**2. ECG third-order cumulant slices classification**

#### **2.1 Background**

Many techniques have been introduced to detect fetal heartbeats during labour. The advances in higher-order statistics, non-linear filtering, and artificial neural networks are exploited to propose a hybrid technique to improve the non-invasive detection of fetal heartbeats during labour. A third-order cumulants (TOCs) technique for non-invasive fetal heartbeat detection has been proposed (Zgallai et al., 1997). The ECG signal is processed using a Volterra filter. To improve the performance of the Volterra filter, quadratic and cubic LMF Volterra filters have been proposed (Zgallai et al., 2007). The proposed system uses the mother and fetal third-order cumulants (TOCs), which carry the signature imprints of their respective QRS-complexes, in the signal processing phase.

Quadratic and cubic Volterra filters with LMF updates have been employed to synthesise the signal into linear, quadratic, and cubic parts, and retain only the linear part. The classification phase employs an LMS-based single-hidden-layer perceptron. This section proposes incorporating an adaptive cubic LMF Volterra and an artificial neural network classifier to improve the detection rate (Zgallai, 2010). The cubic Volterra filter has been shown to improve the performance of some biomedical signals such as electromyographic signals during labour (Zgallai et al., 2009). Cross validation has been done by comparing the results of the detection to QRS peaks of the fetal heartbeat extracted from the fetal scalp electrode which is the goldstandard.

#### **2.2 Third-order cumulants**

A non-Gaussian signal {X(k)} has TOCs given by (Nikias and Petropulu, 1993):

$$\mathbf{C}\_{3}^{\times}(\tau\_{1}, \tau\_{2}) = \text{Cum}\{\mathbf{X}(\mathbf{k}), \mathbf{X}(\mathbf{k} + \tau\_{1}), \mathbf{X}(\mathbf{k} + \tau\_{2})\} \cdot \tag{2.1}$$

The calculations of the TOCs are implemented off-line due to the large CPU time required to calculate the lags. One way of reducing this load is to use 1-d slices of the TOCs. Onedimensional slices of ),( <sup>21</sup> x c3 can be defined as (Nikias and Petropulu, 1993):

$$\mathbf{r}\_{2,1}^{\times}(\tau) \underline{\operatorname{Cum}}\left(\mathbf{X}(\mathbf{k}), \mathbf{X}(\mathbf{k}), \mathbf{X}(\mathbf{k} + \tau)\right) = \underline{\mathbf{c}}\_{3}^{\times}(0, \tau) \tag{2.2}$$

$$\mathbf{r}\_{\mathbf{i},2}^{\times}(\mathbf{r}) \underline{\operatorname{Cum}}\{\mathbf{X}(\mathbf{k}), \mathbf{X}(\mathbf{k}+\tau), \mathbf{X}(\mathbf{k}+\tau)\} = \underset{\mathbf{C}\_{\lambda}}{\operatorname{C}}(\tau, \tau) \tag{2.3}$$

where r 2,1() and r 1,2() represent 1-d wall and diagonal slices, respectively. The former can be obtained from Eq. (2.1) by assuming 1 = 0. The later obeys the condition 1 = 2. Employing 1-d slices will have the effect of reducing the CPU time by reducing the complexity of the operations. The calculations of TOC slices are comparable to those of autocorrelation and take CPU time of approximately 1 msec unlike TOCs, which take 1 sec to calculate. For a sampling rate of 0.5 KHz and an FHR of the order of 120 bpm, a real-time system can be implemented. An algorithm which calculates any arbitrarily chosen off diagonal and off wall one-dimensional slice, and hence reduce the CPU time by 99%, has been developed (Zgallai, 2007).

Adequate knowledge of the TOC of both the maternal and fetal ECG signals must first be acquired in order to pave the way for fetal QRS-complex identification and detection. There are several motivations behind using TOC in processing ECG signals; (i) ECG signals are predominantly non-Gaussian (Rizk and Zgallai, 1999), and exhibit quadratic and higherorder non-linearities supported by third- and fourth-order statistics, respectively. (ii) Gaussian noise diminishes in the TOC domains if the data length is adequate (Nikias and Petropulu, 1993). It is possible to process the ECG signal in Gaussian noise-free domains. For ECG signals a minimum length of 1 sec is adequately long to suppress Gaussian noise in the TOC domains, whilst not long enough to violate Hinich's criterion of local stationarity (Brockett et al., 1988). In general, ECG signals are non-stationary in the statistical sense, but relatively short data can be successfully treated with conventional signal processing tools primarily designed for stationary signals. When dealing with individual cardiac cycles, nonstationarity is not an issue but when one takes on board the heart rate time series which is chaotic and multi-dimensional then it is not wise to assume stationarity for analysis purposes. (iii) In the TOC 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 (Zgallai et al., 1997). (iv) ECG signals contain measurable quantities of quadratic and, to a lesser extent, cubic non-linearities. Such measurable quantities of non-linearity, if not synthesised and removed before any further processing for the purpose of signal identification and classification, could lead to poor performance with regard to fetal QRS-complex detection rates.

