## 1 Introduction
However, it suffers from an inherent compromise between time and frequency resolution depending on the shape of the analysis window. Longer windows are more suitable for resolving signal components in the frequency domain, while short windows are more suitable for resolving signal components in time. One way to avoid this problem is to shift the power to the center of gravity by using the method of reassignment [23]. The reassignment method achieves a high time-frequency (TF) resolution, but it cannot be used to reconstruct the original signals. The synchrosqueezing transformation estimates the local center of gravity only along the frequency axis [23]. The methods of reassignment and synchrosqueezing have previously been applied to HRV estimation [24; 25]. The separable kernel TEDs have the flexibility to independently adjust the smoothing and thereby the resolutions along the time and frequency axes. Therefore, these methods generally perform better than the spectrogram [26]. The modified B distribution (MBD) uses a time-only smoothing kernel defined as [27]. Previous studies show that, due to sharp cutoff of the MBD kernel in the frequency domain, the MBD achieves high energy concentration for many real-life signals with slow variation in the slope of instantaneous frequencies such as HRV and EEG seizure signals [28].The S-method can be used to compute a TFD that has a high energy concentration for auto-terms with significantly reduced cross-terms as a result, consequently combining the advantages of both the spectrogram and the Wigner distribution [29]. The S-method is a computationally efficient method and has been successfully applied in a number of areas [30; 31]. The MBD and the S-method do not have any parameters to adapt a specific direction of the smoothing window in the TF domain. Therefore, these methods cannot achieve an optimal energy concentration for frequency modulated signals that follow a certain direction in the TFD. The adaptive optimal kernel TFD (AOKTFD) achieves this by locally adjusting the shape of the smoothing window [32]. However, the AOKTFD method fails to perform well when there are a number of signal components of varying amplitudes and directions. In such cases,the AOKTFD optimizes the direction based on the strongest component and therefore degrades the resolution of weaker components [33]. The adaptive directional time frequency distribution (ADTFD) overcomes this limitation by adjusting the direction of the smoothing function locally at each point in the TF plane. Previous studies have shown that the ADTFD offers high resolution and can be used to estimate the instantaneous frequency of closely spaced signal components [33]. Due to its high resolution, the ADTFD has found applications in e.g., classification of EEG signals and direction of arrival estimation [34; 35]. Many measured signals can also be modeled using amplitude modulation frequency modulation (AM-FM) signals, [26; 36]. The measured signal needs to be separated from a given mixture so that the resultant signal is mono-component. This can be achieved by use of the empirical mode decomposition, which breaks down a given signal into a number of intrinsic mode functions [37]. Each intrinsic mode function has only one extreme value between zero-crossings and the mean is zero. The method, called the Hilbert-Huang transform, has found a number of applications in biomedical engineering [38; 39; 40]. The empirical mode decomposition suffers from mode mixing problems for signal components close in the TF domain [23]. Therefore, in order to accurately estimate the instantaneous frequency of multi-component signals, the quadratic TF methods, and particularly the adaptive TFDs, achieve better performance than the empirical mode decomposition based methods [23; 41].