**1. Introduction**

This chapter investigates the application of digital signal processing techniques to ECG signals. The first few sections of this chapter are devoted to definitions and properties of cumulants, their spectra, and associated statistics. This is followed by describing the structural properties of the third-order cumulants of an adult male's chest ECG, maternal chest ECG, transabdominally-measured ECG, as well as fetal ECG signal using scalpelectrode. The non-linearity and non-stationarity of ECG signals are investigated using the bispectrum and bicoherence squared. The third-order cumulants, bispectra, and bicoherence squared of some noise components, namely, the baseline wander, electromyographic (EMG), and motion artefact noise isolated from the MIT/BIH databases are analysed. Finally, concluding remarks are discussed and summarised.

Adequate knowledge of the higher-order statistics (HOS) of both the maternal and fetal ECG signals must be acquired in order to pave the way for fetal QRS-complex identification and detection. There are several motivations behind using higher-order statistics in processing ECG signals. These motivations are:


© 2012 Zgallai, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Zgallai, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

maternal QRS-complex bispectrum (Zgallai, 2007). It is comparatively easy to detect and classify either using the bispectral contour template matching technique.

Second- and Third-Order Statistical Characterization of

1 2

1 2

*n*

(3)

 

3

1 2 0

*n*

*n*

 

  *n*

(1)

 

1 2 0

*n*

*n*

 

 

Non-Linearity and Non-Gaussianity of Adult and Fetal ECG Signals and Noise 27

1 2

 

1 2

Given a set of n real variables {x1, x2, …, xn}, their joint moments of order, r = k1 + k2 + … + kn

*n n k k k*

is their joint characteristic function. E{.}

, i.e., (Nikias and Petropulu, 1993)

(2)

1 2

2 3 m E x . m E x . m E x . 1 12 1 3 1 (4)

 

*<sup>m</sup>* (6)

2 3 1 12 2 1 3 3 21 1 c m . c m m . c m – 3 m m 2 m . (5)

1 2

( , ,, ) [ , ,, ] { , ,, }() *n n*

*jx x x E e n n*

 

*<sup>r</sup> k k k k k k <sup>r</sup> <sup>n</sup>*

 

denotes the expectation operator. Another form of the joint characteristic function is defined

( , , , ) [ ( , , , )] ln

For Gaussian processes, the logarithm of the characteristic function is a polynomial of degree two. Hence, all cumulants of order three and higher will be identically zero. The joint cumulants of order r of the same set of random variables, are defined as the coefficients in the Taylor expansion of the second characteristic function about zero, i.e., (Nikias and

( , ,, ) [ , ,, ] { , ,, }() *n n*

*<sup>r</sup> k k k k k k <sup>r</sup> <sup>n</sup>*

*n n k k k*

Thus, the joint can be expressed in terms of the joint moments of a set of random variables. The moments of the random variable {x1} are defined as (Nikias and Petropulu, 1993)::

For three random variables x1, x2, and x3 the third-order cumulants are defined as

 

3 12 3 12 1 2 1 2 2 2 2 1 1 , , <sup>2</sup> *x x xx x x <sup>x</sup> c m mm m m*

 

1. c [a1x1, a2x2, …, anxn] = a1a2 … an c [x1, x2, …, xn] (Nikias and Petropulu, 1993).

n n

**2. Background and definitions** 

are given by (Kravtchenko-Berejnoi, V. et al. 1995):

1 2 1 2

*Mom x x x E x x x j*

1 2 1 2

where ( , ,, ) 11 22 ( , ,, ) { } 1 2

 

1 2 1 2

 

*Cum x x x E x x x j*

Cumulants are related to moments by (Nikias and Petropulu, 1993)

1 2 1 2

(Kravtchenko-Berejnoi, V. et al. 1995):

**2.2. Properties of cumulants** 

*n*

as the natural logarithm of 1 2 ( , ,, ) *<sup>n</sup>*

 

**2.1. Cumulants** 

 

Petropulu, 1993)


An adaptive third-order Volterra structure (Nam and Powers, 1994) has been used to synthesise the linear, quadratic non-linear, and cubic non-linear components of ECG signals. The removal of non-linearities in the transabdominal ECG signal yields an increase in the fetal heartbeat detection rates by up to 7% in the third-order cumulant matching technique (Zgallai, 2010), and 10% in the bispectral contour template matching technique (Zgallai, 2012 a, Zgallai, 2012 b).

