**4. Third-order statistics of ECG signals**

### **4.1. Cumulants and their slices of cardiac cycles**

The maternal chest ECG is measured using the standard three-lead electrode system. The maternal transabdominally-measured signals are obtained using two surface electrodes. The electrode pair is set over the umbilicus, and lined up with the median vertical axis of the uterus. The ground electrode is located on the woman's hip. The fetal scalp electrode is used when deemed necessary. Multi-dimensional Third-order cumulants TOCs were computed for the above ECG signals as well as for the four segments of the maternal transabdominal cardiac cycles. The four segments were coded as I, II, III, and IV, each of length 250 ms which has been considered short enough as not to satisfy the assumption of non-stationarity, and long enough to meet the threshold of the higher-order statistical variances. The four coded segments ascribe to the following often occurring scenario; (I) Segment I, 0–250 ms; Predominantly maternal QRS-complex (no fetal QRS-complex present), (II) Segment II, 251 –500 ms; The first fetal heartbeat with maternal contribution, (III) Segment III, 501 ms – 750 ms; QRS-free ECG, and (IV) Segment IV, 751 – 1000 ms; The second fetal heartbeat with maternal contribution.

Fig. 4 (a), (b), (c), and (d) each depicts ECG signals (upper panel) and their third-order cumulants (lower panel) for fetal scalp ECG (550 ms), maternal chest ECG (900 ms), and two different and randomly picked transabdominally-measured maternal ECGs (1000 ms each). The subject is at 40 weeks gestation after the water has been broken hence facilitating fetal scalp measurements. The maternal cardiac cycle begins 50 ms before the R-wave and ends 50 ms before the next R-wave.

Second- and Third-Order Statistical Characterization of Non-Linearity and Non-Gaussianity of Adult and Fetal ECG Signals and Noise 33

32 Practical Applications in Biomedical Engineering

**3.2. The second-order cumulants of ECG signals** 

with maternal contribution, and QRS-free ECG contributions.

**3.3. The power spectrum of ECG signals** 

**4. Third-order statistics of ECG signals** 

50 ms before the next R-wave.

**4.1. Cumulants and their slices of cardiac cycles** 

Fig. 2 (a) shows a full maternal transabdominal cardiac cycle (1000 ms) which has been divided into four segments, I, II, III, and IV. These segments represent (I) the predominantly maternal QRS-complex, (II) the first fetal heartbeat with maternal contribution, (III) QRSfree ECG, and (IV) the second fetal heartbeat with maternal contribution, respectively. Fig. 2 (b) shows a typical example of the second-order cumulants (auto-correlation functions) for the segments shown in Fig. 2 (a). Second-order statistics do not show any distinguishable features that could be used to differentiate between maternal QRS-complex, fetal heartbeat

Fig. 3 depicts the power spectrum using the FFT method for (a) fetal scalp electrode ECG signal (data length 500 ms), (b) maternal transabdominal ECG (data length 1000 ms). The maternal cardiac cycle begins 50 ms before the R-wave and ends 50 ms before the next Rwave. The subject is at the first stage of labour (40 weeks gestation). The FFT method reveals a fetal scalp electrode ECG principal spectral peak at 30 Hz. The FFT method for the transabdominal cardiac cycle reveals the maternal principal spectral peak of 15 Hz. However, the FFT does not show fetal spectral peak from the segmented transabdominal

The maternal chest ECG is measured using the standard three-lead electrode system. The maternal transabdominally-measured signals are obtained using two surface electrodes. The electrode pair is set over the umbilicus, and lined up with the median vertical axis of the uterus. The ground electrode is located on the woman's hip. The fetal scalp electrode is used when deemed necessary. Multi-dimensional Third-order cumulants TOCs were computed for the above ECG signals as well as for the four segments of the maternal transabdominal cardiac cycles. The four segments were coded as I, II, III, and IV, each of length 250 ms which has been considered short enough as not to satisfy the assumption of non-stationarity, and long enough to meet the threshold of the higher-order statistical variances. The four coded segments ascribe to the following often occurring scenario; (I) Segment I, 0–250 ms; Predominantly maternal QRS-complex (no fetal QRS-complex present), (II) Segment II, 251 –500 ms; The first fetal heartbeat with maternal contribution, (III) Segment III, 501 ms – 750 ms; QRS-free ECG, and

(IV) Segment IV, 751 – 1000 ms; The second fetal heartbeat with maternal contribution.

Fig. 4 (a), (b), (c), and (d) each depicts ECG signals (upper panel) and their third-order cumulants (lower panel) for fetal scalp ECG (550 ms), maternal chest ECG (900 ms), and two different and randomly picked transabdominally-measured maternal ECGs (1000 ms each). The subject is at 40 weeks gestation after the water has been broken hence facilitating fetal scalp measurements. The maternal cardiac cycle begins 50 ms before the R-wave and ends

signal. There is a shallow peak at 28 Hz and a shifted peak at 42 Hz (Zgallai, 2007).

**Figure 2.** (a). Maternal transabdominal cardiac cycle (1000 ms) divided into four segments. The maternal cardiac cycle begins 50 ms before the R-wave and ends 50 ms before the next R-wave. The subject is at the first stage of labour (40 weeks gestation). (b). Typical examples of the second-order cumulants computed using the segments I, II, III, and IV shown in 2 (a) of maternal transabdominal ECG.

