ECG Denoising Using the Extended Kalman Filtre EKF Based on a Dynamic ECG Model

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1 ECG Denosng Usng the Extended Kalman Fltre EKF Based on a Dynamc ECG Model Mohammed Assam Oual, Khereddne Chafaa Department of Electronc. Unversty of Hadj Lahdar Batna,Algera. assam_oual@yahoo.com Moufd Hadjab Department of Electronc. Unversty of Djllal Labes. Sd Bel Abbes,Algera Abstract In ths paper an Extended Kalman Flter (EKF) has been proposed for the flterng of ECG Sgnals. The method s based on a nonlnear dynamc model, prevously ntroduced for the generaton of synthetc ECG sgnal. The results show that the EKF may be used as a powerful tool for ECG sgnal denosng; our study was performed on artfcal and real ECG sgnals. Keywords-Extended; Kalman Fltre; Nonlnear dynamc model ;Electrocardogram. I. INTRODUCTION An electrocardogram descrbes the electrcal actvty n the heart, and can be decomposed n characterstc components, named P, Q, R, S and T waves []. The cardac electrcal actvty s a convenent non-nvasve tool for the detecton, predcton and montorng of rare cardac events and related anomales such as arrhythmas, prmary concerns n ECG recordng ncluded dstorton due to nose contamnaton, artfacts, and nterference from other sgnals. Hence extracton of pure cardologcal ndces from nosy measurements has been one of the major concerns of bomedcal sgnal processng and needs relable technques to preserve the dagnostc nformaton of the recorded sgnal. On the other hand, n recent, years some research has been conducted towards the generaton of synthetc ECG sgnals. Regardng the physologcal bases of ECG sgnals, a true ECG model should consder the morphology of the PQRST complex, together wth the RR-wave tmng. In prevous wor, a synthetc model has been proposed whch has unfed the morphology and pulse tmng of the ECG sgnal n a sngle nonlnear dynamc model []. In ths paper the EKF based on the nonlnear dynamc ECG model has been used to extract the ECG components contamnated wth the bacground nose. The paper s organzed as follows. Secton II provdes bacgrounds on the EKF theory and summarzes the ECG artfcal model. The thrd secton deals wth the detals of the proposed method. Smulaton results are provded n secton VI. Fnally dscusson and concluson are provded n secton V. II. THEORY A. Extended Kalman Fltre Revew The EKF s a nonlnear extenson of conventonal Kalman flter[3], that has been specfcally developed for systems havng nonlnear dynamc model. For a dscrete nonlnear system wth the state vector and observaton vector,the dynamc model and ts lnear approxmaton near a desred reference pont may be formulated as follows: x f ( x, w,) f ( xˆ, wˆ,)()() A x xˆ F w wˆ y g( x, v,) g( xˆ, ˆ,)()() ˆ ˆ v C x x G v v where: f ( x, wˆ,) A x C g( x, vˆ,) x x xˆ xxˆ f ( xˆ, w,) F w g( xˆ, v,) G v wwˆ vvˆ Here, and are the process and measurement noses, respectvely, wth covarance matrces = { } and = { }. In order to mplement the EKF, the tme propagaton, and the measurement propagaton equaton are summarzed as follows: xˆ ˆ f ( x, w,) w P A P A F Q F T T xˆ ˆ ( ˆ x K y g x, v,) v T T K P C C P C G P P KC P () () (3)

