Performance Optimization of Multipair Massive MIMO Polarized Relay Systems

Abstract

In this paper, we analyze the challenges faced by multipair massive Multiple-input-multiple-output (MIMO) relay channels in 5G wireless communication systems, where high path loss and severe shadow fading between different user nodes can cause poor channel quality. To overcome these limitations, we propose a polarization selection scheme for antenna arrays combining multipair massive MIMO relay and beamforming to improve MIMO relay channel quality. Specifically, our main goal is to exploit the potential of polarization diversity to maintain and improve the link quality while maintaining the compact size of the MIMO antenna array. To achieve this goal, we introduce a dynamic weighted particle swarm optimization algorithm with contraction factor (CF-DWPSO) to select the polarization direction. In addition, we employ distributed beamforming to effectively suppress or eliminate inter-pair interference. The performance of the simulation analysis shows that CF-DWPSO combined with beamforming provides significant performance improvement of the multipair massive MIMO polarized relay channels, which further indicates that it is of great necessity for improving the performance of this system to optimize the polarization selection by combining beamforming techniques.

1. Introduction

With the increasing traffic of modern wireless systems, the demands for higher coverage and data rates are also growing, which makes increasing system capacity and transmission distance one of the most important performance parameters for this system [1]. A massive MIMO base station with relay communication are one of the main methods to achieve this target, which makes it possible to be widely applied in indoor and cluttered urban wireless communication environments, where line-of-sight is not available, through high-angle resolution and low-cost implementation [2]. Research in [3] shows that massive MIMO enhances the capability of the system against fast fading, interference, and noise with spatial diversity and beamforming techniques, thus improving the spectral efficiency and reliability of the system. Compared to previous ones, today’s relays can be used not only to enhance coverage but also to improve system capacity [4], and their importance in massive MIMO systems has been demonstrated in numerous studies [5]. In massive MIMO relay systems, improving the system capacity while suppressing inter-pair interference is the key study problem of system performance [6], which is the focus of this paper. To suppress inter-pair interference in the spatial domain, the introduction of large-scale MIMO in full-duplex (FD) relay systems is a recent active research field [7,8]. The study in [9] shows the elimination of inter-pair interference in multipair massive MIMO relay systems by simple maximum ratio combining/maximum ratio transmission (MRC/MRT). In [10], how to utilize massive MIMO to suppress various types of interference and noise was studied in FD relay systems. In addition, relay beamforming is an effective method for interference suppression [11], which improves the transmission quality of the system by weighting the desired signal to separate the signals of different transceivers to suppress inter-pair interference [12]. The various beamforming methods for AF and DF were studied in [13,14]. A hybrid beamforming algorithm for total rate maximization when designing a relay-assisted MU-MIMO system under the wideband assumption was investigated, and it indicated that beamforming can have an impact on the performance of MIMO relay systems by eliminating interference in [15]. Moreover, the relay beamforming technique combined with massive MIMO can achieve higher beamforming gain, which can further improve the performance of MIMO relay systems [16]. Research in [17] showed that massive MIMO antenna arrays with beamforming techniques can improve the SINR while suppressing channel interference, thereby improving the performance of MIMO relay systems.

On the other hand, to further achieve higher system performance, polarization can be used as a new resource to increase the number of uncorrelated channels to improve the system channel capacity without increasing the scale of the MIMO antenna array [18,19]. In a scattering-rich environment, the polarization characteristics determine the effectiveness, diversity, or severity of polarization mismatch of polarization multiplexing [20]. Research in [21] showed that antenna polarization mismatch between channel transmission and antenna will result in 10–20 dB power loss, which significantly impacts the channel capacity of the system. The study in [22] showed that polarization can provide independent degrees of freedom for information transmission, and a 6-fold increase in system channel capacity can eventually be achieved by utilizing the polarization technique. In [23], the effect of different polarization configurations on the capacity of a point-to-point MIMO relay system was studied, and the results showed that the polarization optimization of the antenna can achieve higher channel capacity compared to the conventional optimization approaches on the time and spectrum domains. The research on antenna polarization is mainly focused on literature [24,25] at present, and there are few studies on optimizing the polarization configuration of antennas. Research in [26] showed that optimizing the polarization configuration is an effective way to improve the channel transmission performance. In brief, it is necessary to study polarization configuration optimization in combination with massive MIMO relay beamforming to improve the performance of MIMO relay channel systems.

