Learning-Based Massive Beamforming

Consider a single cell multi-user massive MIMO system where the BS is equipped with $N_{T}$ transmit antennas and serves $K$ users each equipped with $N_{R}$ antennas [14]. Let $\mathbf{V}_{k}\in\mathbb{C}^{N_{T}\times d_{k}}$ denotes the transmit beamforming that the BS employs to send the signal ${\mathbf{s}}_{k}\in\mathbb{C}^{d_{k}\times 1}$ to user $k$. The BS signal is given by,

where it is assumed $\mathbb{E}\left[{\mathbf{s}}_{k}{\mathbf{s}}_{k}^{H}\right]=\mathbf{I}$.

Assuming a flat-fading channel model, the received signal $\mathbf{y}_{k}\in\mathbb{C}^{N_{R}\times 1}$ at user $k$ can be written as

where matrix $\mathbf{H}_{k}\in\mathbb{C}^{N_{R}\times N_{T}}$ represents the channel matrix from the BS to user $k$, while $\mathbf{n}_{k}\in\mathbb{C}^{N_{R}\times 1}$ denotes the additive white Gaussian noise with distribution $\mathcal{CN}(0,\sigma_{k}^{2}\mathbf{I})$. We assume that the signals for different users are independent from each other and from receiver noises. In this paper, we treat the multi-user interference as noise and employ linear receive beamforming strategy, i.e., $\mathbf{U}_{k}\in\mathbb{C}^{d_{k}\times N_{R}},\forall k$, so that the estimated signal $\hat{{\mathbf{s}}}_{k}\in\mathbb{C}^{d_{k}\times 1}$ is given by $\hat{{\mathbf{s}}}_{k}=\mathbf{U}_{k}^{H}\mathbf{y}_{k},\forall k.$

A basic problem of interest is to find the transmit beamformers $\{\mathbf{V}_{k}\}$ such that the system weighted sum-rate is maximized subject to a total power constraint due to the BS power budget. Mathematically, it can be written as follows

where $P_{max}$ denotes the BS power budget, the weight $\alpha_{k}$ represents the priority of user $k$ in the system, and $R_{k}$ is the rate of user $k$ given by

where ${\textbf{A}}_{k}\triangleq\sum_{j\neq k}\mathbf{H}_{k}\mathbf{V}_{j}\mathbf{V}_{j}^{H}\mathbf{H}_{k}^{H}$.

Under the independence assumption of ${\mathbf{s}}_{k}$’s and $\mathbf{n}_{k}$’s, the MSE matrix $\mathbf{E}_{k}$ can be written as,

Followed by [6], we can obtain the equivalent WMMSE form as

Furthermore, inspired by the structure of ZF beamforming [8], to reduce the complexity in the massive antenna scenario, we also restrict $\mathbf{V}_{k}$’s to the range space of $\mathbf{H}^{H}$, i.e., let it satisfy $\mathbf{V}_{k}=\mathbf{H}^{H}\mathbf{X}_{k}$ with some $\mathbf{X}_{k}\in\mathbb{C}^{KN_{R}\times d_{k}}$, where $\mathbf{H}\triangleq\left[\mathbf{H}_{1}^{H}~{}~{}\mathbf{H}_{2}^{H}~{}~{}\ldots\mathbf{H}_{K}^{H}\right]^{H}\in\mathbb{C}^{KN_{R}\times N_{T}}$. As a result, by defining $\mathbf{M}_{k}\triangleq\mathbf{U}_{k}\mathbf{W}_{k}\mathbf{U}_{k}^{H}$ and $\bar{\mathbf{H}}\triangleq\mathbf{H}\mathbf{H}^{H}\in\mathbb{C}^{KN_{R}\times KN_{R}}$, we can derive the three main steps of the corresponding WMMSE algorithm as follows

The algorithm repeats the above three steps until convergence. For ease of exposition, it is termed as reduced-WMMSE (R-WMMSE).

