Massive MIMO signal transmission in spatially correlated channel environments
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Abstract
Employment of massive multi-input multi-output (m-MIMO) transmission techniques has recently been considered as a key technology for the provision of high capacity to a large number of users. Zero-forcing beamforming (ZFBF) techniques can be employed to maximize the transmission capacity, but they may require high implementation complexity in multi-user m-MIMO transmission environments. In this paper, we consider multi-user m-MIMO signal transmission with flexible complexity in spatially correlated channel environments. We initially set the beam weight for conventional maximum ratio transmission (MRT), which may experience interbeam interference. Then, we adjust the beam weight to remove the interbeam interference, while taking into account the trade-off between the implementation complexity and the performance. To this end, we sequentially adjust the beam weight to remove the interbeam interference in a descending order of interference power. The more interference sources are removed, the closer the performance of proposed scheme approaches to that of ZFBF. The proposed beamforming (BF) technique can provide performance close to that of ZFBF in highly correlated m-MIMO channel environments, while significantly reducing the implementation complexity.
Keywords
Massive multi-input multi-output Zero-forcing beamforming Maximum ratio transmission Complexity Interbeam interference1 Introduction
Wireless data traffic has dramatically been increasing with vitalization of wireless multimedia services in cellular communication systems. As a consequence, mobile data traffic has grown 4000-fold over the past 10 years and almost 400 million-fold over the past 15 years [1]. To support explosively increasing wireless traffic demand, there have been an extensively large number of ongoing research works including the improvement of spectral efficiency and the increase of bandwidth and network density. Employment of massive multi-input multi-output (m-MIMO) antenna techniques has been considered as one of key techniques for the improvement of spectral efficiency in advanced cellular communication systems. It may also provide high energy efficiency, making it quite feasible for being applied to green communication systems [2, 3, 4].
M-MIMO techniques can transmit signal with the high degree of freedom, enabling to simultaneously serve a large number of users with high transmission capacity without the increase of transmission resource. Employment of maximum ratio transmission (MRT) or zero-forcing beamforming (ZFBF) technique has been considered for multi-user m-MIMO signal transmission. The MRT technique can be implemented with low implementation complexity, but it may seriously suffer from inter-beam interference, providing capacity much lower than the ZFBF technique [5, 6, 7]. On the other hand, the ZFBF technique can remove the inter-beam interference at the expense of very high implementation complexity, making it unfeasible in m-MIMO environments [8]. There exist many precoding or beamforming (BF) schemes for the downlink transmission of multi-user MIMO systems [9, 10]. In [9], the multi-user MIMO downlink channel is decomposed into multiple parallel independent single-user channels. In [10], the general design of multi-user MIMO precoding matrices is proposed considering various circumstances such as closely spaced antennas and the arbitrary number of equipped antennas. Since these schemes use singular value decomposition (SVD) technique, they suffer from extensively high complexity when they are applied in the m-MIMO environment. The low-complexity BF scheme based on QR decomposition was introduced in [11, 12], whose implementation complexity is, however, still high if a massive number of antennas are used. In order to reduce the implementation complexity further, the BF scheme based on the iterative QR decomposition (IQRD) was proposed in [13]. However, in case that the multiplexing order is not full, the IQRD technique may require an additional combining process resulting in the increase of implementation complexity. Thus, a BF scheme with an affordable implementation complexity became an critical issue in the area of research and development for multi-user m-MIMO systems, and the demand for designing such a BF scheme is continuously growing.
In this paper, we propose a flexible complexity BF scheme for a multi-user m-MIMO transmission system, which is designed by considering the trade-off between the implementation complexity and the transmission performance. Unless the power of interference sources to other users is equally distributed, we can achieve the desired performance by removing major interference sources instead of all. We consider the use of a beam weight determined by the MRT as an initial beam weight. Then, we adjust the beam weight to remove major interference sources generated by the MRT. For ease of implementation, we sequentially adjust the beam weight to remove the interbeam interference in a descending order of interference power. The larger the number of inter-beam interference sources to be removed, the closer performance to that of ZFBF we can get. In practice, m-MIMO antennas may need to be installed in a small space, which may result in the presence of a high correlation in the m-MIMO channel. When the m-MIMO channel experiences high correlation, the proposed technique can provide performance close to that of ZFBF by only removing a small number of strong interference sources.
