# Analysis and Design of Rateless Two-way Relay Networks Based on a Multiply-and-Forward Scheme
## 5 Simulation Results  
In this section, we offer simulation results to validate our work on the design of rateless MF-TWRN. In our simulation, the transmit power of all nodes is normalized and the distances between the relay node and source nodes are identical with the value of 1. Moreover, assume the path loss exponent is of 2. The channel fading coefficients are with unit power, i.e., \(g_{i,\text{R}}\sim\text{Rayleigh}(0.5)\), \(i\in\{\text{A},\text{B}\}\).  
For an uncoded TWRN system with BPSK modulation, Figure 3 demonstrates the SER performance of two forwarding schemes, namely, MF and AF, over the flat Rayleigh fading channel, regarding theoretical and simulation results. The solid lines in Figure 3 depict numerical results of MF and AF, respectively, while the dotted lines are plotted using theoretical results in (22) and (35). It is implied that the numerical SERs can be approximated by theoretical ones, coinciding in the sufficiently high SNR regime especially. Just like the conclusion drawn in the Section 3, numerical results also show that the MF scheme outperforms the AF scheme in the flat Rayleigh channel.  
Using the optimization model proposed in (66), Table 11 offers the optimized result with parameters \(\sigma_{\text{n}}^{2}=10^{-2}\), \(d_{c}=200\), \(y_{0}=5\times 10^{-4}\) and \(\varepsilon_{0}=5\), which is denoted by \(\Omega_{\text{opt}}(x)\). The authors in [25] studied the design of fountain codes in a three-stage TWRN using joint network-channel coding. Their optimized distribution was obtained with the channel condition \(\sigma_{\text{n}}^{2}=10^{-2}\) as well and is denoted by \(\Omega_{\text{JNC}}(x)\) for short. In the flat Rayleigh fading channel with the transmit SNR of 20 dB, Figures 4 and 5 demonstrate the BER performance of MF-TWRN at node A with different lengths of source information, when using the optimized distribution \(\Omega_{\text{opt}}(x)\). It is clear that the BER decreases as the source length increases from 2000 to 4000, and approaches the theoretical lower bound especially in the high overhead regime. This observation made on Figures 4 and 5 also validate the theoretical analysis carried out in Section 4. To compare the overall performance of a TWRN system thoroughly, Figures 6-8 show the performance of MF-TWRN from the perspectives of frame-error-rate (FER) and throughput, respectively. Figures 6 and 7 compare the mean FER of two source nodes using different optimized distributions and the comparison results imply that our optimized distribution \(\Omega_{\text{opt}}(x)\) can improve the decoding performance of the whole TWRN system. Furthermore, Figure 8 describes the throughput of MF-TWRN using different degree distributions, in which the numerical results are obtained for the source length of 4000 while the theoretical results correspond to upper bounds on the throughput derived in (62). Along with the increase of the SNR, the actual throughput of TWRN approaches the theoretical upper bound. One can get a higher throughput using the optimized distribution \(\Omega_{\text{opt}}(x)\) rather than \(\Omega_{\text{INC}}(x)\) proposed in [25], which  
Figure 5: BER performance of source node A when using \(\Omega_{\text{opt}}(x)\) with \(k=4000\).  
Figure 6: Frame error rate (FER) performance of systematic Luby transform (SLT)-coded multiply-and-forward two-way relay network (MF-TWRN) using different degree distributions with \(k=2000\).