research article noise suppression in ecg signals through...

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 763903 13 pageshttpdxdoiorg1011552013763903

Research ArticleNoise Suppression in ECG Signals through Efficient One-StepWavelet Processing Techniques

E Castillo D P Morales A Garciacutea F Martiacutenez-Martiacute L Parrilla and A J Palma

Department of Electronics and Computer Technology ETSIIT University of Granada C Daniel Saucedo Aranda18071 Granada Spain

Correspondence should be addressed to D P Morales diegopmugres

Received 16 March 2013 Accepted 13 May 2013

Academic Editor Chang-Hwan Im

Copyright copy 2013 E Castillo et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper illustrates the application of the discrete wavelet transform (DWT) for wandering and noise suppression inelectrocardiographic (ECG) signals A novel one-step implementation is presented which allows improving the overall denoisingprocess In addition an exhaustive study is carried out defining threshold limits and thresholding rules for optimal waveletdenoising using this presented techniqueThe systemhas been tested using synthetic ECG signals which allow accuratelymeasuringthe effect of the proposed processing Moreover results from real abdominal ECG signals acquired from pregnant women arepresented in order to validate the presented approach

1 Introduction

Electrocardiogram (ECG) is a valuable technique that hasbeen in use for over a century not only for clinic applications[1] ECG acquisition from the human skin involves the useof high gain instrumentation amplifiers This fact makes theECG signal to be contaminated by different sources of noise[2] This circumstance is highlighted when the target is themeasurement of fetal ECG signals acquired over the motherrsquosabdomen [3] For processing ECG signals it is necessary toremove contaminants from these signals that make visualinspection and ECG feature extraction difficult In generalECG contaminants can be classified into different categoriesincluding power line interference electrode pop or contactnoise patient-electrode motion artifacts electromyographic(EMG) noise and baseline wandering Among these noisesthe power line interference and the baseline wandering(BW) are the most significant and can strongly affect ECGsignal analysis Except for these two noises other noisesmay be wideband and usually involve a complex stochasticprocess which also distorts the ECG signal The powerline interference is narrow-band noise centered at 50Hz or60Hz with a bandwidth of less than 1Hz Usually the ECGsignal acquisition analog hardware can remove the powerline interference However the baseline wandering and other

wideband noises are not easy to be suppressed by analogcircuits Instead the software scheme is more powerful andfeasible for portable ECG signal processing Thus denoisingthis type of signals is decisive for further parameter extractionin clinic applications

Wavelet transform (WT) [4] is a useful tool for a variety ofsignal processing [5 6] and compression applications [7 8]its primary and most advantageous application areas arethose that have to generate or process wideband signals Thistransform produces a time-frequency decomposition of thesignal under analysis which separates individual signal com-ponentsmore effectively than the traditional Fourier analysisThis fact makes WT one of the most used tools for biosignalprocessing with ECG being an obvious candidate for thistype of analysis Discrete wavelet transform (DWT) providesamultiresolution analysis which allows representing a signalby a finite sum of components at different resolutions so thateach component can be processed adaptively based on theobjectives of the application

This paper proposes an arrangement of discrete wavelettransform structures for ECG signal processing on portableembedded computing real-time implementations [9] focus-ing on the suppression of different types of noise includingDC levels andwanderingThis suppression is carried out witha one-stepwavelet processing which reduces computing cost

2 Journal of Applied Mathematics

Appropriate software models and parameter selection arepresented based on mathematical analysis for the mentionedECG digital processingMoreover these softwaremodels willalso enable very quick tests of the required signal processingIn order to select an appropriate method for the developmentof the software modeling noise suppression and featureextraction procedures on ECG signals are studied in thefollowing

2 Materials and Methods

21 DWT Fundamentals Wavelet transform (WT) is one ofthe most used tools in multiresolution signal analysis due toits ability to decompose a signal at various resolutions whichallow observing high-frequency events of short duration innonstationary signals [10] The continuous WT of a signal119891(119905) isin 1198712(119877) is defined as [11]

119882(119886 119887) = int+infin

minusinfin

119891 (119905) 120595119886119887

(119905) 119889119905 (1)

120595119886119887

(119905) =1

radic119886120595lowast (

119905 minus 119887

119886) (2)

In (2) lowast denotes complex conjugate and 119886 is a scale factorand 119887 a translation factor The normalization factor 119886minus12 isincluded so that 120595

119886119887 = 120595 Thus 120595

119886119887(119905) represents a

shifted and scaled version of the so-called mother wavelet120595 which is a window function that defines the basis for thewavelet transformation A mother wavelet 120595 of order 119898 isa function 120595 119877 rarr 119877 which satisfies the following fourproperties

(1) If119898 gt 1 then 120595 is (119898 minus 1) times differentiable(2) 120595 isin 119871infin(119877) If 119898 gt 1 for each 119895 isin 1 119898 minus 1

120595(119895) isin 119871infin(119877)(3) 120595 and all its derivatives up to order (119898 minus 1) decay

rapidly for each 119903 gt 0 there is a 120574 gt 0 such that

10038161003816100381610038161003816120595(119895)

(119905)10038161003816100381610038161003816 lt

1

119905 119895 isin 1 119898 minus 1 for each |119905| gt 120574 (3)

(4) For each 119895 isin 0 1 119898 int 119905119895120595(119905)119889119905 = 0

Practical applications rely on the discrete wavelet transform(DWT) since it provides enough informationwhile requiringreasonable computation time and resources The DWT of adiscrete function 119891(119899) is defined as

119882(119886 119887) = 119862 (119895 119896) = sum119899

119891 (119899) 120595119895119896

(119899) (4)

where

120595119895119896

(119899) = 2minus1198952120595 (2minus119895119899 minus 119896) (5)

and 119886 = 2119895 and 119887 = 1198962119895 with 119895 119896 isin Z Thus DWT iscomputed in practice through a set of two FIR-like filterslowpass and highpass at each decomposition level 119894 followedby a downsampling by 2 which implies a reduction in

the sampling frequency The result of the low-pass filter isusually known as the approximation and is used as inputto the following decomposition level while the result of thehigh-pass filter is called the detail Thus the usual DWTcorresponds to the scheme in Figure 1(a) with

119886(119894)119899

=119873minus1

sum119896=0

119892119896119886(119894minus1)2119899minus119896

119894 = 1 2 119869

119889(119894)119899

=119873minus1

sum119896=0

ℎ119896119886(119894minus1)2119899minus119896

119886(0)119899

equiv 119909119899

(6)

for an 119873-sample input sequence The coefficients of thelow-pass and high-pass filters are defined by the family ofwavelet functions used as basis for the transformation HaarDaubechies Quadratic Spline and so forth [11]

Since every detail is the result of high-pass filtering of theprevious approximation and approximations are the resultof low-pass filtering these details and approximations atevery decomposition level contain information at differentfrequencies and time scales this reflects the multiresolutionanalysis of theWT Moreover it is possible to reconstruct theoriginal signal from the set of details and approximationsthrough the inverse DWT so

119886(119894minus1)119898

=

(1198732)minus1

sum119896=0

1198922119896119886(119894)(1198982)minus119896

+(1198732)minus1

sum119896=0

ℎ2119896119889(119894)(1198982)minus119896

119898 even(1198732)minus1

sum119896=0

1198922119896+1

119886(119894)((119898minus1)2)minus119896

+(1198732)minus1

sum119896=0

ℎ2119896+1

119889(119894)((119898minus1)2)minus119896

119898 odd(7)

where once more the corresponding details are high-passfiltered and the approximations are low-pass filtered thistime through the appropriate reconstruction filtersThus thereconstructed approximation at level 119894 minus 1 is obtained byaddition of the output of the reconstruction filters and anupsampling by 2 as shown in Figure 1(b)

22 DWT-Based ECG Processing The special characteristicsof ECG signals and their frequency spectrummake the use oftraditional Fourier analysis for the detection of ECG featuresdifficult [12] Wavelet transform can be applied in manyfields but its primary and most advantageous applicationareas are those that have generate or process widebandsignalsThis is due to themultiresolution analysis that waveletprovides which allows representing a signal by a finite sum ofcomponents at different resolutions so that each componentcan be processed adaptively based on the objectives of theapplication In this way this technique represents signalscompactly and in several levels of resolution which is idealfor decomposition and reconstruction purposes Thus the

Journal of Applied Mathematics 3

H(z)

H(z)

G(z)

G(z)

G(z)

H(z)

a(0)n

a(1)n

d(1)n

d(2)n

a(2)n a

(3)n

d(3)n

(J)

darr2

darr2

darr2

darr2

darr2

darr2

middot middot middot

(a)

+

+

+

(J)

a(0)n

a(1)n

d(1)n

d(2)n

a(2)n

a(3)n

d(3)n H(z)

G(z)

H(z)

G(z)

H(z)

G(z)

uarr2

uarr2

uarr2

uarr2

uarr2

uarr2


(b)

Figure 1 Block diagramof themultiresolution decomposition usingthe wavelet transform (a) analysis filter bank and (b) reconstructionfilter bank

discrete wavelet transform is an effective way to digitallyremove noises within specific subbands for ECG signals

Several wavelet families have been proposed for ECGprocessing [12] Selecting the right wavelet for this applicationis a task requiring at least a brief discussion The wavelettype to use for the discrete wavelet analysis is an importantdecision for this processing Singh and Tiwari [13] studieddifferent wavelet families and analyzed the associated prop-erties We can distinguish between two types the orthogonalwavelets as Haar Daubechies (dbxx) Coiflets (coifx) Symlets(symx) and biorthogonal (biorx x) where 119909 indicates theorder of the wavelet and the higher the order the smootherthe wavelet The orthogonal wavelets are not redundant andare suitable for signal or image denoising and compressionA biorthogonal wavelet is a wavelet where the associatedwavelet transform is invertible but not necessarily orthogo-nal Designing biorthogonal wavelets allows more degrees offreedom than orthogonal wavelets One of these additionaldegrees of freedom is the possibility to construct symmetricwavelet functionsThe biorthogonal wavelets usually have thelinear phase property and are suitable for signal or imagefeature extraction

221 Wavelet-Based Wandering Suppression Baseline wan-dering (BW) usually comes from respiration at frequenciesvarying between 015 and 03Hz Different methods forwandering suppression exist will the common objective ofthem to make the resulting ECG signals contain as littlebaseline wandering information as possible while retainingthemain characteristics of the original ECG signal One of theproposed methods consists in high-pass digital filtering forexample a Kaiser Window FIR high-pass filter [14] could bedesigned where appropriate specifications of the high-passfilter should be selected to remove the baseline wandering

In addition to digital filters the wavelet transform canalso be used to remove the low frequency trend of a signalThis wavelet-based approach is better because it introducesno latency and less distortion than the digital filter-basedapproach The required steps for the application of thiswavelet-based processing for baseline wander correction arethe following

(1) Decomposition apply wavelet transform to the signalup to a certain level 119871 in order to produce thewavelet approximation coefficients 119886(119871)

119899that captures

the baseline wander(2) Zeroing approximation coefficients replace the

approximation vector 119886(119871)119899

for an all-zero vector tosubtract this part from the raw ECG signal andremove the wandering

(3) Reconstruction compute wavelet reconstructionbased on these zeroing approximation coefficients oflevel 119871 and the detail coefficients of levels from 1 to 119871in order to obtain the BW corrected signal

The main idea is to remove the low frequency componentswhich better estimate the baseline wander This processing iseasy to carry out using wavelet decomposition for which it isnecessary to select the proper wavelet function and resolutionlevel

222 Wavelet-Based Denoising The goal in ECG denoisingis to try to recover the clean ECG from the undesiredartifacts with minimum distortion The recovered signal iscalled denoised signal and it allows further ECG processingsuch as in the case of separation of fetal ECG [3] QRScomplex detection and parameter estimation (such as thecardiac rhythm) The underlying model for the noisy signalis basically of the following form

