# Energy-Efficient User Pairing for Downlink NOMA in Massive MIMO Networks
## 3 Energy Efficiency  
Considering the energy efficiency as the performance metric of interest, defined as the ratio between throughput in (bit/s/cell) and the total power consumption in (W/cell) as follows [19]:  
\[\mathrm{EE}=\frac{Throughput(bit/s/cell)}{power\, consumption(W/cell)}=\frac{R}{\gamma}. \tag{2}\]  
The power consumption \(\gamma\) for a downlink NOMA-based scheme consists of the effective transmitted power \(ETpower\) and the circuit power consumption \(Cpower\). The \(ETpower\) takes the efficiency of power amplifier \(\xi\) into consideration, such that  
\[\gamma=ETpower+Cpower, \tag{3}\]  
\[ETpower=\frac{1}{\xi}\sum_{u=1}^{U}q_{u}. \tag{4}\]  
where \(q_{u}\) is the BS transmitted power to user \(u\). The circuit power consumption \(Cpower\) is the sum of different components according to [19]. Additionally, \(Cpower\) represents the  
Figure 1: NOMA-based scheme network model.  
power consumed by encoding and decoding \(P_{cd/dec}\), power consumed by the signaling of backhaul \(P_{bh}\), and power consumed by digital signal processing \(P_{sp}\).  
\[Cpower=P_{fix}+P_{tc}+P_{ce}+P_{cod/dec}+P_{bh}+P_{sp}. \tag{5}\]  
where \(P_{fix}\) stands for the load independent power of infrastructure and power of control signaling, and \(P_{tc}\) represents the power consumed by transceiver chains, which is calculated according to the following equation:  
\[P_{tc}=AP_{BS}+P_{lo}+UP_{UE}. \tag{6}\]  
where \(P_{BS}\) is the power required to operate the circuit components of the BS, e.g., analog-to-digital converters, I/Q mixers, and modulators/demodulators. \(P_{lo}\) is the power consumed by the local oscillator. \(P_{UE}\) is the power consumed by circuit components in the user equipment. For the downlink, \(P_{cod}\) is the power in watt/bit/s required to encode throughput \(R\) in bit/s at BS, and \(P_{dec}\) is the power required to decode \(R\) at the user equipment.  
\[P_{cod/dec}=(P_{cod}+P_{dec})R. \tag{7}\]  
where \(P_{bh}\) is the power required by the load-dependent backhaul as follows:  
\[P_{bh}=P_{bt}.R. \tag{8}\]  
where \(P_{bt}\) is the power of the backhaul traffic in watt/bit/s. Additionally, \(P_{ce}\) stands for the power consumed in channel estimation:  
\[P_{ce}=\frac{3B}{ML_{bs}}(U)\Big{(}A\tau+A^{2}\Big{)}. \tag{9}\]  
We have \(B/M\) coherence blocks/s, where \(B\) is the communication bandwidth and \(M\) is the number of samples per coherence block. \(L_{bs}\) is the computational efficiency of BS measured in flops/s. The power consumed by digital signal processing \(P_{sp}\) is computed as  
\[P_{sp}=P_{sp-t}+P_{sp-c}^{DL}+P_{sp-pairing}^{DL}. \tag{10}\]  
\(P_{sp-t}\) stands for the power consumed in the downlink transmission.  
\[P_{sp-t}=\frac{3B}{ML_{bs}}AU\tau_{d}. \tag{11}\]  
where \(\tau_{d}=(1-U/M)\) is the part used for downlink data in each coherence interval since uplink data are not considered here. \(P_{sp-c}^{DL}\) is the power required for computing the beamforming vector.  
\[P_{sp-c}^{DL}=\frac{4B}{ML_{bs}}A\Big{(}\frac{U}{2}\Big{)}. \tag{12}\]  
For the NOMA-based scheme, only \(\frac{U}{2}\) beamforming vectors are computed. The power consumed in applying the Gale-Shapley algorithm for pairing, which is executed every coherence block, which means \(\frac{B}{M}\) times/s. The factor \(\Big{(}\frac{U}{2}\Big{)}^{2}\) is according to the computational complexity calculations.  
\[P_{sp-pairing}^{DL}=\frac{3B}{ML_{bs}}\bigg{(}\frac{U}{2}\bigg{)}^{2}. \tag{13}\]