Second- and Third-Order Statistical Characterization of

Non-Linearity and Non-Gaussianity of Adult and Fetal ECG Signals and Noise 35

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, computing any other arbitrary slice requires the development of an auxiliary algorithm (Zgallai, 2007). 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.

**Figure 3.** The power spectrum using the FFT method for (a) fetal scalp electrode ECG (data length 500 ms), and (b) maternal transabdominal ECG signal (data length 1000 ms). The maternal cardiac cycle begins 50 ms before the R-wave and ends 50 ms before the next R-wave. The subject is at the first stage of labour (40 weeks gestation).

A quick glance at the similarities of the four cumulant patterns in Fig. 4 (a), (b), (c) and (d), gives a little hope of successful detection of the fetal presence in the maternal cardiac cycle. To complicate the matter further, the two transabdominal cumulants in Fig. 4 (c) and (d) look dissimilar even though both contain two fetal QRS-complexes. However, the best way to distinguish between those patterns is to slice them and look for discriminant features.

Fig. 5 shows the third-order cumulants and their diagonal (l.h.s.) and wall (r.h.s.) slices of one transabdominal cardiac cycle which is segmented into four segments of 250 ms each for (I) predominantly maternal QRS, (II) the first fetal heartbeat with maternal contribution, (III) QRS-free ECG, and (IV) the second fetal heartbeat with maternal contribution. The diagonal and wall TOC slices of the maternal segment (I) are easily distinguished from the diagonal and wall TOC slices of segments (II), (III), and (IV). Furthermore, the diagonal and wall TOC slices of fetal QRS segments (II) and (IV) are distinguishable from the diagonal and wall TOC slices of the QRS-free ECG segment (III) in that there is a distinguishable and wellformed peak at the origin in both diagonal and wall TOC slices. The peak of the QRS-free ECG segment is much narrower and more related to motion artefact than a signal.

Note that having computed the three-dimensional TOCs, either the diagonal or the wall slice could be 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, computing any other arbitrary slice requires the development of an auxiliary algorithm (Zgallai, 2007). 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.

34 Practical Applications in Biomedical Engineering

of labour (40 weeks gestation).

**Figure 3.** The power spectrum using the FFT method for (a) fetal scalp electrode ECG (data length 500 ms), and (b) maternal transabdominal ECG signal (data length 1000 ms). The maternal cardiac cycle begins 50 ms before the R-wave and ends 50 ms before the next R-wave. The subject is at the first stage

A quick glance at the similarities of the four cumulant patterns in Fig. 4 (a), (b), (c) and (d), gives a little hope of successful detection of the fetal presence in the maternal cardiac cycle. To complicate the matter further, the two transabdominal cumulants in Fig. 4 (c) and (d) look dissimilar even though both contain two fetal QRS-complexes. However, the best way to distinguish between those patterns is to slice them and look for discriminant features.

Fig. 5 shows the third-order cumulants and their diagonal (l.h.s.) and wall (r.h.s.) slices of one transabdominal cardiac cycle which is segmented into four segments of 250 ms each for (I) predominantly maternal QRS, (II) the first fetal heartbeat with maternal contribution, (III) QRS-free ECG, and (IV) the second fetal heartbeat with maternal contribution. The diagonal and wall TOC slices of the maternal segment (I) are easily distinguished from the diagonal and wall TOC slices of segments (II), (III), and (IV). Furthermore, the diagonal and wall TOC slices of fetal QRS segments (II) and (IV) are distinguishable from the diagonal and wall TOC slices of the QRS-free ECG segment (III) in that there is a distinguishable and wellformed peak at the origin in both diagonal and wall TOC slices. The peak of the QRS-free

ECG segment is much narrower and more related to motion artefact than a signal.

Note that having computed the three-dimensional TOCs, either the diagonal or the wall slice could be 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

Second- and Third-Order Statistical Characterization of

Non-Linearity and Non-Gaussianity of Adult and Fetal ECG Signals and Noise 37

**Figure 5.** (a) Maternal transabdominal ECG signal (upper panel) and the synchronised fetal ECG signal measured using fetal scalp electrode (lower panel). (b), (c), (d) and (e) are the third-order cumulants and their diagonal (l.h.s.) and wall (r.h.s.) slices for segments I, II, III, and IV, respectively, each segment is 250 ms. Segment I: pre-dominantly maternal QRS-complex, segment II, the first fetal heartbeat with maternal contribution, segment III: QRS-free ECG, and segment IV: the second fetal heartbeat with maternal contribution. The maternal cardiac cycle begins 50 ms before the R-wave and ends 50 ms before the next R-

wave.

**Figure 4.** ECG signals (upper panel) and their third-order cumulants (lower panel) for (a) fetal cardiac cycle using fetal scalp electrode ( data length 550 ms), (b) maternal chest cardiac cycle using one surface electrode and a reference electrode (data length 900 ms), (c) and (d) are two maternal transabdominal cardiac cycles measured using twin surface electrodes (data length 1000 ms each). The maternal cardiac cycle begins 50 ms before the R-wave and ends 50 ms before the next R-wave. The subject is at the first stage of labour, 40 weeks.