2 Z where = {,,, } s the a pror estmate of the state vector,, at the hupdate,usng the observatons, to, and = {,,, } s the a posteror estmate of the state vector after addng the h observatons. and are defned n the same manner to be the estmatons of the covarance matrces n the h stage, before and after usng the h observaton, respectvely. B. ECG Dynamc Model McSharry et al. have proposed s synthetc ECG generator, whch conssts of a three dmensonal state equaton, whch generate a trajectory wth the Cartesan coordnate (,, )[].Ths model has several parameters, whch maes t adaptable to many normal and abnormal ECG sgnals. As t may be seen n (4) the dynamc model conssts of a three dmensonal state equaton. x x y y y z a exp() z z P, Q, R, S, T b (4) R.5 P T.5 S Q Fgure. The synthetc ECG trajectory generated by (4). Where: = +, = ( ) ( ) = (, ) ( the four quadrant arctangent of the real parts of the elements of and,wth (, ), and s the angular velocty of the trajectory as t moves around the lmt cycle. The baselne wander of the ECG sgnal has been modeled wth, whch s coupled wth the respratory frequency : = ( ) =.5 =.5 (5) The values of the parameters of (4) are lsted n Table I TABLE I PARAMETRES OF THE ECG MODEL OF (4) Inde() P Q R S T Tmes (Sec) (rads) -π/3 -π/ π/ π/ In fact the three dmensonal trajectory whch s generated from (4),conssts of a crcular lmt cycle whch pushed up and down when t approaches one of the P,Q,R,S or T ponts. The projecton of these trajectory ponts on the z axs gves a synthetc ECG sgnal. The three dmensonal trajectory and a typcal synthetc ECG generated from the values of Table I may be seen n fg. and temps (sec) Fgure. A typcal synthetc ECG sgnal generated by (4). III. METHOD A. Descretzaton of the Nonlnear Dynamc ECG Model The nonlnear dynamc ECG model (4) s a contnuoustme model, and snce the Kalman flter s a dscrete algorthm, then, a dscretzaton of the nonlnear dynamc ECG model (4) s necessary. Usually the dscretzaton s done usng the Euler method[4]. Thus, the nonlnear dynamc ECG model (4) n ts dscrete form s gven by: x( )()()() h x hy y( )()()() h y hx z( ) a h exp(( )()) h z hz P, Q, R, S, T b Where h s a samplng tme. B. Quas real ECG Model The nonlnear dscrete ECG model (6), can be rewrtten n the followng compact form: X f () X (7) (6) Where s the state vector gven by = [ ].

3 x( )()()() h x hy y( )()()() h y hx z( ) ah exp(( )()) h z hz P, Q, R, S, T b The vectoral equaton (8) represents the state equaton wthout nose of the dscrete ECG model. To represent a more real ECG sgnal, we need to ntroduce some random noses to the model (7) as follows: X f ( X,) w (9) Where = [ ] s a random vector of the addtve, normal and Gaussan nose, then, (8) becomes as follows: x( ) ()()()() h x hy w y( )()()()() h y hx w z( ) ah exp(( )())() h z hz w3 PQR,,, S, T b The measurement equaton correspondng to the state representaton () can be joned to the state vector = [ ] by the followng relaton: s g( X,) v X v () wth s the consdered measure, and, s the measurement addtve, normal and Gaussan nose. C. Lnearzaton of the Nonlnear dynamc ECG model In order to use an EKF t s necessary to drve a lnear approxmaton of (). Therefore, the second step was to lnearze the nonlnear model usng () and (). Accordng (), (3) and (4) represent a lnearzed verson of () and () wth respect to the state varable x,y and z. x( )((),(),(),()) F x y z w y( )((),(),(),()) G x y z w z( )((),(),(),()) H x y z w3 s g( X,) v (3) (8) () () F ()() hx hy x x()() y F hx()() y h y x()() y h F z G hx()() y x x()() y h G hx() () hy y x()() y G z h H ahy() exp (4) x P, Q, R, S, T x()() y b b H ahx() exp y P, Q, R, S, T x()() y b b H h z F G H w w w 3 F F G G H H w w w w w w 3 3 g g g g ; h ; x y z v D. Evaluaton For evaluatng the performance of the proposed method we have used the Sgnal to nose rato (SNR), SNR mprovement (mp) and Mean squared error (MSE) measures by the means of the expressons: powerof thesgnal x SNRdb log powerof thenose b mpdb log x x n d x x x x (5) (6) ()() n MSE (7) N Where denotes the clean ECG, s the denosed sgnal and represents the nosy ECG. The power of the sgnal s gven by /,and the power of the nose s merely hs varance and s the number of the samples.

4 IV. RESULTS In what follows, we wll test the proposed method for flterng an electrocardogram sgnals. In a frst step, the sgnal consdered to be fltered wll be the sgnal generated by the quas real ECG model () (Flterng of an artfcal ECG), then a real ECG sgnal wll be fltered drectly. A. Artfcal ecg sgnal Denosng The consdered ECG sgnal s generated by the model () and ( ). The perod of ths sgnal s sec, whch the parameters are gven n the TABLE.I. Wth random nose as follows: process nose: ( ) =. Measurement nose : ( ) = The smulaton results are gven n fg.3, 4 and 5, where we notce although the flterng operaton has been well establshed (denosed ECG follows the morphology of the sgnal generated). To see the qualty of flterng numercally and evaluate the performances, the formulas (5),(6) and (7) are used, whch gves us the followng results n TABLE.II Fgure 5. The EKF output for ( ) = and ( ) = TABLE II PERFORMANCE RESULTS ( ( ) = ) and ( ) = Nosy ECG Denosed ECG SNR [db] MSE MSE Imp[db] SNR[db] The results presented n TABLE II approve that the proposed method mnmzed the effect of the noses on the generated ECG sgnal. The vsualzaton of these results s presented n the Fg.6 and 7, where we have gven the curves correspondng errors (errors before and after flterng) x Fgure 3. A typcal synthetc ECG sgnal generated by (4) Fgure 4. A typcal synthetc ECG sgnal generated by (4) wth Whte Gaussan nose x Fgure.6 Error before Flterng Fgure 7. Error after Flterng

