**5. Discussion and conclusions**

The objective of this chapter is to introduce the subject of higher-order statistics (HOS) and its applications to the non-linear / non-Gaussian ECG signals to pave the way for employing HOS-based techniques as the solution to the formidable problem of transabdominal fetal heartbeat detection during labour. High detection rates can be accomplished by invoking the HOS-based techniques, namely, the third-order cumulant template matching and the bispectral contours template matching and which utilises a set of different levels of bispectral contours.

The key question that was attempted is why do HOS-based techniques yield the highest possible Fetal Heart Rate FHR? The reasons behind achieving high FHRs when using the HOS-based well-refined techniques are; (1) Under broad signal and noise conditions, higher-order cumulants and their spectra become high signal-to-noise ratio domains where detection, parametric estimation and signal classification can be performed. (2) The Gaussian noise diminishes in the HOS domains if the data length is adequate. For ECG signals, a minimum length of 1 sec is sufficiently long to suppress Gaussian noise and maintain a low level of HOS variances in the HOS domains, whilst not sufficiently long to violate Hinich's criterion for local stationarity. (3) In the third-order domain all sources of noise with symmetric probability density functions (pdfs), e.g., Gaussian and uniform, will vanish. The ECG signals are retained because they have non-symmetric distributions. This implies that it is more than adequate to utilise only the TOCs and their bispectra. There is no need to seek higher-than-third-order statistics as implicated in all the Independent Component Analysis (ICA) applications to FHR detection. (4) The maternal and fetal QRSbispectral contours, which are used as the discriminant patterns in the identification and classification, only overlap with the bispectra of the baseline wander and that of the EMG at very low levels (around –20 dB normalised to the peak of the maternal QRS-complex bispectrum). Therefore, it is comparatively easy to detect and classify QRS-complexes in ECG signals utilising either the TOC or the BIC template matching techniques.

#### **5.1. Direct computations of individual 1-d TOC slices**

It is also shown that, having computed the two-dimensional TOC, either the diagonal or the wall slice or a combination of the diagonal and wall slices is used in the detection /

classification process. Therefore, computing the full multi-dimensional TOC and then extracting individual slices is an unnecessary waste of the CPU time. So, why not compute any arbitrary 1-d slice directly without firstly having to compute the two-dimensional TOC and secondly extract the 1-d slice? The TOC-diagonal and the TOC-wall slices are straightforward to compute directly, by freezing one of the two cumulant lags and changing the other one. However, to compute any other arbitrary slice requires the development of an auxiliary algorithm. It has been found that performing direct computations of the 1-d TOC slices instead of computing the 2-d TOC firstly, and secondly extracting individual 1-d slices results in saving of more than 99% of the CPU time. The same applies to the 2–d bispectrum. However, it has to be borne in mind that it is the matching of the horizontal bispectral contours that is used in the Bispectral Contour (BIC) template matching technique instead of the 1-d polar bispectral slices. Because in order to use the 1-d polar bispectrum slices effectively one needs to use a minimum of 24 polar slices so as not to miss the capturing of rapid changes or null features in the bispectrum that could be used as discriminant patterns. Whereas for BIC contours the number of discriminant horizontal slices required for detection / classification does not exceed 10.

Second- and Third-Order Statistical Characterization of

Non-Linearity and Non-Gaussianity of Adult and Fetal ECG Signals and Noise 51

linearities that could be due to higher-order coupling of the maternal and fetal own harmonics and the concomitant mixing of the ECG signals and the non-linear uterine contraction interference signal. This so called coupling between maternal and fetal ECGs (Rizk et al., 2001) is manifested in a newly formed bicoherence squared peak(s) which did not exist in either of the isolated maternal bicoherence squared or the isolated fetal bicoherence squared computed from their respective ECG signals. The non-linear second- or third-order Volterra structure has been used (Rizk et al., 2001) to quantify the effect of this coupling, in part. The rest of the quantification process is carried out using the bicoherence

It is worth mentioning that, depending on the bispectrum estimation method employed, the techniques for the detection and quantification of quadratic phase coupling are divided into two categories: the conventional and the parametric. The conventional techniques are based on the bicoherence spectrum and they are better qualifiers of the phase coupling (Kim and Powers, 1978; Kim and Powers, 1995). However, their resolution is limited by the uncertainty principle of the Fourier transform. On the other hand, the parametric techniques are based on the auto-regressive (AR) modelling of the third-order cumulants. Although the parametric AR methods are not good quantifiers, they possess a high resolution capability, much higher than the frequency resolution of the conventional methods (Nikias and Raghuveer, 1987; Raghuveer and Nikias, 1986). The so called coupling results in nonstationarity in the transabdominal ECG signal. This is evidenced by the filling of the bispectrum OT region which is used as a measure of non-stationarity in non-Gaussian

For noise identification and characterisation in the third-order domain, the MIT/BIH databases were utilised (MIT/BIH 1997). Apart from Gaussian noise, there exist three types of non-Gaussian noise in ECG signals, namely, baseline wander, electromyographic (EMG), and motion artefact noise. 10,000 samples of each of these three types of noise are analysed.

