## 5 Performance Analysis of MSD Algorithm Based on Hybrid SC  
### Computational Accuracy of MSD Algorithm  
In order to assess the performance of the MSD circuit constructed using the hybrid SC approach, this study initially carried out an experiment to evaluate the overall computational accuracy of the MSD algorithm. The sampling factor of the receiver was set to four. Leveraging the previously designed fundamental SC units, the computational accuracy was verified for the three-, five-, and seven-symbol variants of the MSD algorithm.  
Additionally, the experiments also involved comparing the computational accuracy of the hybrid SC approach proposed in this study with that of a pure SC approach; specifically, the pure SC method employed a re-randomization technique in the symbol decision stage, as outlined in [41].  
Figure 14 illustrates the computational accuracy across diverse computation schemes and different symbol lengths. The results show that the hybrid SC achieved higher computational accuracy than pure SC. This can be attributed to two primary reasons: First, although re-randomization allows a new random sequence to be regenerated based on an existing random sequence, it does not fully eliminate the correlation between the two sequences. Second, re-randomization largely maintains the probability values of the sequences. When computational errors exist in the sequences undergoing re-randomization, these errors are propagated to the newly generated sequences, resulting in increased computational errors. In light of these observations, this study adopts a hybrid SC method to develop the MSD algorithm, aiming to enhance computational accuracy while ensuring computational efficiency.  
Figure 14 also illustrates the computational accuracy under different bit quantization schemes. The error in binary computation arises from two primary sources: (1) quantization loss determined by the bit width; (2) the scaling operations that are necessary to prevent data overflow during summation processes. As the observation window of the MSD algorithm expands from three to seven symbols, the summation of input data increases proportionally. This necessitates a larger scaling factor to prevent overflow in adder operations, which  
\begin{table}
\begin{tabular}{l l l l l l l l l} \hline \hline
**2-Input Adder** & **Proposed** & **[36]** & **[37]** & **[38]** & **[39]** & **[40]** & **8-Bit** & **9-Bit** \\ \hline LUT & 4 & 2 & 3 & 5 & 3 & 5 & 5 & 5 \\ \hline FF & 4 & 1 & 3 & 7 & 3 & 5 & 8 & 9 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Hardware resource overhead of 2-input adders.  
\begin{table}
\begin{tabular}{l l l l l l l l l l l l l} \hline \hline \multirow{2}{*}{**Adder**} & \multicolumn{2}{c}{**3-Input Adder**} & \multicolumn{2}{c}{**4-Input Adder**} & \multicolumn{2}{c}{**5-Input Adder**} & \multicolumn{2}{c}{**7-Input Adder**} \\ \cline{2-13}  & **k = 2** & **8-Bit** & **9-Bit** & **k = 2** & **8-Bit** & **9-Bit** & **k = 3** & **8-Bit** & **9-Bit** & **k = 4** & **8-Bit** & **9-Bit** \\ \hline LUT & 6 & 23 & 26 & 6 & 29 & 32 & 7 & 45 & 51 & 7 & 59 & 66 \\ \hline FF & 6 & 27 & 30 & 6 & 28 & 31 & 6 & 54 & 60 & 6 & 64 & 71 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Hardware resource overhead of different adders.  
introduces greater computational errors. Consequently, the seven-symbol MSD exhibited a greater computational error than the three-symbol MSD.  
Figure 14 provides a comparative analysis of computational accuracy between hybrid SC and binary computing paradigms. However, it is critical to recognize that error manifestations differ fundamentally between these architectures: Binary computation introduces errors through floor operations, while the errors from hybrid SC derive from inherent variance in probabilistic bit-stream representations. Consequently, direct numerical comparison of error metrics does not provide an equitable reflection of algorithmic performance across different computational frameworks..