### Staff Members of the Project

## 2. Low-Complexity Linear Precoding

Although BF precoding has a low complexity and achieves a high performance as the number of antennas goes to infinity for a fixed number of users, in practice, the maximum number of antennas will be limited and the number of users in a cell may be large [9]. For this scenario, it has been shown that the MMSE precoder achieves a considerable performance gain compared to the BF precoder [12, 13]. The MMSE precoding matrix is given by

$\boldsymbol{\mathrm{V}}_\mathrm{MMSE} = \xi_\mathrm{MMSE} \boldsymbol{\mathrm{H}}^\mathrm{H} \left( \boldsymbol{\mathrm{H}} \boldsymbol{\mathrm{H}}^\mathrm{H} + \sigma_n^2 \boldsymbol{\mathrm{I}}_K \right)^{-1},$

where $\boldsymbol{\mathrm{H}}$, $\sigma_n^2$, and $\boldsymbol{\mathrm{I}}_K$ denote the channel matrix, the noise variance, and the $K \times K$ identity matrix, respectively. $\xi_\mathrm{MMSE}$ is a normalization factor, which ensures that the transmit power constraint $\rm{tr}\left( \boldsymbol{\mathrm{V}}_\mathrm{MMSE} \boldsymbol{\mathrm{V}}_\mathrm{MMSE}^\mathrm{H} \right)$=$P_\mathrm{TX}$ is met, where $P_\mathrm{TX}$ is the total transmit power at the base station. The MMSE precoder still entails a high complexity due to the required matrix inversion, especially for the large matrices typical for massive MIMO systems. To overcome this problem, in our recent work, we approximated the matrix inversion in the MMSE precoder by a matrix polynomial, which leads to the following polynomial expansion (PE) precoder matrix [15]

$\boldsymbol{\mathrm{V}}_\mathrm{PE} = \xi_\mathrm{PE} \boldsymbol{\mathrm{H}}^\mathrm{H} \sum_{l=0}^{L} \omega_l \left( \boldsymbol{\mathrm{H}} \boldsymbol{\mathrm{H}}^\mathrm{H} \right)^l,$

where $\xi_\mathrm{PE}$ is a normalization factor, which ensures that the transmit power constraint $\rm{tr}\left( \boldsymbol{\mathrm{V}}_\mathrm{PE} \boldsymbol{\mathrm{V}}_\mathrm{PE}^\mathrm{H} \right)$=$P_\mathrm{TX}$ is fulfilled and the coefficients $\omega_l$ in $\boldsymbol{\mathrm{V}}_\mathrm{PE}$ can be determined by solving an optimization problem, e.g., for sum-MSE minimization or sum rate maximization. Moreover, in our recent work [15], we have shown that the optimal coefficients $\omega_l$ do not depend on the instantaneous channel realizations and can be determined easily using tools from random matrix theory. Furthermore, we have shown that for large matrix polynomial orders $L$, the proposed PE precoder approaches the sum rate of the MMSE precoder. The sum rate performance of the PE precoder for a system with $N=100$ base station antennas is depicted in Fig. 2. As can be seen, if the number of users is small compared to the number of antennas, all precoding schemes achieve almost the same sum rate which confirms that in such scenarios BF precoding is preferable, since it requires a lower computational complexity than all other schemes. With increasing number of users, the performance gap between the proposed PE precoding scheme and the BF precoding scheme increases and has a peak at $K = N$. From Fig. 2, it can also be concluded that for $K/N = 1$, even for $L = 1$ a sum rate improvement of more than $40\%$ compared to BF precoding can be achieved with the proposed PE precoder. For the same $K/N$ and $L = 2$, the proposed PE precoder achieves more than $86\%$ of the sum rate of the MMSE precoder.

### Research Challenges and Open Problems

Some open issues in design of PE precoders include:

**Consideration of multi****-cell interference and pilot contamination**The PE precoder in [15] has been designed for a single-cell scenario. However, multi-cell networks are more realistic application scenarios and will be considered in our future work. Furthermore, the coefficients of the precoder matrix can be optimized taking into account the effect of pilot contamination. Here, we expect to achieve a higher performance than that of the MMSE precoder, since the PE precoder offers more degrees of freedom for optimization.**Analysis of the impact of multiple antennas at the user terminals**Most works on massive MIMO assume single-antenna user terminals (UT). However, it would be interesting to investigate the improvement of the sum rate when several antennas are employed at the UTs. In particular, the rate gap between multiple-antenna UTs and single-antenna UTs could be investigated.

## 3. Physical-layer security

While traditionally data security in wireless communication systems was addressed only at the application layer using cryptographic methods, recently, there has been a paradigm shift and data security is already taken into account in the design of the physical (PHY) layer of communications systems [16]. In practice, PHY layer security can either serve as a complement to conventional security mechanisms or completely replace them, thereby avoiding complicated mechanisms and protocols for key distribution and management. For small-scale (conventional) MIMO systems, it has been shown that PHY layer security is facilitated by exploiting multiple transmit antennas and introducing artificial noise at the transmitter [17]. The combination of PHY layer security and massive MIMO is highly promising since massive MIMO systems offer more degrees of freedom and simpler transceiver designs compared to small-scale MIMO systems. While a large system PHY layer secrecy analysis of multi-user MIMO systems employing regularized channel inversion precoding has recently been reported in [18], a comprehensive analysis and design of PHY layer security mechanisms under the framework of massive MIMO systems is missing in the literature. For example, issues such as effective artificial noise generation and precoding for secure massive MIMO systems as well as the impact of pilot contamination on PHY security have not been addressed. Some of our recent results on this topic can be found in [19].

