**3.1 Background**

This section describes a hybrid system using the mother and fetal ECG bispectral contours (BIC), which carry the signature imprints of their respective QRS-complexes, in the signal processing phase. The classification phase employs LMS-based single-hidden-layer classifiers. The maternal chest ECGs and the fetal scalp electrode ECGs have been used as templates or the HOS representatives in the classification phase. The bispectral contour matching technique is used to identify the signatures of both the maternal and fetal QRScomplexes. It will be shown that the highest achievable Fetal Heartbeat (FHB) classification rate using the BIC template matching technique is 90.12% with reduced false positives and negatives associated with the power spectrum-based FHB classification rate of 70%. Furthermore, the BIC has a marginally improved classification performance over and above the TOC during episodes of overlapping fetal QRS-complexes and maternal T-waves. This is achieved at the expense of complexity and computation time. The hybrid bispectral contour matching technique is an extension to the hybrid cumulant matching technique. Therefore, the choice of the NN classifier is based on the general discussion presented previously. *Prior information* remain as valuable assets and are exploited herein. It is the matching of the horizontal 2-d bispectral contours that has been used in the 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 to facilitate capturing the most rapid changes in the bispectrum including null features that could be used as discriminant patterns. Whereas for BIC contours, provided that they are horizontally cut at a maximum number of 10 levels, a good quality discriminant picture can be made available for the neural network classifier. For example, it is very unlikely that maxima and troughs are missed because of any changes in their respective positions.

The same procedure of Section 2 is applied with the replacement of the third-order cumulant slices by the bispectral contours (usually 10 contours including the tip of the peak and are spaced by approximately 1 dB). The CPU time for the bispectrum computation is almost twice that for cumulants and 2000 times that for individual TOC slices. The Detection key operations are exactly the same as those described in Section 2 except that the third-order cumulant slices are now going to be replaced by the bispectral contours (Zgallai, 2012).

#### **3.2 ECG bispectrum**

The nth-order cumulant spectrum of a process {x(k)} is defined as the (n-1)-dimensional Fourier transform of the nth-order cumulant sequence. The nth-order cumulant spectrum is thus defined as (Dogan and J. M. Mendel, 1993):

$$\mathbf{C}\_{n}^{\mathbf{x}}(\alpha\_{1}, \alpha\_{2}, \dots, \alpha\_{n-1}) = \sum\_{\mathbf{r}\_{1} = -\infty}^{+\infty} \cdots \sum\_{\mathbf{r}\_{n-1} = -\infty}^{+\infty} \mathbf{c}\_{n}^{\mathbf{x}}(\mathbf{r}\_{1}, \mathbf{r}\_{2}, \dots, \mathbf{r}\_{n}, \mathbf{r}\_{n-1}) \ e^{-j(\alpha\_{1}\mathbf{r}\_{1} + \alpha\_{2}\mathbf{r}\_{2} + \dots + \alpha\_{n}\mathbf{r}\_{n-1})} \tag{3.1}$$

where

$$\left| \left| \alpha\_{i} \right| \leq \pi \qquad \text{for} \qquad \text{i} = 1, 2, \ldots \text{n} - 1, \text{ and } \qquad \left| \left| \alpha\_{1} + \alpha\_{2} + \ldots + \alpha\_{n-1} \right| \right| \leq \pi.$$

The bispectrum, n = 3, is defined as:

234 Recurrent Neural Networks and Soft Computing

This section describes a hybrid system using the mother and fetal ECG bispectral contours (BIC), which carry the signature imprints of their respective QRS-complexes, in the signal processing phase. The classification phase employs LMS-based single-hidden-layer classifiers. The maternal chest ECGs and the fetal scalp electrode ECGs have been used as templates or the HOS representatives in the classification phase. The bispectral contour matching technique is used to identify the signatures of both the maternal and fetal QRScomplexes. It will be shown that the highest achievable Fetal Heartbeat (FHB) classification rate using the BIC template matching technique is 90.12% with reduced false positives and negatives associated with the power spectrum-based FHB classification rate of 70%. Furthermore, the BIC has a marginally improved classification performance over and above the TOC during episodes of overlapping fetal QRS-complexes and maternal T-waves. This is achieved at the expense of complexity and computation time. The hybrid bispectral contour matching technique is an extension to the hybrid cumulant matching technique. Therefore, the choice of the NN classifier is based on the general discussion presented previously. *Prior information* remain as valuable assets and are exploited herein. It is the matching of the horizontal 2-d bispectral contours that has been used in the 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 to facilitate capturing the most rapid changes in the bispectrum including null features that could be used as discriminant patterns. Whereas for BIC contours, provided that they are horizontally cut at a maximum number of 10 levels, a good quality discriminant picture can be made available for the neural network classifier. For example, it is very unlikely that

maxima and troughs are missed because of any changes in their respective positions.

are now going to be replaced by the bispectral contours (Zgallai, 2012).

1 1

*n x x <sup>j</sup> C c n n n n <sup>e</sup>*

 

 

thus defined as (Dogan and J. M. Mendel, 1993):

 

The same procedure of Section 2 is applied with the replacement of the third-order cumulant slices by the bispectral contours (usually 10 contours including the tip of the peak and are spaced by approximately 1 dB). The CPU time for the bispectrum computation is almost twice that for cumulants and 2000 times that for individual TOC slices. The Detection key operations are exactly the same as those described in Section 2 except that the third-order cumulant slices

The nth-order cumulant spectrum of a process {x(k)} is defined as the (n-1)-dimensional Fourier transform of the nth-order cumulant sequence. The nth-order cumulant spectrum is

12 1 12 1 ( , ,, ) (,, , ) *n n*

12 1 1,2,... 1, ... *<sup>i</sup> <sup>n</sup>*

*for i n and*

 

  (3.1)

11 22 1

 

( )

 

**3. ECG bispectrum contour classification** 

**3.1 Background** 

**3.2 ECG bispectrum** 

where

 

$$\mathbf{C}\_{3}^{\chi}(o\_{1}, o\_{2}) = \sum\_{\mathfrak{r}\_{1} = -\infty}^{+\infty} \sum\_{\mathfrak{r}\_{2} = -\infty}^{+\infty} c\_{3}^{\chi}(\mathfrak{r}\_{1}, \mathfrak{r}\_{2}) \ e^{-j(o\_{1}\mathfrak{r}\_{1} + o\_{2}\mathfrak{r}\_{2})} \tag{3.2}$$

where 312 (,) *<sup>x</sup> <sup>c</sup>* is the third-order cumulant sequence. The computational complexity of the bispectrum is of the order of N3.