This paper focuses on the optimization of antenna polarization selection in a multipair massive MIMO polarized relay channel system considering only the space domain and polarization domain. To reduce the complexity of the system, this paper only utilizes massive MIMO beamforming technology in the space domain at the full-duplex AF relay node to distinguish between different transceivers to suppress inter-pair interference, and the transceivers are equipped with uniform planar arrays (PLA) for polarization selection optimization in the polarization domain to maximize the system channel capacity. To this end, our contribution is to propose a dynamic weighted particle swarm optimization algorithm with contraction factor (CF-DWPSO) combining massive MIMO relay beamforming with polarization direction selection optimization and to analyze the impact of the method on improving system capacity and suppressing or eliminating inter-pair interference. To effectively evaluate polarization optimization schemes for multipair massive MIMO relay links in more realistic scenarios, the channel model is described based on the geometric stochastic channel model (GSCM) by introducing polarization. In this work, the numerical simulations use the data generated by this model under typical real-scene measurements, and the performance evaluation of the system will be based on the channel matrix constructed from the measured data from Lund University [27,28].

The paper is organized as follows: In Section 2, the multipair massive MIMO relay channel modeling based on COST 2100 channel model is introduced. Section 3 introduces the polarization optimization schemes of multipair MIMO relay links. The performance evaluation results of the different polarization optimization schemes are provided in Section 4. Section 5 summarizes the work.

2. Channel Modeling for a Multipair Massive MIMO Relay System

In this work, we describe a two-hop multipair MIMO relay model, as shown in Figure 1, where the BSs act as the relaying nodes whose number is R, and the MSs act as the transmitters and receivers whose numbers are both K. Each node of this model is equipped with multiple antennas. The transceiver is equipped with a dual-polarized antenna array, and the relay is equipped with a massive MIMO antenna array. As assumed in [29], there is no direct link between transmitter and receiver, and each transmitter communicates with its corresponding receiver with the aid of all relays with amplify-and-forward (AF) mode. Without loss of generality, the same transmitter and receiver pairs can be called the same users, whereas the different transmitter and receiver pairs can be called different users. All relay nodes are independent of one another and have no signal transmission, and all nodes apply a half-duplex communication mode.

The signals of any two nodes are transmitted in a complex scattering environment. To suppress or eliminate inter-pair interference from different transceivers, the relay nodes utilize an ideal linear beamforming technique and AF mode to forward the signals sent from the transmitter. In the first hop channel, the received signals at the t relay nodes are described by

$$y_t=\sqrt{E_s}·\sum _{i=1}^K{W}_i·R_{it}{s}_i+n_{tr},$$

(1)

where $E_s$ is the transmitter power of transmitter nodes, $s_i$ is the signal vector from the ith transmitter, $R_{it}$ is the channel matrix between the ith transmitter and the tth relay, $W_i$ is the ith transmitter beamforming vector, and $n_{tr}$ is the additive white Gaussian noise (AWGN) of the first hop channel. Similarly, the receiver signals of all relay nodes are

$$Y_T={\left[y_1,\cdots ,y_T\right]}^T=\sqrt{E_s}·\left[W_1^s,\cdots ,W_K^s\right]·\left[\begin{array}{cccc}{R}_{11}& R_{12}& \cdots & R_{1K}\\ R_{21}& R_{22}& \cdots & R_{2K}\\ ⋮& ⋮& \ddots & ⋮\\ R_{K1}& R_{K2}& \cdots & R_{KK}\end{array}\right]·{\left[s_1,\cdots ,s_K\right]}^T+n_{tr},$$

(2)

