1. Introduction
The electrocardiogram (ECG) signal, in a non-invasive method, incorporates of the maternal ECG (MECG) signal, the fetal ECG (FECG) signal, and several sources of interference, such as power line interference, baseline wander, motion artifact, fetal brain activity, muscle artifact, as well as noise, such as instrumentation noise [
1,
2,
3]. FECG signal is used to monitor the health status of the fetus by determining its maturity level, reactivity, development, and existence of fetal distress [
4].
FECG extraction and enhancement method requires the elimination of the MECG as well as optimal detection of the FECG. The frequencies of both signals are a few Hertz’s and are possibly overlapping. Thus, separating them using the conventional linear filter fails. To address this problem, a large number of FECG extraction algorithms have been proposed over the past few decades. Some of these algorithms were based on the blind source separation (BSS) or blind source extraction (BSE) techniques [
5,
6]. In general, the extraction algorithms can be classified as either spatial (non adaptive) or temporal (adaptive) algorithms [
7]. Examples of the BSS/BSE based non-adaptive algorithms include principal component analysis (PCA) [
8], independent component analysis (ICA) [
8,
9], time scale image (TSI) and singular value decomposition (SVD)/ICA [
10], periodic component analysis [
11], parallel linear predictor (PLP) filters [
12,
13], template subtraction (TS) [
14], artificial neural network (ANN) [
15], support vector regression (SVR) [
16], tensor decomposition (TD) [
17], deflation [
18], adaptive comb filter (ACF) [
19], and null space component (NCA) [
20]. Examples of the adaptive algorithms include the multi-sensory adaptive noise canceller (MSANC) [
7], fast adaptive orthogonal group ICA [
21], adaptive Volterra filter (AVF) [
22], adaptive neuro fuzzy inference system (ANFIS) and wavelet transform [
23], Kalman filtering [
24], event synchronous canceller [
25], and type-2 adaptive neuro-fuzzy inference systems [
26].
The PCA method is a standard statistical technique focused on finding a transformation matrix that transforms the input data to another set of data that are uncorrelated, yielding an estimate of the unknown source signals. The method is highly affected by noise and the extraction is weak if the set of input data are statistically independent [
8]. The quality of the separation was improved by employing ICA that eliminates the higher-order dependence, rather than imposing second order dependence in PCA [
8]. The ICA is affected by significant noise, like PCA, and it has some limitations when used alone, unless it is combined with another method. For instance, the work in [
9] used ICA to separate FECG signals and detect and classify of mother and fetal heart beats based on compressive sensing (CS) theory. ICA has been also used together with the wavelet decomposition [
27], TSI and SVD [
10], and adaptive noise cancellation [
28], to extract FECG signals.
The work in [
11] was intended to remove the MECG artifacts from FECG recording, by extracting the most periodic linear mixtures of a recorded ECG. Some extraction merits were recorded using this method. In [
12], the work presented a novel BSE algorithm using a class of PLP filters whose input is the covariance matrix of the whitened data, while the estimated source signals being considered as the parallel filter coefficients. The method has the merits of solving the power level ambiguity, and has a fast convergence. The work in [
13] employed the BSE based PLP to extract FFEC signals from ECG recording. The template subtraction technique was applied in [
14] to remove MECG from abdominal signals using an approximation of the current MECG segment based on a linear combination of previous MECG segments aligned on the R-peak.
The work in [
15] analyzed the ECG signal based on three different ANN classifiers and combined based (discrete wavelet transform and morphology) features. In [
16], the SVR technique was applied to approximate the nonlinear mapping of the MECG component from thoracic signal then extract the FECG signal by subtracting the mapped thoracic signal from an abdominal signal. The technique could obtain good results for the small sample training set. The work in [
17] employed a robust tensor decomposition and extended Kalman filter (EKF) to extract the FECG signal and estimate its R-peaks locations. The results showed efficient estimation of the R peaks. In [
18], an online version of an iterative subspace denoising procedure was proposed for removing MECG from abdominal signals.
BSS-based ACF was presented in [
19] to estimate quasi-periodic component from physiologic signal, such as ECG, by adjusting the temporal variations in fundamental frequency. The method has large FECG extraction failure and can obtain reasonable estimates of the FECG signals for approximately 60% of the abdominal signals in the database used in the paper.
The comparison between the relative performances of these algorithms is a challenging task due to the absence of a large public database and of also the absence of a defined evaluation methodology. However, it is possible to highlight the strengths and weaknesses of limited algorithms, evaluated on the same database and using the same methodology [
29].
The NCA was proposed in 2007 by R. B. Chena and Y. NianWub [
30] to solve the over-complete BSS problem. The solution space of the source signals were characterized by the null space of the mixing matrix using SVD. The problems were formulated in the framework of the Bayesian latent variable model. The work was only applied to three sound signals. There is no information about the performance of this approach when the number of signals is increased. The computational complexity (CC) of this algorithm was not provided. In addition, there were no comparisons with other methods. Another NCA algorithm was presented in [
31] for noisy mixture. This algorithm used a transformation matrix to resolve the rotation ambiguity and extract the source signals that were assumed to be linearly independent. The initial guess of this algorithm depends more heavily on the solutions as compared with ICA. In addition, it has higher complexity than many existing ICA methods. The work in [
20] presented an extension of NCA framework, named the constraint NCA (c-NCA) approach. This approach was considered as an alternative approach to the c-ICA. The c-NCA used signal-dependent semidefinite operators, which is a bilinear mapping, as signatures for operator design. A prior knowledge of how the data are prepared, collected, and mixed, is needed in this approach. This method has many issues. First, the algorithm requires a little knowledge about the sources during initialization such as imposing sparsity constraints on representing source signals [
20]. This is not suitable for real-life cases. Second, the condition for convergence requires the calculation of maximal eigenvalues of the Hessian matrix of the objective function. The calculation of eigenvalues is numerically intractable. Third, the complexity of the algorithm is high and approaches
, where
is the number of iterations,
is the number of proximal splitting iterations, and
N is the number of samples. Thus, designing new null space separating operator with less computational complexity, with no initialization constraint, and fast convergence, is crucial.