119904 (119899) = 119891 (119899) + 120590119890 (119899) (8)

where 119904(119899) represents the noisy signal 119891(119899) is an unknowndeterministic signal time 119899 is equally spaced and 120590 is a noiselevel In the simplestmodel we suppose that 119890(119899) is a Gaussianwhite noise 119873(120583 1205902) = 119873(0 1) The denoising objectiveis to suppress the noise part of the noisy ECG signal andto recover the clean ECG From a statistical viewpoint themodel is a regression model over time and the method canbe viewed as a nonparametric estimation of the function 119891using orthogonal basis

Wavelet denoising has emerged as an effective methodrequiring no complex treatment of the noisy signal [15] Itis due to the sparsity localitym and multiresolution natureof the wavelet transformThe wavelet transform localizes themost important spatial and frequential features of a regularsignal in a limited number of wavelet coefficients Moreoverthe orthogonal transform of stationary white noise resultsin stationary white noise Thus in the wavelet domain therandom noise is spread fairly uniformly among all detailcoefficients On the other hand the signal is representedby a small number of nonzero coefficients with relativelylarger values This sparsity property assures that wavelet


shrinkage can reduce noise effectively while preserving thesharp features (peaks of QRS complex) The general waveletdenoising procedure basically proceeds in three steps [11]

(1) Decomposition apply wavelet transform to the noisysignal 119904(119899) to produce the noisy wavelet coefficients tothe level119873 by which we can properly distinguish thepresence of partial discharges

(2) Threshold detail coefficients for each level from 1 to119873 select appropriate threshold limit and apply softor hard thresholding to the detail coefficients at someparticular levels to best remove the noise

(3) Reconstruction compute wavelet reconstructionbased on the original approximation coefficients oflevel 119873 and the modified detail coefficients of levelsfrom 1 to 119873 in order to obtain estimatedsmoothedsignal 119891(119899)

Although the application of this denoising method is notconceptually complex some issues are carefully studied andaddressed in the following for getting satisfactory results

3 One-Step DWT-Based Baseline Wandering(BW) and Noise Suppression Proposal

Due to the similar wavelet structure for the application of BWand noise suppression we propose here to apply in only onestep both wavelet-based techniques It will save importantresources andor time which would facilitate any futureportable hardware implementation [16 17] The requiredsteps for the application of this approach are as follows

(1) Decomposition the wavelet decomposition is applieddown to a certain level 119871 in order to produce theapproximation coefficients 119886(119871)

119899that capture the BW

(2) Zeroing approximations the approximation 119886(119871)119899

isreplaced by all-zero vector

(3) Threshold details the level 119872 (with 119872 lt 119871) allow-ing to properly distinguish the presence of partialdischarges in the noisy details must be selectedAdditionally for each level from 119894 = 1 to 119872 theappropriate threshold limit and rule (soft or hard)are applied to the detail coefficients 119889(119894)

119899for better

removing the noise

(4) Reconstruction the wavelet reconstruction based onthe zeroing approximations of level 119871 the modifieddetails of levels 1 to 119872 and original details of levelsfrom 119872 + 1 to 119871 is computed to obtain the BWcorrected and denoised signal

Thus simultaneous BWandnoise suppressions are easy to getusing this wavelet-based technique for which it is necessaryto select the proper parameters One of the aims of this workis to develop a mathematical processing model oriented toportable hardware implementation Thus a deep analysis ofthe involved parameters is detailed in the following

31 Analysis of Parameters

311 Wavelet Family The selection of the wavelet familyhas to be based on the similarities between the ECG basicstructure and thewavelet functions and the type of processingto apply For ECGwandering suppression the selection of thewavelet family is based on the study of the wavelet that bestresembles the most significant and characteristic waveformQRS of the ECG signalThus the detail sequences at differentlevels of decomposition from 1 to 119871 can capture and keep thedetail features that are of interest for properly reconstructingthe ECG without baseline wander This is achieved throughthe elimination of the approximation sequence at level 119871 Theselection of the wavelet family for ECG denoising is madein a similar way to that for wandering removing it is alsobased on the different types of wavelets and their correlationto different signals The order-6 Daubechies wavelet is afunctionwell suited because of its similarity to an actual ECGOther features include order-10 Symlets

312 Resolution Level for Wandering Another importantparameter for wavelet-based wandering suppression is thedecomposition level 119871 so that 119886(119871)

119899can capture the baseline

wander without oversmoothing To select this resolutionlevel it is important to take into account the maximumnumber of decomposition levels 119873 which is determined bythe length of the sampled ECG signal Decomposition levelmust be a positive integer not greater than log

2(119871119904) where

119871119904is the length of the signal two different methods can be

employed for selecting the resolution level

(i) Visual inspection it consists of plotting the approxi-mation sequences for different decomposition levelsand to select the resolution level whose approxima-tion better captures the baseline wandering ECGsignal from lead 4 of the DaISy dataset [18] is chosento illustrate this methodThis dataset contains 8 leadsof skin potential recordings of a pregnant womanThelead recordings three thoracic and five abdominalwere sampled at 250 sps rate and are 10-second longFigure 2 plots the ECG signal and approximationsequences 119886(119894)

119899for 119894 = 5 to 9 This figure shows that

119886(6)119899

and 119886(7)119899

are the approximation sequences bettercapturing the baseline wander However this methodis not effective for real-time processing

(ii) Analytical calculation this method is based on thecalculation of the resolution level for wander suppres-sion as followsWavelet decomposition uses half bandlow-pass and high-pass filters As it was commentedin Section 2 multiresolution decomposition allowsrepresenting a signal by a finite sum of componentsat different resolutions Each decomposition levelcontains information at different frequency bandsand time scales The approximation sequence at level119894 119886(119894)119899 is decomposed to obtain approximation and

detail at level 119894 + 1 119886(119894+1)119899

and 119889(119894+1)119899

Specificallyapproximation 119886(119894+1)

119899is the result of low-filtering

approximation 119886(119894)119899

and a downsampling by 2 so that


500 1000 1500 2000 2500

minus08minus06minus04minus02

002040608

ECG

dat

a

(a)

10 20 30 40 50 60 70

02

02

02

02

5 10 15 20 25 30 35 40

2 4 6 8 10 12 14 16 18 20

2 3 4 5 6 7 8 9 10 11

1 15 2 25 3 35 4 45 5 55 6

02

4

4

Samples

minus2

minus2

minus2

minus2

minus2

A5

A6

A7

A8

A9

(b)

Figure 2 ECG data with BW (lead 4 DaIsy dataset) and approxima-tion sequences at decomposition levels from 5 to 9

frequencies in 119886(119894)119899

that are above half of the highestfrequency component are removed This means thatat every decomposition level the frequency band ofthe approximation is reduced to the half Figure 3shows a study of the frequency subbands of waveletdecomposition of DaISy dataset lead 4 This figuredisplays approximation FFTs for resolution levelsfor 119894 = 1 to 7 Let us consider that the mostimportant frequency bands in baseline wander arebelow a certain frequency 119891

119888 For example 119891

119888=

1Hz for wandering coming from respiration (015ndash03Hz) [19] Other wandering components such asmotion of the patients and instruments have higherfrequencies components To remove wandering itshould be necessary to select the resolution level suchas the approximation captures the ECG componentsfor frequencies lower than this119891

119888The decomposition

level for wandering suppression can be calculated asfollows

119871BW = int [log2(119865max119891119888

)] (9)

10

0

0

0

0

0

0

0 20 30 40 50 60

10 20 30 40 50 60

10 20 30 40 50 60

10 20 30 40 50 60

10 20 30 40 50 60

10 20 30 40 50 60

10 20 30 40 50 60

005025

01

0501

03

102

0208

02

14

0515

Frequency (Hz)|F

FTA

1|

|FFT

A2|

|FFT

A3|

|FFT

A4|

|FFT

A5|

|FFT

A6|

|FFT

A7|

0 01 02 03 04 05 06 07 08 09

15

05

Frequency (Hz)

|FFT

A7|

01

05

0 01 02 03 04 05 06 07 08 09Frequency (Hz)

|FFT

A7|

Figure 3 FFT of ECG DaIsy dataset lead 4 ECG1 and FFTs ofapproximation sequences at decomposition levels from 1 to 7 andzoom to FFTs approximation and detail level 7

where 119865max is the highest frequency component Forexample the sampling frequency for DaISy dataset is250Hz so 119865max=125Hz (Nyquist Theorem) and for119891119888= 1Hz from (9) we can obtain 119871BW = 7 After

applying seven low-pass filters and downsamplingprocesses 119886(7)

119899captures frequencies from 0Hz to

0977Hz and is a good estimation of baseline wanderThe visual inspection method also determined that119886(7)119899

captures the baseline wander The BW correctedsignal can be obtained using wavelet reconstructionbased on the detail coefficients of levels from 1 to 7 andzeroing 119886(7)

119899 Figure 4 illustrates the proposed method

for wavelet-based wandering suppression This figureshows the original ECG signal (DaISy dataset lead 4)the estimated baseline wander with 119886(7)

119899 and the ECG

signal obtained after zeroing 119886(7)119899

313 Level of Decomposition for Denoising The maximumlevel for detail thresholding 119872 depends on several factorssuch as the SNR in the original signal or the sample rateThe level of decomposition specifies the number of levelsin the discrete wavelet analysis to take into account fordetail thresholding Unlike conventional techniques waveletdecomposition produces a family of hierarchically organizeddecompositions The selection of a suitable level for thehierarchy will depend on the signal and experience Mostoften the level is chosen based on a desired low-pass cut-offfrequency Measured signal having lower SNR usually needsmore levels of wavelet transform to remove most of its noiseOn the other hand a factor that also influences the optimal


500 1000 1500 2000 2500

ECG signal

2 4 6 8 10 12 14 16 18 20

0123

500 1000 1500 2000 2500

005

BW corrected ECG

Estimated BW using A7

minus05

minus3minus1minus2

005

minus05

Figure 4 Original ECG signal (DaIsy dataset lead 4) estimated BWfrom 119860

7 and BW corrected signal

level decomposition for denoising is the sampling rate of ECGsignal Sharma et al [15] use a study of the spectrum of thedetail coefficient at each level for estimating the optimumlevel decomposition for denoisingThey concluded that noisecontent is significant in high frequency detail subbands whilemost of the spectral energy lies in low frequency subbandsFor an ECG signal sampled at 119865

119904= 500Hz noise content

is significant in detail subbands 119889(1)119899 119889(2)119899 and 119889(3)

119899 Thus

in order to avoid losing clinically important components ofthe signal such as PQRST morphologies only detail 119889(1)

119899

119889(2)119899 and 119889(3)

119899should be treated for denoising Manikandan

andDandapat [20] present a wavelet energy-based diagnosticdistortion measure to assess the reconstructed signal qualityfor ECG compression algorithmsTheirwork includes a studyshowing for a given sampling frequency the information ofthe ECG signal and its energy contribution at each frequencysubbandThis helps to understand how the noise perturbs thedetail coefficients and the effect over the energy spectrumThe authors also show several examples that clarify theeffect of applying ldquozeroingrdquo of detail coefficients at differentdecomposition levels As an example to clarify the correctselection of this parameter Figure 5 shows 4th level waveletdecomposition for ECG signal DaISy dataset lead 4 It canbe observed that small subbands reflect the high frequencycomponents of the signal and large subbands reflect thelow frequency components of the signal The effect of highfrequency artifacts can be seen in detail subbands 119889(1)

119899and

119889(2)119899 These bands are weighted with small values because the

energy contribution to the spectrum is low For this exampleit should be advisable to treat only detail subbands 119889(1)

119899and

119889(2)119899

for wavelet denoising in order to maintain the mainfeatures of the ECG signal It is due as the figure shows tothe fact that detail subbands higher than level 2 contain mostof the significant information for diagnostic A more detailedstudy for the determination of the optimum level for wavelet-based denoising is shown in Section 4

