# Uplink Transmit Power Control for Single-Carrier Grouped FDMA with Iterative Multiuser Detection
## 2 System Model
### SNR-Variance Density Evolution  
In this study, a semi-analytic method called SNR-variance density evolution is used to evaluate the SNR of users during the IMD [7; 27; 28]. With SNR-variance density evolution, the process of the IMD is modeled by using two local processors; an equalizer and a decoder (Figure 3 in [7]). The equalizer outputs an SNR by considering the variance from the output of the decoder as an input parameter and conversely, the decoder outputs a variance by using the SNR from the equalizer output as an input value. The SNR-variance density evolution models the input-output relation of each local processor as a function that can be numerically obtained through simulation.  
Let \(\hat{x}_{u,m}(n)\) be the time-domain equalized symbol at time \(n\), i.e., \(\hat{\mathbf{x}}_{u,m}=\mathbf{F}_{K}^{-1}\hat{\mathbf{x}}_{u,m}\) with \(\hat{\mathbf{x}}_{u,m}\triangleq[\hat{x}_{u,m}(1)\ \hat{x}_{u,m}(2)\cdots\hat{x}_{u,m}(K)]^{T}\). Then, from [7], the SNR of \(\hat{x}_{u,m}(n)\) is given by  
\[\Gamma_{\hat{x}_{u,m}(n)}=\frac{\mu_{u,m}^{2}}{\mu_{u,m}-\
u_{u,m}\mu_{u,m}^{2}}. \tag{13}\]  
As \(\Gamma_{\hat{x}_{u,m}(n)}\) is invariant with respect to \(n\), we use the notation: \(\Gamma_{u,m}\triangleq\Gamma_{\hat{x}_{u,m}(n)}\). Moreover, as the variance is calculated based on the _a priori_ log-likelihood ratios from the MAP decoder [7], the variance is given by a function of SNR, which is denoted as \(\
u_{u,m}(n)=f(\Gamma_{u,m})\). As \(\Gamma_{u,m}\) is invariant with respect to \(n\), we have  
\[\
u_{u,m}=f(\Gamma_{u,m}). \tag{14}\]  
Similarly, the bit error rate (BER) performance is also expressed as a function of SNR as follows:  
\[BER=h(\Gamma_{u,m}). \tag{15}\]  
Because the functions \(f(\cdot)\) and \(h(\cdot)\) do not have closed-form expressions, they are generally estimated by using Monte Carlo simulation. That is, they should be numerically obtained offline by simulating the decoder performance over an AWGN channel for quantized values of SNR; one may construct a lookup table for the quantized values in the domain of \(f(\cdot)\) and \(h(\cdot)\), or use a curve-fitting method [27; 28]. Using (13), (14), and the numerically estimated \(f(\cdot)\), the SNR-variance density evolution updates the SINRs recursively at the \(i\)th iteration as  
\[\Gamma_{u,m}^{(i+1)}=\frac{\mu_{u,m}^{2}}{\mu_{u,m}-f(\Gamma_{u,m}^{i})\mu_{u, m}^{2}}. \tag{16}\]  
Then, the variance is updated as  
\[\
u_{u,m}^{(i+1)}=f(\Gamma_{u,m}^{(i+1)}) \tag{17}\]  
such that we converge on both \(\Gamma_{u,m}\) and \(\
u_{u,m}\) by repeating (16) and (17). The \(i\)th SINR, \(\Gamma_{u,m}^{(i)}\), analytically tracks the SINR of the signal of user \(u\) at chunk \(m\), obtained at the \(i\)th iteration of the IMD. As we do not have a priori information from the decoder at the initial iteration, we assume \(f(\Gamma_{u,m}^{(0)})=1\). If the IMD is finished, we can estimate the BER performance with (15) and, based on the SNR-variance function \(f(\cdot)\), we can accurately estimate the final SINRs of the users. (The noise part of \(\Gamma_{u,m}\) is defined in (13) to include both the multi-user interference term and AWGN. Thus, to emphasize the existence of the user interference, we call it SINR in the following sections).