## 2 Materials and Methods
### Short-Time Fourier Transform Theory  
Fourier transform (FT) is a mathematical tool used to perform the linear decomposition of a signal into its constituent spectral components. The resulting decomposition provides spectral information of the analyzed signal but no time localization of the spectral content [18; 19]. To apply FT, it is necessary the signal to be stationary.  
Short-time Fourier transform is a Fourier-related transform able to deal with the non-stationarity characteristics of a signal, albeit in a limited way due to the length of the signal segment to be analyzed. The results yield information about the time localization of the spectral components, which are computed by performing a Fourier Transform over overlapped windowed segments of a signal to reduce spectral leakage [33]. The continuous form of the STFT is expressed in Equation (1):  
\[STFT[x(t)]=X(\tau,f)=\int_{-\infty}^{+\infty}x(t)w(t-\tau)e^{-\beta\pi ft}dt \tag{1}\]  
where \(x(t)\) is a time-varying signal, \(w(t-\tau)\) is a window function shifted by the parameter \(\tau\), \(e^{-\beta\pi ft}\) is the Euler identity, and \(X(\tau,f)\) is the Fourier transform of the windowed segment of \(x(t)\). The discrete form of the STFT is expressed in Equation (2):  
\[STFT[x(n)]=X(m,k)=\sum_{n=0}^{L-1}x(n)w(n-m)e^{-\beta\pi n}\frac{k}{L} \tag{2}\]  
where \(x(n)\), \(w(n-m)\), \(e^{-\beta\pi n}\frac{k}{L}\), and \(X(m,k)\) represent the same variables as those in the continuous form but dependent on the discrete variable \(n\) and \(k\).  
The results of the transformation can be displayed using a spectrogram, which provides a graphical representation of the spectral power distribution over time. This is obtained by squaring the absolute value of Equation (2):  
\[S_{x}^{w}(m,k)=\left|X(m,k)\right|^{2}=\left|\sum_{n=0}^{L-1}x(n)w(n-m)e^{- \beta\pi n}k\right|^{2} \tag{3}\]  
The spectrogram obtained with the STFT is distorted by the spectral leakage resulting from the windowing operation and the [PERSON]'s uncertainty principle. In signal processing, this principle states that an accurate estimation of both time and frequency cannot be obtained simultaneously. This implies that a wide window gives better frequency resolution but poor time resolution, while a narrower window gives good time resolution but poor frequency resolution.  
Since the length of the window is fixed and therefore, cannot be modified during the transformation, a balance must be determined a priori between the desired time and frequency resolutions according to the needs of the analysis. To determine the frequency resolution, we took into account the lower frequency limit of the VLF band, which is 3 mHz. To represent one full cycle of this frequency, we need 333.33 s. However, for nondeterministic signals (as is the case of the HRV), it is recommended to use analysis epochs longer than the slowest spectral component [13]. In a previous work, it is shown that the spectral power estimation of a multicomponent signal decreases as the epoch increases with respect to the slowest component [21]. Based on that work, we decided to use 10 min epochs. This decision also considered that the number of segments to be analyzed during HD decreases as the epoch increases, and becomes less representative of the dynamics of the estimated spectral power. Once we determined the time resolution, we calculated the frequency resolution obtained based on the uncertainty principle, which states that:  
\[\Delta t\Delta f\geq\frac{1}{2} \tag{4}\]  
where \(\Delta t\) is the time resolution and \(\Delta f\) is the frequency resolution. For \(\Delta t=600s\), \(\Delta f\) is at least 0.833 mHz, which is lower than the lower limit of the VLF band.