massive antenna array. Due to antenna elements in close proximity, the MCE is caused by a part of data in one antenna element outflow to other antennas when the antennas operate simultaneously. The performance degradations of a massive MIMO wireless system due to the SCE and MCE have been reported in [710], respectively. Recently, optimally finding the antenna deployment and antenna location in massive MIMO systems has been considered in the literature. In [11], the uplink transmission of a single-cell multiuser distributed massive MIMO system with a large number of distributed antennas deployed at BS which receives information from multiple single-antenna users is considered. The radius of the distributed circular BS antenna array can be optimally designed for maximizing the average achievable rate of the distributed massive MIMO system. Later on, Koyuncu [12] considers the single-cell multi-user MIMO uplink transmission systems with several single antenna transmitters/users and one BS with massive antennas. For a massive MIMO uplink where the BS antennas are evenly distributed to n different locations of the cell. It has been shown that a per-user rate is achievable by optimizing the n different antenna locations of the cell. However, there are practically no papers concerning the design problem of beamforming and spatial precoding for the massive MIMO systems with QOSTBC transmission and proposing effective approaches to mitigate the SCE and MCE simultaneously.

In this paper, we consider the robust beamforming and spatial precoding to enhance the BER performance of massive MIMO systems with QOSTBC transmission in the presence of MCE as well as SCE. A robust method is developed to incorporate the cooperative coevolutionary particle swarm optimization (CCPSO) [13] with the beamforming-selection spatial precoding (BSSP) scheme [4]. Recently, there are few works concerning quantized feedback-based precoding for multiple-input single-output (MISO) systems [1416]. Quantized feedback method is developed to achieve performance gain in MIMO system by transmitting a few bits feedback about CSI from receiver to transmitter for reducing feedback overhead. Quantized feedback-based precoding is a closed-loop system proposed to alleviate the performance degradation due to erroneous CSI feedback. It exploits the channel information available by transmitting few bits feedback about the CSI from the receiver to transmitter. The research works [1416] focus on the imperfect quantized feedback-based diagonal precoding for orthogonal/non-orthogonal space-time block codes in MISO systems with only one receive antenna. The spatial precoding matrices used are all set to diagonal matrices. However, the robust method proposed in this paper explores the joint design of beamforming and precoding for QOSTBC massive MIMO systems with any number of transmit antennas and receive antennas to deal with the SCE and MCE. Moreover, the spatial precoding matrix designed by the proposed method is not restricted to be a diagonal matrix.

The main contributions of the robust method over the work of [4] are briefly described as follows. An efficient scheme is developed to incorporate the joint beamforming and spatial precoding with QOSTBC transmission. To achieve the optimal power distribution to the beam groups partitioned by the BSSP scheme, we introduce an appropriate manner to decide the optimal power parameters during the CCPSO optimization process. Moreover, to mitigate the SCE and MCE simultaneously, we present an efficient approach to optimally adjust the positions of the antenna array elements by maximizing the average mutual information (AMI) associated with the downlink channel. Based on an approximated average BER (AA-BER) of wireless MIMO systems proposed for the fitness function of the CCPSO, the robust method finds the optimal beamforming and precoding coefficients by minimizing the AA-BER of wireless MIMO systems. The robust method not only can provide better average BER performance than the BSSP method, but also preserves the advantages of the BSSP method having lower overheads of the downlink training and the CSI feedback in massive wireless MIMO systems. Moreover, the robust method makes QOSTBC MIMO communications with full transmission rate for any number of transmit antennas achievable. Simulation examples confirm the effectiveness of the robust method in the QOSTBC wireless MIMO channel scenarios with SCE as well as MCE. The main contributions of this paper are summarized as follows:

(I) We develop a robust joint beamforming and spatial precoding method to effectively deal with QOSTBC MIMO communication systems in the presence of mutual coupling and spatial correlation effects due to using massive antennas. Through the proposed beam selection spatial precoding scheme in conjunction with the appropriate optimization formulation, the considered problems can be effectively alleviated as shown by simulation examples.

(II) The proposed joint beamforming and spatial precoding method can preserve the advantage of reducing the overheads for downlink training and feedback of channel state information between the base station (BS) and user equipments (UEs) for the frequency-division duplex (FDD) mode of downlink transmission in MIMO systems.

(III) In the literature, it is well known that the QOSTBC can achieve 1/2 of the full transmission rate for any number of transmission antennas as well as the codes with 3/4 of the full transmission rate for the specific cases of three and four transmit antennas. There are practically no papers developing the spatial precoding for QOSTBC MIMO communications deployed massive antennas with full transmission rate. Through the proposed beam selection spatial precoding scheme in conjunction with the QOSTBC strategy, the proposed joint beamforming and spatial precoding method provides a novel manner to make QOSTBC MIMO communications with full transmission rate for any number of transmit antennas achievable.

This paper is organized as follows. In Section 2, we briefly describe the downlink transmission model in wireless MIMO systems with/without spatial correlation as well as mutual coupling effects. The principle of wireless MIMO communication systems using QOSTBC transmission is also summarized. In Section 3, we propose a robust method based on the CCPSO algorithm for QOSTBC MIMO communication systems. Appropriate fitness functions for implementing the CCPSO algorithm to tackle the considered problem are also presented. Section 4 provides the simulation results for illustration and comparison. Finally, we conclude this paper in Section 5.

We use the following notation in the paper: Superscript.H denotes the transpose-conjugate operation. Superscript.T denotes the transpose operation. S(:,j) stands for the jth column vector of matrix S. The Frobenius norm of a matrix S is denoted by ||S||.

System model and fundamentals

Downlink transmission model

For the downlink transmission of a wireless MIMO channel, Fig. 1 depicts the basic system model of MIMO channels for the BS with NB antennas and each UE with NU antennas. Consider that the BS is equipped with a linear array of NB antennas and K UEs with a linear array of NU antennas deployed at each UE, we assume that the linear arrays of BS and each UE have the reference point located at the first antenna element as shown in Fig. 2. A typical NU×NB MIMO channel matrix H between the BS and each UE for the corresponding MIMO channel can be written as follows [17]:

Fig. 1
figure1

Downlink block diagram of MIMO channels for the BS with NB antennas and each UE with NU antennas

Fig. 2
figure2

A uniform linear array with signal direction angle θ

$$ {\begin{aligned} \textbf{H}\!&=\! \left[ \begin{array}{cccc} h_{1,1} & {\dots} & h_{1,N_{B}} & \\ \vdots & {\ddots} & \vdots & \\ h_{N_{U},1} & {\dots} & h_{N_{U},N_{B}} &\end{array}\right] \!\\&\quad=\!\sum_{l=1}^{L}a_{l}\sqrt{N_{U}N_{B}/L}\textbf{e}_{r}(\theta_{r,l})\textbf{e}_{t}(\theta_{t,l})^{H}, \end{aligned}} $$
(1)

where θr,l and θt,l denote the corresponding direction angles of arrival and departure, respectively. Moreover, the vectors er(θr,l) and et(θt,l) represent the receive and transmit array response vectors associated with the angles of θr,l and θt,l, respectively. al is the complex gain of the lth downlink transmission path. L is the total number of transmission paths. Moreover, the unit-norm array response vector of an M-element linear array with direction angle = θ off array broadside, irrespective of transmit and receive antennas, can be expressed as follows:

$$ \textbf{e}(\theta)=\sqrt{1/M} \left(\begin{array}{cc} 1\\ e^{j\pi (A_{1} sin \theta)}\\ \vdots \\ e^{j\pi (A_{M} sin \theta)} \end{array} \right), $$
(2)

where Ai,i=1,2,...,M represents the location normalized by the half signal wavelength of the ith antenna element. On the other hand, if the BS is equipped with a circular array of NB antennas and K UEs with a circular array of NU antennas deployed at each UE. The corresponding NU×NB MIMO channel matrix H between the BS and each UE is given by [17]:

$$ {\begin{aligned} \textbf{H}&= \left[ \begin{array}{cccc} h_{1,1} & {\dots} & h_{1,N_{B}} & \\ \vdots & {\ddots} & \vdots & \\ h_{N_{U},1} & {\dots} & h_{N_{U},N_{B}} &\end{array} \right] \\&\quad=\sum_{l=1}^{L}a_{l}\sqrt{N_{U}N_{B}/L}\textbf{e}_{r}\left(\phi_{r,l},\theta_{r,l}\right)\textbf{e}_{t}\left(\phi_{t,l},\theta_{t,l}\right)^{H}, \end{aligned}} $$
(3)

where (ϕr,l,θr,l) and (ϕt,l,θt,l) denote the corresponding (azimuth, elevation) angles of arrival and departure, respectively. Moreover, the vectors er(ϕr,l,θr,l) and et(ϕt,l,θt,l) represent the receive and transmit array response vectors associated with the angles of (ϕr,l,θr,l) and (ϕt,l,θt,l), respectively. Finally, al is the complex gain of the lth downlink transmission path. Assume that the circular arrays of BS and each UE have the reference point located at the center of the circular array and radius equal to R as shown in Fig. 3, then the unit-norm array response vector of an M-element circular array with azimuth angle = ϕ and elevation angle = θ, irrespective of transmit and receive antennas, can be expressed as follows:

Fig. 3
figure3

A uniform circular array with signal direction angle (θ,ϕ)

$$ \textbf{e}(\phi,\theta)=\sqrt{1/M} \left(\begin{array}{cc} e^{j\pi \rho (sin(\theta) cos(\phi - \varphi_{1}))}\\ \vdots \\ e^{j\pi \rho (sin(\theta) cos(\phi - \varphi_{M}))} \end{array} \right), $$
(4)

where φk represents the angle difference between the first and the kth antenna elements of the circular array. For a uniform circular array, \(\varphi _{k}=\frac {2\pi (k-1)}{M}\). \(\rho =\frac {1}{\sqrt {2(1-cos(2\pi /M))}}\) represents the radius normalized by the half of the signal wavelength.

For the downlink of a multiuser MIMO wireless system with K UEs, the received downlink training signal matrix \(\textbf {Y}_{k}^{DT}\) at the kth UE can be written by [4]:

$$ \textbf{Y}_{k}^{DT}=\textbf{H}_{k}\textbf{S}_{k}+\textbf{N}_{k}, $$
(5)

where Sk denotes the NB×NB training signal matrix and Nk the NU×NB background noise matrix received by the kth UE. Moreover, the NU×1 received signal vector \(\textbf {y}_{k}^{DL}\) at the kth UE can be written as [17]:

$$ \textbf{y}_{k}^{DL}=\textbf{H}_{k}\textbf{W}_{k}\textbf{x}_{k} + \sum_{i\neq k}^{K} \textbf{H}_{k} \textbf{W}_{i}\textbf{x}_{i} + \textbf{n}_{k}, $$
(6)

where xk denotes the data symbol vector and nk the NU×1 background noise vector. Wk denotes the corresponding precoding matrix with appropriate size and Hk with NU×NB the downlink channel matrix associated with the kth UE.

In the literature, a point-to-point wireless MIMO system with spatial correlation effect only was considered in [18]. In the presence of the spatial correlation and mutual coupling effects for a downlink communication, according to the analysis presented in [19, 20], the channel matrix H for the non line of sight (NLOS) propagation condition (i.e., L>1) can be simplified as the following expression:

\(\textbf {H}= \left [ \begin {array}{cccc} h_{1,1} & {\dots } & h_{1,N_{B}} & \\ \vdots & {\ddots } & \vdots & \\ h_{N_{U},1} & {\dots } & h_{N_{U},N_{B}} & \\ \end {array} \right ] \)

$$ =\sum_{l=1}^{L}a_{l}\sqrt{N_{U}N_{B}/L}\left(\textbf{R}_{U}^{MC}\right)^{1/2}\textbf{H}_{w,l}\left(\textbf{R}_{B}^{MC}\right)^{1/2}, $$
(7)

where Hw,l denotes the matrix with i.i.d. Gaussian entries for the lth downlink transmission path. \(\textbf {R}_{B}^{MC}\) and \(\textbf {R}_{U}^{MC}\) denote the matrices containing the normalized spatial correlation coefficients in the presence of mutual coupling between the antenna elements deployed at BS and UEs, respectively. Moreover, the (i,j)th elements of \(\textbf {R}_{U}^{MC}\) and \(\textbf {R}_{B}^{MC}\) are respectively given as follows [20]:

$$ R_{U}^{MC}(i,j)=\frac{1}{\sqrt{P_{Ui}P_{Uj}}} \sum_{m=1}^{N_{U}}\sum_{n=1}^{N_{U}}C_{U}(i,m)C_{U}(j,n)^{*}R_{U}(m,n), $$
(8)

and

$$ {{} \begin{aligned} R_{B}^{MC}(i,j)\,=\,\frac{1}{\sqrt{P_{Bi}P_{Bj}}} \sum_{m=1}^{N_{B}}\sum_{n=1}^{N_{B}}C_{B}(i,m)C_{B}(j,n)^{*}R_{B}(m,n), \end{aligned}} $$
(9)

where CU(i,m) and CB(j,n) represent the (i,m)th and (j,n)th elements of the mutual coupling matrices CU and CB associated with the receiver antenna array and the transmitter antenna array, respectively. They can be calculated by [21]:

$$ \textbf{C} = ({Z}_{A} + {Z}_{T})+{(\textbf{Z} + {Z}_{T} \textbf{I})}^{-1}, $$
(10)

where

$$ \textbf{Z}= \left[ \begin{array}{cccc} Z_{1,1} & {\dots} & Z_{1,M} & \\ \vdots & {\ddots} & \vdots & \\ Z_{M,1} & {\dots} & Z_{M,M} & \\ \end{array} \right] $$
(11)

and M is the number of antennas. Consider the dipole length is half wavelength \(\left (l=\frac {\lambda }{2}\right)\). The (a,b)th element of Z is generated as follows:

$$ {{} \begin{aligned} Z_{a,b} = \left\{ \begin{array}{ll} \frac{\eta}{4 \pi} (\zeta + \ln(2kl) -C_{i}(2kl)) + j \frac{\eta}{4 \pi} S_{i}(2kl) = Z_{A} \\, a=b\\ \\ \frac{\eta}{4 \pi} \left\{ (2C_{i}(k d_{a,b})) - C_{i}\left(k\left(\sqrt{d_{a,b}^{2} + l^{2}} + l\right)\right)\right. \\ \left. - C_{i}\left(k\left(\sqrt{d_{a,b}^{2} + l^{2}} - l\right)\right) \right\} \\ - j \frac{\eta}{4 \pi} \left\{ (2S_{i} (k d_{a,b})) - S_{i}\left(k\left(\sqrt{d_{a,b}^{2} + l^{2}} + l\right)\right)\right. \\ \left.- S_{i}\left(k\left(\sqrt{d_{a,b}^{2} + l^{2}} - l\right)\right) \right\}, a \neq b \end{array}, \right. \end{aligned}} $$
(12)

where

$$Z_{T}=Z_{A}^{*}, $$
$$S_{i}(x) = \int_{0}^{x} \frac{\sin(\tau)}{\tau} d \tau, $$
$$C_{i}(x) = - \int_{x}^{\infty} \frac{\cos(\tau)}{\tau} d \tau, $$

η denotes intrinsic impedance 120π(ohms), ζ= ln(1.781)0.5772, \(k=\frac {2 \pi }{\lambda }\), and da,b is the distance between the ath and the bth antennas (in λ), a and b =1,2,....,M. Moreover, PUi and PBi represent the average powers of the ith antennas at UEs and BS, respectively. According to [20], they can be computed as follows:

$$ {\begin{aligned} P_{Ui}&={\Re} \left [\sum_{m=1}^{N_{U}}\sum_{n=1}^{N_{U}}C_{U}(i,m)C_{U}(i,n)^{*}R_{U}(m,n) \right]\\&\qquad\sum_{n=1}^{N_{U}}|C_{U}(i,n)|^{2} R_{U}(n,n), \end{aligned}} $$
(13)

and

$$ {\begin{aligned} P_{Bi}&={\Re} \left[\sum_{m=1}^{N_{B}}\sum_{n=1}^{N_{B}}C_{B}(i,m)C_{B}(i,n)^{*}R_{B}(m,n) \right ]\\&\qquad\sum_{n=1}^{N_{B}}|C_{B}(i,n)|^{2} R_{B}(n,n), \end{aligned}} $$
(14)

respectively, where [x] denotes the real part of x. RU(m,n) and RB(m,n) represent the (m,n)th elements of the RU and RB matrices containing the normalized spatial correlation coefficients associated with the antennas deployed at UEs and BS, respectively. Moreover, the formulas for computing RU(m,n) and RB(m,n) regarding circular or linear antennas are provided by [20].