Similarly, in the second hop channel, the received signals at all receiver nodes are described by

$$Y_K=\sqrt{E_r}·\left[W_1^d,\cdots ,W_K^d\right]·\left[\begin{array}{cccc}{G}_{11}& G_{12}& \cdots & G_{1K}\\ G_{21}& G_{22}& \cdots & G_{2K}\\ ⋮& ⋮& \ddots & ⋮\\ G_{K1}& G_{K2}& \cdots & G_{KK}\end{array}\right]·{\left[y_1,\cdots ,y_T\right]}^T+n_{rd},$$

(3)

where $E_r$ is the transmitter power of relay nodes, $G_{ij}$ is the channel matrix between the ith relay and the jth receiver, $n_{rd}$ is the AWGN of the second hop channel, and $W_i^d$ is the ith receiver beamforming vector. The beamforming vector ($W^s$ or $W^d$) by the relay node for the transceiver can be given as

$$W^{s/d}=\left[{\mathrm{b}}_1^{s/d}\left({\omega }_0\right),{\mathrm{b}}_2^{s/d}\left({\omega }_0\right),\cdots {\mathrm{b}}_n^{s/d}\left({\omega }_0\right),\dots ,{\mathrm{b}}_N^{s/d}\left({\omega }_0\right)\right],$$

(4)

where ${\omega }_0$ is the signal frequency, ${\omega }_0=2\pi f=2\pi \frac{c}{\lambda }$, N is the number of columns of the antenna array, and $b_n^{s/d}$ is the steering vector of the desired signal of the nth at the receiver or transmitter, as given by

$$b_n^{s/d}\left({\omega }_0\right)={\left[e^{-j{\omega }_0{\tau }_{1n}},e^{-j{\omega }_0{\tau }_{2n}}\cdots e^{-j{\omega }_0{\tau }_{mn}}\cdots e^{-j{\omega }_b{\tau }_{Mn}}\right]}^T,$$

(5)

where M is the number of rows of the antenna array, and ${\tau }_{mn}$ is the delay between the nth column antenna element and the mth row antenna element of the antenna array at the transmitter. The steering vector of the antenna array has a strong correlation with the distance between the antenna units in the array. Assuming the position of the kth antenna is ($x_k$,$y_k$), the ${\tau }_{kn}$ is given by

$${\tau }_{kn}=\frac{d}{c}=\frac{1}{c}·\left(x_{k}cos{\theta }_{n}cos{\psi }_n+y_{k}sin{\theta }_{n}cos{\psi }_n\right),$$

(6)

where d is the inter-element spacing of the antenna array, and ${\theta }_n$ and ${\psi }_n$ are the elevation and the azimuth of the nth antenna, respectively. Based on Equations (2) and (3), $Y_K$ is rewritten as

$$Y_K=\sqrt{E_r{E}_s}·{\left[{\mathbf{W}}^d\right]}^T·\mathbf{H}·{\mathbf{W}}^s·S_K+{\mathbf{n}}_{\mathbf{s}},$$

(7)

where ${\mathbf{n}}_{\mathbf{s}}$ is the AWGN of this system, and $\mathbf{H}$ is the channel matrix of the multipair relay channel model in multiple transceivers, which can be written specifically as

$$\mathbf{H}=\left[\begin{array}{cccc}{H}_{11}& H_{12}& \cdots & H_{1K}\\ H_{21}& H_{22}& \cdots & H_{2K}\\ ⋮& ⋮& \ddots & ⋮\\ H_{K1}& H_{K2}& \cdots & H_{KK}\end{array}\right],$$

(8)

where $H_{ij}$ is the channel matrix from the jth transmitter to the ith receiver, which is composed of the channel matrix of the first and second hops. When $i=j$, $H_{ij}$ can be defined as the effective links from the same users, noted as $H_{ii}$, whereas it is defined as the interference links when $i\ne j$, noted as $H_{ij}$, $i,j\in [1,K]$.