## **2.3 LMF quadratic and cubic volterra**

226 Recurrent Neural Networks and Soft Computing

This chapter investigates the application of recurrent neural networks to the detection and classification of ECG signals. Third-order cumulants, bispectrum, polyphase, and embedded Volterra are utilised. The chapter develops methodology for adult ECG detection using the higher-order statistics. It extends that to non-invasive fetal heartbeat detection, during labour. The work is also extended to classify adult heart abnormalities based on the phases of the higher-order statistics. The chapter is organised as follows; Sections 2 and 3 employ third-order cumulant slices and bispectrum contours, respectively, to detect adult and fetal ECG signals. Section 4 introduces a method of ECG abnormality detection using polyphase. Section 5 shows how third-order cumulants could be utilised for the detection of late

Many techniques have been introduced to detect fetal heartbeats during labour. The advances in higher-order statistics, non-linear filtering, and artificial neural networks are exploited to propose a hybrid technique to improve the non-invasive detection of fetal heartbeats during labour. A third-order cumulants (TOCs) technique for non-invasive fetal heartbeat detection has been proposed (Zgallai et al., 1997). The ECG signal is processed using a Volterra filter. To improve the performance of the Volterra filter, quadratic and cubic LMF Volterra filters have been proposed (Zgallai et al., 2007). The proposed system uses the mother and fetal third-order cumulants (TOCs), which carry the signature imprints

Quadratic and cubic Volterra filters with LMF updates have been employed to synthesise the signal into linear, quadratic, and cubic parts, and retain only the linear part. The classification phase employs an LMS-based single-hidden-layer perceptron. This section proposes incorporating an adaptive cubic LMF Volterra and an artificial neural network classifier to improve the detection rate (Zgallai, 2010). The cubic Volterra filter has been shown to improve the performance of some biomedical signals such as electromyographic signals during labour (Zgallai et al., 2009). Cross validation has been done by comparing the results of the detection to QRS peaks of the fetal heartbeat extracted from the fetal scalp

)}k(X),k(X),k(X{Cum),( <sup>21</sup> <sup>1</sup> <sup>2</sup>

The calculations of the TOCs are implemented off-line due to the large CPU time required to calculate the lags. One way of reducing this load is to use 1-d slices of the TOCs. One-

c ),0()}k(X),k(X),k(X{Cum)(r <sup>x</sup>

c3 can be defined as (Nikias and Petropulu, 1993):

c3 . (2.1)

1,2 (2.2)

3

A non-Gaussian signal {X(k)} has TOCs given by (Nikias and Petropulu, 1993):

potentials. Section 6 summarises the conclusions.

electrode which is the goldstandard.

x

x

x

**2.2 Third-order cumulants** 

dimensional slices of ),( <sup>21</sup>

**2.1 Background** 

**2. ECG third-order cumulant slices classification** 

of their respective QRS-complexes, in the signal processing phase.

The Volterra structure is a series of polynomial terms (Schetzen, 1980) which are formed from a time sequence {y(k). The output of the filter is expressed as

$$\mathbf{y(n)} = \sum\_{i\_{\mathbf{l}}=\mathbf{l}}^{N} \mathbf{a}\_{i\_{\mathbf{l}}}^{\dagger} \mathbf{x}\_{\mathbf{k}-i\_{\mathbf{l}}+\mathbf{l}} + \sum\_{i\_{\mathbf{l}}=\mathbf{l}}^{N} \sum\_{i\_{\mathbf{l}}=\mathbf{l}}^{N} \mathbf{a}\_{i\_{\mathbf{l}},i\_{\mathbf{l}}}^{2} \mathbf{x}\_{\mathbf{k}-i\_{\mathbf{l}}+\mathbf{l}} \mathbf{x}\_{\mathbf{k}-i\_{\mathbf{l}}+\mathbf{l}} + \dots \tag{2.4}$$

Adaptive conventional Volterra is updated using the Least-Mean Squares (LMS) criterion. The LMS algorithm minimises the expected value of the squared difference between the estimated output and the desired response. A more general case is to minimise E{e(n)2N} (Wallach and Widrow, 1984). N = 2 is the Least-Mean-Fourth (LMF). The LMF algorithm updates the weights as follows:

Detection and Classification of Adult and

cycles.

3.75 sec.