Second- and Third-Order Statistical Characterization of

Non-Linearity and Non-Gaussianity of Adult and Fetal ECG Signals and Noise 37

36 Practical Applications in Biomedical Engineering

stage of labour, 40 weeks.

**Figure 4.** ECG signals (upper panel) and their third-order cumulants (lower panel) for (a) fetal cardiac cycle using fetal scalp electrode ( data length 550 ms), (b) maternal chest cardiac cycle using one surface electrode and a reference electrode (data length 900 ms), (c) and (d) are two maternal transabdominal cardiac cycles measured using twin surface electrodes (data length 1000 ms each). The maternal cardiac cycle begins 50 ms before the R-wave and ends 50 ms before the next R-wave. The subject is at the first

**Figure 5.** (a) Maternal transabdominal ECG signal (upper panel) and the synchronised fetal ECG signal measured using fetal scalp electrode (lower panel). (b), (c), (d) and (e) are the third-order cumulants and their diagonal (l.h.s.) and wall (r.h.s.) slices for segments I, II, III, and IV, respectively, each segment is 250 ms. Segment I: pre-dominantly maternal QRS-complex, segment II, the first fetal heartbeat with maternal contribution, segment III: QRS-free ECG, and segment IV: the second fetal heartbeat with maternal contribution. The maternal cardiac cycle begins 50 ms before the R-wave and ends 50 ms before the next Rwave.

Fig. 6 shows four selected slices of the third-order cumulants computed using one cardiac cycle for each of the following; (a) and (b) an adult male and female chest, respectively, (c) maternal transabdominal, and (d) fetal scalp electrode ECG signal.

Second- and Third-Order Statistical Characterization of

Non-Linearity and Non-Gaussianity of Adult and Fetal ECG Signals and Noise 39

**4.2. The bispectrum, contour maps and slices for cardiac cycles** 

ECG bispectrum over and above the power spectrum one should regress,

Fig. 7 shows the bispectrum magnitudes (left panel) and the corresponding contours (right panel) using one cardiac cycle for; (a) fetal scalp electrode ECG, (b) maternal chest ECG, and (c) maternal transabdominal ECG signal. Before attempting to assess any advantages of the

**Figure 7.** The bispectrum magnitude (left panel) and contour map (right panel) for (a) a fetal cardiac cycle using fetal scalp electrode (data length 550 ms), (b) a maternal chest cardiac cycle (data length 1000 ms), and (c) a maternal transabdominal cardiac cycle (data length 1000 ms). The maternal cardiac cycle begins 50 ms before the R-wave and ends 50 ms before the next R-wave. The subject is at the first stage

of labour, 40 weeks gestation. The direct method is used to calculate the bispectrum.

**Figure 6.** Third order cumulant slices at 0o (wall), 11.25o , 22.50o, and 45o (diagonal) for (a) male chest cardiac cycle using one surface electrode (data length 1180 ms), (b) maternal chest cardiac cycle using one surface electrode (data length 900 ms), (c) maternal transabdominal cardiac cycle using twin surface electrodes (data length 1000 ms), and (d) fetal cardiac cycle using fetal scalp electrode (data length 550 ms). The maternal cardiac cycle begins 50 ms before the R-wave and ends 50 ms before the next R-wave. The female subject is at the first stage of labour, 40 weeks gestation.

#### **4.2. The bispectrum, contour maps and slices for cardiac cycles**

38 Practical Applications in Biomedical Engineering

Fig. 6 shows four selected slices of the third-order cumulants computed using one cardiac cycle for each of the following; (a) and (b) an adult male and female chest, respectively, (c)

**Figure 6.** Third order cumulant slices at 0o (wall), 11.25o , 22.50o, and 45o (diagonal) for (a) male chest cardiac cycle using one surface electrode (data length 1180 ms), (b) maternal chest cardiac cycle using one surface electrode (data length 900 ms), (c) maternal transabdominal cardiac cycle using twin surface electrodes (data length 1000 ms), and (d) fetal cardiac cycle using fetal scalp electrode (data length 550 ms). The maternal cardiac cycle begins 50 ms before the R-wave and ends 50 ms before the next R-wave.

The female subject is at the first stage of labour, 40 weeks gestation.

maternal transabdominal, and (d) fetal scalp electrode ECG signal.

Fig. 7 shows the bispectrum magnitudes (left panel) and the corresponding contours (right panel) using one cardiac cycle for; (a) fetal scalp electrode ECG, (b) maternal chest ECG, and (c) maternal transabdominal ECG signal. Before attempting to assess any advantages of the ECG bispectrum over and above the power spectrum one should regress,

**Figure 7.** The bispectrum magnitude (left panel) and contour map (right panel) for (a) a fetal cardiac cycle using fetal scalp electrode (data length 550 ms), (b) a maternal chest cardiac cycle (data length 1000 ms), and (c) a maternal transabdominal cardiac cycle (data length 1000 ms). The maternal cardiac cycle begins 50 ms before the R-wave and ends 50 ms before the next R-wave. The subject is at the first stage of labour, 40 weeks gestation. The direct method is used to calculate the bispectrum.

for a moment, to view the power spectrum and locate the frequency ranges for adult and fetal QRS-complexes. The power spectrum of appropriately sampled ECG showed the QRScomplex 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. 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. An alternative spectrum estimator was used instead, namely, the multiple signal classification (MUSIC) pseudo-spectrum. The MUSIC-based pseudo-spectrum also 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).

