# Averting BER Floor with Iterative Source and Channel Decoding for Layered Steered Space-Time Codes
## 4 Simulation Results and EXIT Analysis  
This section presents the performance analysis of the schemes in Table 2 employing \(N_{t}=4\) transmit AAs and 4 receive antennas. The simulations of the proposed three-stage system are performed using the IT++ signal processing and communication library which is coded in C++. The IT++ supports several free and open-source libraries and is mainly utilized in numerical analysis, signal processing and digital communications. For the purpose of fair analysis, we have accordingly set the overall code rate of different SBC schemes to be persistent. The source encoded bits-to-SP symbols were carried using the anti-Gray mapping (AGM). For convenience, we refer to the AGM as the one which is different from Gray mapping. For enriching reliability in our experimental setup, we averaged the results after repeating the transmission experiment 160 times. It is to be noted that the SP scheme was considered for \(L=16\) modulated symbols as in [14]. For investigating the convergence behaviour of the schemes presented in Table 2, we will be utilizing the EXIT chart analysis as proposed in [51]. EXIT chart is based on the sharing of mutual information (MI) between constituent decoders [41]. For the system proposed in Figure 1, the MI at each stage of the three decoders is distinguished from each other by replacing the subscript (.) with 1, 2, and 3 for the SBC, URC, and SP-demapper, respectively. The URC decoder yields two extrinsic outputs, labelled as \(L_{e}^{2}(s^{\prime})\) and \(L_{e}^{2}(r)\), and therefore its EXIT functions can be written as \(F_{s}^{i}[I_{2,A}(s^{\prime}),\ I_{2,A}(r)]\) and \(F_{r}[I_{2,A}(s^{\prime}),\ I_{2,A}(r)]\) as discussed in [7]. As the SBSD and SP demapper each deal with a single input and output from and to the URC decoder, their EXIT functions are given as \begin{table}
\begin{tabular}{c c c} \hline \hline
**SBC Type** & **Symbols in Decimals** & \(d_{H,\text{min}}\) \\ \hline Un-optimized \(\text{SBC}_{[2,6]}^{*}\) & 0, 16, 32, 48 & 1 \\ \hline \(\text{SBC}_{[2,6]}\) & 0, 22, 41, 63 & 3 \\ \hline \(\text{SBC}_{[3,9]}\) & 0, 78, 149, 219, 291, 365, 438, 504 & 4 \\ \hline \(\text{SBC}_{[4,12]}\) & 0, 286, 557, 819, 1099, 1365, 1638, & \\ \(\text{SBC}_{[4,12]}\) & 1912, 2183, 2457, 2730, 2996, 3276, & 5 \\  & 3538, 3809, 4095 & \\ \hline \(\text{SBC}_{[5,15]}\) & 0, 1086, 2141, 3171, 4251, 5285, 6342, & \\  & 7416, 8471, 9513, 10570, 11636, & \\  & 12684, 13746, 14801, 15855, 16911, & 6 \\  & 17969, 19026, 20076, 21140, 22186, & \\  & 23241, 24311, 25368, 26406, 27461, & \\  & 28539, 29571, 30653, 31710, 32736 & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Rate-1/3 un-optimized and optimized SBC schemes with associated symbols and \(d_{H,\text{min}}\). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Value** & **Parameters** & **Value** \\ \hline Source Code & H.265/HEVC & Transmitter AAs & 4 \\ \hline Source Bit-Rate & 64 kbps & Receiver Antennas & 4 \\ \hline Frame Rate & 15 fps & Intermediate Code & URC \\ \hline Slices per Frame & 9 & Interleaving Bits & 10,000 \\ \hline Overall Code Rate & 1/3 & Video Sequence & QCIF Akiyo \\ \hline Number of CTUs per slice & 60 &
\begin{tabular}{c} Normalized Doppler \\ Frequency \\ \end{tabular} & 0.01 \\ \hline Frames to be Encoded & 300 & Modulation Scheme & SP \\ \hline Error Protection Scheme & SBC & Channel & Rayleigh Fading \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Proposed system parameters. \(F_{s}[I_{1,A}(s)]\) and \(F_{r}\)\([I_{3,A}(r^{\prime}),\ E_{b}/N_{0}]\) respectively. For simplicity, we refer to the intermediate URC decoder and the SBSB as combined outer soft input soft output (SISO) module. The EXIT chart curves of the proposed SBC schemes presented in Table 2 are depicted in Figures 2 and 3. It can be observed in Figure 2 that the outer EXIT curve constituting the un-optimized \(\text{SBC}^{*}_{[2,6]}\) scheme is unable to produce the required open EXIT tunnel for convergence.