RIS-Assisted MIMO Communication Systems: Model-based versus Autoencoder Approaches

Reconfigurable intelligent surface (RIS) has been recently emerged as a cost-effective paradigm that can tackle the complexity issues [1]. Each RIS device comprises a number of reflecting elements that can be controlled to manipulate the incoming signals in the desired manner. With a proper phase shift design, the reflected signal can be added constructively to enhance the signal strength [1]. The authors in [2] proposed a joint active and passive beamforming design in MISO multi-user system to minimize the total transmit power consumption. In [3], an alternating maximization approach has been applied to jointly optimize precoding matrix and RIS phase-shifts in order to boost energy efficiency of the system. Consider a RIS-assisted point-to-point MIMO system, the authors in [4] investigated the optimization of RIS phase-shift in OFDM MIMO system to enhance to capacity of the cascaded channel. However, in [4], a precoding matrix at the transmitter has not been considered. In [5], a low complexity design of precoding matrix and phase-shift of RIS has been proposed to maximize the spectral efficiency of a point-to-point MIMO system. Related to the communication reliability, there are only a few works. In [6], an RIS-based Vertical Bell Labs layered space-time (VBLAST) system has been proposed to enhance the bit error rate (BER) performance of the MIMO system. These previous works suffer from high computational complexity since the algorithms are implemented in iterative manners. The local solutions from the model-based approaches might be much worse than the global optimum, motivating for a better design.

Various applications of machine learning in wireless communications such as resource management, channel estimation, and signal detection [7] have been proposed to address intractable non-convex optimization problems and high-complexity issues. Related to the RIS system, the idea of applying machine learning in designing RIS phase-shift has been widely studied. In [8], the authors proposed a deep neural network (DNN) to learn the optimal RIS phase-shift from the users’ positions in order to maximize the through put of the multi-user MISO system. In [9], a framework which is comprised of two convolutional neural networks (CNNs) is utilized to jointly optimize the RIS phase-shift and precoding matrix in an unsupervised fashion with the goal of maximizing the sum rate of all users. With the same optimization objective with [9], in [10], a reinforcement learning-based framework is proposed to predict the optimal precoding matrix and RIS phase-shift with the given channel realizations. Although different machine learning approaches have been successfully applied in RIS systems, there is very limit work that focuses on improving signal detection performance and reliability of the system. In [11], the authors considered a jointly transmitter, receiver and RIS phase-shift design. In this framework, an autoencoder approach is proposed to enhance the BER performance of RIS-assited SISO system. Furthermore, the authors in [12] proposed a autoencoder approach-based design for RIS-assited MIMO systems to improve data detection performance. However, the works in [11] and [12] are limited since the autoencoders are trained based on one deterministic channel realization. This not only makes the autoencoder can work only for one specific channel but also does not really reflex the practical aspects of physical radio channels.¹¹1The previous works [11, 12] assumed sufficiently large coherent time, so their autoencoder architectures may not enable them to cope with the rapid changes of propagation channels in practice. Motivated by this drawback, we propose an autoencoder approach that can encode data through different channel realizations to enhance the BER.

Paper contributions: In this paper, for the model-based approach, we first study the two different MIMO communication system models that optimize radio resource and smartly control the propagation environments with the BER minimization as the utility metric. For the data-driven approach, we propose an autoencoder design for RIS-assisted MIMO communication systems, which consider the practical conditions of fading channels, and therefore close to the real applications than previous works. We jointly design the RIS phase-shift and the transceiver in order to reduce the BER at the receiver. Following the end-to-end framework, the transmitter, receiver, and the RIS device are modeled by individual deep neural networks with a unique loss function that minimizes the bit detection error on average. Our proposed data-driven approach establishes a unified framework to learn the entire system in order to enhance the communication reliability. Numerical results demonstrates the benefits of the data-driven approach that provides much better BER performance than the other model-based approaches. Moreover, we numerically show that the proposed framework can learn to encoder data that are robust to the fluctuation of propagation channels, and therefore retains a good BER performance under imperfect channel state information (CSI).

Notation: The upper and lower bold letters are utilized to denote the matrices and vectors. The superscript $(\cdot)^{T}$ and $(\cdot)^{H}$ are the regular and Hermitian transpose. $\mathbf{I}_{N}$ denotes an identity matrix of size $N\times N$ and $\mathrm{arg}(\cdot)$ is the argument of a complex number. $\|\cdot\|$ and $\|\cdot\|_{F}$ denote the Euclidean and Frobenius norm. The expectation of a random variable is denoted by $\mathbb{E}\{\cdot\}$, while $\mathcal{CN}(\cdot,\cdot)$ is a circularly symmetric Gaussian distribution. Finally, $\mathcal{O}(\cdot)$ is the big-$\mathcal{O}$ notation.

We consider a point-to-point MIMO system where a transmitter equipped with $N_{t}$ antennas transmits $N_{s}$ data streams to the receiver. The receiver has $N_{r}$ antennas to enhance the received signal strength. Furthermore, the system performance is boosted by the support of an RIS comprising $K$ passive reflecting elements as illustrated in Fig. 1. The reflection matrix $\boldsymbol{\Theta}\in\mathbb{C}^{K\times K}$ is formulated as