Downlink communication using QOSTBC transmission

Space-time block coding (STBC) was presented by [22] to combine coding, modulation, and signal processing for achieving transmit diversity of a wireless communication. The space-time block encoder generates the simplest type of spatial temporal codes to exploit the transmission diversity provided by multiple transmit antennas in wireless communications. For downlink transmission, the STBC at BS with M antennas encodes the data to be transmitted into M transmitted code symbols. At each UE, the signal received by each receive antenna is the M transmitted signals plus noise. It was shown in [22] that full diversity can be achieved and a very simple maximum-likelihood decoding algorithm can be employed at the decoder with reduced data rates. However, it was shown in [23] that the data rate of a full-diversity code is less than or equal to one. To achieve the goal of a full-rate full-diversity STBC, a novel transmit diversity scheme was proposed by [24]. This so-called two-branch transmit diversity scheme is developed based on a 2×2 block code carried out in sets of two modulated symbols. As a result, the transmission rate can be one. Nevertheless, this STBC scheme is only suitable for two transmit antennas.

In order to search for similar schemes available for MIMO systems with more than two transmit antennas to achieve diversity level higher than two, orthogonal space-time block code (OSTBC) was then introduced by [25]. OSTBC is a generalization of the scheme of [24] to arbitrary number of transmit antennas. Moreover, OSTBC preserves the advantage of having linear maximum-likelihood decoding with full transmit diversity. Unfortunately, it is also shown [25] that a complex orthogonal design and the corresponding space-time block code which provides full diversity and full transmission rate is not possible for more than two transmit antennas. Moreover, a large MIMO system using OSTBC transmission usually requires an OSTBC of very large dimension in order to take the advantages of OSTBC for transmission. Nevertheless, employing a large dimension OSTBC is not possible for complex constellations, and the data transmission rate of the resulting OSTBC MIMO system will be significantly reduced even if it can be achieved. Recently, an OSTBC transmission scheme embedding the OSTBCs of small dimensions based on the concept of null space to achieve good rate for large MIMO systems was proposed in [26]. It is shown by simulation of [26] that the code rate can be improved at the cost of degraded performance. For practical applications, as mentioned by [5], when large-scale or massive MIMO systems are considered, the BS is deployed with massive antennas to simultaneously serve a large number of UEs within the same frequency band and achieve higher spectral and energy efficiency. However, it has been shown in [26] that the proposed scheme of [26] is not work if the number of transmit antennas is larger than the number of receive antennas. Based on the similar concept of embedding the small OSTBCs, a code transmission scheme for large MIMO system was presented in [27]. Pilot matrices are employed in the proposed scheme to increase the data rate. Another work [28] considered the drawbacks caused by using STBCs and OSTBCs. A space-time transmission scheme (STTS) is then proposed in [28] for large MIMO communication systems. In this work, the proposed scheme compromises diversity order to obtain good data rate and decouple decoding. It was also mentioned that the proposed scheme may be useful for base station (BS) to BS communication. In the literature, some more works employ differential/non-differential orthogonal/non-orthogonal space-time block code (STBC) on the subject of fastly varying channels for arbitrarily correlated Rayleigh/Ricean channels have been presented in [29, 30]. On the other hand, a precoder design for orthogonal/non-orthogonal STBC over correlated Ricean MIMO channels with invertible correlation matrices has been developed in [31].

This drawback of the OSTBC motivates the research work [32] on developing the so-called quasi-orthogonal space-time block code (QOSTBC). It relaxes the orthogonality constraints due to OSTBC by constraining subsets of data symbols orthogonal to each other instead of constraining every single symbol orthogonal to any other. It has been shown in [32] that the QOSTBC can achieve 1/2 of the full transmission rate for any number of transmission antennas as well as the codes with 3/4 of the full transmission rate for the specific cases of three and four transmit antennas. A typical example of a full rate quasi-orthogonal code that achieves full transmission rate for four transmit antennas is as follows:

$$ \textbf{S}= \left[ \begin{array}{cccc} S_{1,2} & S_{3,4} \\ S_{3,4} & S_{1,2} \end{array} \right] = \left[ \begin{array}{cccc} s_{1} & s_{2} & s_{3} & s_{4} \\ -s_{2}^{*} & s_{1}^{*} & -s_{4}^{*} & s_{3}^{*} \\ -s_{3}^{*} & -s_{4}^{*} & s_{1}^{*} & s_{2}^{*} \\ s_{4} & -s_{3} & -s_{2} & s_{1} \\ \end{array} \right] $$
(15)

where si, i=1,2,3,4, are the symbols selected based on the input bits. Consider the simplest mobile communication system where BS is equipped with four transmit antennas and the UE is equipped with one receive antenna. Assume that the system has a slow fading channel. The signal vector received at the UE can be modeled as:

$$ {{}\begin{aligned} \textbf{y}\,=\, \left[ \begin{array}{cc} y(1) & \\ y(2) & \\ y(3) & \\ y(4) &\end{array} \right] &\,=\, \left(\left[ \begin{array}{cccc} h_{1,1} & h_{1,2} & h_{1,3}& h_{1,4} \end{array} \right] \textbf{W} \left[ \begin{array}{cccc} s_{1} & s_{2} & s_{3} & s_{4} \\ -s_{2}^{*} & s_{1}^{*} & -s_{4}^{*} & s_{3}^{*} \\ -s_{3}^{*} & -s_{4}^{*} & s_{1}^{*} & s_{2}^{*} \\ s_{4} & -s_{3} & -s_{2} & s_{1} \\ \end{array} \right] \right)^{T} \\&+ \left[ \begin{array}{cccc} n(1) & \\ n(2) \\ n(3) & \\ n(4) \end{array} \right] \end{aligned}} $$
(16)

where y(i) and n(i) represent the signal and the noise received by the receive antenna at the ith time slot. W denotes the corresponding 4×4 spatial precoding matrix.

The robust beamforming and spatial precoding method

In this section, we present a robust method to design beamforming and spatial precoding for dealing with the mutual coupling effect (MCE) and the spatial correlation effect (SCE) for QOSTBC massive wireless MIMO channels. To construct the spatial precoding matrix, the optimization of beamforming coefficients taking into account the spatial correlation as well as the mutual coupling effects between the antennas is established. To tackle the SCE and MCE which are due to antenna spacing and array geometry, we construct an appropriate optimization problem which can be solved by using the CCPSO algorithm to find the optimal beamforming coefficients for the spatial precoding matrix and the optimal locations for the antenna elements at the BS and UEs simultaneously. As shown below, the proposed method makes a QOSTBC massive MIMO wireless channel that achieves full-rate transmission for any number of transmit antennas achievable.