Second- and Third-Order Statistical Characterization of

Non-Linearity and Non-Gaussianity of Adult and Fetal ECG Signals and Noise 41

improving the temporal resolution prior to the bispectral calculations for both fetal and maternal chest segmented QRS-complexes. However, looking at the maternal chest and transabdominal bispectral diagonal slices, lowering of the QRS peak frequency from 17 Hz

**Figure 8.** The bispectra of (a) a fetal scalp and (b) a maternal chest ECG signal (left panel) and the corresponding contour maps (right panel). The maternal cardiac cycle begins 50 ms before the R-wave and ends 50 ms before the next R-wave. The subject is at the first stage of labour, 40 weeks gestation. The bispectrum is calculated using the direct method. A Hanning window is used to calculate the

A possible cause of this shifting in the QRS-complex frequency peak is the susceptibility and lack of predictability of the bispectral representation of highly-complex multi-frequency signals. During labour contractions, the presence of very strong deterministic and chaotic signals emanating from the uterus, and the accompanying motion artefacts result in highly dimensional transabdominal signals (Rizk et al., 2000) is shown. Consequently it is very difficult to isolate with integrity the maternal and fetal QRS- complex spectral peaks without first resorting to

The non-linearity in the ECG signal can be detected using the bicoherence squared. Fig. 10 depicts the bicoherence squared and their corresponding contour maps using one cardiac cycle for a fetal scalp electrode, maternal chest, and maternal transabdominal ECG. The bicoherence squared has peaks at the frequency pairs of (6 Hz,15 Hz) and (14 Hz,14 Hz) for the fetal scalp cardiac cycle, (15 Hz,15 Hz) for the maternal chest cardiac cycle, and (7.5

super-resolution algorithms using eigenvector-based projections (Zgallai, 2007).

to 15 Hz is observed.

bispectrum which is averaged for smoothing.

**4.3. Non-linearity of ECG signals** 

It is clearly seen in Fig. 7 that all significant twin-frequency peaks occur at frequencies lower than the p-wave and QRS-complex frequencies. It is very difficult to observe any p-wave or QRS-complex frequencies. The only thing that could be construed from these results is that the combined effect of the low temporal resolution resulting from using the whole cardiac cycle and the low spectral resolution inherent in the bispectrum formation, the QRScomplex twin peaks which should occur at frequency ranges from (15 Hz, 15 Hz) to (20 Hz,20 Hz) for adults and from (25 Hz,25 Hz) to (40 Hz,40 Hz) for fetal scalp electrode ECG are completely masked and cannot be found even at –30 dB normalised to any significant low frequency peak. Instead, only low frequencies predominate (Zgallai, 2012 b).

Fig. 8 shows the bispectra of fetal scalp electrode and maternal chest ECG signals (left panel) and the corresponding contour maps (right panel). The maternal cardiac cycle begins 50 ms before the R-wave and ends 50 ms before the next R-wave. The subject is at the first stage of labour, 40 weeks gestation. The bispectrum is calculated using the direct method which involves calculating a two-dimensional Fourier transform. Ten Hanning windows is used in calculating the bispectrum which are averaged for smoothing. The bispectral peaks of the fetal scalp electrode and maternal chest QRS-complexes exist at (40 Hz,40 Hz) and (13 Hz,13 Hz), respectively. However, they are shifted, shallow and inconclusive even though they are centred near the right frequency pairs, (30 Hz,30 Hz) for the fetal scalp electrode and (17 Hz,17 Hz) for the maternal chest ECG.

The temporal resolution could be improved by applying appropriate segmentations to the QRS-complexes. Instead of taking one cardiac cycle for an adult, which is on average 1000 ms, the 250 ms QRS-complex segment which is centred on the R-wave and runs 125 ms in opposite directions is considered. This also applies to the fetal scalp electrode ECG signal but with a reduced QRS-complex length of typically 60 ms.

Fig. 9 (top) depicts bispectral slices of the fetal QRS-complex which shows the correct position of a spectral peak at 30 Hz but only on the diagonal slice. Fig. 9 (middle and bottom) show maternal chest and transabdominal QRS-complex **bispectrum** slices. The maternal chest and abdomen both exhibit spectral frequencies of 17 Hz and 15 Hz, respectively, but only on the diagonal slice. Considerable improvement has resulted due to improving the temporal resolution prior to the bispectral calculations for both fetal and maternal chest segmented QRS-complexes. However, looking at the maternal chest and transabdominal bispectral diagonal slices, lowering of the QRS peak frequency from 17 Hz to 15 Hz is observed.

**Figure 8.** The bispectra of (a) a fetal scalp and (b) a maternal chest ECG signal (left panel) and the corresponding contour maps (right panel). The maternal cardiac cycle begins 50 ms before the R-wave and ends 50 ms before the next R-wave. The subject is at the first stage of labour, 40 weeks gestation. The bispectrum is calculated using the direct method. A Hanning window is used to calculate the bispectrum which is averaged for smoothing.

