## 4 Energy Efficiency
### Power Consumption by Signal Detection  
We next analyze the complexity and the resulting circuit power (signal processing-related) consumption for different signal detectors. We first consider the MMB-LR-SIC detector. With MMB-LR-SIC, the overall complexity arises from the generation of multiple LR-SIC detectors per channel realization and the calculations of multiple signal estimates per channel use. The former is performed only once for a coherent interval where \(\mathbf{H}\) is fixed, while the latter is conducted for each received symbol in (6). The power consumption of the MMB-LR-SIC at the receiver is estimated in [3; 12] as  
\[\mathrm{P}_{\mathrm{SD}}^{\mathrm{MMB-LR-SIC}}=B\frac{\mathrm{C}_{\mathrm{ detection}}^{\mathrm{MMB-LR-SIC}}}{\mathrm{LBS}}+\lambda\frac{\mathrm{C}_{\mathrm{Common}}^{ \mathrm{MMB-LR-SIC}}}{\mathrm{LBS}} \tag{35}\]  
where \(B\) is the bandwidth, \(\mathrm{C}_{\mathrm{detection}}^{\mathrm{MMB-LR-SIC}}\) is the number of flops required for detection in one channel use, assuming that the detector coefficients are already computed and stored; \(\mathrm{C}_{\mathrm{Common}}^{\mathrm{MMB-LR-SIC}}\) represents the number of the calculations required to obtain to perform the QR decomposition, CLLL, filter calculations, etc, which are shared in a coherent block. \(L_{BS}\) represents the computational efficiency on the BS in [45]. The coefficient \(\lambda\) characterizes the number of coherence blocks (i.e., the number of times that the MMB-LR-SIC receiver is updated) in a second, which can be computed as  
\[\lambda=\frac{B}{B_{c}T_{c}}. \tag{36}\]  
In the following complexity analysis, we ignore lower order terms for simplicity, which do not contribute significantly to the overall complexity. Let one flop refer to one operation of real-valued numbers, and thus each complex-valued multiplication and addition costs 6 and 2 flops, respectively. All complexity calculations are based on [46]. The QR decomposition of \(\mathbf{H}\) costs  
\[\mathrm{C}_{\mathrm{QR}}=8K^{2}\left(N-\frac{K}{3}\right) \tag{37}\]  
flops. The computational complexity \(\mathrm{C}_{\mathrm{clll}}\) of the CLLL varies with channel realization and can be analyzed following [25]. In this paper, we track \(\mathrm{C}_{\mathrm{clll}}\) through numerically by summing up the complexities of the basic operations in the CLLL algorithm. Computing \(\mathbf{T}^{-1}\) costs about  
\[\mathrm{C}_{\mathbf{T}^{-1}}=8(K^{3}+K^{2}) \tag{38}\]  
flops. Let \(t=K-i+1\) be the number of columns of the extended channel matrix \(\widetilde{\mathbf{G}}_{i}^{e}\) in (23) after the interference cancellation. Then calculating (27) costs \(8t^{2}K+4t^{3}\) flops and computing the filter \(\mathbf{w}_{i}^{e}\) costs about \(8 Kt\) flops. Therefore, finding the \(K\) MMSE-SIC filters costs  
\[\mathrm{C}_{\mathrm{filter}}=8K^{3}+8K+\sum_{t=1}^{K}(4t^{3}+8t^{2}+12 Kt) \tag{39}\]flops, assuming that explicit inversion of \((\widetilde{\mathbf{G}}_{i}^{\varepsilon}(\widetilde{\mathbf{G}}_{i}^{ \varepsilon})^{H})\) is conducted. As such, the MMSE-based ordering can be achieved with no extra cost. The detector construction part requires a complexity of  
\[\text{C}_{\text{common}}^{\text{MMB-LR-SIC}}=\text{C}_{\text{QR}}+\text{C}_{ \text{clll}}+\text{C}_{\text{T}^{-1}}+L\text{C}_{\text{filter}}. \tag{40}\]  
In order to detect the transmitted symbol, we first calculate \(\mathbf{Q}^{H}\mathbf{y}_{q}\), which costs about \(8 KN\) flops. In the LR domain, detecting one transmitted symbol costs about \(16K\) flops and cancelling the interference due to a single symbol costs another \(16K\) flops. After obtaining the a symbol estimate in the LR domain, computing its ML metric costs \(8K^{2}\) flops. Finally a single transmit symbol estimate is chose and converted to the original domain, with \(8K^{2}\) flops. Based on the above analysis,  
\[\text{C}_{\text{detection}}^{\text{MMB-LR-SIC}}\approx 8 KN+40 LK^{2}+8K^{2}, \tag{41}\]  
which increases linearly with the number of branches \(L\). The LR-SIC detector is a special case of MMB-LR-SIC with \(L=1\).