# Uplink Transmit Power Control for Single-Carrier Grouped FDMA with Iterative Multiuser Detection
## 2 System Model
### Problem Formulation  
The main objective of this study is to design an appropriate resource allocation model for uplink SC-GFDMA. As each chunk in the GFDMA structure is shared by multiple users in a non-orthogonal manner, the corresponding users must be carefully determined. In this study, only users who are expected to satisfy pre-determined QoS are permitted to share a chunk. The ideal objective is to maximize the spectral efficiency in each chunk by determining an optimal user set and the optimal transmit powers of the users in the set. That is, we intend to solve the following problem:  
\[\underset{P_{u,m},L_{m}}{\operatorname{argmax}} \eta_{m}\] subject to \[BER_{u,m}<BER_{th},\;P_{u,m}\leq 1,\;\forall u\in\mathcal{U}_{m}, \tag{18}\]  
where \(\eta_{m}\) is the uplink spectral efficiency in chunk \(m\). In the GFDMA framework,  
\[\eta_{m}=\sum_{u\in\mathcal{U}_{m}}r_{u,m}(1-BER_{u,m})\quad(\text{bps/Hz}), \tag{19}\]  
where \(r_{u,m}\) is the single-user data rate of user \(u\) at chunk \(m\). In practical wireless communication, QoS in terms of BER generally satisfies \(BER_{th}<10^{-3}\). Hence, we approximate \(\eta_{m}\) as \[\eta_{m}\approx\sum_{u\in\mathcal{U}_{m}}r_{u,m} \tag{20}\]  
We further simplify the problem by assuming that all the users in each chunk use the same index of the modulation and coding scheme (MCS), i.e., \(r_{u,m}=r_{m}\); note that all the users scheduled in the same chunk use the same MCS, but users in different chunks can use different MCSs. Thus,  
\[\eta_{m}=r_{m}U_{m}. \tag{21}\]  
As BER is a monotonically decreasing function of SINR, we can obtain the value of the SINR, \(\Gamma_{th}\), corresponding to the value of \(BER_{th}\). Hence, with (21), the problem (18) can be modified to  
\[\underset{P_{u,m},U_{m}}{\operatorname{argmax}} U_{m}\] subject to \[\Gamma_{u,m}^{(I)}>\Gamma_{th},\ P_{u,m}\leq 1,\ \forall u\in \mathcal{U}_{m}, \tag{22}\]  
where \(\Gamma_{u,m}^{(I)}\) is the SINR at the \(I\)-th iteration of the IMD, obtained using the SNR-variance density evolution (16), and \(I\) denotes the maximum number of iterations; thus, \(\Gamma_{u,m}^{(I)}\) is the expected SINR of user \(u\) at chunk \(m\), at the final iteration of the IMD.  
The maximization problem of spectral efficiency is now modified to the maximization problem of the number of users who share a chunk while satisfying the maximum power constraint and the QoS constraint given in terms of SINR. Owing to the QoS constraint, specific scheduling is mandatory to check whether a user will be satisfied with the QoS; the scheduler should use SNR-variance density evolution to estimate the SINR of user \(u\) at chunk \(m\), that is achieved when the IMD is completed. Moreover, adjusting the transmit powers after determining the uplink user set \(\mathcal{U}_{m}\) barely increases the spectral efficiency achievable in each chunk, as the spectral efficiency is proportional to the number of users as shown in (21). It is desirable that the transmit powers of all the users should be adjusted before scheduling, for supporting the scheduler to select more QoS-satisfied users in each chunk. Thus, in this study, power control is performed for all the users before the scheduler finally determines the appropriate users to share each chunk (Figure 2). The proposed power control algorithm is derived independent of the scheduler, and it also functions as a pre-scheduler because it may allocate zero power to inappropriate users to exclude them from being candidates for further scheduling. As a scheduler, the pattern-aware scheduling method proposed in [23] is considered because it was demonstrated to achieve near-optimal performance for the given transmit power.