1. Capacity-Approaching Linear Precoding with Low-Complexity for Multi-User Large-Scale MIMO systems
Xinyu Gao1, Linglong Dai1, Jiayi Zhang1, Shuangfeng Han2, and Chih-Lin I2
Good afternoon everyone! My name is Xinyu Gao from Tsinghua University. Today, it’s my great honor to make this presentation. The title of this presentation is Capacity-Approaching Linear Precoding with Low-Complexity for Multi-User Large-Scale MIMO systems.
1Department of Electronic Engineering, Tsinghua University
2Green Communication Research Center, China Mobile Research Institute

2. Contents 1 Technical Background 2 Proposed Solution 3
Complexity Analysis 4
Simulation Results
This presentation will consist of 5 parts, that is technical background, proposed solution, complexity analysis, simulation results, and conclusion. At the beginning, let’s have a view of the technical background 5
Conclusions

3. Advantages and challenges of large-scale MIMO
What is large-scale MIMO?
Large antenna array at BS
Advantages
Improve the spectrum and energy efficiency by orders of magnitude
key technology for 5G
Challenges [Rusek’13]
Efficient pilot pattern design under pilot contamination
Reliable channel estimation and channel feedback
Low-complexity capacity-approaching precoding
We are familiar with the large-scale MIMO. Unlike the traditional small-scale MIMO, large-scale MIMO equips a very large number of antennas (e.g., 256 antennas or even more) at the base station (BS) to simultaneously serve multiple users
The main advantages of LS-MIMO is it can improve both the spectrum ['spektrəm] efficiency and energy efficiency by orders of magnitude. It is considered as the promising key technology for 5G. However, realizing the attractive merits of large-scale MIMO in practice faces some challenging problems as we listed in this slide, one of which is the low-complexity precoding in the downlink.
[1] F. Rusek, etc, “Scaling up MIMO: Opportunities and Challenges with Very Large Arrays”, IEEE Signal Processing Magazine, vol. 30, no. 1, pp , Feb

4. Precoding schemes: linear vs. nonlinear
Nonlinear precoding (DPC)
Eliminate interferences by successive encoding and decoding
Optimal but prohibitively high complexity for large-scale MIMO
Linear precoding
Eliminate interferences by linear precoding matrix
Low complexity compared to nonlinear schemes
Near optimal when (4 times or more)
The existing preocoding schemes can be divided into two categories, that is the nonlinear precoding and the linear precoding.
The nonlinear precoding schemes, such as dirty paper precoding, can achieve the optimal channel capacity by utilizing successive encoding and decoding to eliminate the interferences. However, the nonlinear precoding schemes usually involves high complexity when the dimension of MIMO system is large or the modulation order is high. This fact makes them difficult to be realized for large-scale MIMO.
By contrast, the linear precoding schemes utilize the linear precoding matrix to eliminate the multi-user interferences. Their complexity is much lower than the non-linear schemes. What’s more, it has been proved that when number of BS antennas N is much larger than the number of users K, linear precoding schemes can achieve the near-optimal capacity as shown by the figure in this slide.
Linear precoding is more attractive for large-scale MIMO!

5. Classical linear ZF precoding
Precoding matrix
Transmitted signal vector after precoding
where
Problem when dimension is large
Complicated matrix inversion with a large number of divisions
Existing method: Neumann-based precoding [Prabhu’13]
Convert matrix inversion into matrix-matrix/vector multiplications
Complexity reduction is not obvious if high accuracy is required
Here is a briefly introduction of the classical linear zero forcing precoding. We define the matrix W, which need to be inversed, in this form. It can be observed that the ZF precoding involves complicated matrix inversion of large size, together with a large number of divisions.
The existing method to solve the matrix inversion in ZF precoding is Neumann-based precoding. This method can convert the matrix inversion into a series of matrix to matrix or matrix to vector multiplication. However, when high accuracy is required, only marginal complexity reduction can be achieved.
[1] H. Prabhu, etc, “Approximative matrix inverse computations for large-scale MIMO and applications to linear pre-coding systems,” in Proc. IEEE Wireless Communications and Networking Conference (WCNC’13), Apr. 2013, pp. 2710–2715.

