# Transient Time Reduction in Time-Varying Digital Filters via Second-Order Section Optimization
## 1 Introduction  
Digital filters are fundamental tools in signal processing, widely used in applications ranging from telecommunications and control engineering to audio engineering. Traditionally, filters are designed with fixed coefficients optimized to meet specific performance criteria such as frequency response and stability. However, in dynamic systems or time-varying environments, static filters may fail to deliver sufficient responsiveness to changing signal characteristics [1, 2, 3]. To address this, researchers have developed both adaptive filtering techniques--such as LMS, RLS, or Wiener filters--and time-varying filter architectures that modify filter characteristics over time [4, 5, 6]. Adaptive filters typically update their coefficients in real time, guided by an error signal derived from system feedback. While powerful, they are often computationally intensive and may require high memory throughput, making them less suitable for real-time or embedded systems with strict resource constraints. In contrast, time-varying filters with precomputed parameter trajectories offer a fixed, feedforward method for enhancing transient performance without incurring significant runtime/online cost.  
Time-varying filters, characterized by coefficients that change over time, offer enhanced flexibility and adaptability in signal processing. Despite their advantages, time-varying filters introduce new challenges, such as increased computational complexity and potential stability issues. One critical concern is transient time--the period required for a filter to settle into its steady-state response (typically within 2%). This aspect is particularly important in real-time applications, where delays can degrade system performance. Higher-order filters, while providing better selectivity, are especially prone to prolonged transients, further complicating the design process.  
This paper investigates the use of numerical optimization of second-order section (SOS) coefficients to minimize transient times in time-varying filters, offering a pathway to more efficient signal processing.  
Transient time is a crucial factor in filter performance, representing the duration required for a filter to reach its steady-state response after a change in input or configuration. Additionally, a filter's initial conditions (hot/cold start) significantly influence transient duration. In many applications, especially real-time systems, long transient times can introduce unacceptable delays, reducing overall system responsiveness and accuracy. Higher-order filters, while providing precise control over signal characteristics, often exhibit longer transient times, making their practical implementation more challenging. Reducing transient time without compromising filter stability and performance is a significant design challenge that demands innovative solutions.  
The primary objective of this paper is to minimize transient time in digital filters with time-varying coefficients. To achieve this, we focus on decomposing higher-order filters into second-order sections (SOSs) to improve numerical stability and parameter control. A numerical optimization approach is proposed to fine-tune the coefficients of one or more SOSs, effectively reducing transient behavior. The goal is to strike a balance between transient time reduction and other critical performance metrics, including stability, steady-state accuracy, and memory consumption. By demonstrating the effectiveness of this method, this study aims to provide a practical framework for designing time-varying filters with improved responsiveness while maintaining efficient memory usage for storing time-varying coefficients.