For noise identification and characterisation in the third-order statistical domain, use is made of the recorded normal ECG signals contained in the MIT/BIH databases (MIT/BIH, 1997). The third-order cumulants, bispectra, and bicoherence squared of some noise components, namely, the baseline wander, electromyographic (EMG) (Zgallai, 2009), and motion artefact noise isolated from the MIT/BIH databases are analysed. Knowing the statistics of those noise components, would facilitate the detection of ECG signals against a cocktail of background noise in either the cumulant or the bispectrum domain. Higher detection rate of fetal QRS-complex can be achieved in the enhanced fetal QRS-complex bispectrum domain against both maternal and motion artefact bispectral contribution (Zgallai, 2012 a). Bispectral enhancement has been carried out after removing the baseline wander, and in difficult cases, after linearisation (removing non-linearity from the noise contaminated maternal transabdominal signal).

#### **2. Background and definitions**

#### **2.1. Cumulants**

26 Practical Applications in Biomedical Engineering

a, Zgallai, 2012 b).

contaminated maternal transabdominal signal).

maternal QRS-complex bispectrum (Zgallai, 2007). It is comparatively easy to detect

iv. 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

v. ECG signals contain quadratic and cubic non-linearities (Rizk et al., 1998). 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

An adaptive third-order Volterra structure (Nam and Powers, 1994) has been used to synthesise the linear, quadratic non-linear, and cubic non-linear components of ECG signals. The removal of non-linearities in the transabdominal ECG signal yields an increase in the fetal heartbeat detection rates by up to 7% in the third-order cumulant matching technique (Zgallai, 2010), and 10% in the bispectral contour template matching technique (Zgallai, 2012

For noise identification and characterisation in the third-order statistical domain, use is made of the recorded normal ECG signals contained in the MIT/BIH databases (MIT/BIH, 1997). The third-order cumulants, bispectra, and bicoherence squared of some noise components, namely, the baseline wander, electromyographic (EMG) (Zgallai, 2009), and motion artefact noise isolated from the MIT/BIH databases are analysed. Knowing the statistics of those noise components, would facilitate the detection of ECG signals against a cocktail of background noise in either the cumulant or the bispectrum domain. Higher detection rate of fetal QRS-complex can be achieved in the enhanced fetal QRS-complex bispectrum domain against both maternal and motion artefact bispectral contribution (Zgallai, 2012 a). Bispectral enhancement has been carried out after removing the baseline wander, and in difficult cases, after linearisation (removing non-linearity from the noise

because they have non-symmetric distributions (Zgallai, et al., 1997).

to poor performance with regard to fetal QRS-complex detection rates.

and classify either using the bispectral contour template matching technique. iii. In the HOS domain, the Gaussian noise diminishes if the data length is adequate (Nikias and Petropulu, 1993; Nam and Powers, 1994). This implies that it is possible, under certain conditions, to process the ECG signal in Gaussian noise-free domains. It was found (Rizk and Zgallai, 1999) that for ECG signals a minimum length of 1 sec is adequately long to suppress Gaussian noise in the higher-order statistical domains, whilst not long enough to violate Hinich's criterion of local stationarity (Brockett et al. 1988). Hinich tests for Gaussianity and linearity were performed on ECG signals (Zgallai, 2007). 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. For example, when dealing with individual cardiac cycles, non-stationarity 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 (Rizk et al. 2002).

Given a set of n real variables {x1, x2, …, xn}, their joint moments of order, r = k1 + k2 + … + kn are given by (Kravtchenko-Berejnoi, V. et al. 1995):

$$\text{Mom} \quad \{\mathbf{x}\_1^{k\_1}, \mathbf{x}\_2^{k\_2}, \dots, \mathbf{x}\_n^{k\_n}\} \quad \stackrel{\nabla}{=} \quad E(\mathbf{x}\_1^{k\_1}, \mathbf{x}\_2^{k\_2}, \dots, \mathbf{x}\_n^{k\_n}) = (-j)^r \left. \frac{\hat{\mathcal{C}} \not{\phi}(a\_1, a\_2, \dots, a\_n)}{\hat{\mathcal{C}} a\_1^{k\_1} \hat{\mathcal{C}} a\_2^{k\_2} \dots \hat{\mathcal{C}} a\_n^{k\_n}} \right|\_{a\_1 = a\_2 = \dots = a\_n = 0} \tag{1}$$

