### Signal Extraction: Efficient Estimation, Unit Root-Tests and Early Detection of Turning Points

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Marrosero et al. By detecting fHR from both real and synthetic records, they showed that the Mermaid method outperforms the other two algorithms.

It also appears that this method is more efficient in batch and on-line operating modes, which is vital for real-time implementation. Sameni et al. In their work, the interpretation of the independent components obtained from multiple leads was closely related to the representation of vectorcardiograms of individual signals.

The subspaces of the fetal and maternal ECG signals are not completely different. By using dynamic filters, it should be possible to use a dynamic model that is time-synchronous with mECG and thereby to remove the maternal component while preserving the fetal one. The main aim of this investigation was to process aECG signals in order to extract fECG signals from them by applying and comparing two popular non-adapative signal processing ICA and PCA methods and then successfully determine the fetal Heart Rate fHR, which is one of the main indicators used to detect and diagnose fetal hypoxia Hyvarinen and Oja, Block diagram of the experimental setup for fECG signal extraction and quality assessment.

The FSE recordings were quality assured by clinical experts resulting in a set of reference markers precisely indicating the time of occurrence of the R-waves. Additionally, the results were statistically evaluated by means of the Bland-Altman method. We also provide the graphical interpretation of the results to facilitate the visual evaluation of the fECG signal separation process. We analyzed the estimated as well as the reference fECG signals and also carried out the evaluation of the MA as well.

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Non-adaptive methods have been applied to the fECG signal extraction problem Kahankova et al. Nevertheless, performance verification of the applied methods using real data is a more complex task than testing them with synthetic signals. In this research, we used abdominal ECG signals from a publicly available database as well as data acquired in Poland for details please see section 5. Each recording in the Polish data included four aECG signals. One recording could be tested by: 1 using 11 possible combinations of electrodes; 2 6 possible combinations when utilizing 2 signal channels; 3 4 possible combinations when using 3 channels; and 4 1 combination when using all of the channels simultaneously.

The pre-processing block includes a bandpass FIR filter with a bandwidth from 3 to Hz for isoline drift baseline wander correction. Moreover, it was necessary to equalize the amplitudes of both components in order to ensure that the maternal components had the same amplitude. This led to a significant increase in the fECG extraction performance. There are two signal sources in the maternal body, maternal and fetal heart, producing signals that can be measured by means of electrodes placed on the maternal abdomen.

Abdominal electrodes measure both maternal and fetal signals.

The signal from the first electrode is marked as x 1 t and the second as x 2 t. Each of these recorded signals was generated by weighing the signals s 1 t and s 2 t of the individual signal sources hearts. The relationship between the signals obtained from the electrodes x i t and the signals generated by the individual hearts s i t can be expressed as follows:.

Principle of the ICA method: linear mixture of the unobserved independent source signals - s 1 t , s 2 t : source signals; and x 1 t , x 2 t : recorded signals.

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The values of the signals are considered samples instantiations of the random variables, not functions of time. The aim of ICA is to return the linear unmixing matrix W in order to acquire the estimated independent components y such that:. We assumed here that the number of independent components s is equal to the number of observed variables; this is a simplifying assumption that is not completely necessary.

Otherwise, the best unmixing matrix, that separates sources as independent as possible, is given by the generalized inverse Penrose-Moore matrix:. Whitening can be performed by using the eigenvalue decomposition of the covariance matrix:. A new whitened vector is created as follows:. There are four basic limitations of the ICA method Lee et al. Only one original independent component can have Gaussian distribution. If multiple Gaussian sources exist, the ICA method is not able to extract these sources independent components from the data x. If we have an n -dimensional data vector, then we can find a maximum of n -independent components using the ICA method.

Some or all calculated y components can be multiplied by -1 with respect to the original components. The order of the original independent components cannot be determined by the ICA method. It is a way of identifying patterns in data, and expressing the data in such a way as to highlight their similarities and differences. Since patterns can be hard to find in data of high dimension, where the luxury of graphical representation is not available, PCA is a powerful tool for analyzing data.

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Additionally, further morphological analysis MA is provided for both reference and estimated data. The other main advantage of the PCA method is that we can find patterns in the data and then we can compress the data, i. We have a set of data that consists of m input vectors. One of the approaches to this solving problem is the projection of the data along orthogonal basis vectors through eigen decomposition of the covariance estimate of the input.