5 To evaluate and see the performance of the proposed method, ncreasng the power of the measurement nose as, ( ) =, The results are shown n fg. 8,9 and and Fg. and and TABLE.III. TABLE III PERFORMANCE RESULTS ( ( ) = ) and ( ) = Nosy ECG Denosed ECG SNR [db] MSE MSE Imp[db] SNR[db] Fgure. Error before Flterng.4 x Fgure 8. A typcal synthetc ECG sgnal generated by (4) Fgure. Error after Flterng Fgure 9. A typcal synthetc ECG sgnal generated by (4) wth Whte Gaussan nose. B. Real ecg sgnal Denosng The ECG conssted s a normal snus rhythm, taen from the PhysoNet ECG database[6], referenced by 6786.dat. The samplng frequency of ths sgnal s 8 Hz, The values of the parameters of (4) for ths real ECG sgnal are lsted n Table IV. TABLE IV PARAMETRES OF THE ECG MODEL (4) FOR THE REAL ECG SIGNAL 6786.DAT. Inde() P Q R S T Tmes (Sec) (rads) -π/3 -π/ π/ π/ The smulaton results are gven n fg.3, 4 and 5, where we notce although the flterng operaton has been done (denosed ECG s very smooth compared to the real nosy ECG) Fgure. The EKF output for ( ) = and ( ) = on a tme nterval of.5 seconds

6 ECG réel bruté ECG débruté ECG réel bruté ECG débruté ECG réel bruté ECG débruté ECG réel bruté ECG débruté Nosy ECG Denosed ECG ECG réel bruté ECG débruté The results of ths paper approve the applcablty of the Extended Kalman Flter (EKF),for the flterng of nosy ECG sgnals ECG réel bruté ECG débruté 3 ECG réel bruté ECG débruté Ampltude ECG Ampltude ECG Fgure 3. The EKF output for ( ) = 4 on a tme nterval of 6seconds ECG réel bruté Nosy ECG Denosed ECG ECG débruté Ampltude ECG Ampltude ECG Ampltude ECG Ampltude ECG Fgure 4. The EKF output for ( ) = 4 on a tme nterval of seconds ECG réel bruté ECG débruté ECG réel bruté ECG débruté ECG réel bruté Nosy ECG Denosed ECG ECG débruté Ampltude ECG Ampltude ECG ECG réel bruté ECG débruté 8 ECG réel bruté ECG débruté Ampltude ECG 4 Ampltude ECG Fgure 5. The EKF output for ( ) = 4 between tme nstants 4 and 4.8 seconds. To valdate the proposed method, several ECG sgnal have been chosen from the PhysoNet ECG database. The results are presented n the fg.6. The curves suggest that the proposed method functoned and can be appled to anyone ECG sgnals. I. CONCLUSION In ths paper an EKF was desgned for the flterng of ECG sgnals. The EKF s dynamc model was based on a three dmensonal nonlnear dynamc model prevously ntroduced for the generaton of synthetc ECG sgnals. Ths nonlnear model was descretzed and lnearzed n order to be used n an EKF, The desgned flter was later appled to artfcal and real ECG sgnals Fgure 6. The EKF output for several real ECG sgnals. REFERENCES [] Z.Abedn, R. Conner, ECG Interpretaton-The Self-assessment Approach Texas Tech Unversty Health Scences Centre 8. [] P.E.Mcsharry, G.D.Clfford,Tarasseno, L.A Smth, A Dynamc Model for Generatng Synthetc Electrocardogram Sgnals, IEEE Trans. Bomed. Eng,vol.5,No.3,Mar.3,pp [3] M.S.Grewal,A.P.Andrews. Kalman Flterng: Theory and Practce Usng Matlab,second Edton John Wley &Sons, Inc. [4] A.Fortn," Analyse Numérque," deuxème édton, Ecole Polytechnque de Montréal,. [5] physonet,http// -