# Online Algorithm for Deriving Heart Rate Variability Components and Their Time-Frequency Analysis
###### Abstract  
Heart rate variability (HRV) containing four components of high (HF), low (LF), very low (VLF), and ultra-low (ULF) frequencies provides insight into the cardiovascular and autonomic nervous system functions. Classical spectral analysis is most often used in research on HRV and its components. The aim of this work was to develop and validate an online HRV decomposition algorithm for monitoring the associated physiological processes. The online algorithm was developed based on variational mode decomposition (VMD), validated on synthetic HRV with known properties and compared with its offline adaptive version AVMD, standard VMD, continuous wavelet transform (CWT), and wavelet package decomposition (WPD). Finally, it was used to decompose 36 real all-night HRVs from two datasets to analyze the properties of the four extracted components using the Hilbert transform. The statistical tests confirmed that the online VMD (VMDon) algorithm returned results of comparable quality to AVMD and CWT, and outperformed standard VMD and WPD. VMDon, AVMD, and CWT extracted four components from the real HRV with frequency content slightly exceeding the previously recognized ranges, suggesting the possibility of their modes mixing. Their ranges of variability were assessed as follows: HF: 0.11-0.40 Hz; LF: 0.029-0.14 Hz; VLF: 4.7-31 mHz; and ULF: 0.002-3.0 mHz. 202515210  
by increasingly complex integral pulse frequency modulation (IPFM) models [11; 12]. A well-studied relationship is the change in heart rate (HR) due to increased SNS or decreased PNS activities [13; 14; 15]. However, much more information is hidden in HRV spectral properties and, more specifically, in its four frequency bands. The high-frequency (HF) component (0.15-0.4 Hz) is mostly influenced by the PNS activity via the vagus nerve, being synchronized with the respiratory cycle [13; 14; 16; 17]. The low-frequency (LF) component (0.04-0.15 Hz) is modulated by the SNS, PNS, and baroreflex system [13; 14; 16; 17]. Both very-low-frequency (VLF) (0.0033-0.04 Hz) and ultra-low-frequency (ULF) components (\(<\)0.0033 Hz) are not well understood, and not as often used as HF or LF. VLF and ULF are generally thought to be associated with the circadian rhythm [14; 16; 18], temperature regulation [14; 17; 19], renin-angiotensin system [17; 20], physical activity [21], and vasomotor activity [14]. In summary, these properties are not well known (especially for VLF and ULF), and their derivation may contribute to further research in the area of network physiology focusing on physiological interactions within the human organism. HRV is a non-stationary, irregularly time-sampled signal and is mostly processed after interpolation to the evenly spaced samples, usually using cubic spline interpolation with a 4 Hz sampling rate [22; 23; 24]. The short-term analysis can minimize the impact of signal non-stationarity using fragments of 5 min or less [23; 25]. This is a common approach but does not allow for the separation of lower-frequency components like VLF and ULF. To extract ULF, a long-term analysis is needed, a minimum of several hours (typically 24 h) [16], but then the problem of stationarity must be considered. Another issue is the presence of outliers in the signal. The typical approach for short-term analysis contains the removal of the whole corrupted fragments [4; 26; 27] and modification of erroneous samples [28; 29; 30], but this can result in the modification of the phase of the frequency components, especially VLF and ULF. Another method, rarely used, is online analysis, often performed similarly to short-term analysis, extracting only HF and LF with sliding windows of a few or tens of seconds [29; 31]. There are many methods by which the HRV spectrum can be derived [32]. Among the most popular are autoregressive modeling, which requires finding the optimal model order [5; 9; 24; 33] and the short-time Fourier transform (STFT) that can determine local frequency components from short quasi-stationary fragments of a signal [1; 29; 33; 34].