# Rateless Coded Uplink Transmission Design for Multi-User C-RAN
## 4 Decoding Performance Analysis and Degree Profile Design
### Degree Profile Optimization  
With the above EXIT analysis, we optimize the degree profile of the Raptor code adopted at each user. The task is to find a uniformly good output degree profile for each user under all possible channel states, with the knowledge of the probability distribution of each channel state. Recall that the channel matrix \(\mathbf{H}\) remains invariant within each round of transmission and varies independently round by round. For each round, both users continuously transmit the Raptor codeword  
Figure 5: Extrinsic information transfer (EXIT) chart for the multi-user (MU) detector. until the BBU pool decodes both messages successfully. Hence, the length of the Raptor codeword for each user is related to the channel matrix \(\mathbf{H}\) in each round, which can be expressed as:  
\[L_{i}(\mathbf{H})=\frac{K}{R_{p}}\overline{\alpha}_{i}(\mathbf{H})\sum\
olimits _{d=1}^{d_{c}}\frac{\omega_{i,d}}{d}, \tag{23}\]  
where \(\overline{\alpha}_{(i)}(\mathbf{H})\) is the average degree of the LT input nodes for user \(i\) under channel matrix \(\mathbf{H}\). Note that during each round of transmission, the two users start and stop transmitting simultaneously, therefore we have \(L_{1}(\mathbf{H})=L_{2}(\mathbf{H})\triangleq L(\mathbf{H})\). In general, the channel matrix space is continuous. In practice, we discretize the space into \(Q\) states: \(\mathbf{H}_{q},q=1,\ \dots,\ Q\). The probability of each state is denoted as \(\Pr\bigl{(}\mathbf{H}_{q}\bigr{)}\). The average Raptor code length over all \(Q\) channel states can be expressed as:  
\[E\bigl{\{}L_{i}\bigl{(}\mathbf{H}_{q}\bigr{)}\bigr{\}}=\sum\
olimits_{q=1}^{Q} \Pr\bigl{(}\mathbf{H}_{q}\bigr{)}\frac{K}{R_{p}}\overline{\alpha}_{i}\bigl{(} \mathbf{H}_{q}\bigr{)}\sum\
olimits_{d=1}^{d_{c}}\frac{\omega_{i,d}}{d}. \tag{24}\]  
Apparently, \(E\bigl{\{}L\bigl{(}\mathbf{H}_{q}\bigr{)}\bigr{\}}=E\bigl{\{}L_{1}\bigl{(} \mathbf{H}_{q}\bigr{)}\bigr{\}}=E\bigl{\{}L_{2}\bigl{(}\mathbf{H}_{q}\bigr{)}\bigr{\}}\).We can approximate \(\overline{\alpha}_{i}\bigl{(}\mathbf{H}_{q}\bigr{)}\) as \(\overline{\alpha}_{i}\bigl{(}\mathbf{H}_{q}\bigr{)}=\overline{\alpha}_{i,0} C_{i}^{-1}\bigl{(}\mathbf{H}_{q},P\bigr{)}\)[45], where \(\overline{\alpha}_{i,0}\) is a constant and is independent of the channel matrix. \(C_{i}\bigl{(}\mathbf{H}_{q},P\bigr{)}\) is the theoretically maximal achievable rate of user i with transmit power \(P\) under channel matrix \(\mathbf{H}_{q}\). \(C_{i}\bigl{(}\mathbf{H}_{q},P\bigr{)}\) can be calculated as follows. According to (1), the two-user C-RAN can be regarded as a two-user multiple-input-multiple output (MIMO) system with two antennas at the receiver. Note that to be precise, the latter is the ideal case of the former wherein the quantization error at RRHs is ignored. The capacity region of the two-user MIMO system is depicted in Figure 6, and can be expressed as:  
\[R_{1}\leq I(x_{1};\boldsymbol{y}|x_{2})=H(\boldsymbol{y}|x_{2})-H( \boldsymbol{y}|x_{1},x_{2})\triangleq B_{1}, \tag{25}\]  
\[R_{2}\leq I(x_{2};\boldsymbol{y}|x_{1})=H(\boldsymbol{y}|x_{1})-H(\boldsymbol{ y}|x_{1},x_{2})\triangleq B_{2}, \tag{26}\]  
\[R_{1}+R_{2}\leq I(x_{1},x_{2};\boldsymbol{y})=H(\boldsymbol{y})-H(\boldsymbol{ y}|x_{1},x_{2})\triangleq B_{12}, \tag{27}\]  
where \(R_{1}\) and \(R_{2}\) denote the achievable rates of user 1 and user 2, respectively. The derivation of each entropy term is not difficult, thus it is omitted here. Note that in our setting, both users have the same message length and transmit simultaneously (i.e., the times of channel usage for both users are the same). Therefore, the achievable rates of the two users are always the same.