500 1000 1500 2000 2500

020

ECG

200 400 600 800 1000 1200

20

100 200 300 400 500 600

50 100 150 200 250 300

20 40 60 80 100 120 140

20 40 60 80 100 120 140

minus20minus40

minus20minus60

20minus20minus60

20minus20minus60

20minus20minus60

20minus20minus60

CD 1

CD 2

CD 3

CD 4

CA 4

Figure 5 fourth level wavelet decomposition for DaISy dataset lead4

314 Threshold Limits Most algorithms are based on theprevious threshold definition established in Donogorsquos uni-versal theory [21] Since this work modified versions of theuniversal threshold and new thresholds have been proposedMatlab functions for denoising (wdenm thselectm) [22]and Labview blocks (Wavelet Denoise Express VI) [23]establish some of these thresholds as predefined optionsrigsure sqtwlog heursure and minimax Taking this intoaccount the aim of this work is to develop processingmodels for portable computing implementations so a studyof the proposed threshold based on the following criteriawas made computational complexity delays latency andclock frequency The evaluation of these criteria derives thefollowing threshold classification

PrefixedThresholds For a given signal length119873 and a level ofdecomposition for wavelet denoising 119869 it will be possible toknow the coefficient length for each level 119895 denoted by 119899

119895 and

thus some thresholds proposed in the literature [13] could becalculated using software tools and stored inmemory for laterusage Some of them are the following

(i) universal threshold

Thuni = radic2 log 119873 (10)

(ii) universal threshold level dependent

Thuni119895 = radic2 log 119899119895 (11)

(iii) universal modified threshold level dependent

Thuni mod 119895 =radic2 log 119899

119895

radic119899119895

(12)


(iv) exponential threshold

Thexp = 2((119895minus119869)2)radic2 log 119873 (13)

(v) exponential threshold level dependent

Thexp119895 = 2((119895minus119869)2)radic2 log 119899119895 (14)

(vi) minimax threshold (RefMatlab)

Thminimax = 03936 + 01829 lowast (log (119899

119895)

log (2)) (15)

Minimax threshold uses the minimax principle toestimate the threshold [24]

Nonprefixed Thresholds There are more threshold proposalsthat have been positively evaluated [15 22] and in some casesget better results than the prefixed thresholds However aprevious software calculation would not be possible sincesamples andor coefficient data which are initially unknownare neededOn the other hand the estimation of these thresh-olds could imply complex operations and thus an importantincrement of latency The following are two examples of thistype of thresholds

(i) maxcoef threshold [22]

Thmaxcoef = 2119899minus119869

119899 = round [log2(max 10038161003816100381610038161003816119888119895

10038161003816100381610038161003816)] (16)

(ii) Kurtosis and ECE-based thresholds [15]

ThDF119895 =1

120598119895

timesmax (119888

119895)

119865jSN

ThDF119895 =1

120598119869

timesmax (119888

119895)

119865jSN

(17)

where 120598119895is the energy contribution efficiency of

119895th detail subband 120598119869is detail energy contribution

efficiency of 119895th wavelet subband and 119865jSN is the ratiobetween the Kurtosis value of the signal at subband 119895to the Kurtosis value of Gaussian noise

120598119895=

119864119895

119864119905

times 100

120598119869=

119864119895

119864119905

times 100

1198701015840 =1198984

11989822

=(1119873)sum

119873

119894=1(119909119894minusmedia (119909))

2

((1119873)sum119873

119894=1(119909119894minusmedia (119909))2)

2

(18)

Analyzing this type of thresholds their data and coef-ficients dependence makes them different from prefixedthresholds On the hand the implementation of the involvedoperations of these nonprefixed thresholds requires a highnumber of operations division between them Thus increas-ing implementation complexity of these thresholds On theother hand it will consume a big number of clock cyclesand at least two accesses to each stored sample It couldhave devastating effects over the total latency and real-timeprocessing

315 Threshold Rescaling For signal denoising once thethreshold to be applied is selected this threshold is rescaledusing noise variance

th119895rescaled = 120590

119895sdot th119895 (19)

The noise variance is used to rescale the threshold at eachlevel so other important setting is related to the method forestimating the noise variance at each level If the noise iswhite the standard deviation from the wavelet coefficients atthe first level can be used and the thresholds can be updatedusing this value If the noise is not white best results fordenoising are obtained when estimating the noise standarddeviation at each level independently and using each one torescale the associated wavelet coefficients

The 120590 is calculated based on median absolute deviation(MAD) [15 25 26]

120590 = 119862 sdotMAD (119888119895) (20)

where 119862 is a constant scale factor which depends on distri-bution of noise For normally distributed data 119862 = 14826is the 75th percentile of the normal distribution with unityvariance On the other hand for a univariate dataset ofwavelet details at 119895th level 119888

1198951 1198881198952 119888

119895119899 the MAD is defined

as [27]

MAD = median (10038161003816100381610038161003816119888119895 minusmedian (119888

119895)10038161003816100381610038161003816) (21)

Thus estimated noise variance can be written as

120590 = 14826 sdotmedian (10038161003816100381610038161003816119888119895 minusmedian (119888

119895)10038161003816100381610038161003816) (22)

Some predefined applications or functions such as thewnoisestmMatlab function use a simpler expression for 120590

120590 = 14826 sdotmedian (10038161003816100381610038161003816119888119895

10038161003816100381610038161003816) (23)

It must be noted that the rescale factor 120590 is not a prefixedvalue since as (21) and (22) show it depends on the waveletcoefficient values However memory access is reduced ifexpression (22) is used (expression (21) implies threememoryaccesses while (22) requires only one) Depending on the


Table 1 BW suppression analysis

119865119904(Hz) 119873 119865

119908(Hz) SNR for estimated BW

1198604

1198605

1198606

1198607

1198608

1198609

11986010

11986011

11986012

250 5400

015 1866 6563 12579 25986 25484 7826 minus7957 minus8948 minus9199

019 1866 6563 12570 25636 22445 minus0216 minus8630 minus9072 minus9193




500 10800

015 0612 4491 10172 16583 30755 24674 5845 minus9762 minus10373


023 0612 4491 10168 16485 26036 14864 0825 minus10778 minus10796



1000 21600

015 0104 1040 5684 11679 18344 33993 23436 4831 minus10805

019 0104 1040 5684 11678 18319 31447 20792 minus3154 minus11545

023 0104 1040 5684 11677 18271 26531 13913 minus0238 minus11897

027 0104 1040 5684 11677 18255 24680 7822 minus10512 minus11771

031 0104 1040 5684 11679 18300 25393 3629 minus11258 minus11807

Table 2 Denoising evaluation using synthetic ECG signals

M Threshold Wavelet SNR SNRIMP M Threshold Wavelet SNR SNRIMPSoft thresholding 15 dB Hard thresholding 15 dB

3 Thexp coif3 212631 64015 4 Thminimax sym7 208648 593193 Thexp db7 214798 63944 4 Thminimax sym7 210110 592563 Thexp coif3 214773 63919 4 Thminimax sym7 207633 590173 Thexp db7 212360 63743 5 Thminimax sym7 208288 589583 Thexp sym7 214286 63432 5 Thminimax sym7 207219 586033 Thexp db5 214274 63420 5 Thminimax sym7 209024 581703 Thexp db5 212008 63391 3 Thminimax sym7 208818 57964

Soft thresholding 25 dB Hard thresholding 25 dB3 Thexp bior39 295524 44960 3 Thminimax bior68 295025 458093 Thexp bior39 292865 43649 3 Thminimax bior68 296610 455343 Thexp sym7 292657 43440 4 Thminimax bior68 295801 447253 Thexp sym7 293678 43114 4 Thminimax bior68 293823 446073 Thexp bior39 293712 42784 3 Thminimax bior68 295049 444853 Thexp bior68 292479 41915 3 Thminimax sym6 295035 444713 Thminimax bior39 290937 41720 5 Thminimax bior68 295436 44360

system architecture and data processing a higher number ofmemory accesses could increase the system latency and thusjeopardize real-time processing

316 Thresholding Rules Thresholding can be done usingsoft or hard thresholding Hard thresholding is the simplestmethod It can be described as the usual process of settingto zero the elements whose absolute values are lower thanthe threshold Soft thresholding is an extension of hardthresholding first setting to zero the elements whose absolutevalues are lower than the threshold and then shrinking thenonzero coefficients towards 0 The hard procedure createsdiscontinuities while the soft procedure does not The soft

and hard thresholding is shown in (24) and (25) respectively

119889(119895)119899

= sign (119889(119895)

119899) (

10038161003816100381610038161003816119889(119895)

119899

10038161003816100381610038161003816 minus th119895) if 10038161003816100381610038161003816119889

(119894)

119899

10038161003816100381610038161003816 ge th119895

0 if 10038161003816100381610038161003816119889(119894)

119899

10038161003816100381610038161003816 lt th119895

(24)

119889(119895)119899

=

119889(119895)119899

if 10038161003816100381610038161003816119889(119894)

119899

10038161003816100381610038161003816 ge th119895

0 if 10038161003816100381610038161003816119889(119894)

119899

10038161003816100381610038161003816 lt th119895

(25)

where 119889(119895)119899

and 119889(119895)119899

represent the modified values of 119895th leveldetail coefficient based on the selected threshold and th

119895and

are an approximation of the detail coefficients of the de-noised transformThenewdetail coefficients119889(119895)

119899or119889(119895)119899 have


0 1 2 4 5

0

05

1Lead 1 DaIsy dataset

Time (s)

BW and noise corrected signal

minus05

minus1

0 1 2 3

Time (s)3

4 5

0

05

1

minus05

minus1

(a)

BW and noise corrected signalTime (s)

Time (s)

30 1 2 4 5

0

05


minus05

0 1 2 3 4 5

0

05

1

minus05

(b)

Figure 6 DaIsy database signals (a) lead 1 and noise and BW corrected signal (b) lead 2 and noise and BW corrected signals

ecgca 746 abdominal lead 1 (Physionet Dataset)

0 1 2 3 4 5

0

05

1

minus05

Time (s)

(a)

Time (s)


0 1 2 3 4 5

0

05

1

minus05

(b)

minus01215 22 225 23

0

01

02ecgca 746 abdominal lead 1 (Physionet Dataset)


minus01215 22 225 23

0

01

02

Time (s)

Time (s)

(c)

Figure 7 ecgca 746 abdomen lead 1 Noninvasive Fetal ECG Database (Physionet Dataset) (a) signal (b) BW and noise corrected signal (c)signals detail


ecgca 906 abdomen lead 1 (Physionet Dataset)

Time (s)


0 1 2 3 4 5

0

05

1

minus05

0 1 2 3

Time (s)

4 5

0

05

1

minus05

(a)



Time (s)

0 1 2 3 4 5

0

05

1

minus1

minus05

0 1 2 3 4 5

0

05

1

minus05

Time (s)

(b)



Time (s)

0 1 2 3 4 5

0

05

1

minus05

0 1 2 3 4 5

0

05

1

minus05

Time (s)

(c)

Figure 8 Noninvasive Fetal ECGDatabase (Physionet Dataset) (a) ecgca 906 lead 1 signal BW and noise corrected signal (b) ecgca 473 lead1 signal BW and noise corrected signal (c) ecgca 840 lead 1 signal BW and noise corrected signal

to be calculated for the wavelet transform levels consideredfor denoising as it was pointed previously

4 Results

This section is devoted to analyze the one-step proposedmethod through quantitative parameters When workingwith real noisy ECG signals it is not trivial to calculate aparameter that provides a quantitativemeasure of the benefitsof the applied technique In order to better analyze ourproposed one-step model a separate study of BW and noisesuppression has been carried out using synthetic ECG signalsFor a quantitative evaluation of the BW suppression wehave employed three types of synthetic ECG signals Thesesynthetic signals were elaborated from the ecgm Matlabfunction [28] and are 216 second long and contain 5400

