FETAL HEART RATE VARIABILITY EXTRACTION BY FREQUENCY TRACKING

Allan Kardec Barros��� �

, Noboru Ohnishi��� �

���BMC, RIKEN, Japan. � � Nagoya University, Japan. � UFMA, Brazil.

E-mail: akbarros@ieee.org.

ABSTRACT

In this work, we propose an algorithm to extract the fe-tal heart rate variability from an ECG measured from themother abdomen. The algorithm consists of two methods:a separation algorithm based on second-order statistics thatextracts the desired signal in one shot through the data, anda hearth instantaneous frequency (HIF) estimator. The HIFalgorithm is used to extract the mother heart rate whichserves as reference to extract the fetal heart rate. We car-ried out simulations where the signals overlap in frequencyand time, and showed that the it worked efficiently.

Keywords: Source separation, Independent componentanalysis, Analytic Signal, A priori information, Second or-der statistics, Auto-correlation.

1. INTRODUCTION

The fluctuations of the heart beating or heart rate vari-ability (HRV) is a useful tool for assessing non-invasivelythe status of the autonomic nervous system (ANS). Anda special interest is shown by the scientific community inthe analysis of fetal HRV, with the aim of understandingthe intra-uterin ANS, or detecting eventual cardiac malfunc-tions.

HRV is usually calculated from an electrocardiogram (ECG),after detecting the regular peak that appears in the ECGwaveform due to heart beating, called R-wave (see Fig. 1),and computing the time difference between two consecutiveR-waves. The HRV signal is the sequence of these differ-ences. However, this method has the disadvantage of need-ing more memory for storage and being more sensitive tonoise, specially in the case of fetal HRV, as the fetal ECG(FECG) appear corrupted by strong cardiac artifacts fromthe mother, as shown in Fig. 1.

Recently, using powerful tools of statistical signal pro-cessing, a great development was reached through the con-cept of blind source separation (BSS) and independent com-ponent analysis (ICA). These concepts were successfullyused for separating mutually independent signals in a num-ber of areas, including biomedical signal processing [24, 18,22, 4]. BSS is based on the following principle. Assum-ing that the original (or source) signals have been linearlymixed, and that these mixed sensor signals are available,

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Fig. 1. Example of an ECG signal from a pregnant woman.(1) No fetal influence appears (stronger and slower). (2) Thefetal influence can be noticed (weaker and faster).

BSS finds in a blind manner a linear combination of themixed signals which recovers the original source signals,possibly re-scaled and randomly arranged in the outputs.

However, extracting all the source signals from the sen-sors may not be of interest to the user. Rather, one can usesome a priori information available about the signal in orderto find an important signal. Thus, Barros and Cichocki [2]proposed a quite simple algorithm based on second orderstatistics which was shown in theory and experimentallythat could extract a given signal using temporal informa-tion. On the other hand, Barros and Ohnishi[3] proposeda new method called heart instantaneous frequency (HIF)which showed to be an efficient estimator of HRV using thespectral response of the cardiac signal.

Here we propose an algorithm which extracts the fetalheart rate by combining the above concepts of blind sourceseparation and heart instantaneous frequency. Our methodis designed by using, instead of real signals, the analytic sig-nal along with the exponential notation. The idea is to usethe HIF calculated from the mother ECG to extract the fetal

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heart rate. Another contribution of this work is that thereis no need to have various sensor measurements, as usuallyneeded by the BSS community, because we use only part ofthe spectral response of the sensors, diminishing thereforethe possibility of various sources contributing at the sametime to the mixing process. An advantage of the presentapproach over the one of Barros and Cichocki [2] is thatwe now assume that the signal to be extracted can be non-stationary and have a time-varying frequency.

Thus, we divide this manuscript in the following form.Firstly, we present the method, composed by HIF and theproposed BSS algorithm. In particular, this algorithm usesthe time-varying mother heart instantaneous frequency toextract the fetal contribution to the ECG. Then, we showsimulations and some experimental results. In the next sec-tion, we discuss the results and carry out the conclusions.

