## 6 Conclusions and Future Directions
### Future Directions  
To tackle the algorithm complexity, one potential direction is to simplify HPSO. This involves analyzing the algorithm's components and identifying areas where complexity can be reduced without compromising performance. Streamlining the algorithm can make it more accessible and easier to implement in practical scenarios, enabling wider adoption of HPSO for adaptive equalization.  
Addressing hybridization overhead is another crucial future direction. Researchers can explore methods to optimize the integration of different optimization techniques in HPSO. This optimization can minimize computational overhead by intelligently determining when and how to employ local search mechanisms. By optimizing the hybridization process, the overall efficiency of HPSO can be improved, making it more suitable for real-time adaptive equalization applications.  
Automated parameter selection is another promising direction to overcome the challenges associated with algorithm selection and tuning. By developing automated methods, such as metaheuristic optimization or machine learning algorithms, the task of selecting appropriate parameter values can be automated. This enables HPSO to adapt and optimize its parameters based on the specific adaptive equalization problem at hand, reducing the manual effort and subjectivity involved in parameter tuning.  
Integration compatibility is a significant limitation that can be addressed through future research efforts. Investigating ways to enhance the compatibility of HPSO withdifferent equalization schemes and systems is essential. Developing adaptive mechanisms that seamlessly integrate HPSO with diverse equalization techniques and architectures can significantly improve its effectiveness and versatility in adaptive equalization tasks. Measuring the stability of HPSO is another important avenue for future work. We intend to incorporate Monte Carlo to measure the stability of HPSO during the convergence process by obtaining the standard deviation and mean value of the MSE curve.  
Enhancing the robustness and generalizability of HPSO is another crucial direction. Research can focus on reducing the algorithm's sensitivity to problem characteristics and environmental conditions. By developing mechanisms to handle diverse channel conditions, noise levels, and signal variations, HPSO can become more reliable and applicable in real-world adaptive equalization scenarios. Efforts can be directed toward reducing the development effort required for HPSO implementation. This can involve the creation of user-friendly software frameworks, libraries, or toolkits that provide pre-defined implementations of HPSO for adaptive equalization. By simplifying the development process, researchers and practitioners can more readily adopt and utilize HPSO, accelerating its application and impact in the field of adaptive equalization.  
Conceptualization, A.K., I.S.; methodology, S.G.K.; software, M.A.L.F., J.C.G.; validation, I.A.; formal analysis, A.K., S.G.K.; investigation, I.d.I.T.D., J.C.G.; resources, M.A.L.F.; data curation, I.S., S.G.K.; writing--original draft preparation, A.K., I.S.; writing--review and editing, I.A.; visualization, M.A.L.F., J.C.G.; supervision, I.A.; project administration, I.d.I.T.D.; funding acquisition, I.d.I.T.D. All authors have read and agreed to the published version of the manuscript.  
This study is supported by the European University of Atlantic.  
Not applicable.  
The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.