Instead of using a preset fixed beamforming coefficients as shown in [3] for the B beam groups of the NB antennas at the BS, we set a unit-norm steering vector of the bth beam group according to the BSSP scheme [4] as follows:

$$ \textbf{v}_{b}=\frac{1}{\sqrt{a_{b,1}+a_{b,2}+...+a_{b,N_{B}}}}\left(\begin{array}{cc} a_{b,1}e^{j{\theta}_{b,1}} \\ a_{b,2}e^{j{\theta}_{b,2}} \\ \vdots \\ a_{b,N_{B}}e^{j{\theta}_{b,N_{B}}} \end{array} \right), $$
(17)

where b=1,2,,B, ab,k and θb,k, k=1,2,,NB, are the variables to be adjusted by the CCPSO. In order to provide an appropriate precoding and reduce the unacceptable feedback overhead in the downlink transmission of massive MIMO channels, each UE selects N beams from the precoding matrix P=[v1,v2...,vB] as its precoder and feeds the selected beam indices back to the BS. Therefore, the proposed method employs beamforming to estimate channel state information (CSI) by using a small number of downlink training symbols for FDD massive MIMO systems. The is due to that partitioning all the antennas of the BS into several beam groups enables each antenna in one beam group to transmit the downlink training signal to UEs simultaneously. Since UEs only need to feed the selected beam indices back to the BS, the CSI feedback overhead in bits required by the proposed method is almost the same as that of the original BBSP method of [3]. It can be written as Nf=K×N×log2B. Moreover, the UE selects a beam according to the channel gain of each beam. For simplicity, assume that the kth UE selects the first N beams from the B beam groups as its serving beams. Accordingly, the signal vector received at the kth UE in the downlink transmission with power constraints on the selected beams becomes:

$$ \textbf{y}_{k}^{DL}=\textbf{H}_{k} \textbf{W}_{pk}\textbf{T}_{k}\textbf{x}_{k} + \sum_{i\neq k}^{K} \textbf{H}_{k}\textbf{W}_{pi}\textbf{T}_{i}\textbf{x}_{i} + \textbf{n}_{k}, $$
(18)

where xk denotes an N×1 data symbol vector and K the number of UEs. The precoder Wpk is given by:

$$ \textbf{W}_{pk} = \left[ \textbf{v}_{1} \textbf{v}_{2}... \textbf{v}_{N} \right]. $$
(19)

Moreover, Tk represents the power constraint matrix associated with the selected N beams and is given by:

$$ \textbf{T}_{k}= \left[ \begin{array}{cccc} \sqrt{t_{k1}} & {\dots} & 0 & \\ \vdots & {\ddots} & \vdots & \\ 0 & {\dots} & \sqrt{t_{kN}} &\end{array} \right], $$
(20)

where tki denotes the power distributed to the ith beam selected by the kth UE and the total power transmitted to the N beams is constrained by:

$$ \sum_{i}^{N} t_{ki} = P_{t}. $$
(21)

To incorporate the advantages of full transmission rate and low ML decoding complexity over the conventional multiuser beamforming methods (e.g., zero-forcing beamforming and MMSE beamforming) by using a simple 4×4 QOSTBC with the proposed joint beamforming and spatial precoding method, we can easily set the number of beam groups of the proposed beamforming scheme to 4, i.e., B=4. As a result, we can see that the proposed method preserves the advantages of computational complexity reduction (low ML decoding complexity) and full-rate transmission as compared to using some arbitrary linear dispersion code for massive MIMO communication systems.

Next, consider a QOSTBC that achieves full transmission rate for four-transmit antennas as described in (16) for downlink transmission, we note from (18) and (16) that the signal matrix received at each UE can be reformulated as follows:

$$ \textbf{Y}_{k}^{DL}=\textbf{H}_{k}\textbf{W}\textbf{T}\textbf{S} + \textbf{n}_{k}, $$
(22)

where \(\textbf {y}_{k}^{DL}\) represents the NU×4 data matrix. The ith row of \(\textbf {y}_{k}^{DL}\) is the 1×4 data vector received at the ith antenna of the kth UE. Hk is still the NU×NB channel matrix associated with the transmission. W represents the NB×4 spatial precoding matrix and given by W=[w1,w2,w3,w4], where each wi,i=1,2,3,4, represents the unit-norm steering vector of the ith beam group under the QOSTBC transmission. T denotes the corresponding 4×4 power constraint matrix given by:

$$ \textbf{T}= \left[ \begin{array}{cccc} \sqrt{t_{1}} & 0 & 0 & 0 \\ 0 & \sqrt{t_{2}} & 0 & 0 \\ 0 & 0 & \sqrt{t_{3}} & 0 \\ 0 & 0 & 0 & \sqrt{t_{4}} \end{array} \right] $$
(23)

and S the 4×4 QOSTBC symbol matrix as shown by (15). This formulation reveals that the resulting QOSTBC transmission system can provide a significant advantage of a full transmission rate for massive wireless communication systems using NB>>4 transmit antennas.

The formulation of the optimization problem

In this section, we present the problem formulation for the robust method based on the results developed above. The resulting optimization problem will be solved by using the CCPSO algorithm under appropriate fitness functions. The CCPSO algorithm demonstrates its potential for effectively solving multimodal and nonconvex highly nonlinear optimization problems with significantly higher dimensions [13]. As shown by (17), there are 2NB parameters, namely, \(\phantom {\dot {i}\!}a_{b,1}, a_{b,2},..., a_{b,N_{B}}, {\theta }_{b,1}, {\theta }_{b,2},...., {\theta }_{b,N_{B}}\) which can be adjusted during the optimization process. To mitigate the SCE and MCE, we aim at adjusting the normalized locations of the antenna elements deployed at BS and UEs. For the case of using linear arrays with M antennas, Ai,i=1,2,...,M are to be optimally decided. In contrast, the parameters φk,k=1,2,...,M are to be varied optimally when using a circular arrays with M antennas. Moreover, the elements ti,i=1,2,...,4 of the power constraint matrix T are to be optimally adjusted. Accordingly, we define a parameter vector which is to be found to achieve the robust beamforming and spatial precoding when NB transmit antennas and NU receive antennas are deployed at BS and each UE, respectively as follows:

$$ {\begin{aligned} \textbf{F}&= \left[ a_{1,1}, a_{1,2},..., a_{B,N_{B}}, {\theta}_{1,2},...., {\theta}_{B,N_{B}}, A^{t}_{1},..., A^{t}_{N_{B}},\right.\\&\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \left. A^{r}_{1},..., A^{r}_{N_{U}}, t_{1},..., t_{4} \right], \end{aligned}} $$
(24)

for the case of using linear antenna arrays, where the sets of \({A^{t}_{1},..., A^{t}_{N_{B}}}\) and \({A^{r}_{1},..., A^{r}_{N_{B}}}\) represent the normalized positions of transmit and receive antenna elements, respectively. In contrast,

$$ {\begin{aligned} \textbf{F}&= \left[ a_{1,1}, a_{1,2},..., a_{B,N_{B}}, {\theta}_{1,2},...., {\theta}_{B,N_{B}},\right.\\&\qquad\qquad\qquad \left. \varphi^{t}_{1},..., \varphi^{t}_{N_{B}}, \varphi^{r}_{1},..., \varphi^{r}_{N_{U}}, t_{1},..., t_{4} \right], \end{aligned}} $$
(25)

for the case of using circular arrays, where the sets of \({\varphi ^{t}_{1},..., \varphi ^{t}_{N_{B}}}\) and \({\varphi ^{r}_{1},..., \varphi ^{r}_{N_{U}}}\) represent the angle differences of transmit and receive antenna elements, respectively. From (24) and (25), we note that the number of entries in the parameter vector F is linearly increased by NB and NU. Hence, the computational complexity required for finding the optimal entries of F becomes huge during the optimization process for massive wireless communication systems. To alleviate this difficulty, a two-step optimization procedure is presented as follows. In the first step, we employ the CCPSO algorithm to find the optimal parameter set \(\left (A^{t}_{1},..., A^{t}_{N_{B}}, A^{r}_{1},..., A^{r}_{N_{U}}\right)\) related to the positions of the linear transmit and receive antenna elements, or the optimal parameter set \(\left (\varphi ^{t}_{1},..., \varphi ^{t}_{N_{B}}, \varphi ^{r}_{1},..., \varphi ^{r}_{N_{U}}\right)\) related to the positions of the circular transmit and receive antenna elements. After obtaining the optimal locations for the antenna elements, the channel matrix HK as shown by (7) can be determined. In the second step, based on the channel matrix HK obtained in the first step, the CCPSO algorithm is utilized to find the optimal parameter set \(\phantom {\dot {i}\!}\left (a_{1,1}, a_{1,2},..., a_{4,N_{B}}, {\theta }_{1,2},...., {\theta }_{4,N_{B}}, t_{1},..., t_{4}\right)\) related to beamforming coefficients and the power distribution. According to our experience, the two-step optimization procedure reduces the computation time significantly when performing simulation examples for illustration.