**2.5.3 Overlapping window** 

**2.5.2 Window minimum length and ECG segmentation** 

overlapping below 90% yielded missed fetal heartbeats.

**2.5.4 Averaged fetal heart rate calculation** 

Fetal ECG Using Recurrent Neural Networks, Embedded Volterra and Higher-Order Statistics 229

The duration of the fetal cardiac cycle varies from 250 msec to 500 msec for a range of fetal heart rate between 240 bpm and 120 bpm. The fetal QRS-complex itself occupies between 50 msec and 70 msec. The fetal heartbeat is detected in a flag window of length 250 msec. This window length serves two criteria; (i) it is the minimum length yielding an acceptable upper threshold of both the deterministic and stochastic noise types inherent in the higher-order statistics of the ECG signals encountered, and (ii) this window length allows the detection of one, two, three, or four fetal heartbeats (FHBs) within one maternal transabdominal cardiac cycle. For example, for maternal heartbeat of 60 bpm, the R-wave-to-R-wave = 1000 msec, and four segments x 250 msec = one maternal cardiac cycle = possible four fetal cardiac

When detecting the fetal heartbeat within the maternal transabdominal cardiac cycle, 90% overlapping windows, each of 250 msec duration, are scanned at a rate of 100 Hz with a sampling rate of 0.5 KHz. The overlapping percentage should be carefully chosen to compensate for the apparent loss of temporal resolution due to a lengthy window which is dictated by the maximum threshold of the variance of the TOCs. The average fetal QRScomplex duration is 60 msec. This may be encountered at the beginning, middle, or end of the flag window. Hence by using a window overlapping of 90%, any fetal QRS-complex which may be missed because it starts to evolve, say, 20 msec before the end of a window, can definitely be picked up by the next one or two overlapping windows when it completes its full duration of 60 msec and has definitely reached its full peak (the R-wave). If this particular QRS-complex has enough strength to be picked up by two successive overlapping windows, the algorithm will count it as one FHB. It has been found that reducing the

The instantaneous fetal heart rate is calculated by measuring the interval between two successive R-waves. This requires pinpointing accurately the R-point of the QRS-complex. Although the ECG TOC template matching technique is very effective in detecting the occurrence of the QRS-complex as a whole even when it is completely buried in noise, it cannot locate the R-wave over a window length of 250 msec which satisfies the criterion for the variance threshold. In most transabdominal ECG recordings (85%), the fetal QRScomplexes cannot be seen as they are completely masked by other signals and motion artefact. This obscurity accounts for the lower success rate of fetal heartbeat detection in other assessed fetal heartbeat detection techniques (Sureau, 1996). The adult heartbeats can be measured accurately and the instantaneous heart rate for adults can be calculated. Hence, by counting the number of fetal heartbeats (FHBs) that have occurred between two successive maternal R-waves, one can easily calculate the averaged FHR within the maternal cardiac cycle. On average, the maternal cardiac cycle is 1000 msec. Two maternal cardiac cycles measure 2 sec. So, detecting and displaying up to eight FHBs will take less than 2.000030 sec which is well within the manufacturers' detection-to-display interval of

$$\mathbf{a}\_{i}(\mathbf{n}+\mathbf{l})=\mathbf{a}\_{i}(\mathbf{n})+2\boldsymbol{\mu}\_{i}\ .\ \text{e}^{\ \text{s}}(\mathbf{n}).\text{x}(\mathbf{n})\ .\tag{2.5}$$

The LMF has a faster convergence than the LMS algorithm. It has generally a lower weight noise than the LMS algorithm, with the same speed of convergence. It was shown to have 3 dB to 10 dB lower mean-squared error (MSE) than the LMS algorithm (Wallach and Widrow, 1984). Adaptive LMF-based quadratic and cubic Volterra structures have been developed and shown to outperform LMS-based Volterra by 6-7 dBs (Zgallai, 2007).

#### **2.4 Neural network classifiers**

A major limitation of the back-propagation algorithms is the slow rate of convergence to a global minimum of the error-performance surface because the algorithm operates entirely on the gradient of the error-performance surface with respect to the weights in the singlehidden-layer perceptron. The back-propagation learning process is accelerated by incorporating a momentum term. The use of momentum introduces a feedback loop which prevents the learning process from being stuck at a local minimum on the errorperformance surface of the single-hidden-layer perceptron. The classifier is a single-hiddenlayer perceptrion based on a modified Back-Propagation technique. The modified backpropagation algorithm has a momentum term which helps to avoid local minima. One hundred and sixty one-dimensional TOC slices have been used as templates for the desired signals in the Artificial Neural Network (ANN) classifier.

## **2.5 The proposed algorithm**