10800 and 21600 samples thus the resulting sample ratesare 250 500 and 1000 sps respectively Signals affected byBW are obtained adding a sine wave plus a DC level usingfrequencies from 015 to 031Hz that fit to the frequency bandin real BW Our study has estimated the BW of the signals asthe approximations from level 1 to 12 and has reconstructedthe signal removing the estimated BW Table 1 resumes themain results showing the SNRs between the synthetic andthe BW corrected signal According to the expression of 119871for signals sampled at 250 500 and 1000 sps the adequatedecomposition level for BW suppression will be 7 8 and 9respectively which is corroborated by Table 1This study alsoreflects that better BW suppression (higher SNR) is achievedif the signal is sampled at higher rate

Synthetic ECG signals were also used to evaluate theperformance of the noise suppression These signals were


205 206 207Time (s)

208 209 210

005

minus05

205 206 207 208 209 210

005

minus05

205 206 207 208 209 210

005

minus05

r1 abdomen 1 (Physionet Dataset)

205 206 207 208 209 210

01

minus1

BW and noise corrected signal(db6 N = 8 M = 3 universal multiple soft)



Figure 9 r1 abdomen 1 Abdominal and Direct Fetal Electrocar-diogram Database (Physionet Dataset) signal and BW and noisecorrected signals using different levels for denoising

Time (s)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

0

1r4 abdomen 1 (Physionet Dataset)

0

05BW and noise corrected (db6 N = 8 M = 4 universal multiple soft)

BW and noise corrected (db6 N = 8 M = 4 universal simple soft)

minus1

minus05

0

05

minus05

Figure 10 r4 abdomen 1 Abdominal and Direct Fetal Electrocar-diogram Database (Physionet Dataset) signal and BW and noisecorrected signals usingmultiple and simple rescaling

contaminated adding Gaussian white noise Thus the noisysignal is processed by the proposed method to obtain thedenoised signal This scenario is used by several authors[29] and allows visual inspection and quantitative evaluationThere are several parameters to measure the quality of thedenoised signal [15 29] as the SNR Improvement Measure

Time (s)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

0


0

05BW and noise corrected (db6 N = 8 M = 3 minimax multilevel soft)

BW and noise corrected (db6 N = 8 M = 3 minimax multilevel hard)

minus1

minus05

0

05

minus05

Figure 11 r7 abdomen 2 Abdominal and Direct Fetal Electrocar-diogram Database (Physionet Dataset) signal and BW and noisecorrected signals using the two types of thresholding rules

205 206 207 208 209 210


205 206 207 208 209 210

205 206 207 208 209 210Time (s)

BW and noise corrected (db6 N = 8 M = 4 universal multiple soft)

BW and noise corrected (db6 N = 8 M = 4 minimax multiple soft)

005

minus05minus1

005

minus05minus1

005

minus05minus1

Figure 12 r10 abdomen 2 Abdominal and Direct Fetal Electrocar-diogram Database (Physionet Dataset) signal and BW and noisecorrected signals using two types of thresholds

SNRIMP [29] A study using 3 4 and 5 as maximum levelsfor wavelet denoising119872 universal exponential andminimaxas threshold limits and simple rescaling and soft and hardthresholding was carried out A total of 12 wavelet functionswere used for this evaluation The study also considers twonoise levels approximately 15 dB and 25 dB and a maximumof three attempts for each case (for each attempt all theparameters are the same including noise level but the noisysignal is different due to the randomdistribution of it over the


original signal) Table 2 shows the best results for denoisingObserving these summarized results and all the generateddata we can conclude that there are no large differencesfor the SNRIMP values of the different wavelet functionsComparing soft and hard thresholding soft gets for bothnoise levels higher SNRIMP using a less number of levelsRegarding thresholds Thexp achieves best denoising if it isused along with soft thresholding as it is the case with thecombinationThminimax and hard thresholding

To study BW suppression and denoising for our one-stepdenoising and BW suppression proposal visual inspectionof the obtained signals is also important and in somecases it is even more conclusive than quantitative measuresDaISy dataset and Physionet Dataset [30] are targeted forevaluating the proposed one-step BW and noise suppressionThe recordings from Noninvasive Fetal ECG Database [30]have two thoracic and four abdominal channels sampled at1 ksps all 60 seconds long The signal bandwidth is 001Hzndash100Hz In addition recordings from the Abdominal andDirect Fetal ElectrocardiogramDatabase [31] have been usedEach recording comprises four differential signals acquiredfrommaternal abdomen and the reference direct fetal electro-cardiogram registered from the fetusrsquos headThe fetal 119877-wavelocations were automatically determined in the direct FECGsignal bymeans of online analysis applied in theKOMPORELsystem [31] The recordings sampled at 1 ksps are 5-minutelong and the signal bandwidth is 1Hzndash150Hz

For these signals the selected parameters were waveletfunction db6 119872 = 3 universal threshold soft thresholdingsimple rescaling for DaIsy dataset and multiple rescalingfor Noninvasive Fetal ECG Database Figure 6 includes theobtained result for lead 1 and lead 2 from DaIsy dataset withBW and noise corrected signals being shown Figure 7 showsan example of results for ecgca 746 signal of NoninvasiveFetal ECG Database including the detail of one of the fetalQRS complexes before and after processing These figuresshow that the abdominal ECG signals are BW corrected anddenoised while retaining their main characteristics as thefetal QRS complexes which are very important for futureparameter extraction [7] Figure 8 shows more examples ofresults for signals of Noninvasive Fetal ECG Database

Figures 9 10 11 and 12 show processing examples forsignals of Abdominal and Direct Fetal ElectrocardiogramDatabase displayed Figure 9 uses signal r1 abdomen 1 tomake a study of the BW and noise corrected signals usingthree different levels for signal denoising 119872 = 3 119872 =4 and 119872 = 5 as this level increases the denoised signalloses its main characteristics (ie fetal QRS complexes)Figure 10 shows the denoising of r4 signals using multiplerescaling and simple rescaling It can be observed that the bestdenoising is obtained using multiple rescaling On the otherhand Figure 11 shows the results for r7 abdomen 2 signalusing soft and hard thresholding soft thresholding gets betterdenoised signal since hard thresholding introduces somediscontinuities into the denoised signal Finally universal andminimax thresholds are used for denoising r10 abdomen 2signal as Figure 12 shows Similar results are obtained forthese two thresholds but observing signal details minimaxthreshold provides better results

5 Conclusion

This paper presents the mathematical bases for electro-cardiogram signal denoising by means of discrete waveletprocessing A novel one-step wavelet-based method has beenintroduced performing both BW and noise suppressionwhich makes computationally feasible real-time implemen-tations The presented approach is performed by only onewavelet decomposition and reconstruction step which isrequired for eliminating both types of perturbations Thisapproach has been linked to an exhaustive study of therelated parameters such as number of decomposition levelsthreshold edges rescaling and rules that allow an optimalsignal denoising and meeting specific ECG signal character-istics including signal shape sample rate and noise levelsThe presented results for synthetic ECG signals validatethis method while applications on real AECG signal fromthree different databases have led to improved signals thatare valid for further analysis and extraction of parameterssuch as heart rate variability The defined algorithm will alsoallow its compact implementation thus fitting portable ECGapplications

Acknowledgment

This work has been partially supported by the CEI BIOTICGranada under project 201381 ldquoCompromiso con la investi-gacion y el desarrollordquo

References

[1] J Lee Y Chee and I Kim ldquoPersonal identification based onvectorcardiogram derived from limb leads electrocardiogramrdquoJournal of Applied Mathematics vol 2012 Article ID 904905 12pages 2012

[2] D P Morales A Garcıa E Castillo M A Carvajal J BanqueriandA J Palma ldquoFlexible ECG acquisition systembased on ana-log and digital reconfigurable devicesrdquo Sensors and Actuators Avol 165 no 2 pp 261ndash270 2011

[3] D P Morales A Garcıa E Castillo M A Carvajal L Parrillaand A J Palma ldquoAn application of reconfigurable technologiesfor non-invasive fetal heart rate extractionrdquoMedical Engineeringand Physics 2013

[4] S G Mallat ldquoTheory for multiresolution signal decompositionthe wavelet representationrdquo IEEE Transactions on Pattern Anal-ysis and Machine Intelligence vol 11 no 7 pp 674ndash693 1989

[5] E-B Lin and P C Liu ldquoA discrete wavelet analysis of freakwaves in the oceanrdquo Journal of Applied Mathematics vol 2004no 5 pp 379ndash394 2004

[6] T Abualrub I Sadek and M Abukhaled ldquoOptimal controlsystems by time-dependent coefficients using cas waveletsrdquoJournal of Applied Mathematics vol 2009 Article ID 636271 10pages 2009

[7] R Sameni and G D Clifford ldquoA review of fetal ECG signalprocessing issues and promising directionsrdquoThe Open PacingElectrophysiology ampTherapy Journal vol 3 no 1 pp 4ndash20 2010

[8] C-T Ku K-C Hung H-S Wang and Y-S Hung ldquoHighefficient ECG compression based on reversible round-off non-recursive 1-D discrete periodized wavelet transformrdquo MedicalEngineering and Physics vol 29 no 10 pp 1149ndash1166 2007


[9] C Ghule D G Wakde G Virdi and N R Khodke ldquoDesignof portable ARM processor based ECG module for 12 leadECG data acquisition and analysisrdquo in Proceedings of the 2ndInternational Conference on Biomedical and PharmaceuticalEngineering (ICBPE rsquo09) pp 1ndash8 December 2009

[10] D Karadaglic M Mirkovic D Milosevic and R StojanovicldquoA FPGA system for QRS complex detection based on IntegerWavelet TransformrdquoMeasurement Science Review vol 11 no 4pp 131ndash138 2011

[11] J C Goswami and A K Chan Fundamentals of WaveletsTheory Algorithms and Applications EEUU John Wiley ampSons Hoboken NJ USA 2nd edition 2011

[12] P S Addison ldquoWavelet transforms and the ECG a reviewrdquoPhysiological Measurement vol 26 no 5 pp R155ndashR199 2005

[13] B N Singh and A K Tiwari ldquoOptimal selection of waveletbasis function applied to ECG signal denoisingrdquo Digital SignalProcessing vol 16 no 3 pp 275ndash287 2006

[14] C B Mbachu G N Onoh E N Ifeagwu and S U NnebeldquoProcessing ECG signal with Kaiser Window-Based FIR digitaldiltersrdquo International Journal of Engineering Science and Tech-nology vol 3 no 8 2011

[15] L N Sharma S Dandapat and A Mahanta ldquoECG signaldenoising using higher order statistics in Wavelet subbandsrdquoBiomedical Signal Processing and Control vol 5 no 3 pp 214ndash222 2010

[16] M I Ibrahimy FAhmedMAMohdAli andE Zahedi ldquoReal-time signal processing for fetal heart rate monitoringrdquo IEEETransactions on Biomedical Engineering vol 50 no 2 pp 258ndash262 2003

[17] J J Oresko Z Jin J Cheng et al ldquoAwearable smartphone-basedplatform for real-time cardiovascular disease detection via elec-trocardiogram processingrdquo IEEE Transactions on InformationTechnology in Biomedicine vol 14 no 3 pp 734ndash740 2010

[18] B De Moor Database for the identification of systems (DaISy)2010 httphomesesatkuleuvenbesimsmcdaisy

[19] J G WebsterMedical Instrumentation Application and DesignJohn Wiley amp Sons 1995

[20] M S Manikandan and S Dandapat ldquoWavelet energy baseddiagnostic distortion measure for ECGrdquo Biomedical SignalProcessing and Control vol 2 no 2 pp 80ndash96 2007

[21] D Donoho and I Johnstone ldquoAdapting to unknown smooth-ness via wavelet shrinkagerdquo Journal of the American StatisticalAssociation vol 90 no 432 pp 1200ndash1224 1995

[22] Inc The MathWorks Denoising Wavelet shrinkage nonpara-metric regression block thresholding multisignal threshold-ing 2013 httpwwwmathworkseseshelpwaveletdenois-inghtml

[23] LabView TM Advanced Signal Processing Toolkit WaveletAnalysis Tools User Manual 2013 httpwwwnicompdfmanuals371533apdf

[24] S Sardy ldquoMinimax threshold for denoising complex signalswith waveshrinkrdquo IEEE Transactions on Signal Processing vol48 no 4 pp 1023ndash1028 2000