During the the first-step optimization process by using the CCPSO algorithm, each particle is treated as a point in the D(=NU+NB)-dimensional optimization problem space and a swarm consists of D random particles. Then, the CCPSO searches for the best position (solution or optimum) by updating generations until obtaining a relatively steady position or reaching the limit of a preset iteration number. At the tth moment or iteration, the position vector of the ith particle is given by xi(t) = \(\left [A^{t}_{1},..., A^{t}_{N_{B}}, A^{r}_{1},..., A^{r}_{N_{U}} \right ]\) for the linear transmit and receive antenna elements or xi(t) = \(\left [\varphi ^{t}_{1},..., \varphi ^{t}_{N_{B}}, \varphi ^{r}_{1},..., \varphi ^{r}_{N_{U}} \right ]\) for the circular transmit and receive antenna elements. The personal best position vector pi(t) and the global best position vector g(t) are obtained by evaluating the performances in terms of the fitness values associated with the current population of particles. Moreover, we update the position and velocity characterizing a particle status on the search space according to the formulas given by [13] and convert a particle position vector into a candidate solution vector by employing an appropriate mapping. The performance of the mapped solution vector evaluated by a massive MIMO channel is viewed as the fitness of the corresponding particle. At the end of iteration, the global best position vector g(tmax) is regarded as the optimal position vector for the linear or circular transmit and receive antenna elements. In general, the optimization process will be terminated if the specified maximum iteration number tmax is reached or the best particle position of the whole swarm remains almost static for a significantly large number of successive iterations.

During the second-step optimization process, each particle is treated as a point in the D(=8NB+4)-dimensional optimization problem space and a swarm consists of D random particles. Then, the CCPSO searches for the best position (solution or optimum) by updating generations until obtaining a relatively steady position or reaching the limit of a preset iteration number. At the tth moment or iteration, the position vector of the ith particle is given by xi(t) = \(\left [a_{1,1}, a_{2,1}, \cdots, a_{4,1}, a_{1,2}, \cdots, a_{4,N_{B}}, {\theta }_{1,2},...., {\theta }_{4,N_{B}}, t_{1},...., t_{4} \right ]\). The personal best position vector pi(t) and the global best position vector g(t) are obtained by evaluating the performances in terms of the fitness values associated with the current population of particles. Moreover, we update the position and velocity characterizing a particle status on the search space according to the formulas given by [13] and convert a particle position vector into a candidate solution vector by employing an appropriate mapping. The performance of the mapped solution vector evaluated by a massive MIMO channel is viewed as the fitness of the corresponding particle. At the end of iteration, the global best position vector g(tmax) is regarded as the optimal beamforming coefficient vector. Again, the optimization process will be terminated if the specified maximum iteration number tmax is reached or the best particle position of the whole swarm remains almost static for a significantly large number of successive iterations.

Fitness functions for CCPSO

Fitness function for the first-step CCPSO

In the first-step optimization process, we aim at finding the optimal positions for antenna elements. As shown by (7), the positions of antenna elements determine the entries of \(\mathbf {R}_{U}^{MC}\) and \(\mathbf {R}_{B}^{MC}\). Hence, the characteristics of the downlink channel matrix is significantly affected by the positions of antenna elements. According to information theory, the mutual information is used to measure how much information the output of a channel contains about the input. It has been shown in the literature like [33] that the average mutual information (AMI) can be viewed as an index for evaluating the characteristics of a channel matrix. Accordingly, it is appropriate to find the optimal positions of transmit and receive antenna elements by maximizing the AMI associated with the downlink transmission channel matrix H. According to [33], the AMI of a channel matrix H with size NU×NB is given by:

$$ \mathbf{AMI} = E\left[\log_{2} \left(\det \left[ \mathbf{I} + \frac{SNR}{N_{B}} \mathbf{H}\mathbf{H}^{H} \right] \right) \right], $$
(26)

where I denotes the NU×NU identity matrix and det[X] the determinant of the matrix X. SNR represents the signal-to-noise power ratio at each receive antenna. Therefore, we define an appropriate fitness for implementing the first-step optimization process by using the CCPSO algorithm as follows:

$$ \mathbf{{Fitness1}} = \frac{1}{Q}\sum_{q=1}^{Q} \log_{2} \left(\det \left[ \mathbf{I} + \frac{SNR}{N_{B}} \mathbf{H}_{q}\mathbf{H}^{H}_{q} \right] \right), $$
(27)

where Q denotes the number of the Monte Carlo random channel realizations for the MIMO channel matrices and Hq the qth channel realization.

Fitness function for the second-step CCPSO

For the second-step optimization process, we develop an appropriate fitness function based on the bit error rate (BER) performance for implementing the CCPSO to search the optimal spatial precoding matrix and the power constraints for downlink communication in massive MIMO channels. For practical applications, massive MIMO channels with linear receivers, such as the minimum mean square error (MMSE) receiver have been considered in several emerging standards, e.g., IEEE 802.11n and 802.16e. We consider the MMSE receiver r at each UE for the case of 4 UEs, i.e., K=4, for simplicity. Based on the downlink transmission model for the received signal shown by (22) and the MMSE receiver, we can express the signal received at the kth UE as follows:

$$ \mathbf{y}_{k}^{DL} = \mathbf{r}\mathbf{Y}_{k}^{DL} = \mathbf{r}(\mathbf{H}_{k}\mathbf{W}\mathbf{T}\mathbf{S} + \mathbf{n}_{k}), $$
(28)

where r is the MMSE receiver. Following the derivation presented in [4], the r with size 1×NU is given by:

$$ \mathbf{r} = \mathbf{w}_{k}^{H}\mathbf{H}^{H}_{k} \left(\sum_{i=1}^{4} \mathbf{H}_{k} \mathbf{w}_{i} \mathbf{w}^{H}_{i} \mathbf{H}^{H}_{k} + (1/{\Gamma}) \mathbf{I}_{N_{U}}\right)^{-1}, $$
(29)

where Γ denotes the signal-to-noise power ratio (SNR) and wk the unit-norm steering vector of the kth beam group selected by the kth UE. Accordingly, the signal received at the kth UE during the four time slots can be obtained from (16) and (28) as follows:

$$ {\begin{aligned} \mathbf{y}_{k}^{DL}&\,=\, \left[ \begin{array}{cc} y_{k}(1) & \\ y_{k}(2) & \\ y_{k}(3) & \\ y_{k}(4) &\end{array} \right] = \left(\left[ \begin{array}{cccc} \hat{h}_{1} & \hat{h}_{2} & \hat{h}_{3}& \hat{h}_{4} \end{array} \right] \left[ \begin{array}{cccc} s_{1} & s_{2} & s_{3} & s_{4} \\ -s_{2}^{*} & s_{1}^{*} & -s_{4}^{*} & s_{3}^{*} \\ -s_{3}^{*} & -s_{4}^{*} & s_{1}^{*} & s_{2}^{*} \\ s_{4} & -s_{3} & -s_{2} & s_{1} \\ \end{array} \right] \right)^{T} \\&+ \left[ \begin{array}{cc} n(1) & \\ n(2) & \\ n(3) & \\ n(4) &\end{array} \right], \end{aligned}} $$
(30)

where

$$ \left[ \begin{array}{cccc} \hat{h}_{1} & \hat{h}_{2} & \hat{h}_{3}& \hat{h}_{4} \end{array} \right] =\mathbf{r}\mathbf{H}_{k}\mathbf{W}\mathbf{T} $$
(31)

Moreover, it is easy to show that (30) can be rewritten as:

$$ {{} \begin{aligned} \mathbf{y}_{k}^{DL}=& \left[ \begin{array}{cccc} y_{k}(1) & \\ {y_{k}}^{*}(2) \\ {y_{k}}^{*}(3) & \\ y_{k}(4) \end{array} \right] = \left(\left[ \begin{array}{cccc} \hat{h}_{1} & -\hat{h}_{2} &-\hat{h}_{3}& \hat{h}_{4} \\ \hat{h}^{*}_{2} & \hat{h}^{*}_{1} & -\hat{h}^{*}_{4}& -\hat{h}^{*}_{3} \\ \hat{h}^{*}_{3} & -\hat{h}^{*}_{4} & \hat{h}^{*}_{1}& -\hat{h}^{*}_{2} \\ \hat{h}_{4} & \hat{h}_{3} & \hat{h}_{2}& \hat{h}_{1} \\ \end{array} \right] \left[ \begin{array}{cccc} s_{1} & \\ s^{*}_{2} & \\ s^{*}_{3} & \\ s_{4} \end{array} \right] \right)^{T} \\&+ \left[ \begin{array}{cccc} n(1) & \\ n^{*}(2) \\ n^{*}(3) & \\ n(4) \end{array} \right] \end{aligned}} $$
(32)

