FECG Extraction Algorithm Based on BSS Using Temporal Structure and

DWT

Rui Li 1,a, Baofeng Chen 2,b 1 College of Science, Henan University of Technology, Zhengzhou 450001, China

2 School of Mathematics and Physics, Anyang Institude of Technology, Anyang 455000, China

aliruilyric@126.com,

bbfchen001@163.com

Keywords: Fetal electrocardiogram (FECG); Blind source extraction (BSE); Discrete wavelet transformation (DWT); Second order statistics (SOS)

Abstract. Fetal electrocardiogram (FECG) blind source extraction (BSE) algorithm based on

temporal structure and discrete wavelet transformation (DWT) in noise is proposed in this paper.

After building the basic blind source separation (BSS) and BSE models for FECG, some

preprocessing procedures based on the temporal structure of the FECG are constructed. Using DWT

we can move the conventional time-domain signals to the wavelet-domain, and then the source

number is detected and the robust noise reduction technique in FECG can be deduced too. According

this preprocessing and second-order statistics (SOS), the proposed robust FECG extraction algorithm

is derived.

Introduction

The extraction of fetal electrocardiogram (FECG) [1-16] is of vital importance from a clinical

point of view, because it provides information about the health and the possible diseases of a fetus.

Before delivery, noninvasive techniques to acquire the FECG are preferred. However, the desired

fetal heartbeat signal appearing at the electrode output is always corrupted by considerable noise, such

as the maternal electrocardiogram (MECG) contributions with extremely high amplitude, the

mother’s respiration, the power line interference, and the thermal noise due to electronic equipment.

Therefore obtaining FECG is a difficult task.

A more recent approach employs blind source separation (BSS), which separate the sources from

their mixtures by assuming the sources are statistically independent [3,4], which is also called

independent component analysis(ICA) in this case. It was reported that the results by BSS were more

satisfactory than those by classical methods [1,5]. However, separating all the sources from a large

number of observed sensor signals takes a long time and is not necessary, so the blind source

extraction (BSE) method is preferred. The basic objective of BSE is estimating one or part of the

source signals from their linear mixtures. Such signals are usually with specified stochastic properties

and bring the most useful information. Many BSE algorithms use the property of sparseness [6] or high-order statistics (HOS) [7] to

extract a specific signal. High-order methods address the BSE problem in a completely blind context, since they require few assumptions aside from the statistical independence of the sources. The second-order statistics (SOS) operate in a semi-blind setting, since their derivation usually requires the certain additional assumptions made on the nature of the source signals. Nonetheless, such information is available in practical applications and should be exploited. Furthermore, the high-order approaches often have higher computational load compared with the second-order methods. Thus, the versatile extraction algorithms based on SOS become popular. Barros et al. [8] presented an objective function by using blind signal extraction along with prior information about the autocorrelation property of the FECG from the noisy free measurements. Then, based on this algorithm, a batch of FECG algorithms have been emerged for the noisy free data [9-15].

But FECG are always corrupted by considerable noise, such as the power line interference, the thermal noise due to electronic equipment and so on. Therefore developing FECG extraction algorithm in noisy environment is very important. Although [16] discussed this problem, the authors didn’t present how to deal with the noise. In this paper, we extend the FECG extraction method based on SOS for the noisy case.

Applied Mechanics and Materials Vols. 571-572 (2014) pp 209-212Online available since 2014/Jun/10 at www.scientific.net© (2014) Trans Tech Publications, Switzerlanddoi:10.4028/www.scientific.net/AMM.571-572.209

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Basic Model for FECG

Let us denote the N source signals by the vector ( )ks , and the observed signals by ( )kx . The mixing

can be expressed as: ( ) ( ) ( )k k k= +x Hs v , where the matrix [ ] M N

ijh RH×= ∈ collects the mixing

coefficients. No particular assumptions on the mixing coefficients are made. The problem is now to

estimate both the source signals ( )js k and the mixing matrix H based on observations of the ( )ix k

alone [3,5]. There are two principal approaches to solve this problem. The first approach is to separate

all sources simultaneously which is called BSS. In the second approach, we extract sources one by one

sequentially rather than separating them all simultaneously and this procedure is called BSE.

The task of BSS is to recover all the original signals from the observations ( )kx simultaneously

without the knowledge of H nor ( )ks . Let us consider a linear feed forward memoryless neural

network which maps the observation ( )kx to ( )ky by the following linear transform,

( ) ( ) ( )k k k= =y Wx WHs , where [ ] N N

ijw R ×= ∈W is a separating matrix, ( )ky is an estimate of the

possibly scaled and permutated vector of ( )ks and also the network output signals whose elements

are statistically mutually independent, so that the output signals ( )ky are possibly scaled estimation

of source signals ( )ks . As shown in [3,5], there are two indeterminacies in BSS: scaling ambiguity

and permutation ambiguity. But this does not affect the application of BSS, because the main

information of the signals is included in the waveform of them.

In BSE, one observes an N dimensional stochastic signal vector ( )kx that is regarded as the linear

transformation of an N dimensional source vector ( )ks , i.e., ( ) ( )k k=x Hs , where H is an unknown

mixing matrix. The goal of BSE is to find a vector w such that , T T( ) ( ) ( )y k k k= =w x w Hs is an

estimated source signal up to a scalar. In applications, any of the source signals could come out as the

first one. For solving this problem, many approaches use a priori information of the desired source

signal.