To better describe the polarization distribution at the multipath cluster level, a polarization distribution model, the COST 2100 channel model [30], is introduced in the current MIMO channel, and this method is also applicable to the polarization expansion of other MIMO channel models. The channel matrix $\mathbf{H}$ is generated by the COST 2100 channel model. The signal transmission of any two nodes is transmitted by the COST 2100 channel transmission structure in Figure 2. The COST 2100 channel model is described by clustered multipath components (MPCs). The channel matrix is composed of a superposition of multipath signals generated by each activation cluster scattering environment, as shown in Figure 3.

The MIMO channel matrix from the jth transmitter node to the ith receiver node introduces the polarization characteristics of the antenna and the channel, as given by

$$H_{ij}=\sum _{u=1}^U\sum _{v=1}^V{A}_{u,v}·{\left[Q^i\left({\psi }_{u,v},{\theta }_{u,v}\right)\right]}^T·{\left[{\mathbf{W}}_{u,v}^i\right]}^T·P_{u,v}·{\mathbf{W}}_{u,v}^j·Q^j\left({\psi }_{u,v},{\theta }_{u,v}\right)·e^{j2\pi D_{u,v}t}·\delta \left(\tau -{\tau }_s\right),$$

(9)

where $A_{u,v}$ is the complex amplitude of the vth MPC of the uth cluster, which includes the path loss, the VR gain, the cluster attenuation, the shadow fading, and the MPC gain. The calculation of $A_M$ can be found in [31]. The expressions v and u are the numbers of MPCs in each activated cluster and activation clusters, respectively, $v\in [1,V]$, $u\in [1,U]$; $\theta$ and $\psi$ are the elevation and azimuth of the MPC; $D_{u,v}$ is the Doppler shift of the vth MPC of the uth cluster; ${\tau }_s$ is the total delay of the channel, including the time delay of the MPC, the cluster delay, the channel geometry delay, etc.; $P_{u,v}$ is the polarization matrix of the MPCs; $Q^i$ and $Q^j$ are the total gain of the radiation pattern and antenna at receiver i and transmitter j, respectively, which are written as

$$Q_{i/j}\left({\theta }_{u,v},{\psi }_{u,v}\right)=G_{i/j}\left({\theta }_{u,v},{\psi }_{u,v}\right)\left[\begin{array}{c}{Q}_x^{i/j}\left({\theta }_{u,v},{\psi }_{u,v}\right)\hfill \\ Q_z^{i/j}\left({\theta }_{u,v},{\psi }_{u,v}\right)\hfill \end{array}\right],$$

(10)

where $G_{i/j}$ is the antenna gain of the receiver or transmitter, for which the calculation can be found in [32]; x and z are the antenna’s horizontal and vertical polarization components, respectively; $Q_x^{i/j}$ and $Q_z^{i/j}$ are the radiation patterns gain of the x and z, respectively, which are related to $\psi$, $\theta$, and $\gamma$; $\gamma$ is the angle between the antenna and the z-axis, as shown in Figure 4. The calculation of $Q_x^{i/j}\left(\psi ,\theta \right)$ and $Q_z^{i/j}\left(\psi ,\theta \right)$ can be found in [23].

In this work, we use the WINNER-extended XPR-CPR model to describe the polarization in the channel, which can better describe the fading in the polarization matrix [33]. The polarization matrix of MPCs can be given by

$$P_{u,v}=\left[\begin{array}{cc}{p}_{u,v}^{xx}\hfill & p_{u,v}^{xz}\hfill \\ p_{u,v}^{zx}\hfill & p_{u,v}^{zz}\hfill \end{array}\right]=\left[\begin{array}{cc}\sqrt{\frac{1}{{\zeta }_{u,v}}}exp\left\{j{\Phi }_{u,v}^{xx}\right\}\hfill & \sqrt{\frac{1}{{\rho }_{u,v,z}}}exp\left\{j{\Phi }_{u,v}^{xz}\right\}\hfill \\ \sqrt{\frac{1}{{\zeta }_{u,v}·{\rho }_{u,v,x}}}exp\left\{j{\Phi }_{u,v}^{zx}\right\}\hfill & exp\left\{j{\Phi }_{u,v}^{zz}\right\}\hfill \end{array}\right],$$

