## 3 Methodology
### CTG Signal Analysis
#### 3.2.3 FHR Signal Decomposition and TV-AR Spectrum Computation  
As mentioned above, the FHR signal decomposition is performed by the CEEMDAN method. The empirical mode decomposition (EMD) [48], which corresponds to a previous version of this technique, has been used in different biomedical engineering approaches [49]. Particularly, in FHR signal analysis it has been used for FHR estimation from Doppler Ultrasound signals [57],  
Figure 4: Signal detrending operation. (**a**) Preprocessed FHR signal presented in Figure 3a (blue) and computed floating-line (red); (**b**) detrended FHR signal obtained by the subtraction operation of signals presented (**a**). Figure 3: (**a**) Preprocessed FHR signal; (**b**) preprocessed UC signal. analysis of the FHR signal components in order test the reliability of the EMD performance [58], FHR baseline estimation with analysis of fetal movements [59], assessment of the high frequency FHR information [60], enhancement of the CTG signal quality by reducing signal artifacts [50], and FHR signal feature extraction and classification [49; 61]. These works have concluded that EMD could be appropriate for the analysis of the FHR signal components in the time-domain. Since EMD was proposed [48], it has developed to overcome different problems associated with the decomposition [46; 62; 63]. The main issue involved in the original EMD method is the _mode mixing_; i.e., more than one mode of oscillation may contribute to one IMF, or one mode can spread across different IMFs. In consequence, it can lead to an unreliable decomposition of the FHR signal because the IMFs may not correctly describe the FHR signal dynamics of interest. In order to solve this issue, in [46], the improve complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN) method has been proposed. This updated version solves the described mode mixing problem and subsequent drawbacks, such as residual noise and _spurious_ modes generated by the operation itself. Therefore, in this work, we performed the FHR signal decomposition based on the CEEMDAN method. Following [61], the IMFs were computed by using a noise standard deviation (Nstd) set to 0.03, and both the number realizations (NR) and the maximum number of sifting iterations (NI) were set to 50. For a more in-depth explanation of CEEMDAN, please refer to [46]. Once the FHR signal is decomposed, the spectral analysis is performed for each IMF individually. As explained at the beginning of Section 3, the stationary version of AR modeling has been studied in several approaches, and revealed different advantages compared with non-parametric, spectral-based methods [64]. An AR model assumes that the signal value \(y[n]\) at the current sample number \(n\) in a data sequence \(y[1],y[2],\ldots,y[N]\) can be modeled in function of the \(p\) most recent sample values \(y[n-1],y[n-2],\ldots,y[n-p]\), and a Gaussian white zero-mean noise \(e[n]\) of variance \(\sigma^{2}\) as presented in Equation (2). In this equation, \(p\) corresponds to the model order, whose value is generally much smaller than the sequence length \(N\), \(a_{k}\{k=1,2,\ldots,p\}\) are the stationary AR coefficients, and \(n\) is the discrete-time index. \[y[n]=-\sum_{k=1}^{p}a_{k}y[n-k]+e[n] \tag{1}\]  
Considering that the AR model is applied for each IMF, Equation (1) can be expressed as presented in Equation (2), where \(i\) represents the index number of the computed IMFs. \[\text{IMF}_{i}[n]=-\sum_{k=1}^{p}a_{i,k}\text{IMF}_{i}[n-k]+e[n] \tag{2}\]  
The z-transform can be applied to Equation (2), whose resulting model transfer function is presented in Equation (3). Here, \(IMF_{i}[z]\) and \(E[z]\) are the z-transforms of the time series \(\text{IMF}_{i}[n]\) and the noise \(e[n]\), respectively.