[25] P J Rousseeuw and C Croux ldquoAlternatives to the median abso-lute deviationrdquo Journal of the American Statistical Associationvol 88 no 424 pp 1273ndash1283 1993

[26] I M Johnstone and B W Silverman ldquoWavelet thresholdestimators for data with correlated noiserdquo Journal of the RoyalStatistical Society B vol 59 no 2 pp 319ndash351 1997

[27] WH Swallow and F Kianifard ldquoUsing robust scale estimates indetecting multiple outliers in linear regressionrdquo Biometrics vol52 no 2 pp 545ndash556 1996

[28] Matlab ECG simulation Using atlab 2013 httpwwwmath-workscommatlabcentralfileexchange10858

[29] H G Rodney Tan A C Tan P Y Khong and V H Mok ldquoBestwavelet function identification system for ECG signal denoiseapplicationsrdquo in Proceedings of the International Conferenceon Intelligent and Advanced Systems (ICIAS rsquo07) pp 631ndash634November 2007

[30] A L Goldberger L A Amaral L Glass et al ldquoPhysioBankPhysioToolkit and PhysioNet components of a new researchresource for complex physiologic signalsrdquo Circulation vol 101no 23 pp E215ndashE220 2000

[31] J Jezewski A Matonia T Kupka D Roj and R CzabanskildquoDetermination of fetal heart rate from abdominal signalsevaluation of beat-to-beat accuracy in relation to the direct fetalelectrocardiogramrdquo Biomedical Engineering vol 57 no 5 pp383ndash394 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Appropriate software models and parameter selection arepresented based on mathematical analysis for the mentionedECG digital processingMoreover these softwaremodels willalso enable very quick tests of the required signal processingIn order to select an appropriate method for the developmentof the software modeling noise suppression and featureextraction procedures on ECG signals are studied in thefollowing

2 Materials and Methods

21 DWT Fundamentals Wavelet transform (WT) is one ofthe most used tools in multiresolution signal analysis due toits ability to decompose a signal at various resolutions whichallow observing high-frequency events of short duration innonstationary signals [10] The continuous WT of a signal119891(119905) isin 1198712(119877) is defined as [11]

119882(119886 119887) = int+infin

minusinfin

119891 (119905) 120595119886119887

(119905) 119889119905 (1)

120595119886119887

(119905) =1

radic119886120595lowast (

119905 minus 119887

119886) (2)

In (2) lowast denotes complex conjugate and 119886 is a scale factorand 119887 a translation factor The normalization factor 119886minus12 isincluded so that 120595

119886119887 = 120595 Thus 120595

119886119887(119905) represents a

shifted and scaled version of the so-called mother wavelet120595 which is a window function that defines the basis for thewavelet transformation A mother wavelet 120595 of order 119898 isa function 120595 119877 rarr 119877 which satisfies the following fourproperties

(1) If119898 gt 1 then 120595 is (119898 minus 1) times differentiable(2) 120595 isin 119871infin(119877) If 119898 gt 1 for each 119895 isin 1 119898 minus 1

120595(119895) isin 119871infin(119877)(3) 120595 and all its derivatives up to order (119898 minus 1) decay

rapidly for each 119903 gt 0 there is a 120574 gt 0 such that

10038161003816100381610038161003816120595(119895)

(119905)10038161003816100381610038161003816 lt

1

119905 119895 isin 1 119898 minus 1 for each |119905| gt 120574 (3)

(4) For each 119895 isin 0 1 119898 int 119905119895120595(119905)119889119905 = 0

Practical applications rely on the discrete wavelet transform(DWT) since it provides enough informationwhile requiringreasonable computation time and resources The DWT of adiscrete function 119891(119899) is defined as

119882(119886 119887) = 119862 (119895 119896) = sum119899

119891 (119899) 120595119895119896

(119899) (4)

where

120595119895119896

(119899) = 2minus1198952120595 (2minus119895119899 minus 119896) (5)

and 119886 = 2119895 and 119887 = 1198962119895 with 119895 119896 isin Z Thus DWT iscomputed in practice through a set of two FIR-like filterslowpass and highpass at each decomposition level 119894 followedby a downsampling by 2 which implies a reduction in

the sampling frequency The result of the low-pass filter isusually known as the approximation and is used as inputto the following decomposition level while the result of thehigh-pass filter is called the detail Thus the usual DWTcorresponds to the scheme in Figure 1(a) with

119886(119894)119899

=119873minus1

sum119896=0

119892119896119886(119894minus1)2119899minus119896

119894 = 1 2 119869

119889(119894)119899

=119873minus1

sum119896=0

ℎ119896119886(119894minus1)2119899minus119896

119886(0)119899

equiv 119909119899

(6)

for an 119873-sample input sequence The coefficients of thelow-pass and high-pass filters are defined by the family ofwavelet functions used as basis for the transformation HaarDaubechies Quadratic Spline and so forth [11]

Since every detail is the result of high-pass filtering of theprevious approximation and approximations are the resultof low-pass filtering these details and approximations atevery decomposition level contain information at differentfrequencies and time scales this reflects the multiresolutionanalysis of theWT Moreover it is possible to reconstruct theoriginal signal from the set of details and approximationsthrough the inverse DWT so

119886(119894minus1)119898

=

(1198732)minus1

sum119896=0

1198922119896119886(119894)(1198982)minus119896

+(1198732)minus1

sum119896=0

ℎ2119896119889(119894)(1198982)minus119896

119898 even(1198732)minus1

sum119896=0

1198922119896+1

119886(119894)((119898minus1)2)minus119896

+(1198732)minus1

sum119896=0

ℎ2119896+1

119889(119894)((119898minus1)2)minus119896

119898 odd(7)

where once more the corresponding details are high-passfiltered and the approximations are low-pass filtered thistime through the appropriate reconstruction filtersThus thereconstructed approximation at level 119894 minus 1 is obtained byaddition of the output of the reconstruction filters and anupsampling by 2 as shown in Figure 1(b)

22 DWT-Based ECG Processing The special characteristicsof ECG signals and their frequency spectrummake the use oftraditional Fourier analysis for the detection of ECG featuresdifficult [12] Wavelet transform can be applied in manyfields but its primary and most advantageous applicationareas are those that have generate or process widebandsignalsThis is due to themultiresolution analysis that waveletprovides which allows representing a signal by a finite sum ofcomponents at different resolutions so that each componentcan be processed adaptively based on the objectives of theapplication In this way this technique represents signalscompactly and in several levels of resolution which is idealfor decomposition and reconstruction purposes Thus the


H(z)

H(z)

G(z)

G(z)

G(z)

H(z)

a(0)n

a(1)n

d(1)n

d(2)n

a(2)n a

(3)n

d(3)n

(J)

darr2

darr2

darr2

darr2

darr2

darr2


(a)

+

+

+

(J)

a(0)n

a(1)n

d(1)n

d(2)n

a(2)n

a(3)n

d(3)n H(z)

G(z)

H(z)

G(z)

H(z)

G(z)

uarr2

uarr2

uarr2

uarr2

uarr2

uarr2


(b)







119899that captures







119904 (119899) = 119891 (119899) + 120590119890 (119899) (8)
















119899for better

removing the noise








2(119871119904) where





119886(6)119899

and 119886(7)119899



detail at level 119894 + 1 119886(119894+1)119899

and 119889(119894+1)119899



approximation 119886(119894)119899



500 1000 1500 2000 2500


002040608

ECG

dat

a

(a)

10 20 30 40 50 60 70

02

02

02

02

5 10 15 20 25 30 35 40

2 4 6 8 10 12 14 16 18 20

2 3 4 5 6 7 8 9 10 11

1 15 2 25 3 35 4 45 5 55 6

02

4

4

Samples

minus2

minus2

minus2

minus2

minus2

A5

A6

A7

A8

A9

(b)


frequencies in 119886(119894)119899



119888=




119871BW = int [log2(119865max119891119888

)] (9)

10

0

0

0

0

0

0

0 20 30 40 50 60

10 20 30 40 50 60

10 20 30 40 50 60

10 20 30 40 50 60

10 20 30 40 50 60

10 20 30 40 50 60

10 20 30 40 50 60

005025

01

0501

03

102

0208

02

14

0515

Frequency (Hz)|F

FTA

1|

|FFT

A2|

|FFT

A3|

|FFT

A4|

|FFT

A5|

|FFT

A6|

|FFT

A7|

0 01 02 03 04 05 06 07 08 09

15

05

Frequency (Hz)

|FFT

A7|

01

05

0 01 02 03 04 05 06 07 08 09Frequency (Hz)

|FFT

A7|









119899 and the ECG




500 1000 1500 2000 2500

ECG signal

2 4 6 8 10 12 14 16 18 20

0123

500 1000 1500 2000 2500

005

BW corrected ECG


minus05

minus3minus1minus2

005

minus05






119899 Thus


119899

119889(2)119899 and 119889(3)



119899and



119899and

119889(2)119899


500 1000 1500 2000 2500

020

ECG

200 400 600 800 1000 1200

20

100 200 300 400 500 600

50 100 150 200 250 300

20 40 60 80 100 120 140

20 40 60 80 100 120 140

minus20minus40

minus20minus60

20minus20minus60

20minus20minus60

20minus20minus60

20minus20minus60

CD 1

CD 2

CD 3

CD 4

CA 4




119895 and





Thuni119895 = radic2 log 119899119895 (11)



119895

radic119899119895

(12)








119895)

log (2)) (15)





119899 = round [log2(max 10038161003816100381610038161003816119888119895

10038161003816100381610038161003816)] (16)


ThDF119895 =1

120598119895

timesmax (119888

119895)

119865jSN

ThDF119895 =1

120598119869

timesmax (119888

119895)

119865jSN

(17)




120598119895=

119864119895

119864119905

times 100

120598119869=

119864119895

119864119905

times 100

1198701015840 =1198984

11989822

=(1119873)sum

119873

119894=1(119909119894minusmedia (119909))

2

((1119873)sum119873

119894=1(119909119894minusmedia (119909))2)

2

(18)




119895sdot th119895 (19)



120590 = 119862 sdotMAD (119888119895) (20)


1198951 1198881198952 119888


as [27]


119895)10038161003816100381610038161003816) (21)



119895)10038161003816100381610038161003816) (22)


120590 = 14826 sdotmedian (10038161003816100381610038161003816119888119895

10038161003816100381610038161003816) (23)




119865119904(Hz) 119873 119865


1198604

1198605

1198606

1198607

1198608

1198609

11986010

11986011

11986012

250 5400






500 10800

015 0612 4491 10172 16583 30755 24674 5845 minus9762 minus10373


023 0612 4491 10168 16485 26036 14864 0825 minus10778 minus10796



1000 21600

015 0104 1040 5684 11679 18344 33993 23436 4831 minus10805

019 0104 1040 5684 11678 18319 31447 20792 minus3154 minus11545

023 0104 1040 5684 11677 18271 26531 13913 minus0238 minus11897

027 0104 1040 5684 11677 18255 24680 7822 minus10512 minus11771

031 0104 1040 5684 11679 18300 25393 3629 minus11258 minus11807








119889(119895)119899

= sign (119889(119895)

119899) (

10038161003816100381610038161003816119889(119895)

119899

10038161003816100381610038161003816 minus th119895) if 10038161003816100381610038161003816119889

(119894)

119899

10038161003816100381610038161003816 ge th119895

0 if 10038161003816100381610038161003816119889(119894)

119899

10038161003816100381610038161003816 lt th119895

(24)

119889(119895)119899

=

119889(119895)119899

if 10038161003816100381610038161003816119889(119894)

119899

10038161003816100381610038161003816 ge th119895

0 if 10038161003816100381610038161003816119889(119894)

119899

10038161003816100381610038161003816 lt th119895

(25)

where 119889(119895)119899

and 119889(119895)119899


119895and


119899or119889(119895)119899 have


0 1 2 4 5

0

05


Time (s)


minus05

minus1

0 1 2 3

Time (s)3

4 5

0

05

1

minus05

minus1

(a)