(11)

where $p_{u,v}$ is the attenuation coefficient between the transmitting side and the receiving side of the vth MPC of the uth cluster, ${\zeta }_{u,v}$ is the co-polarization rate (CPR), ${\rho }_{u,v,x}$ and ${\rho }_{u,v,z}$ represent the cross-polarization rate (XPR) concerning the x and z polarization of the antenna, respectively, and $\Phi$ is the phase for different transmitter (Tx) and receiver (Rx) polarizations, which follow the uniform distribution of [$-\pi$, $\pi$); $\zeta$ and $\rho$ follow a lognormal distribution.

Based on the aforementioned modeling, Equation (9) provides the theoretical basis for our study of performance evaluation of multipair massive MIMO polarized relay links.

3. Polarization Optimization of Multipair Massive MIMO Relay Channels

MIMO channels usually contain complex polarization distribution. Based on Equations (8) and (9), the multipair MIMO relay channels are composed of the effective links ($H_{ii}$) and the interference links ($H_{ij}$), and their performance will affect the overall performance of the multipair relay system. Although the antenna’s polarization configuration and the relay’s beamforming play important roles in the channel matrix, the channel matrix is a critical factor in improving the capacity of this system. Therefore, polarization selection optimization combined with relay beamforming techniques is an important way to optimize the performance of multipair MIMO relay systems.

In this work, the overall polarization selection optimization of the antenna array is considered, as the angle and polarization distributions of the MPCs are consistent with each antenna in the array under the far-field propagation conditions [34]. In general, the optimal relay channel capacity for multipair MIMO relay links can be written as

$$\mathrm{C}=max\left\{c_{ii}\right\},$$

(12)

where $c_{ii}$ is the channel capacity of the relay link between the ith transmitter and the ith receiver, i.e., the effective links. We assume that the transmitting power of the antenna array at the transmitter and receiver end is uniformly distributed and fixed, and the antenna elements in the array are kept at the same spacing distribution. The channel capacity is given by

$$c_{ii}=B{log}_2\left\{I+SIN{R}_{ii}\right\},$$

(13)

where B is the bandwidth, I is the identity matrix, and $SIN{R}_{ii}$ is the SINR of the transmitter, and the calculation is given by

$$SIN{R}_{ii}=\frac{\sqrt{\frac{E_r{E}_s}{K}}·{\left|{\left[{\mathbf{W}}^i\right]}^T·H_{ii}^H·H_{ii}·{\mathbf{W}}^j\right|}^2}{{\sum }_{i\ne j}\left|{\left[{\mathbf{W}}^i\right]}^T·H_{ij}^H·H_{ij}·{\mathbf{W}}^j\right|+{∥{\mathbf{W}}^j∥}^2·{\delta }^2},$$

(14)

where ${\delta }^2$ is the noise power, K is the number of transmitters or receivers, $H_{ii}$ is the channel matrix between the ith transmitter and the ith receiver from the same user, and $W^j$ and $W^i$ are the beamforming matrices at the jth transmitter and ith receiver, respectively.

By Equations (9) and (13), the channel capacity depends on the projection of the polarization components of the antenna array at the transmitter and receiver on the MPC polarization. The polarization component of the antenna array is affected by the angle $\gamma$ of the antenna array, i.e., changing the $\gamma$ of the antenna array will change the magnitude of the polarization component of the antenna array. Therefore, we propose two polarization optimization schemes, which are the regular polarization selection scheme and the dynamic weighted particle swarm optimization scheme with contraction factor (CF-DWPSO).

3.1. Regular Polarization Selection $\gamma$ Optimization

The MIMO antenna arrays’ regular polarization selection $\gamma$ optimization scheme is shown in Figure 5. Each node has two polarizations, referred to as vertical polarization v ($\gamma ={90}^{\circ }$) and horizontal polarization h ($\gamma =0^{\circ }$). By choosing the combination of the polarization directions of different transceivers, the channel capacity of the overall multipair MIMO relay links is maximized by Equation (13). The different branches in Figure 5 represent the polarization components of the antennas between different transceivers, and the figure represents all polarization combinations for the regular polarization selection scheme.