where ( , ,, ) 11 22 ( , ,, ) { } 1 2 *jx x x E e n n n* is their joint characteristic function. E{.} denotes the expectation operator. Another form of the joint characteristic function is defined as the natural logarithm of 1 2 ( , ,, ) *<sup>n</sup>* , i.e., (Nikias and Petropulu, 1993)

$$\tilde{\Psi}(\text{co}\_1\text{"}, \text{o}\_2\text{"}, \dots, \text{o}\_{\text{n}}) \underset{\blacksquare}{\nabla} \ln[\phi(\text{o}\_1\text{"}, \text{o}\_2\text{"}, \dots, \text{o}\_{\text{n}})] \tag{2}$$

For Gaussian processes, the logarithm of the characteristic function is a polynomial of degree two. Hence, all cumulants of order three and higher will be identically zero. The joint cumulants of order r of the same set of random variables, are defined as the coefficients in the Taylor expansion of the second characteristic function about zero, i.e., (Nikias and Petropulu, 1993)

$$\mathbf{Cum} \quad \{\mathbf{x}\_1^{k\_1}, \mathbf{x}\_2^{k\_2}, \dots, \mathbf{x}\_n^{k\_n}\} = \mathbf{E}\{\mathbf{x}\_1^{k\_1}, \mathbf{x}\_2^{k\_2}, \dots, \mathbf{x}\_n^{k\_n}\} = (-j)^r \left. \frac{\hat{\sigma}^r \tilde{\Psi}(a\_1, a\_2, \dots, a\_n)}{\hat{\sigma} a\_1^{k\_1} \hat{\sigma} a\_2^{k\_2} \dots \hat{\sigma} a\_n^{k\_n}} \right|\_{a\_1 = a\_2 = \dots = a\_n = 0} \tag{3}$$

Thus, the joint can be expressed in terms of the joint moments of a set of random variables. The moments of the random variable {x1} are defined as (Nikias and Petropulu, 1993)::

$$\mathbf{m}\_1 = \operatorname{E}\{\mathbf{x}\_1\}.\\\mathbf{m}\_2 = \operatorname{E}\left(\mathbf{x}\_1^2\right).\\\mathbf{m}\_3 = \operatorname{E}\left[\mathbf{x}\_1^3\right].\tag{4}$$

Cumulants are related to moments by (Nikias and Petropulu, 1993)

$$\mathbf{c}\_1 = \mathbf{m}\_1.\mathbf{c}\_2 = \mathbf{m}\_2 - \mathbf{m}\_1^2.\mathbf{c}\_3 = \mathbf{m}\_3 - 3\,\mathbf{m}\_2\mathbf{m}\_1 + 2\,\mathbf{m}\_1^3.\tag{5}$$

For three random variables x1, x2, and x3 the third-order cumulants are defined as (Kravtchenko-Berejnoi, V. et al. 1995):

$$m\_3^x \left(\tau\_1, \tau\_2\right) = m\_3^x \left(\tau\_1, \tau\_2\right) - m\_1^x \left[m\_2^x \left(\tau\_1\right) + m\_2^x \left(\tau\_2\right) + m\_2^x \left(\tau\_2 - \tau\_1\right)\right] + 2 \cdot \left(m\_1^x\right)^3\tag{6}$$

#### **2.2. Properties of cumulants**

1. c [a1x1, a2x2, …, anxn] = a1a2 … an c [x1, x2, …, xn] (Nikias and Petropulu, 1993).

	- 2. Cumulants are symmetric functions in their arguments, e.g., c[x1,x2,x3] = c[x2,x1,x3] = c[x3,x2,x1], and so on (Nikias and Petropulu, 1993)..

Second- and Third-Order Statistical Characterization of

 

11 22 1

 

> 

( ,)

 

 

(14)

11 22

(15)

( )

 

 

Non-Linearity and Non-Gaussianity of Adult and Fetal ECG Signals and Noise 29

2,1 2,1 2,1 2,1 ( ) ( ) { ( ) cos( ) ( ) sin( )} *j t xx x x <sup>R</sup> <sup>r</sup> <sup>s</sup> jq e*

Higher order spectra are defined as the multi-dimensional Fourier transforms of the higher order statistics of the superimposed signals in the presence of noise (Nikias and Raghuveer, 1987; Rosenblatt, 1985; Brillinger, 1965). 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.

