# PCONet: A Trainable Feedback Scheme of Precoder for the Uplink Multi-User MIMO Systems
## 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 precodingscheme 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 3 rd generation partnership project (3 GPP) [15] provides a dedicated specification for the precoding matrix indexes (PMIs). Consequently, this protocol scheme has limited versatility, quantity, and accuracy performance.