# Uplink Transmit Power Control for Single-Carrier Grouped FDMA with Iterative Multiuser Detection
## 2 System Model
### Transmitter and Receiver Principles for Data Plane  
Let \(\mathcal{U}_{m}\) be the set of users allocated to the chunk \(m\) (\(U_{m}=|\mathcal{U}_{m}|\)), and further let \(\mathbf{X}_{u,m}=[X_{u,m}(1)\ X_{u,m}(2)\ \cdots\ X_{u,m}(K)]^{T}\) be the frequency-domain transmit block for user \(u\) at chunk \(m\). As the SC-GFDMA shares each chunk to multiple users (Figure 1), the frequency-domain received signal is given by  
\[\mathbf{Y}=\sum_{m=1}^{M}\sum_{u\in\mathcal{U}_{m}}\sqrt{L_{u}P_{u,m}} \mathbf{\Lambda}_{u}\mathbf{L}_{m}\mathbf{X}_{u,m}+\mathbf{W}, \tag{1}\]  
where \(\mathbf{\Lambda}_{u}\) is an \(N\times N\) diagonal matrix consisting of the channel frequency responses of user \(u\), and \(\mathbf{W}\) is an \(N\times 1\) additive white Gaussian noise (AWGN) vector whose entries are i.i.d. complex Gaussian random variables with zero mean and variance \(\sigma_{N}^{2}\). The subcarrier mapping matrix \(\mathbf{L}_{m}\) for chunk \(m\) is defined as \(\mathbf{L}_{m}=[\mathbf{0}_{K\times(m-1)K}\ \mathbf{I}_{K\times K}\ \mathbf{0}_{K\times(N-mk)}]^{T}\), such that \([\mathbf{L}_{m}\mathbf{X}_{u,m}]_{(m-1)K+k}=X_{u,m}(k)\), for \(k=1,\cdots,K\), where \(N=MK\) is the total number of subcarriers in the frequency domain. \(P_{u,m}\) is the transmit power of user \(u\) at chunk \(m\) and \(L_{u}\) is the free-space pathloss from the BS to user \(u\). We assume that the pathloss has a simple far-field expression, i.e.,  
\[L_{u}=\left(\frac{c}{4\pi d_{u}f_{c}}\right)^{2}, \tag{2}\]  
where \(d_{u}\) is the distance between the BS and the user \(u\), \(c\) is the speed of light, and \(f_{c}\) is the center frequency. The transmit power \(P_{u,m}\) is constrained by the maximum transmit energy \(1W=30\) dBm such that  
\[P_{u,m}\leq 1, \tag{3}\]  
for \(u=1,\cdots,U\), and \(m=1,\cdots,M\). From (1), the frequency-domain received signal \(\mathbf{Y}_{m}\) at chunk \(m\) is expressed as  
\[\mathbf{Y}_{m}=\mathbf{L}_{m}^{T}\mathbf{Y} =\sum_{l=1}^{M}\sum_{u\in\mathcal{U}_{l}}\sqrt{L_{u}P_{u,l}} \mathbf{L}_{m}^{T}\mathbf{\Lambda}_{u}\mathbf{L}_{l}\mathbf{X}_{u,l}+\mathbf{ L}_{m}^{T}\mathbf{W} \tag{4}\] \[\overset{(a)}{=}\sum_{u\in\mathcal{U}_{m}}\sqrt{L_{u}P_{u,m}} \mathbf{\Lambda}_{u,m}\mathbf{X}_{u,m}+\mathbf{W}_{m}, \tag{5}\]  
where \(\mathbf{\Lambda}_{u,m}\triangleq\mathbf{L}_{m}^{T}\mathbf{\Lambda}_{u} \mathbf{L}_{m}\) is a \(K\times K\) diagonal matrix consisting of the channel frequency response of user \(u\) at chunk \(m\); the \((k,k)\)th entry of \(\mathbf{\Lambda}_{u,m}\) is denoted as \(\Lambda_{u,m}(k)\), for \(k=1,\cdots,K\). Furthermore, \(\mathbf{W}_{m}\triangleq\mathbf{L}_{m}^{T}\mathbf{W}\) is a \(K\times 1\) complex AWGN vector with the covariance matrix \(\sigma_{N}^{2}\mathbf{I}_{K}\), and (a) follows because \(\mathbf{L}_{m}=[\mathbf{0}_{K\times(m-1)K}\ \mathbf{I}_{K\times K}\ \mathbf{0}_{K\times(N-mk)}]^{T}\). Our SC-GFDMA structure uses IMD, which performs successive multiuser interference cancelation (MUIC) at each iteration. With MUIC, the frequency-domain received signal from user \(u\) is recovered by canceling MAI at each iteration. Thus, based on successive MUIC, the received signal of user \(u\) at chunk \(m\) can be computed as [7]  
\[\mathbf{Z}_{u,m}=\sqrt{L_{u}P_{u,m}}\mathbf{\Lambda}_{u,m}\mathbf{X}_{u,m}+ \sum_{u^{\prime}\in\mathcal{U}_{m},u^{\prime}\
eq u}\sqrt{L_{u^{\prime}}P_{u^ {\prime},m}}\mathbf{\Lambda}_{u^{\prime},m}(\mathbf{X}_{u^{\prime},m}-\mathbf{ \tilde{X}}_{u^{\prime},m})+\mathbf{W}_{m}, \tag{6}\]  
at each iteration of the IMD, where \(\mathbf{\tilde{X}}_{u,m}=\mathbf{F}_{k}\mathbf{\tilde{x}}_{u,m}\), with \(\mathbf{\tilde{X}}_{u,m}\triangleq[\mathbf{\tilde{X}}_{u,m}(1)\ \mathbf{\tilde{X}}_{u,m}(2)\cdots\mathbf{\tilde{X}}_{u,m}(K)]^{T}\), and \(\mathbf{\tilde{x}}_{u,m}=[\mathbf{\tilde{x}}_{u,m}(1)\ \mathbf{\tilde{x}}_{u,m}(2)\cdots\mathbf{ \tilde{x}}_{u,m}(K)]^{T}\); \(\mathbf{\tilde{x}}_{u,m}(n)\) is the mean of the transmitted symbols and is calculated at the decoder.