This work aims to select a transmission path that maximizes the channel capacity of the effective links from all polarization combinations. More detailed optimization steps can be found in [23].

3.2. Dynamic Weighted Particle Swarm Algorithm with Contraction Factor (CF-DWPSO)

The regular polarization selection $\gamma$ optimization algorithm cannot guarantee strict global optimization. The performance of DWPSO has been proven to achieve better optimization of point-to-point single-user MIMO relay links, while the transmission scenario of multipair MIMO relay links is more complex and the search space is relatively large. The learning factor of PSO is more effective than the inertia weight coefficients and can more effectively control and constrain the speed of particles, and it also enhances the performance of the algorithm. The improvement method of adaptive dynamic weights and contraction factors is considered to improve the original algorithm by effectively combining them. An improved particle swarm algorithm, CF-DWPSO, is proposed to optimize the performance of multipair MIMO relay links.

In Section 3.1, each particle represents a regular polarization combination, so that the combination of all particle positions can be given by

$$O=\left(O_1,O_2,\cdots ,O_p,\cdots ,O_P\right),$$

(15)

where $O_p$ is the polarization combination of the pth particle in P, and P is the number of particles; $s_i$ is the update rate of each particle as given by

$$s_i=\left(v_{p_{11}},v_{p_{12}},\cdots ,v_{p_{nm}}\cdots ,v_{p_{NM}}\right),$$

(16)

where $v_{p_{nm}}$ denotes the iterative update rate, and the update rate of the particle swarm is given by

$$S=\left(S_1,S_2,\cdots ,S_p,\cdots ,S_P\right).$$

(17)

In each update iteration of PSO, each particle will update the local optimum and the global optimum, as given by

$$J_p=argmax\left\{C\left(O_{\eta ,p}\right)\right\},\aleph =argmax\left\{C\left(J_p\right)\right\},$$

(18)

where $\eta$ is the number of updates; $J_p$ and ℵ are the local optimum and the global optimum, respectively. The pseudo-code for CF-DWPSO is given by Algorithm 1.

Algorithm 1: Particle swarm optimization with contraction factor (CF-DWPSO).

In Algorithm 1, $\omega$ is the inertia weight, and $\mathrm{CF}$ is the contraction factor, which is only related to the learning factor (${\epsilon }_1/{\epsilon }_2$). Based on the update equation $s_p^{n+1}$ and $o_p^{n+1}$, $\omega$ is set to exponentially decrease between [0.9, 0.3]. The ${\epsilon }_1$ and ${\epsilon }_2$ correspond to the particle’s individual and colonial learning abilities. When ${\epsilon }_1$ = 0, the particle loses its colonial learning ability and the particle can expand the search space with a faster convergence rate. When ${\epsilon }_2$ = 0, the particle loses its individual learning ability, there is no information exchange between individuals, and the whole population is equivalent to multiple particles performing the blind random search with slow convergence speed. Therefore, it is necessary to optimize the learning factor effectively. The process of inertia weight change with contraction factor is shown in Figure 6. The contraction factor is given by

$$\mathrm{CF}=\frac{2}{\left|2-\Xi -\sqrt{{\Xi }^2-4\Xi }\right|},$$

(19)

where $\Xi$ is the joint learning factor, $\Xi ={\epsilon }_1+{\epsilon }_2$, and $\Xi >4$.

4. Performance Evaluation

In the analysis, the simulated model is a two-transceivers two-hop multipair massive MIMO relay channel model with two relays based on COST 2100 channel model, as shown in Figure 7. This model allows us to obtain realistic channel realizations under different deployment scenarios. The evaluation environment is a narrow band indoor environment with a center frequency of 2.6 GHz and a bandwidth of 40 MHz. Polarization parameters of the channel for this environment are shown in

Table 1. Polarization parameters of MIMO indoor network model.

