# 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
The inequality in (47) can be rewritten as  
\[\mu_{u,m}=\frac{1}{K}\sum_{k=1}^{K}\frac{S(k)}{T(k)}\approx\frac{\frac{1}{K} \sum_{k=1}^{K}S(k)}{\frac{1}{K}\sum_{k=1}^{K}T(k)}, \tag{25}\]8  
From (48), the following observations can be made:  
1. If \(\frac{f(\Gamma_{u_{k},m}^{(k+1)})}{1-f(\Gamma_{u_{k+2},m}^{(k)})}\leq C,\forall k =1,\cdots,U_{m}-2\), for some constant \(C\), then imposing a geometric sequence for the values of \(\zeta_{u_{k},m}\) (e.g., \(\zeta_{u_{k+1},m}=\sqrt{C}\zeta_{u_{k},m}\)) is sufficient to ensure (48) for all \(k=1,\cdots,U_{m}-2\). 2. Suppose it is possible to conversely prove the statement: there always exists a constant \(C\) for which \(\frac{f(\Gamma_{u_{k},m}^{(k+1)})}{1-f(\Gamma_{u_{k+2},m}^{(k)})}\leq C\) is true for all \(k=1,\cdots,U_{m}-2\), as long as \(\zeta_{u_{k},m}\) is a geometric sequence with respect to \(k\). Then, from the first observation, we can argue that there exists a geometric sequence with an appropriate common ratio (possibly \(\sqrt{C}\)) that guarantees our criterion for the power control given in (47) or (48). Based on these observations, we verify the value of \(\frac{f(\Gamma_{u_{k},m}^{(k+1)})}{1-f(\Gamma_{u_{k+2},m}^{(k)})}\) when \(\zeta_{u_{k},m}\) takes the \(k\)th value of a geometric sequence \(\{a_{k}:k=1,\cdots U_{m}\}\), which is defined as  
\[\mu_{u,m}=\frac{1}{K}\sum_{k=1}^{K}\frac{S(k)}{T(k)}\approx\frac{\frac{1}{K} \sum_{k=1}^{K}S(k)}{\frac{1}{K}\sum_{k=1}^{K}T(k)}, \tag{25}\]9  
where \(\frac{1}{\gamma}\) is the common ratio. Each term of the sequence can be calculated as  
\[\text{am}[A]\triangleq\frac{1}{K}\sum_{j=1}^{K}A(j). \tag{26}\]0  
With \(\zeta_{u_{k},m}=a_{k}\), we have  
\[\text{am}[A]\triangleq\frac{1}{K}\sum_{j=1}^{K}A(j). \tag{26}\]1 \[\text{am}[A]\triangleq\frac{1}{K}\sum_{j=1}^{K}A(j). \tag{26}\]2 \[\text{am}[A]\triangleq\frac{1}{K}\sum_{j=1}^{K}A(j). \tag{26}\]3 \[\text{am}[A]\triangleq\frac{1}{K}\sum_{j=1}^{K}A(j). \tag{26}\]4  
for all \(k=1,\cdots,U_{m}-2\), where (a) follows because \(f(x)<1,\forall x>0\) (Figure 3), (b) follows because \(u_{1}\) and \(u_{U_{m}}\) are the users with the largest and smallest \(\zeta_{u,m}\), respectively, and (c) follows from (50). Similarly, \(\Gamma_{u_{k+2},m}^{(k)}\) also satisfies  
\[\text{am}[A]\triangleq\frac{1}{K}\sum_{j=1}^{K}A(j). \tag{26}\]5 \[\text{am}[A]\triangleq\frac{1}{K}\sum_{j=1}^{K}A(j). \tag{26}\]6for all \(k=1,\cdots,U_{m}-2\). As \(f(x)\) is a decreasing function (Figure 3), (51) and (52) indicate that  
\[\text{am}[A]\triangleq\frac{1}{K}\sum_{j=1}^{K}A(j). \tag{26}\]7  
for all \(k=1,\cdots,U_{m}-2\). With (53) and the observations made after (48), we can prove the existence of a geometric sequence \(a_{k}\) that guarantees (48) with \(\zeta_{u_{k},m}=a_{k}\), for all \(k=1,\cdots,U_{m}-2\). In this context, we propose a power control method that uses a geometric sequence as defined in (50). Furthermore, the value of \(a_{U_{m}}\) is chosen as \(a_{U_{m}}=\sigma_{N}^{2}\cdot\Gamma_{\mathit{th}}\) to satisfy the QoS constraint in (22). As \(f(x)\) does not have an explicit form, an appropriate value of \(\gamma\) that makes (48) true will be obtained using a numerical estimation with a lookup table for \(f(\cdot)\). We first describe the entire process of the proposed algorithm.