# Transient Time Reduction in Time-Varying Digital Filters via Second-Order Section Optimization
## 2 Related Work  
### Literature Review  
One early approach to mitigate long transients was to manipulate a filter's initial conditions rather than its coefficients. Preloading the filter output with appropriate non-zero initial values can counteract the natural transient decay. [PERSON] and [PERSON] [PERSON] and [PERSON] (2016) demonstrated this technique for an ECG notch filter, effectively suppressing startup oscillation in a 50/60 Hz notch by initializing the filter's output to cancel the impulse response. Later, [PERSON] and [PERSON] extended this idea by designing an FIR notch filter derived from an IIR prototype with carefully set initial conditions, achieving significant transient suppression ([PERSON] and [PERSON], 2016). Another method is to analytically compute the required initial outputs of internal delay elements (filter memory) for a given initial input, often by projecting the initial portion of the input signal onto the filter's homogeneous response. [PERSON] et al. [PERSON] et al. (2016) introduced an iterative signal-shifting algorithm to automate this process. In their approach, the input signal is shifted in time and fed into the filter repeatedly, adjusting the shift until the filter's internal memory converges to values that produce no transient. These methods treat the filter's transient as a signal to be canceled via initial state injection. Additionally, a more general strategy is to make the filter coefficients time-varying during the transient. Instead of keeping a fixed pole and zero configuration, the filter starts with a gentler configuration (to produce a fast decay) and then smoothly transitions to the final, more selective configuration. [PERSON]'s 2010 work introduced a Q-factor-varying notch filter where the pole radius is initially reduced (broadening the notch but damping the ringing) and then increased exponentially to the desired high Q value [10]. This approach significantly shortened the transient duration while still producing a sharp notch in steady state. Similarly, [PERSON] et al. [11] proposed a pole-radius variation scheme for notch filters, demonstrating transient suppression by dynamically adjusting the pole radius over time. In the low-pass filter domain, [PERSON] and [PERSON] [12] developed a time-varying IIR low-pass filter based on an analog oscillatory model. They discretized a second-order system and temporarily modulated its natural frequency and damping ratio to quickly attenuate the step-response transient. By adjusting these parameters (essentially the pole locations) during the initial response, they suppressed overshoot and reduced settling time. An iterative optimization procedure was applied to fine-tune the parameter trajectory for minimal settling time. This trend has continued in various applications, including sensing and robotics. For instance, a time-variant filter for force/torque sensors modulates its coefficients at contact onset, reducing impact force transients [13]. [PERSON] and [PERSON] [14] demonstrated a second-order low-pass filter that automatically adjusts its parameters (via a nonlinear control loop) immediately after a sudden change in the input. By momentarily widening the filter's bandwidth and/or altering its damping factor, the filter settles much faster than a conventional static design, all while preserving the intended low-pass frequency response once the parameters revert. They also analyzed the stability of this linear time-varying (LTV) filter, showing that it maintains bounded-input bounded-output stability under the parameter adaptation scheme. Building on the idea of time-varying coefficients, [PERSON] et al. [15] proposed a variational approach to design the optimal time-variation trajectory for an IIR filter's cutoff frequency. Using calculus of variations, they derived a closed-form time-course for the filter's pole movement that minimizes the rise time (settling time) of the step response.