Time (s)

30 1 2 4 5

0

05


minus05

0 1 2 3 4 5

0

05

1

minus05

(b)



0 1 2 3 4 5

0

05

1

minus05

Time (s)

(a)

Time (s)


0 1 2 3 4 5

0

05

1

minus05

(b)

minus01215 22 225 23

0

01



minus01215 22 225 23

0

01

02

Time (s)

Time (s)

(c)




Time (s)


0 1 2 3 4 5

0

05

1

minus05

0 1 2 3

Time (s)

4 5

0

05

1

minus05

(a)



Time (s)

0 1 2 3 4 5

0

05

1

minus1

minus05

0 1 2 3 4 5

0

05

1

minus05

Time (s)

(b)



Time (s)

0 1 2 3 4 5

0

05

1

minus05

0 1 2 3 4 5

0

05

1

minus05

Time (s)

(c)



4 Results





205 206 207Time (s)

208 209 210

005

minus05

205 206 207 208 209 210

005

minus05

205 206 207 208 209 210

005

minus05


205 206 207 208 209 210

01

minus1





Time (s)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

0


0



minus1

minus05

0

05

minus05



Time (s)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

0


0



minus1

minus05

0

05

minus05


205 206 207 208 209 210


205 206 207 208 209 210

205 206 207 208 209 210Time (s)



005

minus05minus1

005

minus05minus1

005

minus05minus1








5 Conclusion


Acknowledgment


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










H(z)

H(z)

G(z)

G(z)

G(z)

H(z)

a(0)n

a(1)n

d(1)n

d(2)n

a(2)n a

(3)n

d(3)n

(J)

darr2

darr2

darr2

darr2

darr2

darr2


(a)

+

+

+

(J)

a(0)n

a(1)n

d(1)n

d(2)n

a(2)n

a(3)n

d(3)n H(z)

G(z)

H(z)

G(z)

H(z)

G(z)

uarr2

uarr2

uarr2

uarr2

uarr2

uarr2


(b)







119899that captures







119904 (119899) = 119891 (119899) + 120590119890 (119899) (8)
















119899for better

removing the noise








2(119871119904) where





119886(6)119899

and 119886(7)119899



detail at level 119894 + 1 119886(119894+1)119899

and 119889(119894+1)119899



approximation 119886(119894)119899



500 1000 1500 2000 2500


002040608

ECG

dat

a

(a)

10 20 30 40 50 60 70

02

02

02

02

5 10 15 20 25 30 35 40

2 4 6 8 10 12 14 16 18 20

2 3 4 5 6 7 8 9 10 11

1 15 2 25 3 35 4 45 5 55 6

02

4

4

Samples

minus2

minus2

minus2

minus2

minus2

A5

A6

A7

A8

A9

(b)


frequencies in 119886(119894)119899



119888=




119871BW = int [log2(119865max119891119888

)] (9)

10

0

0

0

0

0

0

0 20 30 40 50 60

10 20 30 40 50 60

10 20 30 40 50 60

10 20 30 40 50 60

10 20 30 40 50 60

10 20 30 40 50 60

10 20 30 40 50 60

005025

01

0501

03

102

0208

02

14

0515

Frequency (Hz)|F

FTA

1|

|FFT

A2|

|FFT

A3|

|FFT

A4|

|FFT

A5|

|FFT

A6|

|FFT

A7|

0 01 02 03 04 05 06 07 08 09

15

05

Frequency (Hz)

|FFT

A7|

01

05

0 01 02 03 04 05 06 07 08 09Frequency (Hz)

|FFT

A7|









119899 and the ECG




500 1000 1500 2000 2500

ECG signal

2 4 6 8 10 12 14 16 18 20

0123

500 1000 1500 2000 2500

005

BW corrected ECG


minus05

minus3minus1minus2

005

minus05






119899 Thus


119899

119889(2)119899 and 119889(3)



119899and



119899and

119889(2)119899


500 1000 1500 2000 2500

020

ECG

200 400 600 800 1000 1200

20

100 200 300 400 500 600

50 100 150 200 250 300

20 40 60 80 100 120 140

20 40 60 80 100 120 140

minus20minus40

minus20minus60

20minus20minus60

20minus20minus60

20minus20minus60

20minus20minus60

CD 1

CD 2

CD 3

CD 4

CA 4




119895 and





Thuni119895 = radic2 log 119899119895 (11)



119895

radic119899119895

(12)








119895)

log (2)) (15)





119899 = round [log2(max 10038161003816100381610038161003816119888119895

10038161003816100381610038161003816)] (16)


ThDF119895 =1

120598119895

timesmax (119888

119895)

119865jSN

ThDF119895 =1

120598119869

timesmax (119888

119895)

119865jSN

(17)




120598119895=

119864119895

119864119905

times 100

120598119869=

119864119895

119864119905

times 100

1198701015840 =1198984

11989822

=(1119873)sum

119873

119894=1(119909119894minusmedia (119909))

2

((1119873)sum119873

119894=1(119909119894minusmedia (119909))2)

2

(18)




119895sdot th119895 (19)



120590 = 119862 sdotMAD (119888119895) (20)


1198951 1198881198952 119888


as [27]


119895)10038161003816100381610038161003816) (21)



119895)10038161003816100381610038161003816) (22)


120590 = 14826 sdotmedian (10038161003816100381610038161003816119888119895

10038161003816100381610038161003816) (23)




119865119904(Hz) 119873 119865


1198604

1198605

1198606

1198607

1198608

1198609

11986010

11986011

11986012

250 5400






500 10800

015 0612 4491 10172 16583 30755 24674 5845 minus9762 minus10373


023 0612 4491 10168 16485 26036 14864 0825 minus10778 minus10796



1000 21600

015 0104 1040 5684 11679 18344 33993 23436 4831 minus10805

019 0104 1040 5684 11678 18319 31447 20792 minus3154 minus11545

023 0104 1040 5684 11677 18271 26531 13913 minus0238 minus11897

027 0104 1040 5684 11677 18255 24680 7822 minus10512 minus11771

031 0104 1040 5684 11679 18300 25393 3629 minus11258 minus11807








119889(119895)119899

= sign (119889(119895)

119899) (

10038161003816100381610038161003816119889(119895)

119899

10038161003816100381610038161003816 minus th119895) if 10038161003816100381610038161003816119889

(119894)

119899

10038161003816100381610038161003816 ge th119895

0 if 10038161003816100381610038161003816119889(119894)

119899

10038161003816100381610038161003816 lt th119895

(24)

119889(119895)119899

=

119889(119895)119899

if 10038161003816100381610038161003816119889(119894)

119899

10038161003816100381610038161003816 ge th119895

0 if 10038161003816100381610038161003816119889(119894)

119899

10038161003816100381610038161003816 lt th119895

(25)

where 119889(119895)119899

and 119889(119895)119899


119895and


119899or119889(119895)119899 have


0 1 2 4 5

0

05


Time (s)


minus05

minus1

0 1 2 3

Time (s)3

4 5

0

05

1

minus05

minus1

(a)


Time (s)

30 1 2 4 5

0

05


minus05

0 1 2 3 4 5

0

05

1

minus05

(b)



0 1 2 3 4 5

0

05

1

minus05

Time (s)

(a)

Time (s)


0 1 2 3 4 5

0

05

1

minus05

(b)

minus01215 22 225 23

0

01



minus01215 22 225 23

0

01

02

Time (s)

Time (s)

(c)




Time (s)


0 1 2 3 4 5

0

05

1

minus05

0 1 2 3

Time (s)

4 5

0

05

1

minus05

(a)



Time (s)

0 1 2 3 4 5

0

05

1

minus1

minus05

0 1 2 3 4 5

0

05

1

minus05

Time (s)

(b)



Time (s)

0 1 2 3 4 5

0

05

1

minus05

0 1 2 3 4 5

0

05

1

minus05

Time (s)

(c)



4 Results





205 206 207Time (s)

208 209 210

005

minus05

205 206 207 208 209 210

005

minus05

205 206 207 208 209 210

005

minus05


205 206 207 208 209 210

01

minus1





Time (s)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

0


0



minus1

minus05

0

05

minus05



Time (s)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

0


0



minus1

minus05

0

05

minus05


205 206 207 208 209 210


205 206 207 208 209 210

205 206 207 208 209 210Time (s)



005

minus05minus1

005

minus05minus1

005

minus05minus1








5 Conclusion


Acknowledgment


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






















119899for better

removing the noise








2(119871119904) where





119886(6)119899

and 119886(7)119899



detail at level 119894 + 1 119886(119894+1)119899

and 119889(119894+1)119899



approximation 119886(119894)119899



500 1000 1500 2000 2500


002040608

ECG

dat

a

(a)

10 20 30 40 50 60 70

02

02

02

02

5 10 15 20 25 30 35 40

2 4 6 8 10 12 14 16 18 20

2 3 4 5 6 7 8 9 10 11

1 15 2 25 3 35 4 45 5 55 6

02

4

4

Samples

minus2

minus2

minus2

minus2

minus2

A5

A6

A7

A8

A9

(b)


frequencies in 119886(119894)119899



119888=




119871BW = int [log2(119865max119891119888

)] (9)

10

0

0

0

0

0

0

0 20 30 40 50 60

10 20 30 40 50 60

10 20 30 40 50 60

10 20 30 40 50 60

10 20 30 40 50 60

10 20 30 40 50 60

10 20 30 40 50 60

005025

01

0501

03

102

0208

02

14

0515

Frequency (Hz)|F

FTA

1|

|FFT

A2|

|FFT

A3|

|FFT

A4|

|FFT

A5|

|FFT

A6|

|FFT

A7|

0 01 02 03 04 05 06 07 08 09

15

05

Frequency (Hz)

|FFT

A7|

01

05

0 01 02 03 04 05 06 07 08 09Frequency (Hz)

|FFT

A7|









119899 and the ECG




500 1000 1500 2000 2500

ECG signal

2 4 6 8 10 12 14 16 18 20

0123

500 1000 1500 2000 2500

005

BW corrected ECG


minus05

minus3minus1minus2

005

minus05






119899 Thus


119899

119889(2)119899 and 119889(3)



119899and



119899and

119889(2)119899


500 1000 1500 2000 2500

020

ECG

200 400 600 800 1000 1200

20

100 200 300 400 500 600

50 100 150 200 250 300

20 40 60 80 100 120 140

20 40 60 80 100 120 140

minus20minus40

minus20minus60

20minus20minus60

20minus20minus60

20minus20minus60

20minus20minus60

CD 1

CD 2

CD 3

CD 4

CA 4




119895 and





Thuni119895 = radic2 log 119899119895 (11)



119895

radic119899119895

(12)








119895)

log (2)) (15)





119899 = round [log2(max 10038161003816100381610038161003816119888119895

10038161003816100381610038161003816)] (16)


ThDF119895 =1

120598119895

timesmax (119888

119895)

119865jSN

ThDF119895 =1

120598119869

timesmax (119888

119895)

119865jSN

(17)




120598119895=

119864119895

119864119905

times 100

120598119869=

119864119895

119864119905

times 100

1198701015840 =1198984

11989822

=(1119873)sum

119873

119894=1(119909119894minusmedia (119909))

2

((1119873)sum119873

119894=1(119909119894minusmedia (119909))2)

2

(18)




119895sdot th119895 (19)



120590 = 119862 sdotMAD (119888119895) (20)


1198951 1198881198952 119888


as [27]


119895)10038161003816100381610038161003816) (21)



119895)10038161003816100381610038161003816) (22)


120590 = 14826 sdotmedian (10038161003816100381610038161003816119888119895

10038161003816100381610038161003816) (23)