A possible cause of this shifting in the QRS-complex frequency peak is the susceptibility and lack of predictability of the bispectral representation of highly-complex multi-frequency signals. During labour contractions, the presence of very strong deterministic and chaotic signals emanating from the uterus, and the accompanying motion artefacts result in highly dimensional transabdominal signals (Rizk et al., 2000) is shown. Consequently it is very difficult to isolate with integrity the maternal and fetal QRS- complex spectral peaks without first resorting to super-resolution algorithms using eigenvector-based projections (Zgallai, 2007).

#### **4.3. Non-linearity of ECG signals**

40 Practical Applications in Biomedical Engineering

motion artefact (Zgallai, et al., 1997).

Hz,17 Hz) for the maternal chest ECG.

but with a reduced QRS-complex length of typically 60 ms.

for a moment, to view the power spectrum and locate the frequency ranges for adult and fetal QRS-complexes. The power spectrum of appropriately sampled ECG showed the QRScomplex 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. 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. An alternative spectrum estimator was used instead, namely, the multiple signal classification (MUSIC) pseudo-spectrum. The MUSIC-based pseudo-spectrum also 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

It is clearly seen in Fig. 7 that all significant twin-frequency peaks occur at frequencies lower than the p-wave and QRS-complex frequencies. It is very difficult to observe any p-wave or QRS-complex frequencies. The only thing that could be construed from these results is that the combined effect of the low temporal resolution resulting from using the whole cardiac cycle and the low spectral resolution inherent in the bispectrum formation, the QRScomplex twin peaks which should occur at frequency ranges from (15 Hz, 15 Hz) to (20 Hz,20 Hz) for adults and from (25 Hz,25 Hz) to (40 Hz,40 Hz) for fetal scalp electrode ECG are completely masked and cannot be found even at –30 dB normalised to any significant

Fig. 8 shows the bispectra of fetal scalp electrode and maternal chest ECG signals (left panel) and the corresponding contour maps (right panel). The maternal cardiac cycle begins 50 ms before the R-wave and ends 50 ms before the next R-wave. The subject is at the first stage of labour, 40 weeks gestation. The bispectrum is calculated using the direct method which involves calculating a two-dimensional Fourier transform. Ten Hanning windows is used in calculating the bispectrum which are averaged for smoothing. The bispectral peaks of the fetal scalp electrode and maternal chest QRS-complexes exist at (40 Hz,40 Hz) and (13 Hz,13 Hz), respectively. However, they are shifted, shallow and inconclusive even though they are centred near the right frequency pairs, (30 Hz,30 Hz) for the fetal scalp electrode and (17

The temporal resolution could be improved by applying appropriate segmentations to the QRS-complexes. Instead of taking one cardiac cycle for an adult, which is on average 1000 ms, the 250 ms QRS-complex segment which is centred on the R-wave and runs 125 ms in opposite directions is considered. This also applies to the fetal scalp electrode ECG signal

Fig. 9 (top) depicts bispectral slices of the fetal QRS-complex which shows the correct position of a spectral peak at 30 Hz but only on the diagonal slice. Fig. 9 (middle and bottom) show maternal chest and transabdominal QRS-complex **bispectrum** slices. The maternal chest and abdomen both exhibit spectral frequencies of 17 Hz and 15 Hz, respectively, but only on the diagonal slice. Considerable improvement has resulted due to

low frequency peak. Instead, only low frequencies predominate (Zgallai, 2012 b).

The non-linearity in the ECG signal can be detected using the bicoherence squared. Fig. 10 depicts the bicoherence squared and their corresponding contour maps using one cardiac cycle for a fetal scalp electrode, maternal chest, and maternal transabdominal ECG. The bicoherence squared has peaks at the frequency pairs of (6 Hz,15 Hz) and (14 Hz,14 Hz) for the fetal scalp cardiac cycle, (15 Hz,15 Hz) for the maternal chest cardiac cycle, and (7.5

Hz,7.5 Hz) for the maternal transabdominal cardiac cycle. These bicoherence peaks support non-linearity.

Second- and Third-Order Statistical Characterization of

Non-Linearity and Non-Gaussianity of Adult and Fetal ECG Signals and Noise 43

**Figure 10.** The bicoherence squared (left panel) and contour map (right panel) for (a) a fetal cardiac cycle using fetal scalp electrode (data length 550 ms), (b) a maternal chest cardiac cycle (data length 1000 ms), and (c) a maternal transabdominal cardiac cycle (data length 1000 ms). The maternal cardiac cycle begins 50 ms before the R-wave and ends 50 ms before the next R-wave. The subject is at the first stage of labour, 40 weeks gestation. The bispectrum is calculated using the direct method. A Hanning

validity of the stationarity assumption in relation to such signals. It is only natural to expect that the proximity of two non-linear signals such as the maternal and fetal QRS-complexes would result in non-linear (quadratic and higher-order) coupling and this in turn would invoke non-stationarity. The above is demonstrated to be true by inspecting the bispectral OT region shown in Fig. 11. The maternal cardiac cycle in Fig. 11 begins 50 ms

window is used to calculate the bispectrum which is averaged for smoothing.

**Figure 9.** Bispectrum slices at 0o (wall), 11.25o , 22.50o, and 45o (diagonal) for 250 ms segments of; fetal cardiac cycle using fetal scalp electrode (upper panel), maternal chest cardiac cycle (middle panel), and maternal transabdominal cardiac cycle (lower panel). The maternal cardiac cycle begins 50 ms before the R-wave and ends 50 ms before the next R-wave. The subject is at the first stage of labour, 40 weeks gestation.