That means that the module of the base vector u i is equal to one. Equation 12 is also called the PCA data model. The value s i is the projection of the input data x to the orthogonal base vectors u i. We can also do the opposite approach. After PCA's main components s i i. The aim of the PCA method is to find a linear orthogonal transformation represented by the U matrix so that the variance of s i projection is maximum.

## Signal Extraction: Efficient Estimation, 'unit Root'-Tests and Early Detection of Turning Points

The variance can be defined as:. From equation 14 it follows that the variance of the s projection is a function of the unit vector u i. The problem about estimating the base vectors u i is based on solving equation 16 :. Respective column vectors u i satisfy the orthogonality condition:. That means that U is an identity matrix 15 and can be expressed as:. On the horizontal axis, the values x 1 are plotted, whereas x 2 values are plotted on the vertical axis. The PCA method caused the rotation of the axes from the original position x 1 and x 2 to a position that is denoted by the letters s 1 and s 2.

The axes rotated in the direction of the vectors of the covariance matrix. The variation of this projection is the maximum of all possible directions rotations. The research material consisted of multichannel fetal electrocardiograms obtained from 12 different women at ITAM in Zabrze, Poland during established labors, between 38 and 41 weeks of gestation and saved in 12 different recordings. In all cases the additional scalp electrode was placed for a clinical indication and all patients consented to participate in this study.

Left : Electrode placement system for recording and analysis of bioelectric signals acquired from the maternal abdominal wall, the direct electrode attached to the fetus head sampling rate of 1 kHz, bandwidth 1 Hz— Hz and resolution of 16 bits. The abdominal electrodes configuration was comprised of four electrodes placed around the navel, a reference electrode positioned above the pubic symphysis and a common mode reference electrode with active-ground signal placed on the left leg. It is important to emphasize that the acquisition of abdominal fECG signals during labor presents additional challenges we had to be aware of.

Firstly, during labor the strongest signal was observed when the fetus was fully developed. On the other hand, the quality of abdominal signals acquired during labor was contaminated by considerable muscular activity of the uterus. Moreover, during signal acquisition the scalp electrode very often lost contact with the fetal head, causing temporary signal loss. Some fragments of the direct fECG signals were also distorted by interferences caused by maternal movements. Consequently, we had to select only those short fragments of the recordings for which there were no signal loss periods in the direct reference fECG signal.

The acquisition of direct fetal electrocardiogram was carried out with a typical spiral electrode, commonly used in the direct fECG channel of popular bedside fetal monitors. The accurate occurrence of these peaks was then verified off-line by a group of cardiologists, resulting in a set of reference markers precisely indicating the location of the R-waves. These markers were then stored together with the direct and indirect fECG signals.

Assuming the maximum width of the maternal QRS complex as ms and that of the fetal as 40 ms, the number of complexes without feto-maternal coincidence was 2, maternal and 5, fetal. We used quantitative measures dedicated evaluation indices related to the Signal-to-Noise Ratio SNR to evaluate the sufficiency and diversity of the research data to clearly distinguish between useful signal components and contaminating interferences in the abdominal signals. Assuming that the signal powers of P and T waves in the fetal and maternal complexes are negligible, the average power of interferences P N in the abdominal signal can be presented as an average signal power outside the QRS complex locations given by:.

Additionally, assuming that there is no correlation between the useful signal and the noise component, the average power of the maternal signal P M is given by:. Based on these values, the WMF index relating the maternal and fetal components can be defined as follows:. From the analysis of these indices based on the results obtained from the four abdominal signal channels, it could clearly be seen that for the research data used, the dominant signal was the mECG, whose amplitude was always more than three times higher than that of the noise component signal mean WM is Similarly, its relation to the fetal signal, as indicated by the WMF index, exceeded 7 dB on average range from 1.

### Associated Data

The WF indicator, informing the relationship between fetal QRSs and the noise component, varied very considerably. Its average value was 3.

In this section, we present our evaluation results of the fECG signals extracted from aECG signals acquired by using 4 abdominal electrodes. This direct signal was used for the verification of the extracted signal by comparing the fHR values determined from each of them. In the reference and extracted fECG signals, the R waves were detected and the intervals between subsequent peaks RR intervals were determined. Fetal HR was then calculated as the number of the RR intervals in 1 min.

It includes the mean fHR waveform calculated for all of the available data. This figure shows that 60 min of ECG data i. This dataset covers different gestation ages, fetal positions, maternal body mass indexes BMI , etc. In this study, we used two types of moving average filters MAF.