6. Contents 1 Technical Background 2 Proposed Solution 3
Complexity Analysis 4
Simulation Results
To solve the matrix inversion in linear precoding with low complexity, in this paper, we propose a Gauss-Seidel based precoding scheme, as will be described in detail next. 5
Conclusions

7. Gauss-Seidel (GS) based precoding
Target
Compute with low complexity, where
Decomposition of
Decompose W into a diagonal, a strictly lower triangular, and a strictly upper triangular component
Utilize the GS method
Approximation of
Precode the transmitted signal vector
Our goal is to compute s_hat with low complexity, where matrix W is defined in this form. The first step of our method is to decompose matrix W into a diagonal, a strictly lower triangular, and a strictly upper triangular component like this. Then, by employing the Gauss-Seidel method, s_hat can be approximated by this iteration procedure, where i is the number of iterations and s_hat zero is the initial solution, which can be set as zero vector without loss of generality. Furthermore, the transmitted signal vector after
precoding can be presented as in this form.
It should be pointing out that W is a Hermitian positive define matrix. Therefore, we can conclude that GS-based precoding is convergent for any initial solution.
Notation
W is an Hermitian positive define matrix
GS-based precoding is convergent for any initial solution!

8. Power constraint of GS-based precoding
The precoding matrix should satisfy
For ZF precoding, we have
Precoding matrix of GS-based precoding
Define iteration matrix as
Precoding matrix
Next we will check whether the proposed GS-based precoding satisfies the power contraint. It is rourequired that the precoding matrix P satisfies this condition to constraint the total transmit power. For our method, the precoding matrix can be presented by this equation, where the matrix BGS is defined as the iteration matrix like this. Then, we can derive the following equation in blue color. This equation indicates that when the number of BS antennas N or the number of iterations i go to infinity, we will have this equation. That means, if we choose beta_GS equaling to beta_ZF, the precoding matrxi P_GS will satisfies this equation, that is the power constraint.
Observation
or we have
If ,

9. Convergence rate: GS vs. Neumann
Definition:
The smaller is, the faster convergence rate will be
GS-based precoding vs. Neumann-based precoding
Iteration matrix
With some standard mathematical operations, we have
Faster convergence rate can be achieved by GS-based precoding
GS:
Neumann:
Next we will verify that our method will enjoy faster convergence rate than Neumann-based precoding. Firstly, we should define what is convergence rate. Based on this inequality, we know that the final approximation error is affected by two items, the first one is the error between the initial solution and the true value, which is independent of the number of iterations i. The second one is the frobenius norm of the iteration matrix B. Based on this observation, we can define frobenius norm of B as the convergence rate.
For our method, the iteration matrix can presented in this form while for Neumann-based precoding, the iteration matrix is denoted by this one. Then, with some standard mathematical operations, we can derive the following inequality in blue color. This inequality implies that faster convergence rate can be achieved by the proposed GS-based precoding.

10. Quantified convergence rate
By utilizing
Matrix W is diagonally dominant
Theory of random matrix
Law of large number
(convergence rate) can be well approximated by
Conclusion
The larger N is, the faster convergence rate will be
GS-based precoding is appropriate for large-scale MIMO
Finally, we will quantified the convergence rate to provide more insights. By utilizing some special properties of matrix W, we can conclude that the frobenius norm of the iteration matrix BGS can be well approximated by this one. The figure in this slide shows the gap between the theoretical value and the practical value is negligible, especially when the number of BS antennas N is large.
Based on this approximation, we can conclude that for our method, the larger N is, the faster convergence rate will be, which means the proposed GS-based precoding is quite appropriate for large-scale MIMO

11. Contents 1 Technical Background 2 Proposed Solution 3
Complexity Analysis 4
Next we will discuss the computational complexity of the proposed scheme.
Simulation Results 5
Conclusions

12. Complexity analysis How to measure the complexity?
Since both ZF precoding and GSP need to compute matrix W, we consider the complexity after we have obtained W.
We evaluate the complexity in terms of required number of multiplications and divisions
Where does the complexity comes from?
Solve the linear equation
Compute
Calculate the factor
Since both the conventional ZF precoding and the proposed GS-based precoding need to compute matrix W, here we compare the computational complexity after the matrix W has been obtained. Besides, we evaluate the complexity in terms of the required number of complex multiplications and divisions. It can be found that the complexity of GS-based precoding comes from 3 parts as listed in this slide. Next, we will discuss them one by one

13. Quantified complexity (1)
Solve the linear equation
The solution of can be presented as
Total required number of multiplications is
according to theory of random matrix, no division is required
Compute
The multiplication of a matrix and a vector
Require times of multiplications
No division is required
The first one originates from solving the linear equation, the total required number of complex multiplications of this part is the square of K. Note that since the diagonal elements of matrix W can be well approximated by N the number of BS antennas, therefore we do not need any divisions for this part.
The second one comes from the computation of this one. It is a multiplication of a matrix and a vector, which totally requires NK times of complex multiplications. Also no division is required for this part.