Parameter/[dB]	${\mathit{\mu }}_{\mathit{\tau }}$	${\mathit{\sigma }}_{\mathit{\tau }}$	${\mathit{\mu }}_{\mathit{\zeta }}$	${\mathit{\sigma }}_{\mathit{\zeta }}$	${\mathit{\mu }}_{\mathit{\rho }}$	${\mathit{\sigma }}_{\mathit{\rho }}$
LOS	8.5	4.8	0	0	9	3
NLOS	8	4	0	0	8	3

, and the other indoor channel parameters were kept consistent with [35].

The transceiver is equipped with a dual-polarized antenna array, i.e., the dual-polarized antenna can transmit both horizontal (x) and vertical (z) polarization, as shown in Figure 8b. The antenna array of each node is a linear antenna, and each antenna unit in the antenna array is uniformly distributed in the same plane, and the distance between antennas is not less than $\lambda$/2 wavelengths to reduce mutual coupling between antennas, as shown in Figure 8a, which represents the antenna configuration equipped by the transceiver and relay nodes. Antenna isolation is guaranteed in this configuration, so the polarization selectivity of the entire antenna array can be spatially configured by superimposing the polarization selectivity of individual antennas. An antenna array of $M\times N$ is composed of M rows and N columns of antenna elements.

The transceiver’s power needs to be normalized, and multiple nodes need to obey the same power constraints because the dynamic power will affect the optimization results. This work focuses on evaluating the importance of polarization on the performance of multipair MIMO polarized relay links. We first verify the convergence and complexity of the CF-DWPSO. Next, we evaluate the performance of CF-DWPSO for improving the channel capacity of this simulated model.

4.1. Convergence and Complexity of the CF-DWPSO

The number of snapshots and the snapshot rate are 50 and 50 Hz, respectively. In this paper, we verify the convergence of CF-DWPSO by comparing the PSO with fixed and dynamic inertia weights as a comparison algorithm after several simulations, as shown in Figure 9.

Considering the efficiency of the algorithm, PSO optimization studies generally limit the number of particles to between 20 and 60 [36]. The number of simulated particles is set to 50 in this work. The result demonstrated that the convergence speed of the algorithm is slow with $\omega$ and $\mathrm{CF}$ increasing, and CF-DWPSO does not fall into local optimization quickly when $\omega$ is small. Moreover, the convergence performance of CF-DWPSO is better than others in multipair MIMO polarized relay channels.

Then, we analyzed the complexity of the CF-DWPSO algorithm, and the average optimal value obtained for different particle numbers with different iterations is recorded in detail in

Table 2. Complexity analysis of CF-DWPSO and DWPSO.

Number of Particles	Number of Iterations	Average Optimization Results
		DWPSO	CF-DWPSO
10	10	4.6 b/s/Hz	4.9 b/s/Hz
	50	5.2 b/s/Hz	5.7 b/s/Hz
	100	5.7 b/s/Hz	5.9 b/s/Hz
	300	4.2 b/s/Hz	5.8 b/s/Hz
50	10	5.1 b/s/Hz	5.2 b/s/Hz
	50	6.1 b/s/Hz	6.4 b/s/Hz
	100	5.8 b/s/Hz	5.9 b/s/Hz
	300	6.0 b/s/Hz	6.1 b/s/Hz
100	10	4.8 b/s/Hz	5.1 b/s/Hz
	50	5.9 b/s/Hz	6.2 b/s/Hz
	100	5.7 b/s/Hz	6.1 b/s/Hz
	300	5.3 b/s/Hz	5.8 b/s/Hz

. Based on Table 2, first, when the optimization used DWPSO and CF-DWPSO to achieve the same optimization results, the CF-DWPSO algorithm requires fewer iterations and fewer particles than DWPSO, which indicates that CF-DWPSO has lower complexity with the same performance optimization. Second, when both the number of particles and the number of iterations are around 50, both DWPSO and CF-DWPSO can be optimized to the best performance.

