## 1 Introduction
Another approach to minimize power consumption and hardware complexity is the antenna selection that can be applied in cell-free massive MIMO systems. Still, this is an undesirable solution due to the loss of the antenna gains, which indicates that the signals transmitted or received by the antennas in a specific direction suffer from weakness. Then, the power of the transmitted or received antennas is lower than that of hypothetical ideal antennas in a similar situation, as well as having insufficient data rates [20]. The authors of [21] proposed an antenna selection scheme in cell-free massive MIMO systems to assign each UE to a subset of APs with the best channel status. However, they proposed a single antenna for each UE without considering when the AP is equipped with a large number of antennas and the hybrid analog/digital approach. Furthermore, because a massive MIMO system can use a large number of receiving antennas, the number of switches necessary to link the antennas to the RF chains is large, and these switches can consume a large amount of power. This problem is overcome in [22] by introducing a new hybrid beamforming architecture for conventional massive MIMO systems whereby each RF chain is connected to a subset of antennas that contribute more to the desired power rather than the interference power. Further, they introduced three low-complexity algorithms compared with the exhaustive search approach to perform the selection process for the switches at the base station. In this work, we investigate the flexibility of including or excluding the subset of antennas in the signal combining design by extending the proposed algorithms in [22] from conventional multi-user mm-Wave MIMO systems to cell-free mm-Wave massive MIMO systems, especially when there are a large number of APs. Each AP has a large number of antennas. Then, we propose a novel antenna selection strategy based on the Hungarian matching method, which provides considerable power reduction while maintaining the system's spectral efficiency. Furthermore, we propose the channel quality for each AP to assign each RF chain to its suitable set of selected antennas. Based on that, the proposed matching scheme can improve system performance and provide a tradeoff between spectral and energy efficiencies and computational complexity. In addition, to the best of our knowledge, no other works consider the Hungarian matching algorithm [23] in the hybrid analog/digital approach for cell-free massive MIMO systems. The main contributions of this paper are summarized as follows:  
* We extend the work in [22], which has been proposed for conventional multi-user massive MIMO, to cell-free massive MIMO systems. We adapt the three low-complexity algorithms proposed in [22] to perform the selection process for the switches at the APs. * For all APs in the cell-free network, we propose an assignment optimization problem to accomplish matching between RF chains and several sets of selected antennas based on channel conditions. Then, we propose the Hungarian method to solve this optimization problem based on maximum weight matching in order to maximize energy efficiency. In contrast to [22], instead of assuming that all RF chains in the AP have the same fixed active switches, we exploit the advantages of the matching theory based on the Hungarian algorithm to assign each RF chain at each AP in the cell-free network to the optimal number of activated switches depending on AP channel condition in order to maximize energy efficiency. * Simulation results demonstrate the performance of the proposed antenna selection strategies under an extensive set of cell-free mm-wave massive MIMO scenarios. In particular, the number of APs, the number of antennas, and the number of users in the network are analysed in terms of energy efficiency. In addition, computational complexity of the proposed algorithms is studied in this work. _Notation:_ The \(n_{th}\) row and \(m_{th}\) column entry of matrix \(A\) is expressed as \(a_{nm}\).