## 4. Transmission Design with Finite Alphabet Inputs

Most works on massive MIMO rely on the critical assumption of Gaussian input signals. Although Gaussian inputs are optimal in theory, they are rarely used in practice. Rather, practical communication signals usually are drawn from finite constellation sets, such as pulse amplitude modulation (PAM), phase shift keying (PSK) modulation, and quadrature amplitude modulation (QAM). These finite constellation sets differ significantly from the Gaussian idealization. Accordingly, transmission schemes designed based on the Gaussian input assumption may result in substantial performance losses when finite alphabet inputs are used for transmission in massive MIMO system. As a result, it is necessary to investigate the transmitter design for massive MIMO systems with finite alphabet inputs.

### Performance of Precoder Design with Finite Alphabet Inputs

In Figures 3 and 4, the sum rate performance of a massive MIMO multiple access channel (MAC) with four users is depicted for different precoder designs and QPSK transmit symbols for the suburban and the urban scenarios of the spatial channel model (SCM) used in 3GPP standardization, respectively. We assume statistical CSI is available at the transmitter and consider the jointly-correlated fading MIMO channel. We employ the Gauss-Seidel algorithm together with stochastic programming to obtain the optimal covariance matrices of the users under the Gaussian input assumption. Then, we decompose the obtained optimal covariance matrices $\left\{ {{\mathbf{Q}}_1 , {\mathbf{Q}}_2, \cdots ,{\mathbf{Q}}_K } \right\}$ as ${\mathbf{Q}}_k = {\mathbf{U}}_k \boldsymbol{\Lambda} _k {\mathbf{U}}_k^H$, and use ${\mathbf{B}}_k = {\mathbf{U}}_k \boldsymbol{\Lambda} _k^{\frac{1}{2}}$, $k = 1,2, \cdots, K$ as the precoders. Finally, we calculate the average sum rate for this precoding design for QPSK inputs. We denote the corresponding sum rate as "GP with QPSK inputs". For the case without precoding , we set the precoders ${\mathbf{B}}_1 = {\mathbf{B}}_2 = {\mathbf{B}}_3 = {\mathbf{B}}_4 = \sqrt{\frac{P}{N_t}} \mathbf{I}_{N_t}$. We denote the corresponding sum rate as "NP with QPSK inputs". The sum rates achieved with the Gauss-Seidel algorithm and without precoding for Gaussian inputs are also plotted, and denoted as "GP with Gaussian input" and "NP with Gaussian input", respectively. We denote the proposed design for the MIMO MAC with finite alphabet inputs and statistical CSI as "FAP with QPSK inputs". We observe from Figures 3 and 4 that, for QPSK inputs, the "FAP with QPSK inputs" design achieves a better performance than the other precoding designs for both scenarios. For a sum rate of $24$ b/s/Hz, the SNR gains of the "FAP with QPSK inputs" design over the "NP with QPSK inputs" design for the suburban and the urban scenarios are about $4.5$ dB and $3.5$ dB, respectively. It is noted that the SNR gain for the suburban scenarios is larger than that for the urban scenarios. This is because the correlation of the transmit antennas is higher in suburban scenarios. Therefore, some subchannels are much stronger than the remaining subchannels. As a result, a proper precoding design will yield a larger performance gain. Also, the "GP with QPSK inputs" design results in a substantial performance loss in both scenarios. This is because the Gauss-Seidel algorithm design implements a "water filling" power allocation policy in this SNR region. As a result, when the SNR is smaller than a threshold, the precoders allocate most of the available power to the strongest subchannels and allocate little power to the weaker subchannels. Therefore, some eigenvalues of ${\mathbf{Q}}_k$ approach zero. For finite alphabet inputs, this power allocation policy may result in allocating most power to the subchannels that are close to saturation. This will lead to a waste of transmit power and impede the further improvement of the sum rate performance. This confirms that precoders designed under the ideal Gaussian input assumption may result in a considerable performance loss when adopted directly in practical systems with finite alphabet constraints and underlines the need for taking the finite alphabet input precoding into account for system design.

### Research Challenges and Open Problems

There are many open research problems for transmission design in massive MIMO system with finite alphabet input:

- The precoder design for the massive MIMO MAC with finite alphabet inputs has been investigated only for some special cases. The general jointly-correlated Rician fading channel has not been considered yet. The impact of having a light-of-sight on the performance is unknown.
- The precoder design for the massive MIMO broadcast channel with finite alphabet inputs has been investigated only for the single-antenna user scenario. The extension to the multiple-antenna user scenario is needed.
- Security issues for massive MIMO systems with finite alphabet inputs has barely been investigated.
- The asymptotic mutual information for massive MIMO relay systems with finite alphabet inputs has been analyzed. However, a viable precoder design is still not known.

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