#### **4.4. Proximity of the maternal and fetal QRS- complexes**

There is a general consensus that individual cardiac cycles are locally stationary. This was substantiated (Zgallai, 2007) by Hinich test (Hinich, 1982). However, when applying a highly dimensional signal such as the transabdominal ECG that have several individual non-linear and deterministic signals overlapping both in the time and frequency domains, all coexisting in a cocktail of noise and motion artefact, it is prudent to re-examine the

Second- and Third-Order Statistical Characterization of Non-Linearity and Non-Gaussianity of Adult and Fetal ECG Signals and Noise 43

42 Practical Applications in Biomedical Engineering

non-linearity.

gestation.

Hz,7.5 Hz) for the maternal transabdominal cardiac cycle. These bicoherence peaks support

**Figure 9.** Bispectrum slices at 0o (wall), 11.25o , 22.50o, and 45o (diagonal) for 250 ms segments of; fetal cardiac cycle using fetal scalp electrode (upper panel), maternal chest cardiac cycle (middle panel), and maternal transabdominal cardiac cycle (lower panel). The maternal cardiac cycle begins 50 ms before the R-wave and ends 50 ms before the next R-wave. The subject is at the first stage of labour, 40 weeks

There is a general consensus that individual cardiac cycles are locally stationary. This was substantiated (Zgallai, 2007) by Hinich test (Hinich, 1982). However, when applying a highly dimensional signal such as the transabdominal ECG that have several individual non-linear and deterministic signals overlapping both in the time and frequency domains, all coexisting in a cocktail of noise and motion artefact, it is prudent to re-examine the

**4.4. Proximity of the maternal and fetal QRS- complexes** 

**Figure 10.** The bicoherence squared (left panel) and contour map (right panel) for (a) a fetal cardiac cycle using fetal scalp electrode (data length 550 ms), (b) a maternal chest cardiac cycle (data length 1000 ms), and (c) a maternal transabdominal cardiac cycle (data length 1000 ms). The maternal cardiac cycle begins 50 ms before the R-wave and ends 50 ms before the next R-wave. The subject is at the first stage of labour, 40 weeks gestation. The bispectrum is calculated using the direct method. A Hanning window is used to calculate the bispectrum which is averaged for smoothing.

validity of the stationarity assumption in relation to such signals. It is only natural to expect that the proximity of two non-linear signals such as the maternal and fetal QRS-complexes would result in non-linear (quadratic and higher-order) coupling and this in turn would invoke non-stationarity. The above is demonstrated to be true by inspecting the bispectral OT region shown in Fig. 11. The maternal cardiac cycle in Fig. 11 begins 50 ms

Second- and Third-Order Statistical Characterization of

Non-Linearity and Non-Gaussianity of Adult and Fetal ECG Signals and Noise 45

complexes in (a1) and (b1), respectively, are separated by 200 ms. The resultant bispectrum in (c1) does not support the OT region. However, the situation is totally different when the two R-waves are as close as 35 ms as shown in Fig. 11 (a2) and (b2). Now the OT region of the bispectrum in (c2) is fully occupied and non-stationary. This means conventional signal processing techniques cannot be used to separate the maternal and fetal QRS-complexes. This problem has been adequately solved by linearising (at least removing quadratic coupling) the transabdominal signal before attempting to separate individual QRS-

The MIT/BIH databases (MIT/BIH, 1997) have recordings of the three main types of noise in ECG signals, namely, (a) baseline wander, (b) electro-myographic (EMG) noise, and (c) motion artefact. The following statistics help in the processing stages of the fetal heartbeat detection. When using super-resolution techniques requirement for Gaussian and non-Gaussian extraction and suppression is eliminated except for the conventional removal of

Fig. 12 depicts second- and third-order statistics of a baseline wander noise segment of 10,000 samples (approximately 30 sec) extracted from the MIT/BIH databases. Both the bispectrum and the bicoherence squared show high peaks at low frequencies (< 5 Hz). This means that the effect of the baseline wander noise on both maternal and fetal QRScomplexes at 15 Hz and 30 Hz, respectively, is not significant. It is prudent to eliminate such noise in the pre-processing stage. One conventional method of eliminating baseline wander employs a high-pass filter such as Butterworth high-pass filter of order 5, cut-off frequency of 1 Hz, a transition period of 1 Hz, a minimum ripple of –50 dB outside the main frequency

Fig. 13 shows some statistics of an electromyographic (EMG) noise segment of 10,000 samples extracted from the MIT/BIH databases. The noticeable feature is that the bispectrum is confined to low frequencies less than (10 Hz,10 Hz). This means that it will not interfere with the isolation of the adult QRS-complex bispectrum peak which occupies frequencies between (15 Hz,15 Hz) and (20 Hz,20 Hz), provided that an appropriate super-resolution technique is employed. But 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 eliminated by linearising the transabdominal signal prior to fetal QRS-complex detection in the third-order statistical domain. Under broad signal and noise conditions, linearisation of the transabdominal ECG signal not only removes to a great extent the signal non-linearity, but also partially eliminates other types of non-linearity due

baseline wander which is embedded in all data acquisition systems (baby monitors).