In previous work like [13], the noise power $\sigma_{k}^{2}$ is often fixed for all scenarios, resulting in the trained network only adapting to this noise level. Here we remove the effect of noise by reformulating the problem. Let us define $\tilde{\mathbf{H}}_{k}=\sqrt{\frac{P_{max}}{\sigma_{k}^{2}}}\mathbf{H}_{k}$ and

Then we have the following proposition.

Because of the above equivalence, we consider problem (7) throughout the rest of this paper. Moreover, for notational simplicity, we drop ‘ $\tilde{}$ ’ in all notations in (7).

Figure 1 presents the CNN-based network architecture for beamforming design followed by the idea of [13], where CL and BN denote the convolutional layer and batch normalization layer respectively, leaky relu is chosen as the activation function and several dense layers are used after the flatten layer. The network architecture is further detailed as follows.

In massive MIMO system, the number of transmit antennas $N_{T}$ could be very large. Hence, if we still directly take $\mathbf{H}$ and $\mathbf{V}_{k}$ (or $\mathbf{X}_{k}$) as input and output as in [13, 9], the input/output of the neural network (NN) would be both high dimensional matrices, making it not easy to train an NN. As a consequence, the NN input and output should be redesigned to reduce the NN input/output size (and thus the training complexity and difficulty). In terms of the R-WMMSE algorithm, we find that beamformer $\mathbf{V}_{k}$ is uniquely determined by $\mathbf{X}_{k}$ while $\mathbf{X}_{k}$ depends on $\mathbf{H}\mathbf{H}^{H}$. Hence, $\mathbf{H}\mathbf{H}^{H}$ can be regarded as the NN input, which has reduced size as compared to $\mathbf{H}$ when $N_{T}>>N_{R}$. Moreover, since $\mathbf{X}_{k}$ can be determined by $\mathbf{H}\mathbf{H}^{H}$ and $\{\mathbf{U}_{k},\mathbf{W}_{k}\}$, we can take $\{\mathbf{U}_{k},\mathbf{W}_{k}\}$ as the NN output in order to reduce the size of NN output. Tables I and II list the dimension of various input/output schemes. It can be seen that different choice of input/output leads to different size of input/output. Note that we generally have $N_{T}>>N_{R}$, $N_{T}\geq KN_{R}$, $K\geq N_{R},d_{k}$ in the massive MIMO case.

Furthermore, due to the conjugate symmetry of $\mathbf{H}\mathbf{H}^{H}$, both the real part and imaginary part of $\mathbf{H}\mathbf{H}^{H}$ depend uniquely on their upper or lower triangular parts. Hence, we can further reduce the input size by combining the real part and the imaginary part in a way as shown in Fig. 2. As a result, the dimension of NN input is finally reduced to $KN_{R}\times KN_{R}$. Similar operation can be done for $\mathbf{W}_{k}$, leading to a further reduced size of NN output.

In practice, each user $k$ in the system may have a different priority with weight $\alpha_{k}$ that often changes with time. While most current methods do not take this into consideration, the NN input or structure should be carefully redesigned when the weights are considered. According to the R-WMMSE algorithm mentioned before, both $\mathbf{X}_{k}$ and $\{\mathbf{U}_{k},\mathbf{W}_{k}\}$ depend on $\alpha_{k}\mathbf{H}\mathbf{H}^{H}$.

Two very intuitive ways to merge the weights into the NN are depicted in Figure 3 and Figure 4. One is to merge weights into input as K channels (see Figure 3), and the other is to concatenate weight after convolution and flatten of the input (see Figure 4). Our simulation results show that these two methods can achieve reasonably good performance.

However, these two methods will bring higher computational complexity to the original network which can lead to extra time and cost. Surprisingly, inspired by the update rule of $\mathbf{X}_{k}$ in (II-B), we find that, just by taking $\tilde{\mathbf{H}}\tilde{\mathbf{H}}^{H}$ as input, where $\tilde{\mathbf{H}}_{k}=\sqrt{\alpha}_{k}\mathbf{H}_{k}$ and $\tilde{\mathbf{H}}\triangleq\left[\tilde{\mathbf{H}}_{1}^{H}~{}~{}\tilde{\mathbf{H}}_{2}^{H}~{}~{}\ldots\tilde{\mathbf{H}}_{K}^{H}\right]^{H}$, we can reach the same performance as the previous two intuitive methods without any need for increasing network complexity.

