# 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
Thus, we determine the following criterion for efficient power control:  
\[S(k)\triangleq L_{u}P_{u,m}|\Lambda_{u,m}(k)|^{2},\ \ T(k)\triangleq\sum_{u^{ \prime}\in\mathcal{U}_{m}}v_{u^{\prime},m}L_{u^{\prime}}P_{u^{\prime},m}| \Lambda_{u^{\prime},m}(k)|^{2}+\sigma_{N}^{2}. \tag{24}\]9  
This inequality can be considered to be a condition for the successful signal detection of \(u_{1}\) before the signal detection of \(u_{2}\) at the second iteration. \[\mu_{u,m}=\frac{1}{K}\sum_{k=1}^{K}\frac{S(k)}{T(k)}\approx\frac{\frac{1}{K} \sum_{k=1}^{K}S(k)}{\frac{1}{K}\sum_{k=1}^{K}T(k)}, \tag{25}\]0 Frequency domain variance \(f(SINR)\) versus SINR. The inequality (41) can be rewritten by using (37) and (39) as follows:  
\[\mu_{u,m}=\frac{1}{K}\sum_{k=1}^{K}\frac{S(k)}{T(k)}\approx\frac{\frac{1}{K} \sum_{k=1}^{K}S(k)}{\frac{1}{K}\sum_{k=1}^{K}T(k)}, \tag{25}\]1 \[\mu_{u,m}=\frac{1}{K}\sum_{k=1}^{K}\frac{S(k)}{T(k)}\approx\frac{\frac{1}{K} \sum_{k=1}^{K}S(k)}{\frac{1}{K}\sum_{k=1}^{K}T(k)}, \tag{25}\]2  
and, from the second equality, we have  
\[\mu_{u,m}=\frac{1}{K}\sum_{k=1}^{K}\frac{S(k)}{T(k)}\approx\frac{\frac{1}{K} \sum_{k=1}^{K}S(k)}{\frac{1}{K}\sum_{k=1}^{K}T(k)}, \tag{25}\]3  
This inequality indicates that the remaining interference from \(u_{1}\) at \(i=2\) should be smaller than the sum of canceled interferences from users \(u_{3},\cdots,u_{L_{lm}}\) at \(i=1\). As we are focusing on the detection of \(u_{1}\) at the first iteration, we assume that the signals of users \(u_{4},\cdots,u_{L_{lm}}\) are barely detected at this iteration. This is a reasonable assumption as the remaining interferences from users \(u_{1}\), \(u_{2}\), and \(u_{3}\) are still large at this stage. Specifically, we assume that  
\[\mu_{u,m}=\frac{1}{K}\sum_{k=1}^{K}\frac{S(k)}{T(k)}\approx\frac{\frac{1}{K} \sum_{k=1}^{K}S(k)}{\frac{1}{K}\sum_{k=1}^{K}T(k)}, \tag{25}\]4  
Then, (43) can be simplified to  
\[\mu_{u,m}=\frac{1}{K}\sum_{k=1}^{K}\frac{S(k)}{T(k)}\approx\frac{\frac{1}{K} \sum_{k=1}^{K}S(k)}{\frac{1}{K}\sum_{k=1}^{K}T(k)}, \tag{25}\]5  
Please note that this condition is derived initially from our main criterion (41) for efficient power control. Now, we consider the case of \(k\geq 2\). By generalizing the criterion in (41), we consider the condition for the successful signal detections for users \(u_{1},\cdots,u_{k}\) (such that their signals were successfully canceled out) before the signal detection of \(u_{k+1}\) at the \((k+1)\)th iteration. In mathematical terms, it can be written as  
\[\mu_{u,m}=\frac{1}{K}\sum_{k=1}^{K}\frac{S(k)}{T(k)}\approx\frac{\frac{1}{K} \sum_{k=1}^{K}S(k)}{\frac{1}{K}\sum_{k=1}^{K}T(k)}, \tag{25}\]6  
That is, the decrease in MAI caused by \(k\)-times-iterated IMD should be larger than the total power of the signals of users, \(u_{1},\cdots,u_{k}\), as they were successfully canceled out before the signal detection of \(u_{k+1}\) at \(i=k+1\). Then, using formulas (41) to (45) as a basis condition and inductively assuming that the signals of \(u_{1},\cdots,u_{k}\) were successfully canceled out before \(i=k+1\), a strong mathematical induction can be applied to obtain the following inequality from (46):  
\[\mu_{u,m}=\frac{1}{K}\sum_{k=1}^{K}\frac{S(k)}{T(k)}\approx\frac{\frac{1}{K} \sum_{k=1}^{K}S(k)}{\frac{1}{K}\sum_{k=1}^{K}T(k)}, \tag{25}\]7  
Thus, (47) also indicates the successful detection of the signals of users \(u_{1},\cdots,u_{k}\) before the detection of \(u_{k+1}\) at \(i=k+1\), and we intend to propose a power control algorithm that achieves this condition for all \(k=1,\cdots,U_{m}-2\). If (47) is consecutively satisfied for all \(k=1,\cdots,U_{m}-2\), then the following detection of the last user, at the subsequent iteration, will be interference-free; as (47) is the condition for the detection of the \((k+1)\)th user, the case \(k=U_{m}-1\) corresponds to the detection of the last user.