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

 

( )

(13)

 

(12)

 

 

The nth-order cumulant spectrum is thus defined as (Nikias and Petropulu, 1993):

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

 

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

2 2 ( ) () *x x <sup>j</sup> C ce*

1 2

 

methods for their estimation was discussed (Zurbenko, 1982).

31 2 <sup>312</sup> (,) (,) *x x <sup>j</sup> <sup>C</sup> c e*

 

estimates are asymptotically unbiased and consistent (Nagata, 1970), and easy to implement using FFT-based methods. The ability to resolve harmonic components is limited by the uncertainty principle of the Fourier transform. There are numerous methods for polyspectra estimation based on parametric methods. MA models have been treated in (Nikias, 1988; Friedlander and Porat, 1988). Spectral estimation methods based on non-causal AR models were developed in (Huzi, 1981). Methods based on ARMA models have been published (Lii, 1982). MA, AR and ARMA methods based on higher-order statistics were described (Mendel, 1988). A review of cumulant spectra and the asymptotic properties of their estimators were given in (Rosenblatt, 1983). Practical considerations for bispectral estimation were given (Subba Rao, 1983). The relationship between the bispectrum and conventional

.

 

> 

is the third-order cumulant. Conventional higher-order statistics (HOS)

 

1 1

*for i n and*

 

**2.4. Cumulant spectra** 

1. Power spectrum (n = 2):

2. Bispectrum (n = 3):

is the covariance.

where

**Special cases** 

where 2 ( ) *<sup>x</sup> <sup>c</sup>* 

where 312 (,) *<sup>x</sup> <sup>c</sup>* 


#### **2.3. One-dimensional third-order cumulant slices**

Since higher-order cumulants and spectra are multi-dimensional functions, their computation may be impractical in some applications due to excessive crunching. This is caused by the large CPU time taken to calculate HOS functions, compared to SOS functions. It was suggested to use 1-d slices of multi-dimensional cumulants, and their 1-d Fourier transforms, as ways of extracting useful information from higher-order statistics of non-Gaussian stationary processes (Nagata, 1970). The third-order cumulants of a non-Gaussian process, {x(k)}, is given by (Nikias and Petropulu, 1993):

$$\mathbf{c}\_{\mathcal{L}\_3}^{\times}(\tau\_1, \tau\_2) = \mathbf{c}\{\mathbf{x}(k), \mathbf{x}(k + \tau\_1), \mathbf{x}(k + \tau\_2)\}\,. \tag{7}$$

One-dimensional slices of 3 1 2 (,) *<sup>x</sup> c* can be defined as (Nikias and Petropulu, 1993):

$$\mathbf{r}\_{2,1}^{\times}(\tau) \underline{\nabla c}(\mathbf{x}(k), \mathbf{x}(k), \mathbf{x}(k+\tau)) = \prescript{\times}{}{\mathcal{C}}\_{\mathcal{I}}(\mathbf{0}, \tau) \tag{8}$$

and

$$\left\{ r\_{1,2}^{\times}(\tau) \underline{\nabla} c(\mathbf{x}(k), \mathbf{x}(k+\tau), \mathbf{x}(k+\tau)) = \prescript{\times}{}{\mathcal{C}}\_{\mathcal{B}}(\tau, \tau) \right\} \tag{9}$$

Define the following even and odd functions (Nikias and Petropulu, 1993):

$$s\_{2,1}^{\times}(\tau) \underline{\nabla} \frac{1}{2} [r\_{2,1}^{\times}(\tau) + r\_{1,2}^{\times}(\tau)] \tag{10}$$

and

$$\eta\_{2,1}^{\times}(\tau) \underbrace{\nabla}\_{=2}^{1} [r\_{2,1}^{\times}(\tau) - r\_{1,2}^{\times}(\tau)] \tag{11}$$

A 1-d spectrum could be defined as (Nikias and Petropulu, 1993):