2. METHODS

We model here the ECG as a quasi-periodic signal witha fundamental plus infinite harmonic frequencies as shownbelow,

����������� �� � � ��������� � ������� ����� (1)

where �! ����� is the fundamental frequency and � � ����� is atime-varying amplitude modulator.

Separation Algorithm HIF

HIF

Mother's HIF

Fetal HIF

x1

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Fig. 2. Block diagram of the proposed method, composed ofa separation algorithm and heart instantaneous frequency(HIF)extractor. From two ECG measurements �#" and �%$ ,where the fetal and maternal signals are mixed, and by usingthe mother’s HIF, the separation algorithm extracts the fetalcontribution & , from which the fetal HIF is calculated andoutput.

Figure 2 shows the block diagram of the proposed algo-rithm. Essentially, it is grounded on two methods: HIF and aseparation algorithm. Firstly, from one of the ECG sensors,

we calculate the mother’s HIF, which serves as reference tothe separation algorithm to extract the fetal ECG & �(')� andfrom this, extract the fetal ECG.

2.1. Heart Instantaneous Frequency

For a given signal ������� , the corresponding analytic signal*������ is given by,*������+�,��������-/.�021 �������43 � 021 �������435�768/9 ����:;���<=:+> : � (2)

where 021 �������43 is the Hilbert transform of ������� . An advan-tage of the analytic signal is that it can define uniquely amodulation, dealing with exponentials.

As carried out in the signal processing literature (e.g.[7]), frequency modulation lead us to the possibility of us-ing the concept of the instantaneous frequency. For signal������� , the instantaneous angular frequency ?! ����� is calcu-lated from the analytic signal and is given by

�! �����+� >A@ ! �����> � � @ ! �����B�DCFE�GIH�CFJLK <M021 �������43�������ONQP (3)

We call the heart rate variability estimated from the ECGspectral response as the heart instantaneous frequency (HIF),which involves first the estimation of the spectrogram. Thespectrogram for a given signal R ����� is defined asS ��� ��T �B�VUUUU 6W 8X9 � � � $ �ZY�� R ������[#����<=:;��UUUU

$ P (4)

We then look for the frequency value corresponding tothe maximum of

S ��� ��T � at each time instant in a given fre-quency range. We call this quantity the driver \ ����� , and it isgiven by \ �����+� TL]F^ C�_`1 S ��� ��T �43ba � �4� " ��c#da � �4� " �4�ed P (5)

After this, we calculate the instantaneous frequency byusing a band-pass filter around a central frequency givenat each time instant by the driver. In particular, we usewavelets to construct the filter. The basic wavelet isf �����B�g6W 8 >> �=hi � �j�%k l4m npo nqsr q G�t�uMK W 8 � 9Av \ ����� > � Nxwy �

\ �����+� 6z � v \ ����� P (6)

wherez

is a short time interval. The filtered signal is givenby �������+� 9 �� f ����� R ����� > : P (7)

The heart instantaneous frequency �! ����� is then calcu-lated using (3).

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2.2. Proposed BSS Algorithm

Let us make some preliminary definition, by denotingthe source signal vector as � �(')�+�s1 �F"��(')� P�P�P ���e�(')�43 and themixed vector as � �(')� � 1 �5"F�(')� P�P�P ���5�(')�43 , where the mix-ture is written as � �(')�+��� � �(')� , ' is the sampling number,and � is an ��� nonsingular matrix.

Because we want to extract only the desired source sig-nal, say � � ����� , we can use a simple processing unit describedas & �(')�M� ��� � �(')� , where & �(')� is the output signal and �is the weight vector. Then, let us first define the followingerror,

� �(')�B� & �(')��< � � ��� �����I� P (8)

The idea is to carry out the minimization of the meansquared error � ��� � ��� 1 � $ 3 . From (8), and dropping theindex ' for convenience, we find,

� ��� �+��� � � 1 ��� � 3�� < W � 1 � � ��� � � � � 3)-�� 1 � � $ ��� � 3 P (9)

This cost function achieves minimum when its gradientreaches zero in relation to � . Thus, taking into account that& ����� � , we find,

� � ��� �� � � W � 1 ��� � 3�� < W � 1 � � ��� � � 35��� P (10)

Now, we can solve this equation by the following algo-rithm,

� ��� 1 ��� � 3 � " � 1 � � ��� � � 3 � (11)

We can also assume that the sensor vector has been prewhitenedso that � 1 ��� �53 �� , with this, (11) sleads to the learningrule,

� ��� 1 � � � ��� � 3 P (12)

We propose to use @ ! or �! as in (3). With some morereasoning, one can notice that this algorithm also avoids theso-called permutation/scaling problem which usually hap-pens in ICA theory, by using a priori information on thephase.