Baseline wander Electromyographic Motion artifact

Frequency < 5 Hz Frequency < 120 Hz Frequency < 5 Hz

Support Support Support

Bispectrum Frequency < 5 Hz Frequency < 10 Hz Frequency < 35 Hz

The effect of the baseline wander noise on both the maternal and the fetal QRS-complexes at 15 Hz and 30 Hz, respectively, is not significant. Table 1 shows that only the bispectrum of

**Table 1.** Third-order statistics of three types of noise in ECG signals; baseline wander,

**5.4. Noise identification in male and non-pregnant female adults** 

A brief summary of their third-order statistics is shown in Table 1.

squared.

signals.

Third-order cumulants

Bicoherence Squared

electromyographic noise, and motion artefact.

### **5.2. Bispectral features of QRS-complexes**

The power spectrum of appropriately sampled ECG showed the QRS-complex principal peak in the frequency range from 15 Hz to 20 Hz, and 25 Hz to 40 Hz, for the maternal chest ECG and fetal scalp electrode ECG, respectively. Unfortunately, the power spectrum has limitations as an estimator in terms of resolution, variance, and clarity of the spectrum to be able to produce clear and distinguishable peaks for the P-waves. Therefore, an alternative spectrum estimator was used instead, namely, the multiple signal classification (MUSIC) pseudo-spectrum. The MUSIC-based pseudo-spectrum showed that the principal peaks for the p-waves occupy a range from 5 Hz to 8 Hz for adults. The principal peaks for the Pwaves of the fetal scalp electrode ECG occupy a range from 8 Hz to 10 Hz. The same MUSIC-based spectral estimators have revealed high local energy peaks around 5 Hz due to motion artefact (Zgallai et al., 1997).

As with cumulants, their bispectra were computed for the above mentioned ECG data samples and segmentations using the direct method which involves calculating a twodimensional Fourier transform. The following bispectral peaks have been observed only on the bispectral diagonal slice at the following frequency pairs;


#### **5.3. Quadratic coupling in transabdominally measured ECG signals**

It has been found in maternal transabdominal ECG signals that close proximity of the maternal and fetal QRS-complexes initiates additional quadratic and higher-order nonlinearities that could be due to higher-order coupling of the maternal and fetal own harmonics and the concomitant mixing of the ECG signals and the non-linear uterine contraction interference signal. This so called coupling between maternal and fetal ECGs (Rizk et al., 2001) is manifested in a newly formed bicoherence squared peak(s) which did not exist in either of the isolated maternal bicoherence squared or the isolated fetal bicoherence squared computed from their respective ECG signals. The non-linear second- or third-order Volterra structure has been used (Rizk et al., 2001) to quantify the effect of this coupling, in part. The rest of the quantification process is carried out using the bicoherence squared.

50 Practical Applications in Biomedical Engineering

detection / classification does not exceed 10.

motion artefact (Zgallai et al., 1997).

transabdominal ECG.

the bispectral diagonal slice at the following frequency pairs;

**5.2. Bispectral features of QRS-complexes** 

classification process. Therefore, computing the full multi-dimensional TOC and then extracting individual slices is an unnecessary waste of the CPU time. So, why not compute any arbitrary 1-d slice directly without firstly having to compute the two-dimensional TOC and secondly extract the 1-d slice? The TOC-diagonal and the TOC-wall slices are straightforward to compute directly, by freezing one of the two cumulant lags and changing the other one. However, to compute any other arbitrary slice requires the development of an auxiliary algorithm. It has been found that performing direct computations of the 1-d TOC slices instead of computing the 2-d TOC firstly, and secondly extracting individual 1-d slices results in saving of more than 99% of the CPU time. The same applies to the 2–d bispectrum. However, it has to be borne in mind that it is the matching of the horizontal bispectral contours that is used in the Bispectral Contour (BIC) template matching technique instead of the 1-d polar bispectral slices. Because in order to use the 1-d polar bispectrum slices effectively one needs to use a minimum of 24 polar slices so as not to miss the capturing of rapid changes or null features in the bispectrum that could be used as discriminant patterns. Whereas for BIC contours the number of discriminant horizontal slices required for