119865119904(Hz) 119873 119865


1198604

1198605

1198606

1198607

1198608

1198609

11986010

11986011

11986012

250 5400






500 10800

015 0612 4491 10172 16583 30755 24674 5845 minus9762 minus10373


023 0612 4491 10168 16485 26036 14864 0825 minus10778 minus10796



1000 21600

015 0104 1040 5684 11679 18344 33993 23436 4831 minus10805

019 0104 1040 5684 11678 18319 31447 20792 minus3154 minus11545

023 0104 1040 5684 11677 18271 26531 13913 minus0238 minus11897

027 0104 1040 5684 11677 18255 24680 7822 minus10512 minus11771

031 0104 1040 5684 11679 18300 25393 3629 minus11258 minus11807








119889(119895)119899

= sign (119889(119895)

119899) (

10038161003816100381610038161003816119889(119895)

119899

10038161003816100381610038161003816 minus th119895) if 10038161003816100381610038161003816119889

(119894)

119899

10038161003816100381610038161003816 ge th119895

0 if 10038161003816100381610038161003816119889(119894)

119899

10038161003816100381610038161003816 lt th119895

(24)

119889(119895)119899

=

119889(119895)119899

if 10038161003816100381610038161003816119889(119894)

119899

10038161003816100381610038161003816 ge th119895

0 if 10038161003816100381610038161003816119889(119894)

119899

10038161003816100381610038161003816 lt th119895

(25)

where 119889(119895)119899

and 119889(119895)119899


119895and


119899or119889(119895)119899 have


0 1 2 4 5

0

05


Time (s)


minus05

minus1

0 1 2 3

Time (s)3

4 5

0

05

1

minus05

minus1

(a)


Time (s)

30 1 2 4 5

0

05


minus05

0 1 2 3 4 5

0

05

1

minus05

(b)



0 1 2 3 4 5

0

05

1

minus05

Time (s)

(a)

Time (s)


0 1 2 3 4 5

0

05

1

minus05

(b)

minus01215 22 225 23

0

01



minus01215 22 225 23

0

01

02

Time (s)

Time (s)

(c)




Time (s)


0 1 2 3 4 5

0

05

1

minus05

0 1 2 3

Time (s)

4 5

0

05

1

minus05

(a)



Time (s)

0 1 2 3 4 5

0

05

1

minus1

minus05

0 1 2 3 4 5

0

05

1

minus05

Time (s)

(b)



Time (s)

0 1 2 3 4 5

0

05

1

minus05

0 1 2 3 4 5

0

05

1

minus05

Time (s)

(c)



4 Results





205 206 207Time (s)

208 209 210

005

minus05

205 206 207 208 209 210

005

minus05

205 206 207 208 209 210

005

minus05


205 206 207 208 209 210

01

minus1





Time (s)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

0


0



minus1

minus05

0

05

minus05



Time (s)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

0


0



minus1

minus05

0

05

minus05


205 206 207 208 209 210


205 206 207 208 209 210

205 206 207 208 209 210Time (s)



005

minus05minus1

005

minus05minus1

005

minus05minus1








5 Conclusion


Acknowledgment


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










500 1000 1500 2000 2500


002040608

ECG

dat

a

(a)

10 20 30 40 50 60 70

02

02

02

02

5 10 15 20 25 30 35 40

2 4 6 8 10 12 14 16 18 20

2 3 4 5 6 7 8 9 10 11

1 15 2 25 3 35 4 45 5 55 6

02

4

4

Samples

minus2

minus2

minus2

minus2

minus2

A5

A6

A7

A8

A9

(b)


frequencies in 119886(119894)119899



119888=




119871BW = int [log2(119865max119891119888

)] (9)

10

0

0

0

0

0

0

0 20 30 40 50 60

10 20 30 40 50 60

10 20 30 40 50 60

10 20 30 40 50 60

10 20 30 40 50 60

10 20 30 40 50 60

10 20 30 40 50 60

005025

01

0501

03

102

0208

02

14

0515

Frequency (Hz)|F

FTA

1|

|FFT

A2|

|FFT

A3|

|FFT

A4|

|FFT

A5|

|FFT

A6|

|FFT

A7|

0 01 02 03 04 05 06 07 08 09

15

05

Frequency (Hz)

|FFT

A7|

01

05

0 01 02 03 04 05 06 07 08 09Frequency (Hz)

|FFT

A7|









119899 and the ECG




500 1000 1500 2000 2500

ECG signal

2 4 6 8 10 12 14 16 18 20

0123

500 1000 1500 2000 2500

005

BW corrected ECG


minus05

minus3minus1minus2

005

minus05






119899 Thus


119899

119889(2)119899 and 119889(3)



119899and



119899and

119889(2)119899


500 1000 1500 2000 2500

020

ECG

200 400 600 800 1000 1200

20

100 200 300 400 500 600

50 100 150 200 250 300

20 40 60 80 100 120 140

20 40 60 80 100 120 140

minus20minus40

minus20minus60

20minus20minus60

20minus20minus60

20minus20minus60

20minus20minus60

CD 1

CD 2

CD 3

CD 4

CA 4




119895 and





Thuni119895 = radic2 log 119899119895 (11)



119895

radic119899119895

(12)








119895)

log (2)) (15)





119899 = round [log2(max 10038161003816100381610038161003816119888119895

10038161003816100381610038161003816)] (16)


ThDF119895 =1

120598119895

timesmax (119888

119895)

119865jSN

ThDF119895 =1

120598119869

timesmax (119888

119895)

119865jSN

(17)




120598119895=

119864119895

119864119905

times 100

120598119869=

119864119895

119864119905

times 100

1198701015840 =1198984

11989822

=(1119873)sum

119873

119894=1(119909119894minusmedia (119909))

2

((1119873)sum119873

119894=1(119909119894minusmedia (119909))2)

2

(18)




119895sdot th119895 (19)



120590 = 119862 sdotMAD (119888119895) (20)


1198951 1198881198952 119888


as [27]


119895)10038161003816100381610038161003816) (21)



119895)10038161003816100381610038161003816) (22)


120590 = 14826 sdotmedian (10038161003816100381610038161003816119888119895

10038161003816100381610038161003816) (23)




119865119904(Hz) 119873 119865


1198604

1198605

1198606

1198607

1198608

1198609

11986010

11986011

11986012

250 5400






500 10800

015 0612 4491 10172 16583 30755 24674 5845 minus9762 minus10373


023 0612 4491 10168 16485 26036 14864 0825 minus10778 minus10796



1000 21600

015 0104 1040 5684 11679 18344 33993 23436 4831 minus10805

019 0104 1040 5684 11678 18319 31447 20792 minus3154 minus11545

023 0104 1040 5684 11677 18271 26531 13913 minus0238 minus11897

027 0104 1040 5684 11677 18255 24680 7822 minus10512 minus11771

031 0104 1040 5684 11679 18300 25393 3629 minus11258 minus11807








119889(119895)119899

= sign (119889(119895)

119899) (

10038161003816100381610038161003816119889(119895)

119899

10038161003816100381610038161003816 minus th119895) if 10038161003816100381610038161003816119889

(119894)

119899

10038161003816100381610038161003816 ge th119895

0 if 10038161003816100381610038161003816119889(119894)

119899

10038161003816100381610038161003816 lt th119895

(24)

119889(119895)119899

=

119889(119895)119899

if 10038161003816100381610038161003816119889(119894)

119899

10038161003816100381610038161003816 ge th119895

0 if 10038161003816100381610038161003816119889(119894)

119899

10038161003816100381610038161003816 lt th119895

(25)

where 119889(119895)119899

and 119889(119895)119899


119895and


119899or119889(119895)119899 have


0 1 2 4 5

0

05


Time (s)


minus05

minus1

0 1 2 3

Time (s)3

4 5

0

05

1

minus05

minus1

(a)


Time (s)

30 1 2 4 5

0

05


minus05

0 1 2 3 4 5

0

05

1

minus05

(b)



0 1 2 3 4 5

0

05

1

minus05

Time (s)

(a)

Time (s)


0 1 2 3 4 5

0

05

1

minus05

(b)

minus01215 22 225 23

0

01



minus01215 22 225 23

0

01

02

Time (s)

Time (s)

(c)




Time (s)


0 1 2 3 4 5

0

05

1

minus05

0 1 2 3

Time (s)

4 5

0

05

1

minus05

(a)



Time (s)

0 1 2 3 4 5

0

05

1

minus1

minus05

0 1 2 3 4 5

0

05

1

minus05

Time (s)

(b)



Time (s)

0 1 2 3 4 5

0

05

1

minus05

0 1 2 3 4 5

0

05

1

minus05

Time (s)

(c)



4 Results





205 206 207Time (s)

208 209 210

005

minus05

205 206 207 208 209 210

005

minus05

205 206 207 208 209 210

005

minus05


205 206 207 208 209 210

01

minus1





Time (s)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

0


0



minus1

minus05

0

05

minus05



Time (s)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

0


0



minus1

minus05

0

05

minus05


205 206 207 208 209 210


205 206 207 208 209 210

205 206 207 208 209 210Time (s)



005

minus05minus1

005

minus05minus1

005

minus05minus1








5 Conclusion


Acknowledgment


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










500 1000 1500 2000 2500

ECG signal

2 4 6 8 10 12 14 16 18 20

0123

500 1000 1500 2000 2500

005

BW corrected ECG


minus05

minus3minus1minus2

005

minus05






119899 Thus


119899

119889(2)119899 and 119889(3)



119899and



119899and

119889(2)119899


500 1000 1500 2000 2500

020

ECG

200 400 600 800 1000 1200

20

100 200 300 400 500 600

50 100 150 200 250 300

20 40 60 80 100 120 140

20 40 60 80 100 120 140

minus20minus40

minus20minus60

20minus20minus60

20minus20minus60

20minus20minus60

20minus20minus60

CD 1

CD 2

CD 3

CD 4

CA 4




119895 and





Thuni119895 = radic2 log 119899119895 (11)



119895

radic119899119895

(12)








119895)

log (2)) (15)





119899 = round [log2(max 10038161003816100381610038161003816119888119895

10038161003816100381610038161003816)] (16)


ThDF119895 =1

120598119895

timesmax (119888

119895)

119865jSN

ThDF119895 =1

120598119869

timesmax (119888

119895)

119865jSN

(17)




120598119895=

119864119895

119864119905

times 100

120598119869=

119864119895

119864119905

times 100

1198701015840 =1198984

11989822

=(1119873)sum

119873

119894=1(119909119894minusmedia (119909))

2

((1119873)sum119873

119894=1(119909119894minusmedia (119909))2)

2

(18)




119895sdot th119895 (19)



120590 = 119862 sdotMAD (119888119895) (20)


1198951 1198881198952 119888


as [27]


119895)10038161003816100381610038161003816) (21)



119895)10038161003816100381610038161003816) (22)


120590 = 14826 sdotmedian (10038161003816100381610038161003816119888119895

10038161003816100381610038161003816) (23)




119865119904(Hz) 119873 119865


1198604

1198605

1198606

1198607

1198608

1198609

11986010

11986011

11986012

250 5400






500 10800

015 0612 4491 10172 16583 30755 24674 5845 minus9762 minus10373


023 0612 4491 10168 16485 26036 14864 0825 minus10778 minus10796



1000 21600

015 0104 1040 5684 11679 18344 33993 23436 4831 minus10805

019 0104 1040 5684 11678 18319 31447 20792 minus3154 minus11545

023 0104 1040 5684 11677 18271 26531 13913 minus0238 minus11897

027 0104 1040 5684 11677 18255 24680 7822 minus10512 minus11771

031 0104 1040 5684 11679 18300 25393 3629 minus11258 minus11807








119889(119895)119899

= sign (119889(119895)

119899) (

10038161003816100381610038161003816119889(119895)

119899

10038161003816100381610038161003816 minus th119895) if 10038161003816100381610038161003816119889

(119894)

119899

10038161003816100381610038161003816 ge th119895

0 if 10038161003816100381610038161003816119889(119894)

119899

10038161003816100381610038161003816 lt th119895

(24)

119889(119895)119899

=

119889(119895)119899

if 10038161003816100381610038161003816119889(119894)

119899

10038161003816100381610038161003816 ge th119895

0 if 10038161003816100381610038161003816119889(119894)

