# Transient Time Reduction in Time-Varying Digital Filters via Second-Order Section Optimization
## 6 Conclusions  
In this paper, we introduced a novel method for introducing time-varying coefficients into a single second-order section (SOS), thereby reducing transient times without incurring excessive computational overhead. By selectively adapting only one SOS, we strike a balance between fully time-varying filtering and a purely static approach. Our analysis indicates that filters with sharper roll-offs, such as elliptic or Chebyshev designs, derive the most significant benefits, although even low-selectivity filters show moderate improvements. Practical demonstrations revealed a pronounced reduction in settling time under varying noise levels and signal conditions, underscoring the technique's robustness. We also highlighted the importance of carefully choosing the horizon and triggering mechanism to maximize performance gains. Overall, our findings confirm that a selective time-varying strategy can markedly enhance responsiveness while preserving the filter's stability.  
The time-varying SOS strategy has immediate relevance in scenarios demanding both rapid settling and stable operation, such as live audio signal processing and real-time communication systems. By reducing the filter's transient period, latency-sensitive applications--like wireless data transmission, virtual reality audio rendering, and adaptive control systems--can benefit from quicker responsiveness. Automotive sensor fusion is another promising domain, where faster convergence aids in smoother integration of rapidly updated sensor inputs. The method's targeted adaptation also suits power-constrained embedded devices that need to optimize transient behavior without expending excessive computational resources. Finally, its flexibility and moderate complexity make it a practical choice for any system where short-lived disturbances must be addressed quickly while maintaining robust long-term performance.  
Future work will involve refining the triggering mechanism for time-varying updates, especially in \"hot start\" scenarios where the filter has already been operating. Another promising line of research is to develop dynamic or adaptive horizons that better handle signals with narrow \"stairs\" or sudden changes. Extending this partial adaptation strategy to higher-order or multidimensional filters also merits investigation, as it may open doors to more sophisticated applications like 3D audio or MIMO communication systems. Real-time implementation is equally important, ensuring that the computational overhead of coefficient updates remains feasible for embedded and low-power devices. Finally, an in-depth exploration of the cost-benefit balance between the extra memory needs for storing time-varying parameters and the performance gains in transient reduction can guide practical design choices.  
Conceptualization, P.O. and J.P.; methodology, P.O. and J.P.; software, P.O.; validation, P.O. and J.P.; formal analysis, J.P.; investigation, P.O. and J.P.; resources, P.O.; data curation, P.O.; writing--original draft preparation, P.O.; writing--review and editing, J.P.; visualization, P.O.; All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Dataset available on request from the authors.  
The authors declare no conflicts of interest.