Let

$$ \hat{\mathbf{H}}_{k}= \left[ \begin{array}{cccc} \hat{h}_{1} & -\hat{h}_{2} & -\hat{h}_{3}& \hat{h}_{4} \\ \hat{h}^{*}_{2} & \hat{h}^{*}_{1} & -\hat{h}^{*}_{4}& -\hat{h}^{*}_{3} \\ \hat{h}^{*}_{3} & -\hat{h}^{*}_{4} & \hat{h}^{*}_{1}& -\hat{h}^{*}_{2} \\ \hat{h}_{4} & \hat{h}_{3} & \hat{h}_{2}& \hat{h}_{1} \\ \end{array} \right] $$
(33)

and

$$ \hat{\mathbf{n}}_{k} = \left[ \begin{array}{cccc} n(1) & \\ n^{*}(2) \\ n^{*}(3) & \\ n(4) \end{array} \right], $$
(34)

(30) becomes

$$ \mathbf{y}_{k}^{DL}= \left[ \begin{array}{cccc} y_{k}(1) & \\ {y_{k}}^{*}(2) \\ {y_{k}}^{*}(3) & \\ y_{k}(4) \end{array} \right] = \hat{\mathbf{H}}_{k}(:,i)s_{i} + \sum_{j=1,j\neq k}^{4} \mathbf{H}_{k}(:,j)s_{j} + \hat{\mathbf{n}}_{k}, $$
(35)

where \(\hat {\mathbf {H}}_{k}(:,i)\) represents the ith column of \(\hat {\mathbf {H}}_{k}\). After performing MMSE receiver operation on \(\mathbf {y}_{k}^{DL}\), the kth UE obtains:

$$ {\hat s}_{i}= \hat{\mathbf{R}}_{k} \left[ \begin{array}{cccc} y_{k}(1) & \\ {y_{k}}^{*}(2) \\ {y_{k}}^{*}(3) & \\ y_{k}(4) \end{array} \right] = \hat{\mathbf{R}}_{k} \hat{\mathbf{H}}_{k} \left[ \begin{array}{cccc} s_{1} & \\ s^{*}_{2} & \\ s^{*}_{3} & \\ s_{4} \end{array} \right] + \hat{\mathbf{R}}_{k} \hat{\mathbf{n}}_{k}, $$
(36)

where \(\hat {\mathbf {R}}_{k}\) with size 1×4 represents the corresponding MMSE receiver which can be derived as follows. According to the orthogonality principle, we have:

$$ E\left[\epsilon \left({\mathbf{y}_{k}^{DL}}\right)^{H}\right]=\mathbf{0}, $$
(37)

where

$$ \epsilon=\hat{\mathbf{R}}_{k}\left(\hat{\mathbf{H}}_{k}(:,i)s_{i} + \sum_{j=1,j\neq k}^{4} \hat{\mathbf{H}}_{k}(:,j)s_{j} + \hat{\mathbf{n}}_{k}\right) - s_{i}. $$
(38)

Moreover, assume that sk, si, and nk are independent of each other and their expectations are 0. From (37) and (38), we can get:

$$ \hat{\mathbf{R}}_{k}\sum_{j=1}^{4} \hat{\mathbf{H}}_{k}(:,j)\hat{\mathbf{H}}^{H}_{k}(:,j)-\hat{\mathbf{H}}^{H}_{k}(:,i) + \hat{\mathbf{R}}_{k}\mathbf{r}\mathbf{r}{^{H}}(1/{\Gamma} \mathbf{I}_{4}) = \mathbf{0}. $$
(39)

where I4 denotes the 4×4 identity matrix. Consequently, the MMSE receiver \(\hat {\mathbf {R}}_{k}\) can be obtained from (39) as follows:

$$ \hat{\mathbf{R}}_{k} = \hat{\mathbf{H}}^{H}_{k}(:,i) \left(\sum_{j=1}^{4} \hat{\mathbf{H}}_{k}(:,j)\hat{\mathbf{H}}^{H}_{k}(:,j) + \mathbf{r}\mathbf{r}{^{H}} \frac{\mathbf{I}_{4}}{{\Gamma}} \right)^{-1}, $$
(40)

After the demodulation of \(\mathbf {y}_{k}^{DL}\), we then compare the demodulated bit and the source bit to decide the BER as follows. Consider the binary phase-shift keying (BPSK) for modulation and the maximum likelihood (ML) criterion for data decision after demodulation. Assume that the bit value of each UE is 0 or 1 with equal probability and the noise has complex Gaussian distribution. Let the 1×4 matrix \(\mathbf {A}_{q} = \hat {\mathbf {R}}_{q} \hat {\mathbf {H}}_{q}\) represent the gain vector of the transmission from BS to each UE in the qth Monte Carlo random channel realization for the MIMO channel matrices by using the channel model (7) for simulation. Moreover, assume that Aq(i) is the ith entry of Aq. It represents the gain for the signal transmitted to each UE. The other three entries Aq(j) of Aq, j=1,2,3,4 and ji, represent the gains of the interference received at each UE. According to the BER calculation developed in [4], the total estimated BER for each UE is written as:

$$ {{}\begin{aligned} P_{e}(q) &=\frac{1}{2^{4-1}} \sum\limits_{a_{1}=0}^{1} \dots \sum\limits_{a_{4-1}=0}^{1} \\&Q \left(\Re (A_{q}(i) \,+\, \sum_{c=1, c \ne i}^{4} (-1)^{a_{c}} \!\times\! A_{q}(c)) \sqrt{\frac{2 \Gamma}{{||{\mathbf{r}_{q}}||^{2}}{||{\hat{\mathbf{R}}_{q}}}||^{2}}} \right), \end{aligned}} $$
(41)

where the 1×NU vector rq denotes the qth Monte Carlo random channel realization of the corresponding MMSE receiver r at the UEs.

Finally, we compute the average of the approximated BERs (AA-BER) as follows:

$$ \mathbf{{Fitness2}} = \frac{1}{Q} \sum_{q=1}^{Q} P_{e}(q), $$
(42)

where Q denotes the total number of the Monte Carlo random channel realizations for the MIMO channel matrices by using the channel model (7) for simulation. The Fitness2 of (42) is set to be the fitness function to be minimized by the second-step CCPSO.

An algorithm form of the proposed method

To make the implementation of the proposed method more comprehensive, we summarize the procedure of implementing the proposed method step by step as follows.

The first-step optimization

Step 1. Determine the numbers of transmit and receive antennas, NB and NU, respectively. Set the number t of iterations to zero.

Step 2. At the tth iteration, set the parameter vectors as shown by (24) and (25), respectively to be the position vector of the ith particle of the CCPSO given by \(\mathbf {x_{i}}(t) \,=\, \left [\! A^{t}_{1},..., A^{t}_{N_{B}}, A^{r}_{1},..., A^{r}_{N_{U}} \! \right ]\) for the linear transmit and receive antenna elements or xi(t) = \(\left [\varphi ^{t}_{1},..., \varphi ^{t}_{N_{B}}, \varphi ^{r}_{1},..., \varphi ^{r}_{N_{U}} \right ]\) for the circular transmit and receive antenna elements.

Step 3. Generate the channel matrix H as shown by (7) for downlink transmission. Step 4. Compute the average mutual information (AMI) associated with the downlink transmission channel matrix H according to (26). Step 5. Compute the fitness function Fitness1 for implementing the first-step optimization process by using the CCPSO algorithm according to (27). Step 6. Perform the CCPSO algorithm of [13] for minimizing the fitness function Fitness1 to obtain the most appropriate parameter set xi(t) = \(\left [A^{t}_{1},..., A^{t}_{N_{B}}, A^{r}_{1},..., A^{r}_{N_{U}} \right ]\) for the positions of the linear transmit and receive antenna elements or xi(t) = \(\left [\varphi ^{t}_{1},..., \varphi ^{t}_{N_{B}}, \varphi ^{r}_{1},..., \varphi ^{r}_{N_{U}} \right ]\) for the positions the circular transmit and receive antenna elements. Step 7. Form the channel matrix H as shown by (7) by utilizing the positions for the antenna elements obtained from Step 6. Step 8. Terminate the process if the specified maximum iteration number tmax is reached or the best particle position of the whole swarm remains almost static for a significantly large number of successive iterations. Otherwise, update the positions of antenna elements based on the results obtained from Step 6 and set the iteration number t to t+1. Then, go to Step 2.