4.2. Performance Comparison in Different Conditions

To compare the proposed optimization schemes’ reliability, we further analyzed the performance of different optimization algorithms for improving the channel capacity of multipair MIMO relay links. This work is divided into two optimization strategies for comparison, i.e., polarization optimization and non-optimization schemes. The polarization optimization schemes are CF-DWPSO and DWPSO. The non-optimization schemes refer to randomly selecting a set of polarization combinations among the regular or general polarization configurations.

The results of comparing the channel capacity of different optimization schemes for different links at different signal-to-noise ratios (SNRs) are shown in Figure 10 and Figure 11.

The effective links refer to the relay links formed by the same users’ transmitter and receiver, i.e., link-1-1 and link-2-2 in the figures, and the interference links refer to the relay links formed by different users’ transmitter and receiver, i.e., link-1-2 and link-2-1 in the figures. The results imply several conclusions. First, all optimization algorithms’ performance is significantly improved compared to the non-optimized schemes. Second, by comparing the optimization results of different optimization algorithms in Figure 10, the results indicate that the performance of DWPSO and CF-DWPSO is better than the other algorithms, but DWPSO shows no significant difference for the regular optimization algorithm, whereas the difference between CF-DWPSO and the regular optimization algorithm is relatively significant. This shows that the performance of CF-DWPSO is better than DWPSO in improving the performance of multipair MIMO relay links. Third, comparing Figure 10 and Figure 11, the performance of CF-DWPSO for effective links is significantly better than other algorithms, whereas for interference links, the performance of CF-DWPSO is inferior to the regular optimization algorithm, which effectively suppresses the improvement of the channel capacity of the interference links. It shows that optimizing the performance of multipair MIMO relay links by CF-DWPSO not only improves the channel capacity of the effective links but also effectively reduces the channel utilization of the interference links, thereby suppressing or eliminating the inter-pair interference and improving the performance of overall multipair massive MIMO relay links.

Then, we also compared the average capacity of different links for different optimization schemes combined with beamforming techniques. The results are shown in Figure 12.

In analyzing the results, first, we found that all polarization optimization algorithms improve the channel capacity of each link in multipair MIMO relay channels, which further indicates the necessity of polarization optimization algorithms to improve the channel capacity. Second, comparing Figure 12a,b shows the performance of CF-DWPSO is better than DWPSO for improving the capacity of multipair MIMO polarized relay links. Finally, comparing Figure 12b,c shows the polarization scheme of CF-DWPSO combined with the beamforming optimization scheme can maximize the performance of multipair MIMO relay links, and this also shows which has the best spatial information segmentation for users. This shows that CF-DWPSO combined with beamforming technology of the relay nodes can balance the polarization matching between multipair MIMO relay links and can effectively improve the channel capacity of the overall relay links while suppressing or eliminating the inter-pair interference.

5. Conclusions

In this paper, we have proposed different polarization optimization strategies and studied the transmission performance of multipair MIMO polarized relay links. The multipair MIMO polarized relay channel model is constructed by introducing polarization and combined with the beamforming technology of the relay nodes. By analyzing the polarization dependence of different links and MPCs, we conclude that the selection of the polarization direction configuration and the beamforming technology of the antenna will affect its gain on different relay links, which significantly affects the channel capacity and the inter-pair interference of multipair MIMO relay links. Therefore, we should consider polarization dependence when optimizing the performance of multipair MIMO relay links. Based on the improved PSO, this paper proposes a CF-DWPSO to improve the performance of multipair massive MIMO polarized relay links and adjusts the polarization combination of antenna arrays of different users by CF-DWPSO combined with beamforming technology, so that the capacity of the effective links is maximized and suppresses or eliminates the inter-pair interference. Compared with other optimization algorithms, CF-DWPSO has excellent performance in improving the capacity of multipair MIMO relay links and in suppressing the inter-pair interference by beamforming technology of the relay nodes with the massive MIMO antennas, and this advantage will be more prominent with an increasing number of transceivers. In the future, we will consider adding multi-polarized antenna arrays or applying intelligent reflecting surface (IRS) and varying the polarization direction of the antenna arrays more finely, which can further enhance the channel capacity and suppress or eliminate inter-pair interference.