119899

10038161003816100381610038161003816 lt th119895

(25)

where 119889(119895)119899

and 119889(119895)119899


119895and


119899or119889(119895)119899 have


0 1 2 4 5

0

05


Time (s)


minus05

minus1

0 1 2 3

Time (s)3

4 5

0

05

1

minus05

minus1

(a)


Time (s)

30 1 2 4 5

0

05


minus05

0 1 2 3 4 5

0

05

1

minus05

(b)



0 1 2 3 4 5

0

05

1

minus05

Time (s)

(a)

Time (s)


0 1 2 3 4 5

0

05

1

minus05

(b)

minus01215 22 225 23

0

01



minus01215 22 225 23

0

01

02

Time (s)

Time (s)

(c)




Time (s)


0 1 2 3 4 5

0

05

1

minus05

0 1 2 3

Time (s)

4 5

0

05

1

minus05

(a)



Time (s)

0 1 2 3 4 5

0

05

1

minus1

minus05

0 1 2 3 4 5

0

05

1

minus05

Time (s)

(b)



Time (s)

0 1 2 3 4 5

0

05

1

minus05

0 1 2 3 4 5

0

05

1

minus05

Time (s)

(c)



4 Results





205 206 207Time (s)

208 209 210

005

minus05

205 206 207 208 209 210

005

minus05

205 206 207 208 209 210

005

minus05


205 206 207 208 209 210

01

minus1





Time (s)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

0


0



minus1

minus05

0

05

minus05



Time (s)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

0


0



minus1

minus05

0

05

minus05


205 206 207 208 209 210


205 206 207 208 209 210

205 206 207 208 209 210Time (s)



005

minus05minus1

005

minus05minus1

005

minus05minus1








5 Conclusion


Acknowledgment


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















119895)

log (2)) (15)





119899 = round [log2(max 10038161003816100381610038161003816119888119895

10038161003816100381610038161003816)] (16)


ThDF119895 =1

120598119895

timesmax (119888

119895)

119865jSN

ThDF119895 =1

120598119869

timesmax (119888

119895)

119865jSN

(17)




120598119895=

119864119895

119864119905

times 100

120598119869=

119864119895

119864119905

times 100

1198701015840 =1198984

11989822

=(1119873)sum

119873

119894=1(119909119894minusmedia (119909))

2

((1119873)sum119873

119894=1(119909119894minusmedia (119909))2)

2

(18)




119895sdot th119895 (19)



120590 = 119862 sdotMAD (119888119895) (20)


1198951 1198881198952 119888


as [27]


119895)10038161003816100381610038161003816) (21)



119895)10038161003816100381610038161003816) (22)


120590 = 14826 sdotmedian (10038161003816100381610038161003816119888119895

10038161003816100381610038161003816) (23)




119865119904(Hz) 119873 119865


1198604

1198605

1198606

1198607

1198608

1198609

11986010

11986011

11986012

250 5400






500 10800

015 0612 4491 10172 16583 30755 24674 5845 minus9762 minus10373


023 0612 4491 10168 16485 26036 14864 0825 minus10778 minus10796



1000 21600

015 0104 1040 5684 11679 18344 33993 23436 4831 minus10805

019 0104 1040 5684 11678 18319 31447 20792 minus3154 minus11545

023 0104 1040 5684 11677 18271 26531 13913 minus0238 minus11897

027 0104 1040 5684 11677 18255 24680 7822 minus10512 minus11771

031 0104 1040 5684 11679 18300 25393 3629 minus11258 minus11807








119889(119895)119899

= sign (119889(119895)

119899) (

10038161003816100381610038161003816119889(119895)

119899

10038161003816100381610038161003816 minus th119895) if 10038161003816100381610038161003816119889

(119894)

119899

10038161003816100381610038161003816 ge th119895

0 if 10038161003816100381610038161003816119889(119894)

119899

10038161003816100381610038161003816 lt th119895

(24)

119889(119895)119899

=

119889(119895)119899

if 10038161003816100381610038161003816119889(119894)

119899

10038161003816100381610038161003816 ge th119895

0 if 10038161003816100381610038161003816119889(119894)

119899

10038161003816100381610038161003816 lt th119895

(25)

where 119889(119895)119899

and 119889(119895)119899


119895and


119899or119889(119895)119899 have


0 1 2 4 5

0

05


Time (s)


minus05

minus1

0 1 2 3

Time (s)3

4 5

0

05

1

minus05

minus1

(a)


Time (s)

30 1 2 4 5

0

05


minus05

0 1 2 3 4 5

0

05

1

minus05

(b)



0 1 2 3 4 5

0

05

1

minus05

Time (s)

(a)

Time (s)


0 1 2 3 4 5

0

05

1

minus05

(b)

minus01215 22 225 23

0

01



minus01215 22 225 23

0

01

02

Time (s)

Time (s)

(c)




Time (s)


0 1 2 3 4 5

0

05

1

minus05

0 1 2 3

Time (s)

4 5

0

05

1

minus05

(a)



Time (s)

0 1 2 3 4 5

0

05

1

minus1

minus05

0 1 2 3 4 5

0

05

1

minus05

Time (s)

(b)



Time (s)

0 1 2 3 4 5

0

05

1

minus05

0 1 2 3 4 5

0

05

1

minus05

Time (s)

(c)



4 Results





205 206 207Time (s)

208 209 210

005

minus05

205 206 207 208 209 210

005

minus05

205 206 207 208 209 210

005

minus05


205 206 207 208 209 210

01

minus1





Time (s)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

0


0



minus1

minus05

0

05

minus05



Time (s)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

0


0



minus1

minus05

0

05

minus05


205 206 207 208 209 210


205 206 207 208 209 210

205 206 207 208 209 210Time (s)



005

minus05minus1

005

minus05minus1

005

minus05minus1








5 Conclusion


Acknowledgment


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119865119904(Hz) 119873 119865


1198604

1198605

1198606

1198607

1198608

1198609

11986010

11986011

11986012

250 5400






500 10800

015 0612 4491 10172 16583 30755 24674 5845 minus9762 minus10373


023 0612 4491 10168 16485 26036 14864 0825 minus10778 minus10796



1000 21600

015 0104 1040 5684 11679 18344 33993 23436 4831 minus10805

019 0104 1040 5684 11678 18319 31447 20792 minus3154 minus11545

023 0104 1040 5684 11677 18271 26531 13913 minus0238 minus11897

027 0104 1040 5684 11677 18255 24680 7822 minus10512 minus11771

031 0104 1040 5684 11679 18300 25393 3629 minus11258 minus11807








119889(119895)119899

= sign (119889(119895)

119899) (

10038161003816100381610038161003816119889(119895)

119899

10038161003816100381610038161003816 minus th119895) if 10038161003816100381610038161003816119889

(119894)

119899

10038161003816100381610038161003816 ge th119895

0 if 10038161003816100381610038161003816119889(119894)

119899

10038161003816100381610038161003816 lt th119895

(24)

119889(119895)119899

=

119889(119895)119899

if 10038161003816100381610038161003816119889(119894)

119899

10038161003816100381610038161003816 ge th119895

0 if 10038161003816100381610038161003816119889(119894)

119899

10038161003816100381610038161003816 lt th119895

(25)

where 119889(119895)119899

and 119889(119895)119899


119895and


119899or119889(119895)119899 have


0 1 2 4 5

0

05


Time (s)


minus05

minus1

0 1 2 3

Time (s)3

4 5

0

05

1

minus05

minus1

(a)


Time (s)

30 1 2 4 5

0

05


minus05

0 1 2 3 4 5

0

05

1

minus05

(b)



0 1 2 3 4 5

0

05

1

minus05

Time (s)

(a)

Time (s)


0 1 2 3 4 5

0

05

1

minus05

(b)

minus01215 22 225 23

0

01



minus01215 22 225 23

0

01

02

Time (s)

Time (s)

(c)




Time (s)


0 1 2 3 4 5

0

05

1

minus05

0 1 2 3

Time (s)

4 5

0

05

1

minus05

(a)



Time (s)

0 1 2 3 4 5

0

05

1

minus1

minus05

0 1 2 3 4 5

0

05

1

minus05

Time (s)

(b)



Time (s)

0 1 2 3 4 5

0

05

1

minus05

0 1 2 3 4 5

0

05

1

minus05

Time (s)

(c)



4 Results





205 206 207Time (s)

208 209 210

005

minus05

205 206 207 208 209 210

005

minus05

205 206 207 208 209 210

005

minus05


205 206 207 208 209 210

01

minus1





Time (s)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

0


0



minus1

minus05

0

05

minus05



Time (s)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

0


0



minus1

minus05

0

05

minus05


205 206 207 208 209 210


205 206 207 208 209 210

205 206 207 208 209 210Time (s)



005

minus05minus1

005

minus05minus1

005

minus05minus1








5 Conclusion


Acknowledgment


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 1 2 4 5

0

05


Time (s)


minus05

minus1

0 1 2 3

Time (s)3

4 5

0

05

1

minus05

minus1

(a)


Time (s)

30 1 2 4 5

0

05


minus05

0 1 2 3 4 5

0

05

1

minus05

(b)



0 1 2 3 4 5

0

05

1

minus05

Time (s)

(a)

Time (s)


0 1 2 3 4 5

0

05

1

minus05

(b)

minus01215 22 225 23

0

01



minus01215 22 225 23

0

01

02

Time (s)

Time (s)

(c)




Time (s)


0 1 2 3 4 5

0

05

1

minus05

0 1 2 3

Time (s)

4 5

0

05

1

minus05

(a)



Time (s)

0 1 2 3 4 5

0

05

1

minus1

minus05

0 1 2 3 4 5

0

05

1

minus05

Time (s)

(b)



Time (s)

0 1 2 3 4 5

0

05

1

minus05

0 1 2 3 4 5

0

05

1

minus05

Time (s)

(c)



4 Results





205 206 207Time (s)

208 209 210

005

minus05

205 206 207 208 209 210

005

minus05

205 206 207 208 209 210

005

minus05


205 206 207 208 209 210

01

minus1





Time (s)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

0


0



minus1

minus05

0

05

minus05



Time (s)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

0


0



minus1

minus05

0

05

minus05


205 206 207 208 209 210


205 206 207 208 209 210

205 206 207 208 209 210Time (s)



005

minus05minus1

005

minus05minus1

005

minus05minus1








5 Conclusion


Acknowledgment


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Time (s)


0 1 2 3 4 5

0

05

1

minus05

0 1 2 3

Time (s)

4 5

0

05

1

minus05

(a)



Time (s)

0 1 2 3 4 5

0

05

1

minus1

minus05

0 1 2 3 4 5

0

05

1

minus05

Time (s)

(b)



Time (s)

0 1 2 3 4 5

0

05

1

minus05

0 1 2 3 4 5

0

05

1

minus05

Time (s)

(c)



4 Results





205 206 207Time (s)

208 209 210

005

minus05

205 206 207 208 209 210

005

minus05

205 206 207 208 209 210

005

minus05


205 206 207 208 209 210

01

minus1





Time (s)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

0


0



minus1

minus05

0

05

minus05



Time (s)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

0


0



minus1

minus05

0

05

minus05


205 206 207 208 209 210


205 206 207 208 209 210

205 206 207 208 209 210Time (s)



005

minus05minus1

005

minus05minus1

005

minus05minus1








5 Conclusion


Acknowledgment


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










205 206 207Time (s)

208 209 210

005

minus05

205 206 207 208 209 210

005

minus05

205 206 207 208 209 210

005

minus05


205 206 207 208 209 210

01

minus1





Time (s)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

0


0



minus1

minus05

0

05

minus05



Time (s)

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

0


0



minus1

minus05

0

05

minus05


205 206 207 208 209 210


205 206 207 208 209 210

205 206 207 208 209 210Time (s)



005

minus05minus1

005

minus05minus1

005

minus05minus1








5 Conclusion


Acknowledgment


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














5 Conclusion


Acknowledgment


References








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra








































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article noise suppression in ecg signals through...

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