In our previous work [
32], we proposed a new null space algorithm for complete and over-complete BSS of auto-regressive source signals. Matrix factorization was used to construct the separation (also called transformation) matrix. The algorithm was tested and showed successful extraction for speech and Gaussian signals. The algorithm has high computational complexity. However, it is less than that in [
20]. An alternative approach of estimating the null space separation matrix with less computational complexity is possible by computing the idempotent transformation matrix (ITM) [
33].
This paper is aimed to develop a non-adaptive FECG detection and extraction algorithm, based on using the null space approach in estimating the FECG and MECG signals from the ITM. The algorithm first reduces the effect of noise and interference using pre-processing filters, computes the ITM, then extracts the FECG and MECG signals from the null space of ITM. The algorithm detects also the fetal heart rate and uses it to improve the quality of extracting the FECG signal. A comparison between the proposed algorithms and other similar algorithms will be provided.
The rest of this paper is organized as follows. In
Section 2, we briefly define the BSS problem and how it can be used in FECG extraction. The ECG signal is also illustrated in this section. A review on the popular FECG extraction methods (PCA, FastICA, and PLP), in the context of BSS, is shown in
Section 3. In
Section 4, we present the proposed FECG and MECG extraction algorithms, and how to detect the R peaks in the QRS complex. The experimental results are demonstrated and discussed in
Section 5 and
Section 6, respectively, as well as some topics for future work. Finally,
Section 7 concludes the paper.
2. Problem Formulation
The biological ECG signal of a pregnant woman is a composite signal between the FECG, MECG, and the noise. It has been proven that the noiseless ECG signals can be modelled using the linear BSS model expressed by [
8]:
where
is the
zero mean recorded ECG mixture signals, from the thorax and the abdominal channels,
is the
unknown full rank mixing matrix,
is the
unknown source signals (the FECG and the MECG signals), recalling that
M is the number of recorded ECG signals,
L is the number of the unknown source signals (
), and
N is the number of samples of each measurement. We assume that both
M and
L are less than
N. The matrices
and
have
M and
L row vector signals, respectively. Typical ECG signal is composed of P wave, QRS complex, S wave, and T wave. Both FECG and MECG signals are quasi-periodic. However, the amplitude and duration of P, QRS, and T waves are different. In addition, the FECG signal has higher frequency than the MECG signal [
2,
4]. The ECG signal is captured by appropriate electrodes placed at the abdominal and thorax.
The estimation of
and
from
is the main goal of the BSS problem. To estimate
, we denote matrix
, having the same dimension of
, as the estimated source matrix, given by
where
is the
estimated transformation matrix.
As the BSS model shown in (1) is affected by scaling, permutation, and rotation ambiguities [
34], several methods has been developed to extract
using (2). This will be discussed in
Section 3.
7. Conclusions
A noninvasive FECG extraction algorithm, referred to as NSITM, has been presented. The design problem has been formulated and an analysis has also been provided. The proposed algorithm computes first the ITM matrix from the original ECG input. Then, the raw FECG and MECG signals are estimated from the Null space of . The clean FECG signal is then extracted by removing the unwanted MECG component from the raw FECG signal. This requires FECG/MECG peak detection and a decision-making algorithm to address the exact locations of the MECG peaks. The computational complexity of the proposed algorithm have shown considerable improvement as compared with the previous NCA algorithm. The proposed algorithm was simulated using real and synthesised ECG data, and compared with PCA, FastICA, and PLP algorithms. Visual results using (DAISY) real data have shown that the proposed algorithm is effective in extracting FECG and MECG signals, when selecting the number of abdominal signals to be 5, with two reference signals taken from the thorax. Visual results using real data from the Physionet Challenge 2013 dataset/set a have shown the existence of MECG R peaks in the FECG signals. The MECG peaks have been removed using ACF, thus extracting clean FECG signals. The robustness of the proposed algorithm over time was checked to address the effectiveness of the algorithm in extracting the FECG and MECG signals.
Results of applying the NSITM algorithm to the Physionet/Fetal ECG Synthetic database (FECGSYNDB) have shown the capability of the algorithm in extraction FECG and MECG signals from all eight data signals used in simulation, and for all selected SNR values (available from the Physionet database from 0 dB to 12 dB), with MHR/FHR acceleration/deceleration plus noise being selected as the event type. The average values of the extraction performance metrics (SIR, SAR, SDR, and SPI for the NSITM algorithm have mostly shown significant improvement compared to other algorithms, when data files are used with SNR from 0 dB to 12 dB. Results on applying the NSITM algorithm to the same synthetic data have shown considerable improvement in qSNR when fmSNR varied from −30 dB to 0 dB. The proposed algorithm was also evaluated using statistical measures (SE, ACC, and PPV). Results on applying the proposed algorithm on the Physionet Challenge 2013 data/set a have shown the highest statistical values of SE, ACC, and PPV, as compared with other algorithms.