## 4 Simulation Results and Discussion
### Comparison of the Processing Time and Computational Complexity  
In this section, we compare the processing time and computational complexity of the proposed semi-blind scheme JISB_DSICED3 and the conventional without feedback scheme SB_DSICED3_W/OF. The processing time is an important factor for performance evaluation since it quantifies the effectiveness of the algorithm, especially in 5G short-packet transmissions and IoT applications.  
In this case, a computer with the hardware configuration of Intel (R) Core (TM) I5-10500 CPU @ 3.10 GHz (12 CPUs) and a memory of 16 GB of RAM was used with MATLAB Version 2020b. For the simulation parameters, we set \(p_{S1}/\sigma_{w}^{2}=30\) dB, \(10^{6}\) transmission frames, \(E=128\) symbols, \(\beta=1/32\) (\(\beta E=4\) symbols), \(i_{max}=4\), and \(j_{max}=1\) for the semi-blind scheme JISB_DSICED3 and \(j_{max}=20\) for the semi-blind without feedback scheme SB_DSICED3_W/OF. Because we fixed the same value of the maximum number of decoding iterations for all levels of the SNR, the processing time was nearly the same. Therefore, this configuration was used to calculate the processing time to achieve the MSE and BER at the particular SNR level, \(p_{B}/\sigma_{w}^{2}=10\) dB. Based on the results in Table 3, it was observed that the semi-blind scheme JISB_DSICED3 could significantly reduce the processing time and showed the fastest result because temporary decoding and encoding were not required in the feedback loop and the number of iterations in the SPA decoding process was also reduced, which took less roughly 10-times that compared to the semi-blind without feedback scheme SB_DSICED3_W/OF. This was due to the semi-blind scheme only needing one decoding iteration (\(j_{max}=1\)) to obtain a good result when achieving the best value of the channel estimations, while the semi-blind without feedback scheme SB_DSICED3_W/OF needed up to 20 iterations (\(j_{max}=20\)) to converge and to reach the saturation floor.  
Moreover, the computational complexity of the two schemes was computed based on the summation of the number of computations in the operations including additions/subtractions, multiplications/divisions, and XOR operations based on [40; 55; 56; 57; 58; 59]. Because of the identity and symmetry at the transmitter side, this paper only considered calculating the total number of computations at the receiver side. The formulas for calculating the relative number of computations for each operation are summarised in detail  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Algorithm** & \
\begin{tabular}{c} **Processing Time** \\ **(in min)** \\ \end{tabular} &
\begin{tabular}{c} **Ratio Respected to** \\ **(SB\_DSICED3\_W/OF)** \\ \end{tabular} \\ \hline SB_DSICED3\_W/OF scheme & 475.7 & 1 \\ JISB_DSICED3 scheme & 48.1 & 0.101 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Processing time.  
in Table 4, where \(\bar{u}_{v},\bar{u}_{c}\) denote the average degree of the variable nodes and the average degree of the check nodes of the parity check matrix \(\mathbf{H}\), respectively.  
Figure 13 shows the number of computations versus various values of the symbols \(E\), which was used to calculate the total number of computations to obtain the MSE and BER at the particular SNR level, \(p_{B}/\sigma_{w}^{2}=10\) dB. For the simulation parameters, we set \(\beta E=4\) symbols, \(p_{SI}/\sigma_{w}^{2}=30\) dB, \(10^{6}\) transmission frames, \(i_{max}=4\), and \(j_{max}=1\) for the proposed feedback schemes JISB_DSICED3 and \(j_{max}=20\) for the conventional without feedback scheme SB_DSICED3_W/OF. The result indicated that the proposed feedback scheme JISB_DSICED3 required less cost for completing the computation than the conventional without feedback scheme SB_DSICED3_W/OF.  
Therefore, this result emphasises the practical implementation of the scheme in 5G short-packet transmissions, especially in IoT transmissions and green communications with low power consumption.