Proposed BSE Algorithm

FECG BSE Algorithm. As mentioned in section 1 and 2, the main FECG signal processing model is

the BSE, so in this part, the FECG extraction methods are based on the BSE model. Let us denote the

source signal vector as ( )ks . In the model, the observed vector ( )kx results by linearly mixing the

source signals. Thus, this mixture can be written as ( ) ( ) ( )k k ky Hs v= + where H is an nonsingular

matrix and ( )kv is a vector of additive noise. In general, the noise ( )kv is assumed to be

uncorrelated with the source signals and its correlation matrix is given by

T0 for 0

{ ( ) ( )}for 0

kE k k k

k

≠− = = v

v vR

(1)

Assume that the source signals have temporal structure and they have different autocorrelation

functions, but they do not necessarily have to be statistically independent. In fact, their mutual

decorrelation in a given range is sufficient (but not necessary) to extract them successfully. In other

words, a simple batch algorithm can guarantee blind extraction of any source signal is that satisfies

the specific time delay iτ the following relations :

[ ( ) ( )] 0,

[ ( ) ( )] 0 .

i i i

i j i

E s k s k

E s k s k i j

τ

τ

− ≠

− = ∀ ≠ (2)

A simple solution is to calculate the autocorrelation [ ( ) ( )]j j iE x k x k τ− of sensor signals as a

function of the time delay and find the one corresponding to the pick of [ ( ) ( )]j j iE x k x k τ− [8]. The

aim of extracting the desired FECG is then converted to seek two sets of appropriate parameters w

and b to minimize the objective function: 2 2 2( ) [ ( )] (1 ) .J E k bε σ= − +w

210 Computers and Information Processing Technologies I

Calculating the derivative of the objective function (7) with respect to w and b , we have the

following updating rule:

T

T T T

T 2

( 1) ( ) 2 {( ( ) ( ))( ( ) ( )) } ( ),

( 1) ( ) { ( ) [ ( )( ( )) ( )( ( ))

2 ( )( ( )) ] ( ) 2 },

.

i i

b i i

i i

k k E k b k k b k k

b k b k k E k k k k

b k k k b

µ τ τ

µ τ τ

τ τ σ

+ ← − − − − −

+ ← − − − − −

+ − − −

←

ww w x x x x w

w x x x x

x x w

w w w

(3)

followed by a normalization of the Euclidean norm of vector ( )kw to unit to avoid the trivial solution

( ) 0k =w . In Eq. (3), µw

and bµ are small positive learning rates. Moreover, we can assume without

losing generality that the sensor data are prewhitened, thus, T[ ( ) ( )]E k k =x x I . The algorithm updates

b together with w to make the objective function (7) decrease and eventually extracts the desired

signals [16].

Robust Noise Reduction Technique. A crucial step in many array signal processing applications is

to accurate estimate the unknown number of sources and the variance 2σ of the noise. Standard

techniques for source number estimation often have shortcomings in practice when mixed signals are

buried in additive noise with unknown parameters.

Normally we further assume that 2

vR Iσ= , where 2

σ I is the variance of the noise. In this paper, we

will estimate 2σ using wavelet method. Consider T variate data according to the model, the

covariance matrix is T( )( ( )) TE t tx s v

R x x HR H R = = + , where T[ ( )( ( )) ]E t ts

R s s= . The problem

addressed here is to determine the number of sources contributing to the mixed signals or equivalently

the signal subspace, i.e. Trank( )Ns

HR H= . In the particular case of Gaussian spatially white noise

with variance 2σ ( 2

vR Iσ= ), the eigenvalues in decreasing order are:

2

1 2 1 2P P P Nλ λ λ λ λ λ σ+ +≥ ≥ ≥ > = = = = .

So, instead of estimating the number of mixed source signals from the time domain noisy mixed

signals ( )tx , we estimate P in the wavelet domain using the discrete wavelet transformation(DWT)

of ( )tx . After threshold the detail coefficients of wavelet transformed individual mixed signal, the

next step is to exploit the spatial information of the de-noised data array in the wavelet domain. The

matrix is T

1 2ˆ ˆ ˆ, , ,L L L L

MV v v v = , where { }1 2ˆ ˆ ˆ ˆˆ , , , , , 1, ,L L L

i i i i i i Nv d d d a= = denotes the N T× array

wavelet de-noised transform of the mixed signals at level L . Then the cross-correlation matrix of LV

is computed as: T( )L L LEV

R V V = .

In the case of spatially correlated noise and low SNR (<0dB) it is not guaranteed that the matrix L

VR

is diagonal. Assuming the most general case where the noise is spatially correlated across all

subbands, subspace analysis techniques, such as the singular value decomposition (SVD), are used as

to diagonalize L

VR . Then, T( )L L LE

VR V V = . This procedure will separate the signal subspace from

the noise subspace taking the following form: 1 2 1 2 0P P P Nλ λ λ λ λ λ+ +> > > > > > > ≈ . This

suggests that estimating P equal to estimate the multiplicity of the smallest eigenvalues. To this end,

we consider the differences between adjacent: , 1, , ; 1, ,i j i j i N j NΛ = λ − λ = = . The difference

i jΛ will be small when iλ and jλ are both noise eigenvalues but large if one or both of iλ and jλ

are source eigenvalues. The test starts checking equality (4) from the smallest difference between

adjacent eigenvalues corresponding to the interface between signal and noise subspaces. When the

difference i jΛ greater than the false alarm probability ε , the detection scheme is over.

Applied Mechanics and Materials Vols. 571-572 211

Conclusions

In this paper, we introduce a robust FECG extraction BSE algorithm in noise. After the basic BSE

model for FECG is built, we give some new ideas to tackle this problem via the DWT. Using DWT

we can move the conventional time-domain signals to the wavelet-domain, as a result, the source

number is detected and the noise in FECG can be removed as clearly as possible. Then, the proposed

robust FECG extraction algorithm is derived. At last, the simulation results verify the validity of the

proposed algorithm.

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