## 2 Data Set Construction
### CCSN GW Signals  
We start our investigation by obtaining numerical data associated with the emergent HFF present in a CCSN GW signal, particularly its slope. To do this, we generated CCSN GW signals [5]. These waveforms resemble a CCSN GW signal with an HFF that changes linearly and continuously over time. Mathematically, CCSN GW signals are constructed by modeling a second-order linear, non-homogeneous differential equation with random forcing [5]. Random forcing is understood as the integration of stochastic influences into the physical system. These stochastic influences can be characterized as external, unpredictable disturbances impacting the system's dynamics. Such disturbances are notably significant in systems exhibiting stochastic behavior, where deterministic models may inadequately capture the actual behavior of the system, as is the case with CCSN GW signals. In oscillatory systems and vibrations, such as the damped harmonic oscillator, random forcing denotes unpredictable external forces--exemplified by thermal noise or vibrations--that are mathematically represented through the equation  
\[\frac{d^{2}h(t)}{dt^{2}}+\frac{\omega(t)}{Q}\frac{dh(t)}{dt}+\omega(t)^{2}h(t) =s(t)\, \tag{1}\]  
where \(h(t)\) denotes the time-dependent solution, \(\omega(t)\) the time-dependent natural angular frequency, and \(Q\) the quality factor, indicating the sharpness of resonance in an oscillatory system. It is crucial in differential equations' modeling damped harmonic oscillators to quantify the underdamping and represent the number of oscillations before significant energy dissipation. Finally, the driving force \(s(t)\) is defined as follows:  
\[s(t)=s_{n}\delta(t-t_{n})\, \tag{2}\]  
\(\delta(t-t_{n})\) denotes a Dirac's Delta function evaluated at \((t-t_{n})\), with \(t_{n}\) being the random activation time of the driving force and \(t\) being an independent variable that defines the dynamical evolution of the system.  
To solve the differential Equation (1), the numerical Euler method can be used. It is feasible to apply Runge-Kutta methods, either RK2 or RK4, to address the second-order linear non-homogeneous differential equation with random forcing, which is implemented to model the CCSN GW signals produced. The discretization of random forcing at each time step is necessary, similar to our approach. This can be achieved using a random number generator or by sampling from a specified distribution, such as the Dirac delta function. The selection between RK2 and RK4 hinges on the desired precision and computational efficiency. RK4 typically offers greater accuracy than other methods, including ours, though it demands more computational work per time step. Once the numerical solution is obtained, we can adjust the set of parameters that control the behavior of the equation. These parameters include the duration of the signal, the initial and final frequencies (which encode the HFF), and the quality factor. By altering these parameters, we can generate different solutions, each of which corresponds to a distinct signal with distinct slope values of the HFF. In our study, we specifically created a set of 100 CCSN GW signals with HFF slopes spanning from 500 Hz/s to 5000 Hz/s. These values are in agreement with the numerical estimates reported in the GW astronomy literature, as referenced in [1, 8, 25, 33]. One of the advantages of these models lies in the low computational cost of producing them, compared to those generated by numerical simulations [6, 34]. On the other hand, the explicit linear behavior of the HFF in the model facilitates the determination of the slope of the HFF, denoted \(s\), by applying the following equation:  
\[s=\frac{\Delta f}{\Delta t}=\frac{f_{final}-f_{initial}}{t_{final}-t_{initial}}\, \tag{3}\]  
where \(\Delta f\) and \(\Delta t\) denote the frequency variation and its duration, respectively. Figure 2 illustrates a sample with different analytical CCSN GW signals.