3. RESULTS

We carried out simulations to test the validity of the pro-posed separation algorithm. Here we show one where wemixed two different source signals: one with increasing andanother with decreasing frequency with time, while theycross one another in frequency. The signals were �Z"������ �! �����"�$# � 1 W 8 ���&%('*)*) - 6 � �43 and � $Z����� � ! �����"�$# � 1 W 8 ����+,)*)`<���&%('*)*)M- W � �43 , where

! ����� is a Bartlett window. They werechosen because they overlap temporarily in frequency, as

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Fig. 3. Spectrogram of the sum of the two source signals.Notice the frequency overlapping. In this case, regular fil-ters shall not work.

shown by the spectrogram of their sum in Fig. 3, and reg-ular algorithms [5, 23] shall not work. It is important tonotice that the source signals are amplitude and frequencymodulated at the same time. We mixed randomly the sourcesignals to generate sensor signals and used the a priori fre-quency information to extract either signal. In all the casesthe proposed separation algorithm worked efficiently.

We tested also our algorithm to the ECG data. We sim-ulated a mixture of two ECGs measured originally fromadults and decimated one of them by a factor of two in or-der to have a simulated fetal ECG. Their spectral responseup to 10 Hz are plotted in Fig. 4. It is important to noticethat the fundamental frequency of the FECG is around 2 Hz,which overlaps with the first harmonic of the mother ECG.This stands a problem for regular filters on which HIF algo-rithm is based on. As we can see from Fig. reffig:res, evenif the signals are mixed, the mother fundamental frequencyremains clean. Thus, we used this frequency information tomake sure we were extracting the mother ECG signal. Asproposed in the block diagram in Fig. 2, we removed thissignal from the sensors, from which we estimated the fetalHIF. As we had the original FECG, from its R-waves wefound the heart rate variability. We show this HRV and theextracted HIF in Fig. 5.

4. DISCUSSION

As we can see from the results, the proposed algorithmworked efficiently in two difficult cases where the sourcesignals overlapped in time and frequency. Moreover, it is

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Fig. 4. Spectral response of two source signals simulating a(1) fetal and (2) a maternal ECG measurements. Notice thespectral overlap around 2 Hz.

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Fig. 5. Comparison between the fetal heart rate variabilitymeasured (1) from the ECG R-wave and (2) by the proposedheart instantaneous frequency (HIF) algorithm.

worth to emphasize that this algorithm, contrary to proposedones, reaches convergence in one shot through the data.

On the other hand, extracting signals from a corruptedenvironment has been an old issue in statistical signal pro-cessing, with different approaches which evolved togetherwith machines computational power. In this context, differ-ent statistical tools were used. Firstly, second order statistics(SOS) by the estimation of correlations, and after, higher or-der statistics (HOS), involving for example the estimation ofskewness and kurtosis, were proposed. However, the calcu-lation of HOS moments are more sensitive to data size thenSOS. Thus, our algorithm shows also this advantage, sinceit needs only the calculation of a second order moment.

It is important to emphasize that the energy of the ma-ternal ECG is much stronger than the fetal ECG, besides,the fundamental frequency of the FECG usually overlapsthe first harmonic of the MECG. Thus, source separationstands as a strong tool for solving this problem.

We also believe that it can be useful in other applications.For example, in biomedical signal processing especially inEEG/MEG, e.g., for some experiments with event relatedpotentials (ERP), where the timing is controlled. Equally,the proposed algorithm can be used in speech/audio andtelecommunication applications.

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