## 3 Single-Cell Multicast Transmission Scheme with Partial Channel Quality Feedback  
As in Reference [14], the two multicast channels are assigned for a single-cell multicast transmission scheme where some of the users in \(\Omega\) receive data only through a basic multicast channel (BMCH),  
Figure 1: An exemplary scenario for single-cell multicast transmission with a supplementary multicast channel (\(\Omega=\{1,2,\ldots,6\}\), \(\Omega_{B}=\{1,4\}\)). Now, given \(|\Omega_{B}|\), the optimal resource allocation problem is formulated to minimize \(B_{B}+B_{S}\), while the users in \(\Omega_{B}\) can obtain a channel capacity higher than or equal to \(R_{T}\) only with the BMCH (see (6b) below) and the other users in \(\Omega\backslash\Omega_{B}\) can do it with both channels (see (6c) below). Then the problem is expressed as  
\[\min(B_{B}+B_{S})\] (6a) subject to \[B_{B}log_{2}(1+\gamma_{i})\geq R_{T},\forall i\in\Omega_{B} \tag{6b}\] \[B_{B}log_{2}\Big{(}1+\gamma_{j}\Big{)}+B_{S}log_{2}\Big{(}1+\gamma_{j}^{S} \Big{)}\geq R_{T},\ \forall j\in\Omega\backslash\Omega_{B}\] (6c) \[B_{S}\geq 0 \tag{6d}\]  
Let us denote \(f_{j}(B_{B})\) as the total bandwidth of user \(j\) (\(\in\Omega\backslash\Omega_{B}\)) needed to obtain target multicast rate \(R_{T}\) with a given \(B_{B}\). By rewriting (6c), \(f_{j}(B_{B})\) can be expressed as  
\[f_{j}(B_{B})=\alpha_{j}B_{B}+\beta_{j}\,\quad B_{|\Omega_{B}|}\leq B_{B}\leq B _{B,j} \tag{7}\]  
where \(\alpha_{j}=1-\frac{\log_{2}\left(1+\gamma_{j}\right)}{\log_{2}\left(1+\gamma_{ j}^{S}\right)},\beta_{j}=\frac{R_{T}}{\log_{2}\left(1+\gamma_{j}^{S}\right)}\) and \(B_{B,j}=\frac{R_{T}}{\log_{2}\left(1+\gamma_{j}\right)}\). Since the multicast channel is common to all assigned users, the total bandwidth, \(B_{B}+B_{S}\), should be \(\max\limits_{j\in\Omega\backslash\Omega_{B}}\left\{f_{j}(B_{B})\right\}\). Then, the minimum total bandwidth ((6a) above) can be rewritten as  
\[\min(B_{B}+B_{S})=\min\left[\max\limits_{j\in\Omega\backslash\Omega_{B}}\left\{ f_{j}(B_{B})\right\}\right] \tag{8}\]  
Figure 2 illustrates \(f_{j}(B_{B})\) and how the proposed algorithm works with six users in \(\Omega\) when \(\Omega_{B}=\{1,4\}\). Note that \(f_{j}(B_{B})\) is an affine function that is both convex and concave. Then, by applying [PERSON]'s inequality, we can easily prove that \(\max\limits_{j\in\Omega\backslash\Omega_{B}}\left\{f_{j}(B_{B})\right\}\) is also convex [16], which corresponds to the shaded feasible region. Note, because \(\max\limits_{j\in\Omega\backslash\Omega_{B}}\left\{f_{j}(B_{B})\right\}\) is the pointwise minimum of a family of affine functions, the optimal point is one of the coordinates of the intersections of the lines given in (7) or boundaries. Instead of examining all the continuous values of \(B_{B}\) in the range of \((B_{|\Omega_{B}|},B_{|\Omega|})\), the optimal point can be found iteratively, as shown in Algorithm 1. In the proposed algorithm, the user index, \(j^{(k)}\), which maximizes the required bandwidth in the \(k\)-th iteration, is defined as  
\[j^{(k)}=\underset{\epsilon\to 0}{\text{Imarg}}\max\limits_{j\in\Omega \backslash\Omega_{B}}\left\{f_{j}\Big{(}B_{B}^{(k)}+\epsilon\Big{)}\right\} \tag{9}\]  
where \(B_{B}^{(k)}\) is the bandwidth of the BMCH at the leftmost point of intersection in the \(k\)-th iteration. Note that if user \(j^{(k)}\) has better channel quality on the BMCH than on the SMCH, the slope of \(f_{j}(B_{B})\), that is, \(\alpha_{j^{(k)}}\), is negative; then, allocating more bandwidth to the BMCH can decrease the total bandwidth. Otherwise, allocating more bandwidth to the BMCH is disadvantageous. This implies that it is the optimal point where the sign of \(\alpha_{j^{(k)}}\) changes from negative to positive, for example, the 3 rd iteration in Figure 2. Then, the total bandwidth of the proposed multicast scheme, \(B_{\text{prop}}\), is \(f_{j^{(3)}}\Big{(}B_{B}^{(3)}\Big{)}\).