# Transient Time Reduction in Time-Varying Digital Filters via Second-Order Section Optimization
## 2 Related Work
### Literature Review
This optimal time-varying design achieved a shorter transient than previous ad hoc parameter variation rules, highlighting the benefit of an optimized coefficient schedule. [PERSON] and [PERSON] [16] recently presented a feedback-structured IIR notch filter with transient suppression achieved by continuously varying the feedback gain. In their design, the pole radius (which determines the notch sharpness and transient length) is kept lower than its final value at the moment of filter startup, thus damping the initial response. [PERSON] et al. [17] took a similar time-varying approach by leveraging a lattice wave digital filter structure for the notch. Their lattice notch filter begins with a relatively wider notch (lower \(Q\)) and progressively reduces the notch bandwidth as time advances, effectively shortening the ringing duration. The lattice wave digital implementation ensures numerical robustness and was shown to produce minimal transient overshoot, which the authors validated through FPGA implementation results. In addition to time-varying strategies, there are also design-time techniques to improve notch filter performance. [PERSON] et al. [18] pursued a minimax optimization approach for a fixed-coefficient notch filter targeting 50 Hz noise in ECG. By formulating the filter design as an optimization problem, they adjusted the pole-zero locations to minimize the worst-case error between the desired ideal notch response and the realized filter response. Introducing time variation in filter coefficients raises important stability considerations. In linear time-invariant (LTI) filters, stability is assured by having all poles inside the unit circle (for discrete-time filters). With time-varying filters, however, poles move over time and traditional LTI stability criteria no longer directly apply [19; 20]. [PERSON]'s work [21] introduced an algebraic theory of poles and zeros for linear time-varying systems, providing a groundwork for understanding how system eigenvaluesgeneralize in the time-varying case. This framework helped define notions of \"instantaneous\" poles or spectral values that evolve with time, which is critical for discussing stability beyond the static pole locations of LTI systems. Notably, [PERSON] [22] presented a necessary and sufficient criterion for exponential stability of linear time-varying (LTV) discrete-time systems. Recent studies have extended time-varying filter techniques across various domains. For instance, [PERSON] and [PERSON] [23] embed a command filter with time-varying gain into a backstepping control scheme, simplifying the controller design for high-order systems while preserving prescribed transient performance. [PERSON] et al. [24] develop an adaptive all-pass filter to track nonstationary propagation delays, employing an LMS-style coefficient update that continuously adjusts a filter's parameters for accurate time-varying delay estimation. Furthermore, [PERSON] et al. [25] introduce a novel time-varying filtering algorithm based on a short-time fractional Fourier transform with time-dependent transform order, achieving effective filtering of multi-component signals whose spectral characteristics change over time. In addition to these algorithmic developments, time-varying filters have proven effective in specific applications. [PERSON] et al. [26] present a Kalman filtering approach for bearing prognostics in which the filter's dynamics evolve with the system's degradation state, allowing the model to automatically adapt across different wear stages and improve remaining-life prediction accuracy. Similarly, in the audio domain, [PERSON] and [PERSON] [27] implement time-varying IIR filtering in binaural hearing aids to accommodate device nonlinearities and dynamically changing acoustic environments, illustrating the broad applicability of time-varying filter design in diverse real-world systems.