# Energy Efficient Power Allocation for the Uplink of Distributed Massive MIMO Systems
## 2 System Model  
Similarly to [26], we consider the uplink of the single-cell multi-user massive MIMO system as shown in Figure 1, where the BS is equipped with large-scale antennas located on a circle with radius \(r\). The BS equipped with \(N\) antennas serves \(K\) single-antenna UEs with the same time frequency resource. On one hand, the path-loss fading can be effectively reduced by the distributed antennas. On the other hand, it is more easy to connect the circularly located BS antennas by optical fibres. Similarly to [9], we assume that \(N\gg K\). All antennas connect with central processing unit through high-speed fiber. The channel fading keeps invariant in one time block, but changes independently from one block to another. We assume that the perfect channel state information is available at the BS by reciprocity in the time division duplex system. Based on the perfect CSI available at the BS, ZF linear technique is used to detect the data from the UEs.  
Assuming that the channel fading between the \(n\)-th antenna of BS to the \(k\)-th UE is  
\[g_{nk}=h_{nk}\sqrt{\beta_{nk}},\quad n=1,2,\ldots,N,\;k=1,2,\ldots,K, \tag{1}\]  
where \(h_{nk}\sim\mathcal{CN}(0,1)\) represents the fast fading, and \(\beta_{nk}\) represents the path-loss given as  
\[\beta_{nk}=l_{nk}^{-v}, \tag{2}\]  
where \(l_{nk}\) is the distance between the \(k\)-th UE and the \(n\)-th antenna of the BS, and \(v\) is the path-loss exponent. The channel vector between the BS and the \(k\)-th UE is  
\[\mathbf{g}_{k}=\Big{[}\begin{array}{ccc}g_{1k}&\cdots&g_{Nk}\end{array} \Big{]}^{\mathsf{T}}. \tag{3}\]  
The channel matrix between the BS and all \(K\) UEs is given as  
\[\mathbf{G}=\Big{[}\begin{array}{ccc}\mathbf{g}_{1}&\cdots&\mathbf{g}_{K} \end{array}\Big{]}. \tag{4}\]  
Figure 1: Massive multiple-input multiple-output (MIMO) system with circularly distributed antennas; \(K\) user equipments (UEs) simultaneously transmit signal to the base station (BS) with \(N\) antennas located on a circle with radius \(r\).  
Denote \(s_{k}\sim\mathcal{CN}(0,1)\) as the normalized data symbol from UE \(k\); i.e., \(\mathbb{E}\{\left\|\mathbf{x}\right\|^{2}\}=1\). \(P\) is the total transmit power of all the UEs, and \(\theta=(\theta_{1},\cdots,\theta_{k},\cdots,\theta_{K})\) is the power allocation vector. That is, the transmit power of the \(k\)-th UE is \(P\theta_{k}\). The \(N\times 1\) vector of the received signal at the BS is  
\[\mathbf{y}=\mathbf{G}\left[\sqrt{P\theta_{1}}x_{1},\cdots,\sqrt{P\theta_{k}}x _{k},\cdots,\sqrt{P\theta_{K}}x_{K}\right]^{T}+\mathbf{n}, \tag{5}\]  
where \(\mathbf{n}\) is the additive white Gaussian noise (AWGN) with variance \(\sigma^{2}\) (i.e., \(\mathbf{n}\sim\mathcal{CN}(\mathbf{0},\sigma^{2}\mathbf{I}_{M})\)). The linear detector matrix is given as \(\mathbf{A}=\left(\mathbf{G}^{H}\mathbf{G}\right)^{-1}\mathbf{G}^{H}\). After linear processing, the received signal can be given as  
\[\begin{split}\mathbf{r}&=\mathbf{A}\mathbf{y}= \left(\mathbf{G}^{H}\mathbf{G}\right)^{-1}\mathbf{G}^{H}\mathbf{y}\\ &=\left[\sqrt{P\theta_{1}}x_{1},\cdots,\sqrt{P\theta_{k}}x_{k}, \cdots,\sqrt{P\theta_{k}}x_{K}\right]^{T}+\left(\mathbf{G}^{H}\mathbf{G} \right)^{-1}\mathbf{G}^{H}\mathbf{n}.\end{split} \tag{6}\]  
The received signal from the \(k\)-th UE is given as  
\[r_{k}=\sqrt{P\theta_{k}}x_{k}+\mathbf{a}_{k}^{H}\mathbf{n}, \tag{7}\]  
where \(\mathbf{a}_{k}\) is the \(k\)-th column of matrix \(\mathbf{A}\). Note that the interference terms are eliminated by ZF detector.