The power spectrum of appropriately sampled ECG showed the QRS-complex principal peak in the frequency range from 15 Hz to 20 Hz, and 25 Hz to 40 Hz, for the maternal chest ECG and fetal scalp electrode ECG, respectively. Unfortunately, the power spectrum has limitations as an estimator in terms of resolution, variance, and clarity of the spectrum to be able to produce clear and distinguishable peaks for the P-waves. Therefore, an alternative spectrum estimator was used instead, namely, the multiple signal classification (MUSIC) pseudo-spectrum. The MUSIC-based pseudo-spectrum showed that the principal peaks for the p-waves occupy a range from 5 Hz to 8 Hz for adults. The principal peaks for the Pwaves of the fetal scalp electrode ECG occupy a range from 8 Hz to 10 Hz. The same MUSIC-based spectral estimators have revealed high local energy peaks around 5 Hz due to

As with cumulants, their bispectra were computed for the above mentioned ECG data samples and segmentations using the direct method which involves calculating a twodimensional Fourier transform. The following bispectral peaks have been observed only on

a. (17 Hz,17 Hz) and (15 Hz,15 Hz) for the maternal chest and the transabdominal ECGs, respectively. So, there is a shift in the bispectral peak from 17 Hz to 15 Hz in the

b. (30 Hz,30 Hz) and less prominently at (20 Hz,20 Hz) for the fetal scalp electrode ECG.

It has been found in maternal transabdominal ECG signals that close proximity of the maternal and fetal QRS-complexes initiates additional quadratic and higher-order non-

**5.3. Quadratic coupling in transabdominally measured ECG signals** 

It is worth mentioning that, depending on the bispectrum estimation method employed, the techniques for the detection and quantification of quadratic phase coupling are divided into two categories: the conventional and the parametric. The conventional techniques are based on the bicoherence spectrum and they are better qualifiers of the phase coupling (Kim and Powers, 1978; Kim and Powers, 1995). However, their resolution is limited by the uncertainty principle of the Fourier transform. On the other hand, the parametric techniques are based on the auto-regressive (AR) modelling of the third-order cumulants. Although the parametric AR methods are not good quantifiers, they possess a high resolution capability, much higher than the frequency resolution of the conventional methods (Nikias and Raghuveer, 1987; Raghuveer and Nikias, 1986). The so called coupling results in nonstationarity in the transabdominal ECG signal. This is evidenced by the filling of the bispectrum OT region which is used as a measure of non-stationarity in non-Gaussian signals.

### **5.4. Noise identification in male and non-pregnant female adults**

For noise identification and characterisation in the third-order domain, the MIT/BIH databases were utilised (MIT/BIH 1997). Apart from Gaussian noise, there exist three types of non-Gaussian noise in ECG signals, namely, baseline wander, electromyographic (EMG), and motion artefact noise. 10,000 samples of each of these three types of noise are analysed. A brief summary of their third-order statistics is shown in Table 1.


**Table 1.** Third-order statistics of three types of noise in ECG signals; baseline wander, electromyographic noise, and motion artefact.

The effect of the baseline wander noise on both the maternal and the fetal QRS-complexes at 15 Hz and 30 Hz, respectively, is not significant. Table 1 shows that only the bispectrum of

the motion artefact and the bicoherence squared of the EMG noise have frequencies that would potentially overlap with those of the QRS-complexes of the mother and the fetal, albeit at –20 dB level. The bicoherence squared of the EMG noise is spread over a wide band of frequencies, up to (120 Hz ,120 Hz). The carpet effect of the non-linearity attributed to the EMG noise will be significantly reduced by linearising the transabdominal signal prior to fetal QRS detection in the third-order statistical domain. Under broad signal and noise conditions, linearisation of the transabdominal ECG signals not only removes to a great extent the signal non-linearity, but also partially eliminates other types of non-linearity due to noise or non-linearity due to strong uterine contractions.

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Non-Linearity and Non-Gaussianity of Adult and Fetal ECG Signals and Noise 53

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It could be deduced from Table 1 that there would be overlapping between the bispectral frequencies of motion artefact and those of the maternal and the fetal QRS-complexes, albeit at around –20 dB level. However, the level of noise at the QRS-complex spectra is comparatively small and by using QRS-complex tailor-made spectral windows, the effect of motion artefact on the detection of the QRS-complexes is not noticeable.
