# Rateless Coded Uplink Transmission Design for Multi-User C-RAN
## 5 Simulation Results  
In this section, we simulate the bit error rate (BER) and the achieved throughput of the proposed transmission scheme. In the simulation, we consider the scenario where two users in C-RAN are located within the coverage of two RRHs. The SNR is defined as the ratio of the user-transmitted power \(P\) to the Gaussian noise variance \(\sigma_{0}^{2}\) at each RRH. We assume that the channel matrix \(\mathbf{H}\) has three states with equal probability: \(\mathbf{H}_{1}=\left[ \begin{array}{cc}1.9781&0.6696\\ 1.2958&0.3872\end{array} \right]\mathbf{H}_{2}=\left[ \begin{array}{cc}0.5826 &1.1291\\ 0.8247&1.5324\end{array} \right]\mathbf{H}_{3}=\left[ \begin{array}{cc}2.8884 &0.4533\\ 0.3361&1.1801\end{array} \right]\). Two cases for the quantizer at the RRHs are considered, wherein the number of quantization bits are three and eight, respectively. Apparently, the latter case generates more fronthaul traffic than the former. We assume that the length of message for each user is \(K=9500\) bits, that both users adopt the same LDPC code with \(R_{p}=0.95\) and that the decoding threshold for the LDPC code is derived as \(m_{th}=13.6359\). (1) BER performance  
Firstly, we simulate the error performance of the proposed scheme versus the decoding overhead under each of the channel states \(\mathbf{H}_{1}\), \(\mathbf{H}_{2}\) and \(\mathbf{H}_{3}\), respectively. We fix \(P=0.84\) and \(\sigma_{0}^{2}=1.2^{2}\). The decoding overhead is defined as \(\textit{overhead}=\frac{C\left(\mathbf{H}_{q},P\right)\cdot N}{K}-1\), where \(C\left(\mathbf{H}_{q},P\right)\) is the theoretical upper bound of the achievable rate of the user (recalling (30)), \(N\) is the actual transmitted code length. A larger overhead means a longer code length \(N\) and a lower achieved rate. The overhead of value \(1\) can serve as the theoretical limit. We optimize the degree profile for each user using the scheme proposed in Section 4.2. The resulted profiles are as follows:  
When an eight-bit quantizer is adopted at each RRH, the optimized degree profiles for the two users are denoted as \(\Omega_{1,opt,8-bit}(x)\) and \(\Omega_{2,opt,8-bit}(x)\):  
\[ \begin{array}{l}\Omega_{1,opt,8-bit}(x)=0.0120x^{1}+0.4054x^{2}+0.2678x^{3} +0.2217x^{6}+0.0931x^{20},\\ \Omega_{2,opt,8-bit}(x)=0.0121x^{1}+0.4011x^{2}+0.2808x^{3}+0.1564x^{6}+0.0401x ^{7}+0.1004x^{18}+0.0090x^{19}.\end{array} \]  
When a three-bit quantizer is adopted, the optimized degree profiles for the two users are:  
\[ \begin{array}{l}\Omega_{1,opt,3-bit}(x)=0.0120x+0.4028x^{2}+0.2693x^{3}+0.2 020x^{6}+0.0231x^{7}+0.0907x^{20},\\ \Omega_{2,opt,3-bit}(x)=0.0125x+0.4151x^{2}+0.2053x^{3}+0.1042x^{5}+0.2020x^{6}+ 0.1025x^{20}.\end{array} \]As for comparisons, we also simulate the performance of the degree profile \(\Omega_{BEC}(x)\) (which is optimal under binary erasure channel (BEC) [43]):  
\[\Omega_{BEC}(x) = 0.0008x+0.494x^{2}+0.166x^{3}+0.073x^{4}+0.083x^{5}+0.056x^{8}+\] \[0.037x^{9}+0.056x^{19}+0.025x^{65}+0.003x^{66},\]  
for both three-bit and eight-bit cases. In Figure 7, the three subfigures (a), (b) and (c) show the BER performance of the optimized profiles and BEC profile under channel matrices \(\mathbf{H}_{1}\), \(\mathbf{H}_{2}\) and \(\mathbf{H}_{3}\), respectively. Interestingly, although the degree profiles are optimized to improve the average performance under the three channel states, they still perform well under each channel state. For both three-bit and eight-bit cases, the optimized profiles achieve a BER under \(10^{-4}\) at a smaller overhead compared with the BEC profile. It is also observed that the eight-bit quantizer case has a lower BER compared with the three-bit quantizer case, which verifies the fact that a quantizer with less quantization bits introduces larger quantization noise.