The rest of this paper is organized as follows. Section 2 describes the system model under consideration. Section 3 describes the proposed multi-user beamforming (BF) technique in spatially correlated m-MIMO channel environments. Section 4 evaluates the performance of the proposed BF technique by computer simulation. Finally, Section 5 concludes the paper.
2 System model
where α _{ k } denotes the large-scale fading coefficient from the BS to user k, h _{ k } denotes the (N _{ T }×1) channel vector from the BS to user k, w _{ k } denotes the (N _{ T }×1) beam-weight vector of user k, s _{ k } denotes the signal of user k with E{|s _{ k }|^{2}}=1, P _{ k } denotes the transmit power assigned to user k, n _{ k } denotes the zero-mean complex circular-symmetric additive white Gaussian noise (AWGN) with variance \({\sigma _{k}^{2}}\) at the receiver of user k, and the superscript H denotes the conjugate-transpose of a matrix or a vector.
3 Proposed multi-user m-MIMO transmission
where H denotes the (N _{ T }×K) channel gain matrix from the BS to K users and the k-th column of H is equal to h _{ k }. It follows that the beam weights of user k based on MRT and ZFBF are obtained by \({\mathbf {w}_{k}^{\text {MRT}} = \mathbf {f}_{k}^{\text {MRT}}/\left \| {\mathbf {f}_{k}^{\text {MRT}}} \right \|}\) and \({\mathbf {w}_{k}^{\text {ZF}} = \mathbf {f}_{k}^{\text {ZF}}/\left \| {\mathbf {f}_{k}^{\text {ZF}}} \right \|}\), respectively. The MRT technique may suffer from severe inter-beam interference. The ZFBF technique can avoid the inter-beam interference, but it may require high implementation complexity [8], making it impractical in multi-user m-MIMO transmission environments. In practice, the MRT is preferred to the ZFBF mainly due to the robustness to imperfect channel information and low complexity [15]. We aim to design a BF scheme which can control the amount of interference cancelation taking into account the implementation complexity.
3.1 Proposed multi-user beamforming
where the n-th element of Z _{ k,N } is denoted by z _{ k,N }.
Computation of \({\mathbf {e}_{{z_{k,i}}}}\), 1≤i≤N+1, i≠n, for a given z _{ k,n }
Initialization |
1: \({\mathbf {e}_{{z_{k,1}}}} = \mathbf {w}_{k}^{MRT}\) |
2: for i=2:N+1 |
3: if i≠n |
4: \(\;\;\;\,\,\,\;{\mathbf {e}_{{z_{k,i}}}} = \frac {{{\mathbf {h}_{{z_{k,i}}}} - \sum \limits _{j = 1,j \ne n}^{i - 1} {{{\left ({\mathbf {h}_{{z_{k,i}}}^{H}{\mathbf {e}_{{z_{k,j}}}}} \right)}^{H}}{\mathbf {e}_{{z_{k,j}}}}} }}{{\left \| {{\mathbf {h}_{{z_{k,i}}}} - \sum \limits _{j = 1,j \ne n}^{i - 1} {{{\left ({\mathbf {h}_{{z_{k,i}}}^{H}{\mathbf {e}_{{z_{k,j}}}}} \right)}^{H}}{\mathbf {e}_{{z_{k,j}}}}}} \right \|}}\;\;\;\) |
5: end |
6: end |
Note that h _{ k } and Δ w _{ k } are orthogonal to each other, i.e., \(\mathbf {h}_{k}^{H}\Delta {\mathbf {w}_{k}} = 0\). For each k∈Ω _{ K }, we repeat the above process to obtain the corresponding beam weight \(\bar {\mathbf {w}}_{k}\).
3.2 Performance analysis of multi-user beamforming
Let \({\lambda _{k}}\left ({ = {\alpha _{k}}{P_{k}}/{\sigma _{k}^{2}}} \right)\) be the signal-to-noise ratio (SNR) of user k including the large-scale fading effect. We assume that all users have the same SNR, i.e., λ _{ k }=λ.
Note that the right-hand side of (21) with N=0 equals to the lower-bound of \(E\left \{ {C_{k}^{\text {prop}}} \right \}\) given by (13).
We measure the implementation complexity of BF schemes in terms of the number of floating-point operations (FLOPs) [17], which is defined by the total number of multiplications and additions of real numbers. Note that the multiplication and the addition of two complex numbers require six FLOPs and two FLOPs, respectively. For (N _{ T }×1) complex vectors a and b, the inner product a ^{ H } b requires (8N _{ T }−2) FLOPs, the normalization a/∥a∥ requires (6N _{ T }−1) FLOPs, and ∥a∥ requires (4N _{ T }−1) FLOPs.