In practical systems, sometimes only one stream is transmitted for some user during communication. This raises a new challenge that the number of streams $d_{k}\ (d_{k}\leq N_{R})$ can vary but the dimension of the network output needs to be fixed. Table III shows the number of valid output elements when $d_{k}$ is different. Thus, to ensure that the network output have fixed dimension, certain positions should be set to zero when there exists a single stream transmission.

There are a few simple and intuitive solutions to this problem. The simplest solution is to directly merge the information of the number of user streams into the original input $\mathbf{H}\mathbf{H}^{H}$. Another solution is to ignore the number of streams used and manually set to zero the positions corresponding to empty streams of the output, which may result in discontinuous loss function. These two methods both result in unsatisfactory performance in our experiments.

To achieve better performance than the solutions mentioned above, we introduce an auxiliary network called Index Network whose function is to softly zero out the corresponding positions given the number of streams used. Specifically, as shown in Figure 5, the Index Network is a network separated from the main network, it takes the number of user streams as input and outputs a soft mask having the same dimension of the output of the main network. The output of the main network is multiplied by the mask element-wisely to produce the final output. We find that this method can effectively stabilize the training and improve the performance.

During the testing stage, to ensure that the network outputs a beamforming with correct number of streams, the output elements at certain positions will be set to zero manually at the last layer (Lambda Layer).

During the unsupervised learning phase, variables of the Index Network should be fixed and specific positions should also be assigned with zero before calculating the unsupervised loss.

The main network consists of one convolutional layer with 4 kernels of size $3\times 3$, followed by batch normalization (BN) layer and activation function layer (leaky relu), then only one dense layer with 32 hidden units. The Index Network is of similar scale as the dense layer before. Our network is much simpler than the previous work [13, 9] with much more layers and hidden units (mostly having more than two convolutional and dense layers).

For weighted sum-rate maximization, the channel matrix $\mathbf{H}$ is generated from the complex Gaussian distribution with pathloss between the users and the BS. The pathloss is set to $128.1+37.6\log_{10}(\omega)$[dB] [15] where $\omega$ is the distance between the user and the BS in km ($0.1\sim 0.3$). The noise power is set to be the same for all users and can be calculated by $\sigma_{k}^{2}=10^{\frac{1}{K}\sum_{k}\log_{10}\frac{1}{N_{R}}\sum_{i,j}\mathbf{H}_{k_{ij}}^{2}}\times 10^{-\frac{\rm SNR}{10}}$, where signal-to-noise ratio (SNR) is set as $20{\rm(dB)}$. The priority coefficients $\alpha_{k}$ of the users are generated randomly and $\sum_{k}\alpha_{k}=K$, and $d_{k}$ is also generated randomly for each user $k$ ($d_{k}=1$ indicates dual stream and $d_{k}=0$ indicates single stream). In the simulation, 45000 samples are generated for training and 5000 are for testing. Table IV lists three main test cases in the following experiment with $N_{R}=2$. The last test case is of great importance in industry.

To test whether the predicted precoder $\mathbf{V}$ is good enough to maximize the weighted sum-rate maximization problem, the predicted output should be put back to the objective function and the performance can be defined as follows.

Figures 6 and 7 show the average execution time and average performance compared with the WMMSE algorithm and ZF algorithm respectively. It can be observed that 1) the proposed method can achieve similar performance as the WMMSE algorithm in most cases, while significantly outperforming ZF algorithm (which is unable to handle the user priority); and 2) the proposed method costs less execution time (in testing stage) than both the WMMSE method including many inversion operations and ZF method which needs extra time to decide which beamforming vector should be used for sending single stream.

In summary, our proposed CMBNN model is superior from the perspective of both performance and efficiency.