## 3 Methodology
### CTG Signal Analysis
#### 3.2.3 FHR Signal Decomposition and TV-AR Spectrum Computation
\[H_{i}[z]=\frac{IMF_{i}[z]}{E[z]}=\frac{1}{1+\sum_{k=1}^{p}a_{i,k}z^{-k}}. \tag{3}\]  
Then, the stationary AR spectrum can be computed by evaluating \(H_{i}[z]\) around the unit circle in the complex plane; i.e., \(z=e^{j2\pi f}\):  
\[SAR_{i}(f)=\frac{\sigma^{2}}{\left|1+\sum_{k=1}^{p}a_{i,k}e^{-j2\pi f}\right|^ {2}}. \tag{4}\]  
Considering that the phenomenon of focus involves dynamics strongly variant time, stationary AR modeling [47] is no longer suitable for the analysis. For this reason, we employ TV-AR modeling as an appropriate method for the proposed spectral analysis. A representation of this time-varying version is presented in Equation (5), where \(a_{i,k}(n)\{k=1,2,...,p\}\) are now the time-dependent coefficients, whose set of values \(a_{i,k}\) is updated sample-by-sample \(n\). \[\text{IMF}_{i}[n]=-\sum_{k=1}^{p}a_{i,k}(n)\text{IMF}_{i}[n-k]+e[n]. \tag{5}\]  
Likewise, this operation results in a time-varying AR model transfer function, which is represented by:  
\[H_{i}[z,n]=\frac{IMF_{i}[z,n]}{E[z,n]}=\frac{1}{1+\sum_{k=1}^{p}a_{i,k}(n)z^{- k}}. \tag{6}\]  
Analogously, the time-varying version of the AR spectrum presented in Equation (4) can be described by:  
\[SAR_{i}(f,n)=\frac{\sigma^{2}}{\left|1+\sum_{k=1}^{p}a_{i,k}(n)e^{-j2\pi fk} \right|^{2}}. \tag{7}\]  
This last equation allows performing the proposed time-varying analysis, because the computed spectrum resultantly depends on frequency as well as on time. Following [13], in this work, the AR coefficients (\(a_{k}(n)\)) were computed by using a recursive least squares algorithm with a forgetting factor (\(\lambda\)). The forgetting factor is a model parameter related to the memory time considered in the studied process. Therefore, considering that our approach focuses on the analysis of the same physiological phenomenon as in [13] (analysis of fetal ANS activity over time), we decided to use the same forgetting factor set to 0.99. For the selection of a suited model order \(p\), we first considered the characteristics of the IMFs. An IMF corresponds to a non-overlapping function modulated in amplitude and frequency [46]. Given that concept, we assumed that in the frequency-domain, for each IMF, only one main frequency component contained the information of interest over time; i.e., in the AR spectrum, only one absolute maximum peak involved such information at each sample \(n\). With this in mind, we studied a set of thirty IMFs (selected randomly from different FHR signals extracted from the database). For each selected IMF, the AR spectrum was computed by using different AR model orders in the range of 4 th and 10 th. Then, we examined which order \(p\) offers a better spectral representation of the frequency component of interest, taking into account that only one main component should be clearly distinguished. After a visual analysis, and following [18], we decided to use a 6 th AR model order \(p\). Examples of the CEEMDAN and TV-AR spectrum computation are shown in Figure 5, in the left and right columns, respectively. This example was obtained by the decomposition of the detrended FHR signal presented in Figure 4b. The TV-AR spectrum is represented by a color map, where the lowest and highest energy level are shown in blue and yellow, respectively. Additionally, for better visualization of the spectral dynamics, the energy values have been normalized between 0 and 1 for each sample \(n\). In addition to the time-varying spectral representation, we computed the average of the TV-AR spectrum (right column), which illustrates the frequency band involved for each IMF. It is important to note that the number of IMFs obtained from the 372 signals of the dataset (see Section 3.1) varies from 12 to 17. Nevertheless, based on the spectral information described above (averaged TV-AR spectrum), we decided to study the first ten IMFs, thereby considering the entire FHR frequency band of interest.