The second-step optimization

After obtaining the optimal locations for the antenna elements, the channel matrix HK as shown by (7) can be determined. In the second step, based on the channel matrix HK obtained in the first step, the CCPSO algorithm is utilized to find the optimal parameter set \(\phantom {\dot {i}\!}(a_{1,1}, a_{1,2},..., a_{4,N_{B}}, {\theta }_{1,2},...., {\theta }_{4,N_{B}}, t_{1},..., t_{4})\) related to the beamforming coefficients and the power distribution. Step 1. Determine the channel matrix HK as shown by (7) by using the optimal locations for the antenna elements, respectively obtained in the first-step optimization. Set the number t of iterations to zero.Step 2. At the tth iteration, set the position vector of the ith particle of the CCPSO to xi(t) = \(\left [a_{1,1}, a_{2,1}, \cdots, a_{4,1}, a_{1,2}, \cdots, a_{4,N_{B}}, {\theta }_{1,2},...., {\theta }_{4,N_{B}}, t_{1},...., t_{4} \right ]\). Step 3. Generate the channel matrix H as shown by (7). Step 4. Compute the matrix \(\hat {\mathbf {H}}_{k}\) associated with the downlink transmission channel matrix H according to (33). Step 5. Compute the MMSE receiver \(\hat {\mathbf {R}}_{k}\) associated with the \(\hat {\mathbf {H}}_{k}\) according to (40). Step 6. Calculate the total estimated BER for each UE according to (41). Step 7. Compute the fitness function Fitness2 for implementing the second-step optimization process by using the CCPSO algorithm according to (42). Step 8. Perform the CCPSO algorithm of [13] for minimizing the fitness function Fitness2 to obtain the most appropriate parameter set xi(t) = \(\left [a_{1,1}, a_{2,1}, \cdots, a_{4,1}, a_{1,2}, \cdots, a_{4,N_{B}}, {\theta }_{1,2},...., {\theta }_{4,N_{B}}, t_{1},...., t_{4} \right ]\) for the beamforming coefficients and the power distribution. Step 9. Terminate the process if the specified maximum iteration number tmax is reached or the best particle position of the whole swarm remains almost static for a significantly large number of successive iterations. Otherwise, update the beamforming coefficients and the power distribution based on the results obtained from Step 8 and set the iteration number t to t+1. Then, go to Step 2.

Computational complexity of the proposed method

From the procedure described above for implementing the proposed method, we note that the proposed method requires the additional computational complexity due to using the CCPSO for finding the optimal precoding coefficients without costing additional feedback overhead as compared with the original BBSP method and the method of [4]. According to the CCPSO [13] employed by the proposed method, its computational complexity is dominated by the number tmax of iterations, the number s of the group sizes, the number M of particles, the computational complexity of the fitness functions, and the computational complexity of the particles’ position updating rule. Therefore, the computational complexity of the CCPSO can be expressed as follows:

$$ O_{CCPSO} = O \left(M \times s \times t_{max} \times O^{U}_{CCPSO} \right), $$
(43)

where \(O^{U}_{CCPSO}\) denotes the total computational complexity of the particles’ position updating rule given by [13] and the fitness functions given by (27) and (42), respectively. As a result, the proposed method requires the additional computational complexity given by (43) as compared with the method of [4]. To further evaluate the computational complexity of the proposed method for some setup like 100×100 system as an illustration, we present the more detail descriptions regarding \(O^{U}_{CCPSO}\). The required computational complexity for calculating the fitness function of the first-step optimization is dominated by the calculations of the AMI of (26) and Fitness1 of (27). Their computational complexities are about O(QNBNU) and \(O\left (Q\left (N_{U}^{2}+B^{2}\right)\right)\), respectively, where Q denotes the total number of the Monte Carlo random channel realizations for the MIMO channel matrices. In addition, the required computational complexity for calculating the fitness function of the second-step optimization is dominated by calculating Fitness2 of (42). To calculate Fitness2 of (42), we have to compute the squared norms of the 1×NU vector rq and the 1×4 vector \(\hat {\mathbf {R}}_{q}\), respectively. Hence, the required computational complexity for calculating Fitness2 of (42) is about \(O(Q(N_{U}^{2}+16))\), where Q denotes the total number of the Monte Carlo random channel realizations for the MIMO channel matrices. As a result, we have \(O^{U}_{CCPSO}\) approximately given by \(O^{U}_{CCPSO} \approx \) the total computational complexity of the particles’ position updating rule given by [13] \(+ O(QN_{B}N_{U}) + O\left (Q\left (N_{U}^{2}+B^{2}\right)\right) + O\left (Q\left (Q(N_{U}^{2}+16\right)\right)\). Consider the case of 100×100(NB×NU) system. The resulting computational complexity is approximately given by \(O_{CCPSO} = O(M \times s \times t_{max} \times O^{U}_{CCPSO}) \approx O(M \times s \times t_{max} \times \) [the total computational complexity of the particles’ position updating rule given by [13] \(+ O(QN_{B}N_{U}) + O\left (Q\left (N_{U}^{2}+B^{2}\right)\right)+ O\left.\left.\left (\left (N_{U}^{2}+16\right)\right)\right ]\right) \approx O(M \times s \times t_{max} \times \) [the total computational complexity of the particles’ position updating rule given by [13] + 3O(10000Q)]).

Simulation results and discussion

This section presents simulation examples for illustration and comparison. The following 2 examples are presented to compare the average BER versus SNR in the scenarios considered above under different numbers of antennas used at BS. Example 1 (a large MIMO case) is presented for the case of NB=8 and NU=2. In contrast, the simulation results for Example 2 (a massive MIMO case) is presented for the case of NB=32 and NU=2. The purpose of presenting Example 1 is to make comparisons between the performance differences of using large MIMO and massive MIMO wireless communications. To simplify the illustration and comparison for the proposed method, we consider the simple case of linear arrays as shown in Fig. 2 and the simple case of circular arrays as shown in Fig. 3 instead of using the complicated antenna configurations for 3D channel model like those employed by the authors [19, 20].

Example 1: The BER performance of wireless MIMO channel with N B=8 and N U=2

In this example, we utilize the spatial channel model similar to (7) for simulation. We employ this channel model to generate 40 links with each link sampled 10 times for obtaining a set of Q=400 channel matrices for illustration and comparison. The parameters used are as follows. B=K=4. For the first-step CCPSO procedure, we combine all of the L=10 multipath components produced by sampling each link to get the qth channel matrix \(\mathbf {H}_{q}=\sum _{l=1}^{L} \mathbf {H}_{l}\) for simulation, where Hl denote the lth multipath component. The dimension sS={1,2,4,8}, swarm number \(V= \frac {8}{s}\), tmax=100, the particle number of each swarm is 450, and the fitness function is given by (27) with Q=400. The value for the parameter Z is set to 0.6. Γ=12, NB=8, NU=2. For the second-step CCPSO procedure, the dimension sS={2,4,17,34}, swarm number \(V= \frac {68}{s}\), tmax=100, the particle number of each swarm is 450, and the fitness function is given by (42) with Q=400. The value for the parameter Z is set to 0.6. Γ=12, NB=8, NU=2. The constraint for the total power Pt is set to 4 watts. For the case of using linear antennas, the signal direction angle θ is assumed to be a truncated Gaussian random variable with mean MOA = 0o and standard deviation σ=30o. For the case of using circular antennas, we set the elevation angle θ = 90o and the azimuth angle ϕ to be a truncated Gaussian random variable with mean MOA = 0o and standard deviation σ=30o. For simplicity, we assume that each UE selects one beam group as its serving beam, i.e., N=1. Figure 4 plots the AMI versus SNR of using the existing method [4] and the proposed method. We note from the figure that the AMI are significantly improved after adjusting the positions of array elements for both cases of using linear and circular arrays. Moreover, the results reveal that using circular antennas provides better AMI than using linear antennas under the same number of antenna elements. Figure 5 plots the average BER versus SNR of using the existing method [4] and the proposed method, respectively. As we can observe from the simulation results, the proposed method outperforms the existing method [4]. Moreover, the results reveal that using circular antennas provides better BER performance than using linear antennas under the same number of antenna elements.

