# Uplink Transmit Power Control for Single-Carrier Grouped FDMA with Iterative Multiuser Detection
## 3 Proposed Power Control for SC-GFDMA
### SINR Analysis for Power Control
Hence, from (31), the SINR of \(u_{1}\) at \(i=1\) is given by  
\[S(k)\triangleq L_{u}P_{u,m}|\Lambda_{u,m}(k)|^{2},\ \ T(k)\triangleq\sum_{u^{ \prime}\in\mathcal{U}_{m}}v_{u^{\prime},m}L_{u^{\prime}}P_{u^{\prime},m}| \Lambda_{u^{\prime},m}(k)|^{2}+\sigma_{N}^{2}. \tag{24}\]0 \[S(k)\triangleq L_{u}P_{u,m}|\Lambda_{u,m}(k)|^{2},\ \ T(k)\triangleq\sum_{u^{ \prime}\in\mathcal{U}_{m}}v_{u^{\prime},m}L_{u^{\prime}}P_{u^{\prime},m}| \Lambda_{u^{\prime},m}(k)|^{2}+\sigma_{N}^{2}. \tag{24}\]1  
The SINRs of users \(u_{2}\) and \(u_{3}\) at \(i=1\) are sequentially expressed as  
\[S(k)\triangleq L_{u}P_{u,m}|\Lambda_{u,m}(k)|^{2},\ \ T(k)\triangleq\sum_{u^{ \prime}\in\mathcal{U}_{m}}v_{u^{\prime},m}L_{u^{\prime}}P_{u^{\prime},m}| \Lambda_{u^{\prime},m}(k)|^{2}+\sigma_{N}^{2}. \tag{24}\]2 \[S(k)\triangleq L_{u}P_{u,m}|\Lambda_{u,m}(k)|^{2},\ \ T(k)\triangleq\sum_{u^{ \prime}\in\mathcal{U}_{m}}v_{u^{\prime},m}L_{u^{\prime}}P_{u^{\prime},m}| \Lambda_{u^{\prime},m}(k)|^{2}+\sigma_{N}^{2}. \tag{24}\]3  
where in (34), we have updated \(v_{u_{1},m}^{(0)}\) to \(v_{u_{1},m}^{(1)}\) and, in (35), we have updated \(v_{u_{1},m}^{(0)}\) and \(v_{u_{2},m}^{(0)}\) to \(v_{u_{1},m}^{(1)}\) and \(v_{u_{2},m}^{(1)}\), respectively, by using (17) with the corresponding SINR values at \(i=1\). During SNR-variance density evolution, the variance \(v_{u_{1},m}=f\left(\Gamma_{u_{1},m}\right)\) decreases exponentially as the SINR \(\Gamma_{u_{1},m}\) increases (Figure 3); hence, if the SINR of \(u_{1}\) is sufficiently large for reliable detection, the interference caused by \(u_{1}\) is considerably reduced during the detection of the signals of other users. The simple idea of the proposed power control scheme is that the interference from the signal of user \(u_{1}\) should be successfully canceled out before the detection of the signal of user \(u_{2}\) at the second iteration, \(i=2\). Generally, the signal of \(u_{j}\) for all \(j\leq k\) should be sequentially canceled out before the detection of the signal of \(u_{k+1}\) at \(i=k+1\). To achieve this, the proposed method will properly control the signal power of each user. To derive a specific condition, we examine the MAI caused by \(u_{1}\) toward \(u_{2}\) at \(i=2\). By proceeding with density evolution, the SINR of user \(u_{2}\) at \(i=2\) is given by  
\[S(k)\triangleq L_{u}P_{u,m}|\Lambda_{u,m}(k)|^{2},\ \ T(k)\triangleq\sum_{u^{ \prime}\in\mathcal{U}_{m}}v_{u^{\prime},m}L_{u^{\prime}}P_{u^{\prime},m}| \Lambda_{u^{\prime},m}(k)|^{2}+\sigma_{N}^{2}. \tag{24}\]4  
The denominators of (34) and (36) represent the MAIs for the detection of \(u_{2}\) at the first and second iterations, respectively. We denote the MAI for the detection of \(u_{k}\) at the \(i\)th iteration as \(\text{MAI}_{u_{k},m}^{(i)}\). Then, we have  
\[S(k)\triangleq L_{u}P_{u,m}|\Lambda_{u,m}(k)|^{2},\ \ T(k)\triangleq\sum_{u^{ \prime}\in\mathcal{U}_{m}}v_{u^{\prime},m}L_{u^{\prime}}P_{u^{\prime},m}| \Lambda_{u^{\prime},m}(k)|^{2}+\sigma_{N}^{2}. \tag{24}\]5 \[S(k)\triangleq L_{u}P_{u,m}|\Lambda_{u,m}(k)|^{2},\ \ T(k)\triangleq\sum_{u^{ \prime}\in\mathcal{U}_{m}}v_{u^{\prime},m}L_{u^{\prime}}P_{u^{\prime},m}| \Lambda_{u^{\prime},m}(k)|^{2}+\sigma_{N}^{2}. \tag{24}\]6 (38) \[S(k)\triangleq L_{u}P_{u,m}|\Lambda_{u,m}(k)|^{2},\ \ T(k)\triangleq\sum_{u^{ \prime}\in\mathcal{U}_{m}}v_{u^{\prime},m}L_{u^{\prime}}P_{u^{\prime},m}| \Lambda_{u^{\prime},m}(k)|^{2}+\sigma_{N}^{2}. \tag{24}\]7  
and the MAI decreases as the IMD proceeds as follows:  
\[S(k)\triangleq L_{u}P_{u,m}|\Lambda_{u,m}(k)|^{2},\ \ T(k)\triangleq\sum_{u^{ \prime}\in\mathcal{U}_{m}}v_{u^{\prime},m}L_{u^{\prime}}P_{u^{\prime},m}| \Lambda_{u^{\prime},m}(k)|^{2}+\sigma_{N}^{2}. \tag{24}\]8  
If we assume that the interference from the signal of \(u_{1}\) is fully canceled out at \(i=2\), then the decrease in MAI caused by twice-iterated IMD should be larger than the signal power \(\zeta_{u_{1},m}\), as it is canceled out from the detection.