where $0\leq\beta_{k}\leq 1$ and $-\pi\leq\theta_{k}\leq\pi$ are the magnitude and phase produced by the $k$-th reflecting element. As a popular assumption because of the recent advances towards lossless metasurfaces, we assume a unit signal reflection, i.e., $\beta_{k}=1,\forall k$. In this paper, we assume that the direct link between the transmitter and the receiver is blocked due to large obstacles. The data is sent from the transmitter to the receiver through the RIS by the indirect link with the cascaded channels. The signal $\mathbf{s}$ is assumed to be $M$-QAM (quadrature amplitude modulation) with $\mathbb{E}\{\mathbf{s}\mathbf{s}^{H}\}=\mathbf{I}_{N_{t}}$. The modulated data message is precoded by a linear precoder $\mathbf{F}\in\mathbb{C}^{N_{t}\times N_{s}}$ with $\|\mathbf{F}\|^{2}_{F}=N_{s}$, and then passed through to the receiver via the RIS. The received signal, $\mathbf{y}\in\mathbb{C}^{N_{r}}$, at the receiver is

where $P$ is the total transmitted power. The channel between the transmitter and the RIS is denoted by $\mathbf{G}\in\mathbb{C}^{K\times N_{t}}$, while $\mathbf{H}\in\mathbb{C}^{K\times N_{r}}$ is the channel between the RIS and the receiver. Additionally, $\mathbf{n}\sim\mathcal{CN}(\mathbf{0},\sigma^{2}\mathbf{I}_{N_{r}})$ denotes additive white Gaussian noise (AWGN). We assume that all the channels are known to the transmitter with the help of feedback or channel reciprocity. From (2), the signal is then decoded as

where $\mathbf{Z}\in\mathbb{C}^{N_{s}\times N_{r}}$ represents the equalizer matrix. The active beamforming matrices, i.e., the precoding matrix $\mathbf{F}$ and the equalizer matrix $\mathbf{Z}$, together with the passive reflection matrix $\boldsymbol{\Phi}$ can be optimized for a given utility metric and practical constraints. However, the strong coupling among these optimization variables leads to a nontrivial procedure to solve the resource allocation problems optimally. In this paper, we separately solve subproblems $\boldsymbol{\Theta}$ and $\mathbf{F}$ to obtain a good solution in polynomial time by, first, maximizing the channel capacity defined for (2). Then, based on (3), the channel impairment is compensated by a proper selection of the equalizer matrix $\mathbf{Z}$.

This section describes the RIS-aided autoencoder-based framework that jointly learns the features of all the active and passive devices as illustrated in Fig. 2, where the transceiver and the RIS device are replaced by neural networks.

We evaluate the performance of our considered frameworks by a $4\times 2$ MIMO system transmitting $N_{s}=2$ data streams with equal transmitted power on each data stream $P=4$ [W]. The RIS device is equipped with eight phase-shift elements. The binary phase-shift keying (BPSK) modulation/demodulation are used at the transmitter and the receiver, respectively. In order for the system to train our deep neural networks, $200000$ different data symbols along with $200000$ channel realizations with $\sigma_{e}=0.1$ are used as data for the training phase. We define SNR as the ratio between the transmit signal power and the noise power. The SNR is fixed at $5\mathrm{dB}$, while it will be varied in the testing phase. The Adam optimizer is selected to train the neural networks. The hyper-parameters are chosen as: the number of epochs is $10$, the mini-bath size is $1000$, and the learning rate is $0.0002$. Conditioned on the phase-shift design, the following benchmarks are considered for comparison: $i)$ RIS-assisted Joint Transmitter and Receiver Design is the model-based approach presented in Sec. II and it is denoted as “Model-based” in the figures; $ii)$ RIS-assisted Autoencoder is the data-driven approach presented in Sec. III and it is denoted as “Autoencoder” in the figures.

In Fig. 3$a$, the cross entropy losses for all data streams are plotted as the function of the training iteration. As can be seen, loss functions of both data streams converge after a hundred training iterations. Thanks to the adaptive weight applied in loss function, the losses for both data streams are close to each other in every iteration. To evaluate our proposed model, in Fig. 3$b$, we plot the BER as a function of the SNR for all the considered benchmarks with the two different number of phase shifts. In addition, we also plot the performance of “Model-based” approach with random phase-shifts as a performance bound. As illustrated, thanks to the cooperation between transmitter, receiver and RIS, “Autoencoder” yields the lowest bit error rate in all SNR values. Moreover, when the number of RIS reflecting elements is increased, the performance of both models are greatly improved. This is very intuitive since “Model-based” can achieve higher channel capacity with more RIS reflecting elements. Even though higher capacity does not lead to optimal bit error rate, as mentioned in Remark 1, higher capacity is expected to attain a higher BER performance. For our model, given the increase in the number of phase-shifts, the end-to-end framework can encode and decode data in a more flexible way to combat noise effect and reduce error rate in decoding data. Additionally, as listed in Table. II, our model is dozens of times faster than model-based approaches with 200 iterations which shows its complexity efficiency.

To illustrate the robustness of our model to imperfect CSI, we test the proposed model with different channel error variance values with $K=16,32$ as in Fig. 3$c$. Surprisingly, in both setups the performance of “Autoencoder” remains almost unchanged with different channel error variance. These results show that “Autoencoder” is very robust to imperfect CSI. This robustness comes from the fact that “Autoencoder” is trained with imperfect CSI. Moreover, since various channel realizations are utilized in training process, the encoder can learn to encode data in a way that is robust to various channel conditions. Hence, the effect of imperfect CSI can be reduced. From the results, we can conclude that our proposed model can guarantee a good performance when the perfect CSI is not available at the transmitter.

In this paper, an autoencoder approach for RIS-based MIMO system has been presented to enhance to bit error rate performance of the system. We replaced the transceiver architecture and the RIS model with three FCNN models and trained them jointly with the objective of minimizing the BER of estimated signal at the receiver. By utilizing the BPSK modulation scheme for two independent data streams, the performance of the proposed framework has been compared with the conventional RIS designs in terms of bit error rate performance. Due to the cooperation between the transmitter, receiver and the RIS operation, our framework showed the superior improvements in detecting signals at the receiver.