Second- and Third-Order Statistical Characterization of Non-Linearity and Non-Gaussianity of Adult and Fetal ECG Signals and Noise 29

$$\mathbf{R}\_{2,1}^{\times}(o\rho) = \sum\_{\tau=-\infty}^{\infty} r\_{2,1}^{\times}(\tau) \mathbf{e}^{-j\alpha t} = \sum\_{\tau=-\infty}^{\infty} \left\{ \mathbf{s}\_{2,1}^{\times}(\tau) \cdot \cos(o\nu\tau) - jq\_{2,1}^{\times}(\tau) \cdot \sin(o\nu\tau) \right\} \tag{12}$$

#### **2.4. Cumulant spectra**

Higher order spectra are defined as the multi-dimensional Fourier transforms of the higher order statistics of the superimposed signals in the presence of noise (Nikias and Raghuveer, 1987; Rosenblatt, 1985; Brillinger, 1965). 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 (Nikias and Petropulu, 1993):

$$\mathbf{C}\_{n}^{\mathbf{x}}(\alpha\_{1}, \alpha\_{2}, \dots, \alpha\_{n-1}) = \sum\_{\tau\_{1} = -\infty}^{+\infty} \cdots \sum\_{\tau\_{n-1} = -\infty}^{+\infty} c\_{n}^{\mathbf{x}}(\tau\_{1}, \tau\_{2}, \dots, \tau\_{n} \tau\_{n-1}) \ \ e^{-j(\alpha\_{1}\tau\_{1} + \alpha\_{2}\tau\_{2} + \dots, \alpha\_{n}\tau\_{n-1})} \,, \tag{13}$$

where

28 Practical Applications in Biomedical Engineering

c[x3,x2,x1], and so on (Nikias and Petropulu, 1993)..

**2.3. One-dimensional third-order cumulant slices** 

process, {x(k)}, is given by (Nikias and Petropulu, 1993):

*x c* 

One-dimensional slices of 3 1 2 (,)

and

and

 

Define the following even and odd functions (Nikias and Petropulu, 1993):

A 1-d spectrum could be defined as (Nikias and Petropulu, 1993):

2. Cumulants are symmetric functions in their arguments, e.g., c[x1,x2,x3] = c[x2,x1,x3] =

3. If the random variables {x1, x2, …, xn} can be divided into any two or more groups which are statistically independent, their nth-order cumulant is identical to zero; i.e. c[x1, x2, …, xn] = 0, whereas in general Mom[x1, x2, …, xn] 0 (Nikias and Petropulu, 1993).. 4. If the sets of random variables {x1, x2, …, xn} and {y1, y2, …, yn} are independent, then c[x1+y1, x2+y2, …, xn+yn] = c[x1, x2, …, xn] + c[y1, y2, …, yn] (Nikias and Petropulu, 1993). 5. If the set of random variables {x1, x2, …, xn} is jointly Gaussian, then all the information about their distribution is contained in the cumulants of order n 2. Therefore, all cumulants of order greater than two (n > 2) have no new information to provide. This leads to the fact that all joint cumulants of order n > 2 are identical to zero for Gaussian random vectors. Hence, the cumulants of order greater than two, in some sense,

measure the non-Gaussian nature of a time series (Nikias and Petropulu, 1993).

Since higher-order cumulants and spectra are multi-dimensional functions, their computation may be impractical in some applications due to excessive crunching. This is caused by the large CPU time taken to calculate HOS functions, compared to SOS functions. It was suggested to use 1-d slices of multi-dimensional cumulants, and their 1-d Fourier transforms, as ways of extracting useful information from higher-order statistics of non-Gaussian stationary processes (Nagata, 1970). The third-order cumulants of a non-Gaussian

> xx x <sup>3</sup> 1 2 1 2 ( , ) { ( ), ( ), ( )} *<sup>x</sup> ck k k c*

xxx 2,1 <sup>3</sup> ( ) { ( ), ( ), ( )} (0, ) *<sup>x</sup> <sup>x</sup> r ck k k c*

xx x 1,2 <sup>3</sup> ( ) { ( ), ( ), ( )} ( , ) *<sup>x</sup> <sup>x</sup> r ck k k c*

 

> 2,1 2,1 1,2 <sup>1</sup> ( ) [ ( ) ( )] <sup>2</sup> *x xx s rr*

> 2,1 2,1 1,2 <sup>1</sup> ( ) [ ( ) ( )] <sup>2</sup> *x xx q rr*

 

