Secure Precoding in MIMO-NOMA:
A Deep Learning Approach

Non-orthogonal multiple access (NOMA) is a promising candidate for connecting massively increasing devices to fifth-generation and beyond wireless networks [1]. NOMA is the optimal transmission strategy for both the single-input, single-output (SISO) and multiple-input, multiple-output (MIMO) cases in a single-cell network. In the SISO case, superposition coding at the transmitter with successive interference cancellation at the receiver is optimal. In contrast, in the MIMO case, dirty-paper coding (DPC) is the optimal solution. Nonetheless, in both cases, the base station (BS) broadcasts a superimposed signal of multiple users. This makes secure communications challenging in the presence of adversarial users as signals can be eavesdropped on by such users.

Physical layer security enables the exchange of confidential messages over a wireless medium in the presence of internal or unauthorized eavesdroppers [2]. Specifically, in two-user MIMO-NOMA networks, both users can transmit their messages concurrently and confidentially via secret dirty-paper coding (S-DPC) [3]. While S-DPC is the most spectral efficient precoding for the two-user MIMO-NOMA, it is excessively complex for practical uses. Alternatively, S-DPC region can be achieved by linear precoding and power allocation schemes. In the past years, various linear precoding schemes have been introduced [4, 5, 6, 7]. Generalized singular value decomposition (GSVD)-based precoder [4] is a fast analytical precoder, but it falls short of getting the capacity region when the users have a single antenna. Also, weighted sum-rate maximization [5] and power-splitting [6] approaches are still too time-consuming to be used in practice. More accurately, these approaches require much higher time than the coherence time of wireless channels which is about a few milliseconds [8].

This letter exploits deep learning (DL) to design the covariance matrices of the channel input vectors–or equivalently, to design precoding and power allocation matrices–for secure MIMO-NOMA transmission. Embedding DL into the mobile and wireless networks is well justified in various cases, e.g., when closed-form solutions require poor approximation or the complexity of existing techniques is high [9, 10, 11].

We leverage supervised deep neural networks (DNNs) for secure communication for the two-user MIMO-NOMA channel (see Fig. 1), resulting in a significantly faster solution while almost reaching the spectral efficiency of the S-DPC. To fulfill the task, we first decompose the two-user MIMO-NOMA with confidential messages into two wiretap channels [6] and use a wiretap channel solution for generating and labeling the training data set. We then build and train DNN models that learn to approximate the function mapping channel matrices, base station power, and required secrecy rate of the users to the covariance matrix of the channel input vector for each user. Simulation results prove the efficacy of the developed model since the proposed DL-based precoding scheme has near-optimal performance and outperforms GSVD-based precoding with a large margin. It also brings mapping time below the coherence time of the wireless channel. Therefore, the channel input signal can be designed before the channel coefficients become stale, which is crucial in practice.

Consider a single-cell two-user MIMO-NOMA network, as shown in Fig. 1. Assume that the transmitter, user $1$, and user $2$ are equipped with $n_{t}$, $n_{1}$, and $n_{2}$ antennas, respectively. The transmitter wishes to send two messages $M_{1}$ and $M_{2}$ to user $1$, and user $2$, respectively. When PHY security is a concern, each message must be kept confidential from the other user [12]. ¹¹1In information theory, this channel is known as the MIMO broadcast channel (BC) with two confidential messages [12]. That is, user $1$ should not be able to decode $M_{2}$ and vice versa. As such, the transmitter securely encodes $M_{1}$ and $M_{2}$ to codewords $\mathbf{x}_{1}\in\mathbb{R}^{n_{t}\times 1}$ and $\mathbf{x}_{2}\in\mathbb{R}^{n_{t}\times 1}$, superimposes them $\mathbf{x}=\mathbf{x}_{1}+\mathbf{x}_{2}$, and broadcasts $\mathbf{x}$ [12, 13]. Let $\mathbf{H}_{1}\in\mathbb{R}^{n_{1}\times n_{t}}$ and $\mathbf{H}_{2}\in\mathbb{R}^{n_{2}\times n_{t}}$ be the channel matrices corresponding to user $1$ and user $2$. Then, the received signals at user $1$ and user $2$, respectively, can be represented as

in which $\mathbf{z}_{1}\in\mathbb{R}^{n_{1}\times 1}$ and $\mathbf{z}_{2}\in\mathbb{R}^{n_{2}\times 1}$ are two independent identically distributed (i.i.d) Gaussian noise vectors with mean zero and identity covariance matrices.