**4.5. Cumulants and bispectra of noise components** 

complexes.

lobe.

to noise.

a. Baseline Wander noise

b. Electromyographic noise

**Figure 11.** (a1), (a2) Two typical examples of maternal transabdominal cardiac cycles, (b1) and (b2) are the corresponding fetal ECG signals using fetal scalp electrode. The first fetal QRS-complex in (b1) is separated from the maternal QRS-complex in (a1) by 200 ms. The first fetal QRS-complex in (b2) is separated from the maternal QRS-complex in (a2) by 35 ms. The corresponding bispectrum contour maps at a level of -30 dB for the two cycles in (a1) and (a2) are shown in (c1) and (c2), respectively. The R-wave of the first fetal QRS-complex in (b1) is separated from the R-wave of the maternal QRScomplex in (a1) by 200 ms. The corresponding bispectrum in (c1) does not show extra activity in the OT region. The R-wave of the first fetal QRS-complex in (b2) is separated from the R-wave of the maternal QRS-complex in (a2) by 35 ms. The corresponding bispectrum in (c2) shows extra activities in the OT region due to non-linear coupling between the mother and the baby.

before the R-wave and ends 50 ms before the next R-wave. The subject is at the first stage of labour, 40 weeks gestation. Fetal cardiac cycle data length is 550 ms, and transabdominal ECG cardiac cycle data length is 1000 ms. Two typical transabdominally measured maternal ECG cycles, ((a1), (a2)), and two synchronised fetal scalp ECG cycles ((b1), (b2)) are shown. The lower parts of the Figure, (c1) and (c2), consist of the corresponding maternal bispectral contour maps at a level of -30 dB. The two R-waves of the maternal and fetal QRS-

complexes in (a1) and (b1), respectively, are separated by 200 ms. The resultant bispectrum in (c1) does not support the OT region. However, the situation is totally different when the two R-waves are as close as 35 ms as shown in Fig. 11 (a2) and (b2). Now the OT region of the bispectrum in (c2) is fully occupied and non-stationary. This means conventional signal processing techniques cannot be used to separate the maternal and fetal QRS-complexes. This problem has been adequately solved by linearising (at least removing quadratic coupling) the transabdominal signal before attempting to separate individual QRScomplexes.

#### **4.5. Cumulants and bispectra of noise components**

The MIT/BIH databases (MIT/BIH, 1997) have recordings of the three main types of noise in ECG signals, namely, (a) baseline wander, (b) electro-myographic (EMG) noise, and (c) motion artefact. The following statistics help in the processing stages of the fetal heartbeat detection. When using super-resolution techniques requirement for Gaussian and non-Gaussian extraction and suppression is eliminated except for the conventional removal of baseline wander which is embedded in all data acquisition systems (baby monitors).

a. Baseline Wander noise

44 Practical Applications in Biomedical Engineering

**Figure 11.** (a1), (a2) Two typical examples of maternal transabdominal cardiac cycles, (b1) and (b2) are the corresponding fetal ECG signals using fetal scalp electrode. The first fetal QRS-complex in (b1) is separated from the maternal QRS-complex in (a1) by 200 ms. The first fetal QRS-complex in (b2) is separated from the maternal QRS-complex in (a2) by 35 ms. The corresponding bispectrum contour maps at a level of -30 dB for the two cycles in (a1) and (a2) are shown in (c1) and (c2), respectively. The R-wave of the first fetal QRS-complex in (b1) is separated from the R-wave of the maternal QRScomplex in (a1) by 200 ms. The corresponding bispectrum in (c1) does not show extra activity in the OT region. The R-wave of the first fetal QRS-complex in (b2) is separated from the R-wave of the maternal QRS-complex in (a2) by 35 ms. The corresponding bispectrum in (c2) shows extra activities in the OT

before the R-wave and ends 50 ms before the next R-wave. The subject is at the first stage of labour, 40 weeks gestation. Fetal cardiac cycle data length is 550 ms, and transabdominal ECG cardiac cycle data length is 1000 ms. Two typical transabdominally measured maternal ECG cycles, ((a1), (a2)), and two synchronised fetal scalp ECG cycles ((b1), (b2)) are shown. The lower parts of the Figure, (c1) and (c2), consist of the corresponding maternal bispectral contour maps at a level of -30 dB. The two R-waves of the maternal and fetal QRS-

region due to non-linear coupling between the mother and the baby.

Fig. 12 depicts second- and third-order statistics of a baseline wander noise segment of 10,000 samples (approximately 30 sec) extracted from the MIT/BIH databases. Both the bispectrum and the bicoherence squared show high peaks at low frequencies (< 5 Hz). This means that the effect of the baseline wander noise on both maternal and fetal QRScomplexes at 15 Hz and 30 Hz, respectively, is not significant. It is prudent to eliminate such noise in the pre-processing stage. One conventional method of eliminating baseline wander employs a high-pass filter such as Butterworth high-pass filter of order 5, cut-off frequency of 1 Hz, a transition period of 1 Hz, a minimum ripple of –50 dB outside the main frequency lobe.

b. Electromyographic noise

Fig. 13 shows some statistics of an electromyographic (EMG) noise segment of 10,000 samples extracted from the MIT/BIH databases. The noticeable feature is that the bispectrum is confined to low frequencies less than (10 Hz,10 Hz). This means that it will not interfere with the isolation of the adult QRS-complex bispectrum peak which occupies frequencies between (15 Hz,15 Hz) and (20 Hz,20 Hz), provided that an appropriate super-resolution technique is employed. But 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 eliminated by linearising the transabdominal signal prior to fetal QRS-complex detection in the third-order statistical domain. Under broad signal and noise conditions, linearisation of the transabdominal ECG signal not only removes to a great extent the signal non-linearity, but also partially eliminates other types of non-linearity due to noise.