14. Quantified complexity (2)
A property
Calculate the factor
When the configuration of MIMO is fixed
is known and constant
requires N multiplications
No division is required
The third one is from the calculation of the factor beta. It has been proved that when N and K go infinity while N over K keeps fixed, beta converges to a deterministic value. And when N and K is finite in practice, the practical beta still meets the theoretical [ˌθiəˈrɛtɪkəl] value. Therefore, beta is known and constant, and we only need N times of complex multiplications to compute this one. Again, no division is required.

15. Comparison of complexity: Table
Computational Complexity
The overall complexity is without any divisions
Complexity comparison
Iteration
number
Neumann-based precoding[Prabhu’14]
GS-based
precoding
Based on the analysis above, the overall complexity of our method can be presented by this formula. The table below compares the complexity of the conventional Neumann-based precoding and our method. we can conclude from the table that the Neumann-based precoding can reduce the complexity from K_cubic to K_square when the number of iterations is i = 2, but its complexity is still O(K3) when i ≥ 3. To ensure the approximation performance, usually a large value of i is required (as will be verified later in simulation results), which means the overall complexity is almost the same as the ZF precoding. By contrast, we can observe that the complexity of the proposed scheme is O(K2) for an arbitrary number of iterations. Even for i = 2, the proposed scheme enjoys a lower complexity than the conventional one.
The complexity can be reduced from to !
[1] H. Prabhu, J. Rodrigues, O. Edfors, and F. Rusek, “Approximated matrix inverse computations for large-scale MIMO and applications to linear pre-coding systems,” in Proc. IEEE Wireless Communications and Networking Conference (WCNC’13), Apr. 2013, pp

16. Contents 1 Technical Background 2 Proposed Solution 3
Complexity Analysis 4
Next, we will show the simulation results to verify the advantage of the proposed GS-based precoding scheme.
Simulation Results 5
Conclusions

17. Simulation results Simulation setup
Parameters: (1) M=256, K=16 (2) M=256, K=32
Modulation schemes: 64 QAM
Channel: Rayleigh fading channel
Initial solution: zero-vector
The two figures show the capacity comparison between the Neumann-based precoding and GS-based precoding with zero-vector initial solution. (The MIMO configuration is N*K=256*16 and N*K=256*32, respectively, and i denotes the number of iterations.) It is clear that the classical ZF precoding is capacity-approaching compared to the optimal DPC precoding. (since the performance gap is within 0.5 dB for the achieved capacity of 220 bps/Hz.) In addition, as shown in first figure, when the number of iterations is small, e.g.,2, the Neumann-based precoding cannot converge, leading to the serious multi-user interferences and the obvious loss in capacity, while the proposed algorithm can achieve much better performance. (For example, when SNR = 30 dB, the proposed scheme can achieve 175 bps/Hz, while only 130 bps/Hz can be obtained by the conventional scheme.) As the number of iterations i increases, the performance of both schemes improves. However, when the same number of iterations i is used, the proposed scheme outperforms the conventional one. (For example, when ${i = 4}$, the required SNR to achieve the capacity of 200 bps/Hz by the proposed scheme is 26 dB, while the conventional one requires the SNR of 30 dB.)
Comparing the two figures, we can also find that with a decreasing value of N over K, the performance of Neumann-based precoding becomes worse. For example, when ${i = 4}$, for the first MIMO system, the Neumann-based precoding can achieve 90 percent of capacity of the DPC precoding at SNR equals to 30 dB, while for the second MIMO system, it can only achieve 64 percent. In contrast, when i equals to 4, the proposed GS-based precoding can achieve 99 percent and 97 percent of capacity of the DPC precoding for the first and the second MIMO systems, respectively. This indicates that the proposed scheme is more robust to N over K. Also it verifies the convergence rate analysis above
The GS-based precoding scheme is capacity-approaching!

18. Contents 1 Technical Background 2 Proposed Solution 3
Complexity Analysis 4
Simulation Results
In the end, we summarize this presentation and draw the conclusions. 5
Conclusions

19. Conclusions
In this report, a GS-based precoding is proposed to achieve the capacity-approaching performance in an iterative way without complicated matrix inversion of large size.
Mathematical analysis shows that:
1) GS-based precoding satisfies the power constraint if we set
2) GS-based precoding enjoys faster convergence rate than Neumann-based precoding;
3) faster convergence rate can be achieved by increasing the number of BS antennas
The complexity analysis shows that the proposed scheme can reduce the complexity from to
The simulation results demonstrate that the proposed scheme outperforms the Neumann-based precoding and achieves the exact capacity- approaching performance of ZF precoding with a small number of iterations.
Firstly…Secondly…Thirdly…Fourthly…Finally…

20. Thanks for your attention !
That’s all. Thanks for your attention!
Thanks for your attention !