# Energy Efficient Power Allocation for the Uplink of Distributed Massive MIMO Systems
## 1 Introduction
### Related Work  
Massive multiple-input multiple-output (MIMO) with a large antenna array is a promising technology to satisfy the high requirement of 5G. In a massive MIMO system, the base station (BS) equipped with a hundreds of antennas simultaneously provides services for tens of UEs in the same time-frequency resource [3]. Benefiting from the large-scale antenna array, the energy can be concentrated into narrow beams towards receivers. Compared with traditional MIMO systems, massive MIMO systems can achieve higher spectral efficiency (SE) and energy efficiency (EE) [4].  
A tight lower bound of achievable rate was derived for the multi-cell multi-user uplink MIMO systems in [5]. [PERSON] et al. derived exact uplink rate, the symbol error rate (SER), and the outage probability for the uplink of massive single-input multiple-output systems (SIMO) [6]. [PERSON] et al. derived a tight lower bound of area spectrum efficiency (ASE) for the downlink of a massive MIMO system with zero-forcing (ZF) precoding [7]. [PERSON] et al. derived an approximate expression of the SE for the downlink with maximum-radio transmission (MRT) precoding [8].  
To satisfy an achievable rate requirement, [PERSON] et al. proved that the transmit power of UEs can be made inversely proportional to the number of antennas when the channel state information (CSI) is available at the BS [9]. By contrast, the power consumption of the RF circuits grew linearly withthe antenna number. Therefore, the EE is an important metric of 5G networks, since it represents the ratio between the SE and the total power consumption. [PERSON] et al. proved that the EE is quasi-concave with respect to (w.r.t.) the number of antennas [10]. [PERSON] et al. analyze how to maximize the EE by optimizing the transmit power of UEs, active users, and antenna number [11]. A power allocation scheme was proposed to maximize the EE of the downlink massive MIMO system with MRT precoding [12]. [PERSON] et al. investigated power allocation, subcarrier allocation, and antenna allocation policies to improve the EE in orthogonal frequency division multiple access (OFDMA) downlink massive MIMO systems. To solve the NP-hard problem, an efficient iterative algorithm was proposed based on fractional programming [13]. Chunk-based resource allocation was proposed through maximizing the throughput under a total transmit power constraint for OFDMA systems [14; 15]. Base on [PERSON]'s method, an energy-efficient resource allocation algorithm was proposed for multi-cell OFDMA systems with limited backhaul capacity [16]. A low complexity distributed algorithm was proposed to solve the energy efficient user association problem in massive MIMO enabled heterogeneous networks [17]. [PERSON] et al. analyzed and optimized the EE of massive MIMO systems with non-ideal hardware [18].  
The achievable rates of cell-boundary users is relatively low due to the path loss. Distributed antenna systems (DAS) can mitigate the near-far problem by using distributed radio remote heads (RRHs) [19]. To map all the benefits of massive MIMO and DAS, the combination of them has recently been hotly studied. The average SE and the the cell edge SE were compared in downlink multi-cell DAS systems [20]. [PERSON] et al. derived closed-form expressions for the signal-to-noise ratio (SNR), the symbol error probability (SEP), and the achievable rate for distributed massive MIMO with zero forcing [21]. The SE of centralized and distributed massive MIMO systems were respectively analyzed and compared [22]. A closed-form optimal precoding scheme was derived to maximize the worst-case SNR for massive distributed MIMO systems [23]. [PERSON] et al. derived an approximate EE expression for both the idealistic and the realistic power consumption models of distributed massive MIMO systems [24]. [PERSON] et al. jointly optimized time allocation and energy beamforming to maximize the EE of distributed massive MIMO system with energy harvesting [25].