# Design of Infinite Impulse Response Filters Based on Multi-Objective Particle Swarm Optimization
## 4 Discussion  
This study has greatly improved the process of the multi-objective particle swarm optimization algorithm. Compared with the particle swarm optimization method proposed by [PERSON] and other scholars [11] and [PERSON] and other scholars [11], by calculating the MMSE, ME, and SD, this study optimized the following two parts. \begin{table}
\begin{tabular}{c c c c} \hline  & **Type** & **Before Filtering (dB)** & **After Filtering (dB)** \\ \hline \multirow{4}{*}{Butterworth} & Low-Pass & \(-12.77\) & \(-9.92\) \\  & High-Pass & \(-11.53\) & \(-6.34\) \\  & Band-Pass & \(-12.39\) & \(-8.56\) \\  & Low-Pass & \(-13.62\) & \(-11.19\) \\ \multirow{2}{*}{Chebyshev Type I} & High-Pass & \(-12.63\) & \(-6.63\) \\  & Band-Pass & \(-10.72\) & \(-8.30\) \\ \hline \end{tabular}
\end{table} 
Table 2: SNR comparison. Figure 6: (a) Low–pass filter comparison chart. (b) Low–pass filter comparison chart passband ripple enlarged view. 1. Optimization of frequency response  
The particle swarm optimization algorithm, combined with Pareto efficiency, can find the best balance between multiple objectives, such as between the frequency response of the passband and stopband, which can help design a design that satisfies the frequency within the passband requirements and can effectively suppress the stopband frequency filter. 2. Optimization of selectivity and bandwidth  
The selectivity (the ability to distinguish the passband and stopband) and bandwidth of the filter are essential considerations in design. The particle swarm optimization algorithm combined with Pareto efficiency can improve the selectivity while appropriately controlling the bandwidth to achieve the best compromise between these two aspects. Comparison with Related Literature  
Table 3 shows the design parameters used by [PERSON] et al. [11] and [PERSON] et al. [12]. \(N\) and \(M\) are the numerator and denominator of the filter transfer function. \(\tau\) is the expected group delay. \(f_{p}\) is the passband frequency edge frequency. \(f_{s}\) is the stopband edge frequency. \(R\) is the maximum value of the pole radius. \(P\) is the number of particles. \(I_{max}\) is the maximum number of iterations. Using the same design parameters as in the literature (as shown in Table 3) to perform filter design, in Tables 4 and 5, our approach demonstrates superior or comparable performance in filter design, with lower error rates and more consistent results. This validates the effectiveness of our PSO-based method for multi-objective optimization in IIR filter design, outperforming the methods proposed by [PERSON] et al. [11] and [PERSON] et al. [12]. In summary, combined with Pareto efficiency, the particle swarm optimization algorithm can more effectively identify the optimal balance point between multiple objectives and provide a set of Pareto optimal solutions, which helps solve multi-objective optimization problems. It is beneficial, and this method is suitable for continuous problems and can effectively solve discrete problems. It does not require making assumptions about the specific form of the problem (such as the linearity or nonlinearity of the objective function), making it very adaptable and powerful. \begin{table}
\begin{tabular}{l c c} \hline \hline  & **[PERSON] et al. [11]** & **[This Work]** \\ \hline MMSE (\(\times 10^{-2}\)) & 2.12 & 1.83 \\ ME (\(\times 10^{-2}\)) & 2.72 & 2.34 \\ SD (\(\times 10^{-2}\)) & 0.83 & 0.03 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of results to those of [PERSON] et al. \begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline
**References** & \(N\) & \(M\) & \(\tau\) & \(fp\) & \(f_{s}\) & \(R\) & \(P\) & \(I_{max}\) \\ \hline [PERSON] et al. [11] & 8 & 6 & 5 & 0.175 & 0.25 & 0.92 & 90 & 5000 \\ [PERSON] et al. [12] & 6 & 4 & 4 & 0.25 & 0.33 & 0.90 & 80 & 10,000 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Design parameters used in the literature.