Fig. 4
figure4

The average mutual information for Example 1 with NB=8 and NU=2

Fig. 5
figure5

Average BER performance versus SNR for Example 1 with NB=8 and NU=2

Example 2: The BER performance of wireless MIMO channel with N B=32 and N U=2

Here, the specifications and parameters used are the same as those used by Example 1 except that the following parameters are reset. The number of array elements deployed at BS is set to NB=32 under the consideration of massive wireless MIMO scenarios. For the first-step CCPSO procedure, the dimension sS={2,4,8,16} and swarm number \(V= \frac {32}{s}\). For the second-step CCPSO procedure, the dimension sS={2,4,5,13} and swarm number \(V= \frac {260}{s}\). Figure 6 plots the AMI versus SNR of using the existing method [4] and the proposed method. Again, we note from the figure that the AMI are significantly improved after adjusting the positions of array elements for both cases of using linear and circular arrays. Moreover, the results reveal that using circular antennas provides better AMI than using linear antennas under the same number of antenna elements. Figure 7 plots the average BER versus SNR of using the existing method [4] and the proposed method, respectively. Again, as we can observe from the simulation results, the proposed method outperforms the existing method [4]. The results also reveal that using circular antennas provides better BER performance than using linear antennas under the same number of antenna elements. From the simulation results showing that using circular antennas provides better BER performance than using linear antennas under the same number of antenna elements, we reason that this is due to the following phenomena: (a) Because only small amount of power transmitted by utilizing circular antenna array would be reflected back, circular antenna array provides better return loss as compared to linear antenna array. (b) Because the percentage of bandwidth and the half-power beamwidth for circular antenna array are wider than those for linear antenna array, circular antenna array provides higher gain than linear antenna array. (c) To mitigate MCE, circular antenna array is a more suitable geometrical arrangement than linear antenna array. The more array elements are being used, the better and higher gain could be achieved. Finally, comparing the results of Examples 1 and 2, we see that the system performance can be considerably enhanced by using massive antennas at BS. From the simulation results, we note that the proposed method is very effective in simultaneously dealing with the mutual coupling and spatial correlation problems in QOSTBC massive wireless MIMO communications.

Fig. 6
figure6

The average mutual information for Example 2 with NB=32 and NU=2

Fig. 7
figure7

Average BER performance versus SNR for Example 2 with NB=32 and NU=2

Conclusion

In this paper, we have presented an optimal joint beamforming and spatial precoding method based on a population-based stochastic optimization for QOSTBC massive wireless MIMO communication systems. Based on the concept of beamforming selection, the proposed method can preserve the advantage of reducing the overheads for downlink training and feedback of channel state information between the base station (BS) and user equipments (UEs). The proposed method utilizes a two-step cooperative coevolutionary particle swarm optimization (CCPSO) to find the optimal beamforming coefficients and the optimal positions of array elements. As a result, the proposed method can significantly mitigate the mutual coupling and spatial correlation problems for downlink transmission, particularly when massive antennas are deployed at the BS. Moreover, appropriate fitness functions based on the average mutual information and an estimated average bit error rate (BER) are proposed for implementing the two-step CCPSO scheme. Finally, the proposed method makes QOSTBC MIMO communications with full transmission rate for any number of transmit antennas achievable. Simulation results show that the proposed method can indeed achieve much better BER performance than the existing methods under the scenarios of massive antennas with antenna mutual coupling and spatial correlation. We reason that the proposed method offers a more appropriate beamforming and spatial precoding scheme for future QOSTBC massive wireless MIMO communications. As to analytical results regarding the proposed method, the exact performance analysis of large or massive MIMO system is not an easy task. In the literature, most of the papers present the bit error rate (BER) performance of QOSTBC-based massive MIMO systems through the numerical results without theoretical analysis. To obtain the complete mathematical analysis regarding the proposed method, we are currently conducting the further research on the exact analytical results on the achievable performance gains, mathematical analysis of the proposed schemes/antenna placements in the presence of mutual coupling and spatial correlation effects.

Abbreviations

AA-BER:

Approximated average bit error rate

AMI:

Average mutual information

BER:

Bit error rate

BS:

Base station

BSSP:

Beam selection spatial precoding

CCPSO:

Cooperative coevolutionary particle swarm optimization

CSI:

Channel state information

FDD:

Frequency-division duplex

MCE:

Mutual coupling effect

MIMO:

Multiple-input multiple-output

MMSE:

Minimum mean square error

OSTBC:

Orthogonal space-time block code

PSO:

Particle swarm optimization

QOSTBC:

Quasi-orthogonal space-time block code

STBC:

Space-time block code

SCE:

Spatial correlation effect

SCF:

Spatial correlation function

UCA:

Uniform circular array

UEs:

User equipments

ULA:

Uniform linear array

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Acknowledgements

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Funding

This work was supported by the Ministry of Science and Technology of Taiwan under Grants MOST 106-2221-E-002-062 and MOST 107-2221-E-002-121-MY3. The Ministry of Science and Technology of Taiwan supports the financial grant required for conducting the research. The authors complete all of the design of the study and collection, analysis, and interpretation of data and writing the manuscript.

Availability of data and materials

The data presented in the manuscript can be obtained easily by performing MATLAB programming according to the formulas provided in this manuscript. There is no need to generate the datasets for storage. Hence, Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Authors’ information

Ju-Hong Lee was born in I-Lan, Taiwan. He received the B.S. degree from National Cheng Kung University, Tainan, Taiwan, in 1975; the M.S. degree from National Taiwan University (NTU), Taipei, Taiwan, in 1977; and the Ph.D. degree from Rensselaer Polytechnic Institute, Troy, NY, USA, in 1984, all in electrical engineering. From September 1980 to July 1984, he was a Research Assistant and was involved in research on multidimensional recursive digital filtering with the Department of Electrical, Computer, and Systems Engineering, Rensselaer Polytechnic Institute. From August 1984 to July 1986, he was a Visiting Associate Professor and, in August 1986, became an Associate Professor with the Department of Electrical Engineering, NTU. Since August 1989, he has been a Professor with NTU. He was appointed as a Visiting Professor with the Department of Computer Science and Electrical Engineering, University ofMaryland, Baltimore, MD, USA, during a sabbatical leave in 1996. His current research interests include multidimensional digital signal processing, multirate signal and image processing, detection and estimation theory, analysis and processing of joint vibration signals for the diagnosis of cartilage pathology, statistical signal processing, and adaptive signal processing for smart antennas with applications in mobile wireless communication systems. Dr. Lee received the Excellence Research Awards from the National Science Council (NSC) of Taiwan in the academic years 1988, 1989, and 1991-1994; the Outstanding Research Awards from the NSC in the academic years 1998-2004; and the NSC Research Fellowships for the academic years 2005-2008 and 2011-2014. In 2015, he received the Merit MOST Research Fellow Award from the Ministry of Science and Technology of Taiwan. He has been appointed as NTU’s Tenured Distinguished Professor since August 2006.

Wei-En Sun was born in Taiwan. He received the B.S. degree in electrical engineering from the National Chung Hsing University, Taichung city, Taiwan, in 2016 and the M.S. degree in communication engineering from the National Taiwan University, Taipei, Taiwan, in 2018. His current research interests are in the areas of multiple-antenna multiple-input-multiple-output systems in communications.

Author information

JHL completed the main work and updated the manuscript, as the first author and the correspondence author of this paper. WES performed all of MATLAB programming and prepared the simulation data and figures for presenting in the manuscript. All authors have read and approved the final manuscript for submission.

Correspondence to Ju-Hong Lee.

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Lee, J., Sun, W. Robust beamforming and spatial precoding for quasi-OSTBC massive MIMO communications. J Wireless Com Network 2019, 58 (2019) doi:10.1186/s13638-019-1373-9

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Keywords

  • Massive MIMO
  • Wireless communication
  • Spatial precoding
  • Spatial correlation
  • Mutual coupling
  • Beamforming
  • Quasi-orthogonal space-time block codes
  • Population-based stochastic optimization
  • Cooperative coevolutionary particle swarm optimization
  • Bit error rate