 

 

> 

> >

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

 

 

. (7)

(8)

(9)

(10)

(11)

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

#### **Special cases**

1. Power spectrum (n = 2):

$$\mathbf{C}\_{2}^{\chi}(\boldsymbol{\alpha}) = \sum\_{\boldsymbol{\tau} = -\boldsymbol{\sigma}}^{+\boldsymbol{\sigma}} c\_{2}^{\chi}(\boldsymbol{\tau}) \ e^{-j(\boldsymbol{\alpha} \cdot \boldsymbol{\tau})} \tag{14}$$

where 2 ( ) *<sup>x</sup> <sup>c</sup>* is the covariance.

2. Bispectrum (n = 3):

$$C\_3^{\chi}(\alpha\_1, \alpha\_2) = \sum\_{\tau\_1 = -\infty}^{+\infty} \sum\_{\tau\_2 = -\infty}^{+\infty} c\_3^{\chi}(\tau\_1, \tau\_2) \cdot e^{-j(\alpha\_1 \tau\_1 + \alpha\_2 \tau\_2)} \tag{15}$$

where 312 (,) *<sup>x</sup> <sup>c</sup>* is the third-order cumulant. Conventional higher-order statistics (HOS) estimates are asymptotically unbiased and consistent (Nagata, 1970), and easy to implement using FFT-based methods. The ability to resolve harmonic components is limited by the uncertainty principle of the Fourier transform. There are numerous methods for polyspectra estimation based on parametric methods. MA models have been treated in (Nikias, 1988; Friedlander and Porat, 1988). Spectral estimation methods based on non-causal AR models were developed in (Huzi, 1981). Methods based on ARMA models have been published (Lii, 1982). MA, AR and ARMA methods based on higher-order statistics were described (Mendel, 1988). A review of cumulant spectra and the asymptotic properties of their estimators were given in (Rosenblatt, 1983). Practical considerations for bispectral estimation were given (Subba Rao, 1983). The relationship between the bispectrum and conventional methods for their estimation was discussed (Zurbenko, 1982).

#### **2.5. Non-stationarity and the OT region of the bispectrum**

The bispectrum of a stationary sampled process must be zero in the triangle region OT, i.e., the region defined by the triangle, OT = { The bispectrum in the OT region will be non-zero if the process is non-stationary. The bispectrum has 12 symmetric regions. The knowledge of the bispectrum in one triangular region is enough for a complete description of the bispectrum of a real process.

Second- and Third-Order Statistical Characterization of

Non-Linearity and Non-Gaussianity of Adult and Fetal ECG Signals and Noise 31

2. The transabdominally-measured ECG signal which contains both maternal and fetal contributions amongst other deterministic and chaotic signals plus noise artefacts (Rizk et al., 2002). This is acquired using twin surface electrodes positioned near the mother's

3. the fetal scalp electrode ECG signal which will always be used as a reference signal in the assessment of any particular QRS detection technique based on non-invasive transabdominally-measured ECG signals. The non-symmetry of the probability density functions (pdfs) of the above mentioned signals is shown in the histograms of Fig. 1 and

**Figure 1.** Histograms of typical templates of (a) a maternal chest ECG, (b) a fetal scalp electrode FECG,

and (c) a maternal transabdominal ECG. They all show non-Gaussian distributions.

umbilicus and synchronised with the maternal chest signal;

supports their third-order cumulants.

#### **2.6. Nth-order coherency function**

A normalised cumulant spectrum or the nth-order coherency index is a function that combines the cumulant spectrum of order n and the power spectrum. It is defined as (Nikias and Petropulu, 1993)

$$P\_n^{\mathbf{x}}(o\_1, o\_2, \dots, o\_{n-1}) \underline{\operatorname{\mathbf{U}}} \frac{\operatorname{\mathbf{C}}\_n^{\mathbf{x}}(o\_1, o\_2, \dots, o\_{n-1})}{\operatorname{\mathbf{C}}\_2^{\mathbf{x}}(o\_1) \operatorname{\mathbf{C}}\_2^{\mathbf{x}}(o\_2) \dots \operatorname{\mathbf{C}}\_2^{\mathbf{x}}(o\_{n-1}) \operatorname{\mathbf{C}}\_2^{\mathbf{x}}(o\_1 + o\_2 + \dots + o\_{n-1})} \,\text{.}\tag{16}$$

