1. Introduction
In recent years, the multi-user multiple-input multiple-output (MU-MIMO) technology has offered great advantages over conventional point-to-point MIMO systems due to its improvement on the spectral and the energy efficiencies [
1,
2]. Specifically, the base station (BS) of a MU-MIMO system communicates with a large number of user terminals in the same time-frequency resource by configuring numerous antennas. Furthermore, multiple antennas bring large improvements in throughput and radiated energy efficiency through focusing energy into ever smaller regions of space [
3]. As a result, the MU-MIMO systems have become a fundamental and integral part of present and future generations of wireless networks. Digital beamforming and hybrid analog/digital beamforming are widely applied for inter-user interference reduction with the evolution and growth of 5G technical standards [
4].
Nowadays, the joint optimization of the transceiver has attracted increasing research activities as an effective interference management technique for the uplink MU-MIMO systems [
5]. Since the bottleneck of hardware cost and power consumption in the millimeter-wave mmWave Massive MIMO system will not appear in the uplink MU-MIMO scenarios, we adopt the digital precoding for its excellent performance in terms of sum rates. Some early works adopt the non-iterative [
6,
7] and iterative methods [
8,
9] to solve the highly non-convex problem of the joint transceiver optimization. The non-iterative precoding schemes are based on matrix decomposition, such as the singular value decomposition [
6] and the QR decomposition [
7], which cannot cope with the mismatch between the numbers of transmitting streams and the antennas. We note that the centralized iterative precoding scheme utilizing the method of Lagrange multipliers can solve the mismatch between the numbers of transmitting streams and the antennas. This iterative precoding scheme has the best end-to-end performance in the joint linear transceiver design but requires a certain feedback overhead [
9]. In another aspect of studies, the high complexity nonlinear detector is unfeasible. Linear detectors such as the zero-forcing (ZF) detector and the minimum mean-squared error (MMSE) detector are widely applied for practical systems.
The online implementation of the centralized solution with the optimal MMSE receiver requires the necessary information feedback for the user equipment (UE) to perform real-time updating, whereas the offline one requires the feedback of the optimal precoding matrix. With the subsequent development of beyond fifth-generation (B5G) or sixth-generation (6G) technologies, the dimensions of the precoding matrices scales with the number of antennas, which gradually become rather large [
10,
11]. The increasing feedback overhead of precoding becomes a challenge for the high spectral efficiency, ultra-low latency, and high reliability requirements of future B5G and 6G wireless networks. Therefore, the design of new feedback architecture that lowers the feedback overhead and maintains high performance is crucial to unlocking the full potential of the uplink MU-MIMO.
Limited feedback of precoding techniques has been intensively investigated to reduce the feedback overhead in the uplink wireless communication systems [
12,
13,
14], which mainly focuses on the design of a low-complexity codebook. An efficient precoding scheme is proposed in [
12], where the optimal precoder is chosen from a finite codebook known to both the transmitter and the receiver. The application of the Lloyd-Max algorithm in [
13] can be viewed as a vector quantization problem of codebook design. A three-dimensional MU-MIMO codebook is proposed in [
14] which adopts the signal-to-noise ratio (SNR) maximization criterion to select the optimal codebook. The 3rd generation partnership project (3GPP) [
15] provides a dedicated specification for the precoding matrix indexes (PMIs). Consequently, this protocol scheme has limited versatility, quantity, and accuracy performance. By contrast, the non-codebook-based centralized scheme has enormous throughput advantages compared to the codebook approaches but with relatively high cost of the feedback of precoding matrices.
Recently, as the success of deep learning reaches more fields, the neural-network-based auto-encoder has been recently applied to enhance the performance of MU-MIMO systems in [
16,
17,
18]. It is worth noting that the auto-encoder is well suited to tackling the vector compression problem because of its robustness to the unstable wireless channel conditions. The deep neural network (DNN) in [
19] takes the place of the conventional zero-forcing detection and offers near-optimal transmission quality with much less computational complexity than the optimal scheme. Motivated by the convolutional neural network (CNN)-based deep learning compression approaches of channel state information (CSI) [
20,
21,
22], we propose a novel compression and quantization network architecture named PCQNet for the joint transceiver optimization. The proposed PCQNet compresses the high-dimensional precoding matrices to the low-dimensional vectors. Considering that only bitstreams can be transmitted in a practical digital system, we also introduce a quantization module to convert the floating-point vector into bitstreams. Our proposed PCQNet can flexibly adjust the compression ratios compared with the codebook scheme in [
13]. These CNN-based methods considerably improve the compression performance of the precoding matrices. At the same time, the data-bearing bitstreams are directly produced during the offline training. Thus, the robustness of the network for practical deployment is effectively improved on the basis of the compression network in [
19]. Moreover, we extend the CNN-based compression network in [
20,
21] to the precoding design of the uplink MU-MIMO scenarios.
Multiple works of the transceiver optimization are limited to single-user scenarios or the particular MU multiple-input single-output (MISO) systems [
23,
24]. We focus on the more general MU-MIMO scenarios considering both the interference of multiple users and the interference from multiple data streams of the same UE. We aim to design a trainable compression architecture for the offline implementation of the centralized precoding with the MMSE receiver. In summary, the major contributions are summarized as follows:
We propose a CNN-based architecture named PCQNet to produce the data-bearing bitstreams for each UE to recover the precoding matrices. It can achieve near-optimal performance and further reduce the feedback overhead compared with the existing 3GPP codebook scheme in certain scenarios.
We develop a general trainable compression and quantization framework for the precoding matrices in the uplink MU-MIMO systems. The proposed PCQNet architecture as well as the Lloyd-Max quantization scheme can flexibly adjust the feedback overhead by training an auto-encoder.
The precoding matrices with different compression ratios (CRs) are evaluated on the performance of the centralized implementation with an optimal MMSE transceiver. As far as we know, the effect of the feedback accuracy on the performance has not been investigated before. Specifically, we explore the trade-off between the block error rates (BLER) and the CRs of the precoding matrices.
The remainder of this article is organized as follows:
Section 2 introduces the system model and the joint transceiver optimization.
Section 3 describes the network architecture and the training strategy of the PCQNet. Three baseline methods are also presented to provide a benchmark for our proposed PCQNet. In
Section 4, experimental evaluations and performance analysis are provided to demonstrate the efficiency of our trainable CNN-based PCQNet. Finally, the concluding statements are given in
Section 5.
Notations: Symbols for matrices (vectors) are denoted by boldface upper (lower) case letters. , and denote the real set, the complex set, and the positive integers, respectively. denotes the dimensional complex matrix space. , , , , and denote the conjugate transpose, the Frobenius norm, the Euclidean norm, the trace operation, and the expectation, respectively. is the identity matrix. is a complex Guassian vector with mean and variance .