# Performance of Dynamic Time Division Broadcast Protocol with Rateless Coding
## 3 Performance Analysis
### Diversity-Multiplexing Trade-off
#### 3.2.2. DMT of Sub-DTDBC-RC and DTDBC-RC
Corollary 1: _The DMT of DTDBC-RC is \(d_{DTDBC-RC}(r)=2\), i.e., its diversity gain is always two for any multiplexing gain \(r\) (\(0\leq r\leq 1\)), independent of \(\alpha\)._  
**Proof. ** By setting \(N\to+\infty\) in Equation (17), the result is obtained directly. **4. Simulation and Numerical Results**  
In the section, Matlab is used for simulation and the simulation process proceeds as described in Section 2. The simulation parameters are set as follows: \(C=0.5\text{ bits}/\text{s}/\text{Hz}\), \(\Omega_{0}=\Omega_{1}=\Omega_{2}=1\), and the number of information exchange rounds is \(10^{9}\). In the simulation process, all the channel coefficients are produced independently and keep unchanged in each round of information exchange. Moreover, if the channels in the DTDBC-RC (or sub-DTDBC-RC) protocol cannot support the target rate, an outage event occurs. The theoretical (Equation (1)) and simulated outage probabilities of DTDBC-RC/TDBC-IR are presented in Figure 4 for various \(\alpha\), where the horizontal axis denotes the transmit SNR (the SNR has the same meaning for the other figures in the paper). From the figure, we see the theoretical and simulated outage probabilities coincide exactly, validating the results in Theorem 1. Moreover, when\(\alpha\leq 0.4\), the outage performance becomes better as \(\alpha\) increases. Based on the result of [14] (i.e., the result in Figure 10 of [14]), the optimal cooperation level (i.e., the ratio of a codeword between the part transmitted by the source and that transmitted by the relay) in reaching the best outage performance for one-way coded cooperation lies between 0.6 and 0.7 when the direct and relaying channels are with the same distribution. For DTDBC-RC/TDBC-IR, when \(\alpha=0.4\), it corresponds to the cooperation level \(\frac{0.4}{0.4+1-2\times 0.4}\approx 0.67\), which lies between 0.6 and 0.7; when \(\alpha=0.45\), it corresponds to the cooperation level \(\frac{0.45}{0.45+1-2\times 0.45}\approx 0.82\) (larger than 0.7). Thus, the outage performance of DTDBC-RC/TDBC-IR when \(\alpha=0.4\) is better than that of \(\alpha=0.45\). The expected rate performance corresponding to Figure 4 for TDBC-IR and DTDBC-RC is shown in Figures 5 and 6, respectively. Figure 5 reveals that on the one hand, the expected rate of TDBC-IR at low-SNR region (when \(\rho\leq 5\) dB) becomes larger as \(\alpha\) increases. This is because its expected rate is mainly decided by the second term on the RHS of Equation (10) at this SNR region, i.e., the expected rate decided by the third-time-slot transmission dominates. On the other hand, its expected rate becomes larger as \(\alpha\) decreases at high-SNR region (when \(\rho>11\) dB). This is because its expected rate is mainly decided by the first term on the RHS of Equation (10) at this SNR region, i.e., the expected rate decided by the first two-time-slot transmission dominates. Moreover, as SNR tends to infinity, the expected rate of TDBC-RC reaches \(\frac{(1-\alpha)C}{\alpha}\) (i.e., the first term on the RHS of Equation (10)) since \(e^{-z_{0}/\Omega_{0}}=1\). For DTDBC-RC, the similar expected rate performance trend is observed at low-SNR region (see Figure 6), i.e., the expected rate becomes larger as \(\alpha\) increases. The reason is also similar, i.e., it is mainly decided by the second term (i.e., \(\Sigma_{4}\)) on the RHS of Equation (9) and thus, the expected rate decided by the third-time-slot transmission dominates. However, when the SNR is large enough, its expected rate is independent of \(\alpha\) (unlike for the TDBC-IR) and becomes infinite as SNR tends to infinity. This is because the expected rate of DTDBC-RC is mainly decided by the first term at the RHS of Equation (9) (i.e., \(\Sigma_{3}\)) and it will always succeed in the first two-time-slot transmission (thus, its transmission time \(T_{DTDBC-RC}\) only depends on the channel distribution of the direct link).