# 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  
From (7) and (9), we have  
\[\mu_{u,m}=\frac{1}{K}\sum_{k=1}^{K}\frac{L_{u}P_{u,m}|\Lambda_{u,m}(k)|^{2}}{ \sum_{u^{\prime}\in\mathcal{U}_{m}}v_{u^{\prime},m}L_{u^{\prime}}P_{u^{\prime},m}|\Lambda_{u^{\prime},m}(k)|^{2}+\sigma_{N}^{2}}. \tag{23}\]  
Let \(S(k)\) and \(T(k)\) be respectively defined as  
\[S(k)\triangleq L_{u}P_{u,m}|\Lambda_{u,m}(k)|^{2},\ \ T(k)\triangleq\sum_{u^{ \prime}\in\mathcal{U}_{m}}v_{u^{\prime},m}L_{u^{\prime}}P_{u^{\prime},m}| \Lambda_{u^{\prime},m}(k)|^{2}+\sigma_{N}^{2}. \tag{24}\]  
For analytical tractability, we consider the following approximation:  
\[\mu_{u,m}=\frac{1}{K}\sum_{k=1}^{K}\frac{S(k)}{T(k)}\approx\frac{\frac{1}{K} \sum_{k=1}^{K}S(k)}{\frac{1}{K}\sum_{k=1}^{K}T(k)}, \tag{25}\]  
where the approximation error decreases if the difference among the values of \(S(k),k=1,\cdots,K\), and the difference among the values of \(T(k),k=1,\cdots,K\), decrease. As the frequency correlation among the subcarriers in each chunk increases with the decrease in \(K\), the approximation error of (25) decreases as \(K\) decreases. To validate the accuracy of this approximation, the root-mean-square errors (RMSEs) are depicted in Section 4 for moderate values of \(K\); the RMSEs are generally small and decrease with a decrease in \(K\). For simplicity, let \(\text{am}[A]\) denote the arithmetic mean of \(A(j),j=1,\cdots,K\), such that  
\[\text{am}[A]\triangleq\frac{1}{K}\sum_{j=1}^{K}A(j). \tag{26}\]  
Then,  
\[\text{am}[S] =L_{u}P_{u,m}\cdot\text{am}[|\Lambda_{u,m}|^{2}], \tag{27}\] \[\text{am}[T] =\sum_{u^{\prime}\in\mathcal{U}_{m}}v_{u^{\prime},m}L_{u^{\prime} }P_{u^{\prime},m}\cdot\text{am}[|\Lambda_{u^{\prime},m}|^{2}]+\sigma_{N}^{2}. \tag{28}\]  
Owing to the QoS constraint, which corresponds to a BER less than \(10^{-3}\), the signal power of each user should be considerably larger than the noise power \(\sigma_{N}^{2}\) when it is supposed to be detected. Thus, it is expected that  
\[\frac{L_{u}P_{u,m}\cdot\text{am}[|\Lambda_{u,m}|^{2}]}{\sigma_{N}^{2}}>\Gamma _{th}\gg 1, \tag{29}\]  
for each user \(u\) at chunk \(m\). That is, the noise power \(\sigma_{N}^{2}\) is relatively negligible during the IMD. Thus, from (25), (27), and (28), we have \[\mu_{u,m} \approx\frac{\text{am}[S]}{\text{am}[T]}=\frac{L_{u}P_{u,m}\cdot\text{am }[|\Lambda_{u,m}|^{2}]}{\sum_{u^{\prime}\in\mathcal{U}_{m}}v_{u^{\prime},m}L_{u^ {\prime}}P_{u^{\prime},m}\cdot\text{am}[|\Lambda_{u^{\prime},m}|^{2}]+\sigma_{N} ^{2}}\] \[\approx\frac{L_{u}P_{u,m}\cdot\text{am}[|\Lambda_{u,m}|^{2}]}{ \sum_{u^{\prime}\in\mathcal{U}_{m}}v_{u^{\prime},m}L_{u^{\prime}}P_{u^{\prime },m}\cdot\text{am}[|\Lambda_{u^{\prime},m}|^{2}]}=\frac{\zeta_{u,m}}{\sum_{u^{ \prime}\in\mathcal{U}_{m}}v_{u^{\prime},m}\overline{\zeta_{u^{\prime},m}}}, \tag{30}\]  
where \(\zeta_{u,m}\triangleq L_{u}P_{u,m}\cdot\text{am}[|\Lambda_{u,m}|^{2}]\). Substituting (30) into (16) yields  
\[\Gamma_{u,m}^{(i+1)}=\frac{\mu_{u,m}}{1-v_{u,m}^{(i)}\cdot\mu_{u,m}}\approx \frac{\frac{\zeta_{u,m}}{\sum_{u^{\prime}\in\mathcal{U}_{m}}v_{u^{\prime},m}^{ (i)}\zeta_{u^{\prime},m}}}{1-v_{u,m}^{(i)}\frac{\zeta_{u,m}}{\sum_{u^{\prime} \in\mathcal{U}_{m}}v_{u^{\prime},m}^{(i)}\zeta_{u^{\prime},m}}}=\frac{\zeta_{ u,m}}{\sum_{u^{\prime}\in\mathcal{U}_{m},u^{\prime}\
eq u}v_{u^{\prime},m}^{(i)} \zeta_{u^{\prime},m}}. \tag{31}\]  
Hereafter, we use this SINR approximation to propose a power control method. We define \(u_{k}\) as the user who has the \(k\)th largest \(\zeta_{u,m}\) in \(\mathcal{U}_{m}\). At the first iteration (\(i=1\)) of the IMD, the signal of \(u_{1}\) is detected first as \(u_{1}\) is expected to have the largest SINR. Subsequent values of SINR during the IMD are analyzed using SNR-variance density evolution (Section II-B). To start the density evolution at \(i=1\), the initial values of variances \(v_{u,m}^{(0)}\) are set to the maximum value 1; the variance decreases as the iteration proceeds because properly detected signals are successively canceled out by the iterative detector.