## 4 Adaptive OFDM-IM Method
### Fuzzy Logic Optimization
#### 4.4.5 Computational Complexity  
The computational complexity of the fuzzy logic method mainly consists of three steps: fuzzification, fuzzy inference, and defuzzification. The complexity of fuzzification is \(O(n_{FL})\), where \(n_{FL}\) is the number of input variables; the complexity of fuzzy inference is \(O(m_{FL})\), where \(m_{FL}\) is the number of fuzzy rules; and the complexity of defuzzification is \(O(k_{FL})\), where \(k_{FL}\) is the number of fuzzy membership functions for the output variable. In this study, the fuzzy logic system includes 3 input variables (BER, spectral efficiency, and energy consumption), 27 fuzzy rules, and 1 output variable. Therefore, the overall complexity is \(O(n_{FL}+m_{FL}+k_{FL})=O(31)\). This low complexity makes it well suited to meet the real-time requirements of vehicular communication scenarios.  
In comparison, the computational complexity of exhaustive search is \(O(C_{ES}d_{ES})\), where \(C_{ES}\) represents the number of candidate parameter values, and \(d_{ES}\) is the parameter dimension. While exhaustive search guarantees that the global optimum will be found, its complexity increases exponentially with the growth of the parameter dimensions, making it unsuitable for real-time systems that require rapid decision-making. The genetic algorithm has complexity of \(O(g_{GA}\cdot p_{GA})\), where \(g_{GA}\) is the number of generations and \(p_{GA}\) is the population size. Although it exhibits strong exploratory capabilities in nonlinear optimization, it requires multiple iterations and parameter tuning, resulting in a significant computational overhead. In contrast, the fuzzy logic method, with its parameter adjustment mechanism centered on a rule base, achieves adaptive optimization for dynamic scenarios at a low complexity level of \(O(n_{FL}+m_{FL}+k_{FL})\). This makes it particularly suitable for real-time communication scenarios that require rapid decision-making. However, the fuzzy logic method may fall short in terms of global optimization capabilities, as it relies on a predefined rule base, making it challenging to extend to large-scale, high-dimensional parameter spaces. As shown in Table 4, the fuzzy logic method demonstrates significant advantages in terms of applicability, real-time performance, and dynamic scenario optimization, all while maintaining the lowest complexity. On the other hand, exhaustive search and genetic algorithms excel in terms of global optimization capabilities.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Method** & **Complexity** & **Advantages** & **Disadvantages** \\ \hline \multirow{4}{*}{Fuzzy Logic Method} & & Efficient, suitable & \\  & & for real-time & Limited \\  & & & scenarios, strong & adaptability in \\  & & & adaptability in & global optimization \\  & & & dynamic scenarios & \\ \hline \multirow{4}{*}{Exhaustive Search} & \multirow{4}{*}{\(O(C_{ES}d_{ES})\)} & \multirow{4}{*}{Guarantees global optimal solutions} & High \\  & & & computational \\  & & & complexity for \\ \cline{1-1}  & & & high-dimensional \\ \cline{1-1}  & & & problems, \\ \cline{1-1}  & & & unsuitable for \\ \cline{1-1}  & & & real-time systems \\ \hline \multirow{4}{*}{Genetic Algorithm} & \multirow{4}{*}{\(O(g_{GA}\cdot p_{GA})\)} & Balances & \multirow{4}{*}{Requires multiple iterations, relatively high computational overhead} \\  & & exploration and & & \\ \cline{1-1}  & & efficiency, suitable & \\ \cline{1-1}  & & for nonlinear optimization problems & \\ \cline{1-1}  & & & \\ \cline{1-1}  & & & \\ \cline{1-1}  & & & \\ \hline \hline \end{tabular}
\end{table}
Table 4: Comparison of complexity, advantages, and disadvantages of three algorithms.