4 Performance evaluation
Comparison of the implementation complexity
Beamforming | FLOP |
---|---|
MRT | K(6N _{ T }−1) |
ZFBF [13] | K{(24(K−1)N _{ T } ^{2}+48(K−1)^{2} N _{ T })+54(K−1)^{3}} |
IQRD [13] | \(\begin {array}{l} 8\left ({3{N_{T}} - 1} \right) + \sum \limits _{i = 2}^{K - 1} \left ({i + 1} \right)\left \{8{N_{T}}\left [ {{N_{T}} - \left ({i - 1} \right)} \right ]\right.\\ ~~~~~~~~~~~~~~~~~~~~~~+ 4\left [ {3\left ({{N_{T}} - \left ({i - 1} \right)} \right) - 1} \right ]\\ \left.~~~~~~~~~~~~~~~~~~~~~~+ 8{N_{T}}\left [ {{N_{T}} - \left ({i - 1} \right)} \right ]\left ({{N_{T}} - i} \right)\right \} \end {array}\) |
Proposed BF | \(\begin {array}{l} K\left [ 2\left ({6{N_{T}} - 1} \right) + N\left (\left ({8{N_{T}} - 1} \right){N^{2}} + \left ({14{N_{T}} - 2} \right)N \right.\right.\\ \left.\left.~~~~~~~~~~~~~~~~~~~~~~~~~~~+ 18{N_{T}} + 8 \right)\right ] \end {array}\) |
The ratio of spectral efficiency and implementation complexity of the proposed BF to those of the ZFBF
N | 0 (%) | 1 (%) | 2 (%) | 3 (%) | 4 (%) | 5 (%) | 6 (%) | 7 (%) | |
---|---|---|---|---|---|---|---|---|---|
Spectral efficiency | ρ=0.9 | 60 | 73 | 81 | 87 | 92 | 95 | 99 | 100 |
ρ=0 | 60 | 73 | 81 | 87 | 92 | 95 | 99 | 100 | |
FLOPs | 0.04 | 0.38 | 1.25 | 3.04 | 6.11 | 10.82 | 17.53 | 26.59 |
The ratio of spectral efficiency obtained by the proposed BF with imperfect CSI to that with perfect CSI, where N _{ T }=64 and ρ=0.7
N | 0 (%) | 1 (%) | 2 (%) | 3 (%) | 4 (%) | 5 (%) | 6 (%) | 7 (%) | |
---|---|---|---|---|---|---|---|---|---|
Spectral efficiency | σ _{mse}=−10 dB | 96 | 95 | 93 | 90 | 88 | 86 | 84 | 83 |
σ _{mse}=−5 dB | 84 | 79 | 75 | 72 | 68 | 66 | 63 | 62 |
5 Conclusions
We proposed a BF scheme for multi-user m-MIMO signal transmission with flexible implementation complexity. The proposed BF technique can flexibly be implemented by taking into consideration of trade-off between the implementation complexity and the transmission performance, making it quite feasible in real m-MIMO deployment environments.. The numerical results show that the proposed BF is quite effective in the presence of channel correlation, providing transmission performance close to that of ZFBF, while significantly reducing the implementation complexity. The proposed BF technique can be applied to the uplink transmission of the multi-user m-MIMO system in the form of a combiner or a receive BF scheme implemented at the BS. The beam weight is initialized as a maximal ratio combiner (MRC), and it is adjusted to remove the inter-beam interference in a descending order of uplink interference power considering the tradeoff between the implementation complexity and the performance.
Notes
Acknowledgements
This work was supported by Institute for Information & communications Technology Promotion(IITP) grant funded by the Korea government(MSIP) (No. R0101-16-244, Development of 5G Mobile Communication Technologies for Hyper-connected Smart Services).
Authors’ contributions
We have considered multi-user beamforming with low implementation complexity in m-MIMO environments. The proposed scheme sequentially cancel out the interbeam interference in an order of the strongest one. Thus, it can compromise the implementation complexity, while maximizing the performance. The numerical and simulation results show that the proposed scheme can significantly reduce the implementation complexity, while providing performance similar to the ZFBF in practical m-MIMO environments.
Competing interests
The authors declare that they have no competing interests.
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