In this section, we build a supervised deep learning model to determine suitable covariance matrices for (2). Put differently, we describe signaling design (precoding and power allocation) for secure transmission over the MIMO-NOMA channel. We present a DNN structure that includes data generation, training methodology, DNN structure, and hyper-parameters.

We next evaluate the performance of the proposed DNN-based covariance matrix design (precoding and power allocation) in different antenna settings. A large dataset was used to have the network generalize for many different channels, and cross-validation data was used to prevent over-fitting. The test dataset was based on 1,000 channel matrices $\mathbf{H}_{1}$ and $\mathbf{H}_{2}$ whose elements were generated randomly based on $\mathcal{N}(0,1)$.

The performance of the proposed solution can be evaluated in different ways. We may find the mean square error (MSE) of elements $\mathbf{Q}_{1}$ and $\mathbf{Q}_{2}$ provided by the DNN and those obtained from a traditional capacity-approaching. The lower the MSEs, the better the regression. Alternatively, we can substitute $\mathbf{Q}_{1}$ and $\mathbf{Q}_{2}$ to (2) and compare the secure rate region achieved by the DNN with analytical methods like GSVD and maximum achievable rates (capacity) for this channel. We also compare the performance of the DL and traditional iterative methods in terms of computation time.

Figures 4 and 4 illustrate the accuracy of the proposed DNN model compared to GSVD and capacity region. The rate region plots were made for $n_{t}=2$ and $n_{t}=3$. The users have a single antenna, and the BS power is 10 Watts. Our DNN is able to find covariance matrices for any given $\alpha\in[0,1]$. To demonstrate this, rate region plots were made. These plots show the different rates simultaneously achieved by user 1 and user 2 as the power splitting factor $\alpha$ is changed. In order to be accurate at different power splitting factors, eleven networks were trained, each at a different alpha starting from $\alpha=0$ with a step size of 0.1. This produces a smooth, piecewise linear secure rate region curve and proves that the neural network can be generalized for any $\alpha$. It is important to note that different values of $\alpha$ correspond to different services and result in very different covariance matrices. For example, $\alpha=0$ implies $\mathbf{Q}_{1}=\mathbf{0}$ which gives $R_{1}=0$, i.e., security is important only for user 2, whereas $\alpha=1$ has the reverse implication. That being said, we choose the value of $\alpha$ and the associated DNN based on the users’ quality of service. Averaging over all $\alpha$s, the DNN achieves %98.9 and %97.7 of the capacity rates for $n_{t}=2$ and $n_{t}=3$, respectively. We have used the same regularization parameters on all networks. A finer tune would increase this accuracy.

As can be seen in Fig. 4 and 4, the DNN highly outperforms analytical methods like GSVD in terms of rate region. Further, the proposed method largely outperforms existing iterative solutions [5, 6] in terms of computation time. Table II shows the input signaling design time gain that we obtain with the DNN. It is several times faster than the rotation algorithm and AOWF. All algorithms are tested on the same machine in Matlab. The time difference becomes much larger as $n_{t}$ increases. In short, the DNN is able to get very close to the capacity region of the MIMO-NOMA and achieves this much faster, and thus, it is a viable solution.

A novel deep learning assisted covariance matrix design for the two-user MIMO-NOMA with confidential messages has been developed, trained, and tested in this paper. The proposed DNN, which is used to approximate the capacity region of this channel, is able to achieve nearly perfect accuracy for maximum secure rates for this channel. Remarkably, using the DNN for signaling design significantly reduces the solution time versus existing iteratively solutions and brings this time low enough that it can be used in practice. It also significantly outperforms GSVD precoding in achievable secure rates.