Second- and Third-Order Statistical Characterization of

Non-Linearity and Non-Gaussianity of Adult and Fetal ECG Signals and Noise 47

**Figure 13.** Characterisation of 10,000 samples of electromyographic noise extracted from the MIT/BIH database and sampled at 360 samples per second. (a) time series, (b) its histogram, (c) third-order cumulants, (d) power spectrum using the averaged periodogram method, (e) the bispectrum (l.h.s.) calculated using the direct method with contour maps (r.h.s.) and (f) the bicoherence squared (l.h.s.)

with contour maps (r.h.s.).

**Figure 12.** Characterisation of 10,000 samples of baseline wander noise extracted from the MIT/BIH database and sampled at 360 samples per second. (a) time series, (b) its histogram showing non-Gaussian pdf, (c) third-order cumulants, (d) power spectrum using the averaged periodogram method, (e) the bispectrum (l.h.s.) calculated using the direct method with contour maps (r.h.s.) and (f) the bicoherence squared (l.h.s.) with contour maps (r.h.s.).

Second- and Third-Order Statistical Characterization of Non-Linearity and Non-Gaussianity of Adult and Fetal ECG Signals and Noise 47

**Figure 12.** Characterisation of 10,000 samples of baseline wander noise extracted from the MIT/BIH database and sampled at 360 samples per second. (a) time series, (b) its histogram showing non-Gaussian pdf, (c) third-order cumulants, (d) power spectrum using the averaged periodogram method, (e) the bispectrum (l.h.s.) calculated using the direct method with contour maps (r.h.s.) and (f) the

bicoherence squared (l.h.s.) with contour maps (r.h.s.).

**Figure 13.** Characterisation of 10,000 samples of electromyographic noise extracted from the MIT/BIH database and sampled at 360 samples per second. (a) time series, (b) its histogram, (c) third-order cumulants, (d) power spectrum using the averaged periodogram method, (e) the bispectrum (l.h.s.) calculated using the direct method with contour maps (r.h.s.) and (f) the bicoherence squared (l.h.s.) with contour maps (r.h.s.).

Second- and Third-Order Statistical Characterization of

Non-Linearity and Non-Gaussianity of Adult and Fetal ECG Signals and Noise 49

Fig. 14 depicts second- and third-order statistics of a motion artefact noise segment of 10,000 samples extracted from the MIT/BIH databases. The bispectrum has many frequencies in the triangle region of (0 Hz,0 Hz), (0 Hz,35 Hz) and (35 Hz,0 Hz). These bispectral frequencies of motion artefact would be overlapping with those of the maternal and fetal QRS-complexes, albeit at around –20 dB level. However, the level of noise at the QRS-complex spectra is comparatively small and the effect of motion artefact on the detection of QRS-complexes is not noticeable. Fig. 14 (f) reveals that the bicoherence squared is rather confined to very low

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

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.

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 /

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

frequencies. As mentioned above, linearisation plays a definitive role.

c. Motion artefact noise

**5. Discussion and conclusions** 

bispectral contours.

**Figure 14.** Characterisation of 10,000 samples of motion artefact noise extracted from the MIT/BIH database and sampled at 360 samples per second. (a) time series, (b) its histogram, (c) third-order cumulants, (d) power spectrum using the averaged periodogram method, (e) the bispectrum (l.h.s.) calculated using the direct method with contour maps (r.h.s.) and (f) the bicoherence squared (l.h.s.) with contour maps (r.h.s.).

#### c. Motion artefact noise

48 Practical Applications in Biomedical Engineering

with contour maps (r.h.s.).

**Figure 14.** Characterisation of 10,000 samples of motion artefact noise extracted from the MIT/BIH database and sampled at 360 samples per second. (a) time series, (b) its histogram, (c) third-order cumulants, (d) power spectrum using the averaged periodogram method, (e) the bispectrum (l.h.s.) calculated using the direct method with contour maps (r.h.s.) and (f) the bicoherence squared (l.h.s.) Fig. 14 depicts second- and third-order statistics of a motion artefact noise segment of 10,000 samples extracted from the MIT/BIH databases. The bispectrum has many frequencies in the triangle region of (0 Hz,0 Hz), (0 Hz,35 Hz) and (35 Hz,0 Hz). These bispectral frequencies of motion artefact would be overlapping with those of the maternal and fetal QRS-complexes, albeit at around –20 dB level. However, the level of noise at the QRS-complex spectra is comparatively small and the effect of motion artefact on the detection of QRS-complexes is not noticeable. Fig. 14 (f) reveals that the bicoherence squared is rather confined to very low frequencies. As mentioned above, linearisation plays a definitive role.