The third-order (n = 3) coherence index is also called bicoherence. The nth order coherence index is useful for the detection and characterisation of non-linearities in time series via phase relations of their harmonic components. The coherency index is used to differentiate between linear non-Gaussian processes and non-linear processes when both have non-zero cumulants. If the coherency index is zero, then the process is linear and Gaussian. If the nth order coherency index is not frequency dependent, then the process is linear non-Gaussian. If the coherency index is frequency dependent, then the process is non-linear (Nikias and Petropulu, 1993).

#### **2.7. Statistical measures**

A statistical measure could be described as an unbiased estimate when the expected value of the estimated statistic is, asymptotically, equal to the true value. An estimate of the cumulant spectra is unbiased if

$$E\left|\mathbf{C}\_3^{\wedge^x}\left(\alpha\_1, \alpha\_2\right)\right| = \mathbf{C}\_3^x\left(\alpha\_1, \alpha\_2\right) \tag{17}$$

The bias is defined as the difference between the true value and the expected value.

#### **3. Second-order statistics of ECG signals**

#### **3.1. The probability density functions (pdfs) of ECG signals**

Three essential ECG signals are considered:

1. the maternal chest ECG signal. This is measured using one surface electrode positioned on the chest and one reference electrode on the thigh;

2. The transabdominally-measured ECG signal which contains both maternal and fetal contributions amongst other deterministic and chaotic signals plus noise artefacts (Rizk et al., 2002). This is acquired using twin surface electrodes positioned near the mother's umbilicus and synchronised with the maternal chest signal;

30 Practical Applications in Biomedical Engineering

**2.6. Nth-order coherency function** 

and Petropulu, 1993)

Petropulu, 1993).

**2.7. Statistical measures** 

cumulant spectra is unbiased if

**3. Second-order statistics of ECG signals** 

Three essential ECG signals are considered:

**2.5. Non-stationarity and the OT region of the bispectrum** 

The bispectrum of a stationary sampled process must be zero in the triangle region OT, i.e., the region defined by the triangle, OT = { The bispectrum in the OT region will be non-zero if the process is non-stationary. The bispectrum has 12 symmetric regions. The knowledge of the bispectrum in one triangular

A normalised cumulant spectrum or the nth-order coherency index is a function that combines the cumulant spectrum of order n and the power spectrum. It is defined as (Nikias

12 1 2

( , ,, ) ( , ,, ) [ ( ). ( ) ( ). ( )] *x*

The third-order (n = 3) coherence index is also called bicoherence. The nth order coherence index is useful for the detection and characterisation of non-linearities in time series via phase relations of their harmonic components. The coherency index is used to differentiate between linear non-Gaussian processes and non-linear processes when both have non-zero cumulants. If the coherency index is zero, then the process is linear and Gaussian. If the nth order coherency index is not frequency dependent, then the process is linear non-Gaussian. If the coherency index is frequency dependent, then the process is non-linear (Nikias and

A statistical measure could be described as an unbiased estimate when the expected value of the estimated statistic is, asymptotically, equal to the true value. An estimate of the

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1. the maternal chest ECG signal. This is measured using one surface electrode positioned

 

*x*

**3.1. The probability density functions (pdfs) of ECG signals** 

on the chest and one reference electrode on the thigh;

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The bias is defined as the difference between the true value and the expected value.

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region is enough for a complete description of the bispectrum of a real process.

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3. the fetal scalp electrode ECG signal which will always be used as a reference signal in the assessment of any particular QRS detection technique based on non-invasive transabdominally-measured ECG signals. The non-symmetry of the probability density functions (pdfs) of the above mentioned signals is shown in the histograms of Fig. 1 and supports their third-order cumulants.

**Figure 1.** Histograms of typical templates of (a) a maternal chest ECG, (b) a fetal scalp electrode FECG, and (c) a maternal transabdominal ECG. They all show non-Gaussian distributions.

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

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 with maternal contribution, and QRS-free ECG contributions.

Second- and Third-Order Statistical Characterization of

Non-Linearity and Non-Gaussianity of Adult and Fetal ECG Signals and Noise 33

**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.

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

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