Effective Extraction and Filtering of Frequency Components in Physiological Signals Using Sum-of-Sinusoids Modelling

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1 Signl Processing Reserch Vol. 2 Iss. 4, Decemer Effective Extrction nd Filtering of Frequency Components in Physiologicl Signls Using Sum-of-Sinusoids Modelling José L. Ferreir *1, Ronld M. Arts 2, Pierre J. M. Cluitmns 3 1,2,3 Deprtment of Electricl Engineering, Eindhoven University of Technology 2 Philips Reserch Lortories Eindhoven 3 Kempenheghe Epilepsy Center P. O MB Eindhoven The Netherlnds *1 j.l.ferreir@tue.nl; 2 r.m.rts@tue.nl; 3 p.j.m.cluitmns@tue.nl Astrct In iologicl signl processing, modelling nd extrction of specific frequency components constitute n importnt procedure for filtering signl components of interest s well s rtefct removl. Under some interference scenrios, stisfctory elimintion of rtefcts from the signl must e even performed y sutrction of n rtefct wveform model or templte, rther thn the use of liner nd-pss filters. Tht is the cse of the grdient rtefct induced in the EEG within the fmri scnner, which cnnot e chrcterized y specific ndwidth or spectrl content. This pper presents simple nd ccurte pproch sed upon sumof-sinusoids modelling for signl nd rtefct frequency components representtion in physiologicl signls. According to the proposed method, ech signl frequency component is pproximted s sinusoid, whose mplitude nd phse prmeters re estimted y mking use of the Discrete Fourier Trnsform (DFT). The proposed pproch revels to perform n effective modelling nd extrction of ECG signl components s well s underlying grdient rtefcts in the EEG signl. Keywords Biologicl Signl Modelling; Sum-of-Sinusoids Modelling; Grdient Artefct Correction; Biologicl Signl Filtering Introduction Filtering nd extrction of signl frequency components re uiquitous nd essentil prctices in iologicl nd digitl signl processing in generl. The usge of filtering methodologies llows removl of undesirle noise components, rtefcts, nd other types of interference, which is fundmentl for otining n enhnced signl. In the sme wy, filtering pproches llow extrction of frequency components nd ndwidths of interest (Oppenheim, 1999; Prokis nd Mnolkis, 1996; Rngyyn, 22). Filtering of signl components is commonly performed y liner digitl filters. Digitl filtering techniques re lso termed frequency-selective filters, nd cn e implemented in time nd frequency-domin. Idelly, filter is liner-time invrint system tht psses certin frequency components, nd ll others locted outside the filter cut-off frequency re rejected. In prctice, rel filters do not chieve complete removl of frequencies outside the filter cut-off frequency. A trnsition region occurs round the cut-off frequency in such wy tht the mgnitude response is not constnt in the entire filter pss-nd s well (Oppenheim, 1999; Prokis nd Mnolkis, 1996).Furthermore, concerning removl of physiologicl nd some other types of interference, liner nd-pss filters could not e effective since the former cnnot e chrcterized y ny specific spectrl content or frequency ndwidth (Rngyyn, 22; Allen et l., 1998; Allen et l., 2). In such cses, the usge of non-liner filtering pproches for modelling noise components constitutes n lterntive method for filtering nd extrction of signl components, like the sutrction in time nd frequency-domin methodologies. According to the sutrction in time-domin pproch, when the noise or rtefct cn e ssumed to possess sttionry wveform (i.e., it cn e considered slowly vrying process), n verge wveform or templte which represents the rtefct cn e estimted. Then, such templte is sutrcted from the signl to chieve the signl restortion (Allen et l., 1998; Allen et l., 2). The sme procedure cn e performed in frequency-domin when it is possile to ssume tht the noise power spectrum does not chnge significntly in-etween the updte periods (Vseghi, 2). 73

2 Signl Processing Reserch Vol. 2 Iss. 4, Decemer 213 In this wy, Allen et l. (2) proposed the usge of n verge rtefct wveform in order to represent nd remove the grdient rtefct. Such n rtefct is induced in the EEG signl y vrying grdient mgnetic fields within the fmri scnner. The frequency components of the grdient rtefct cnnot e clened up simply y pplying usul low-pss filtering ecuse of the underlying frequencycomponents which remins in the clinicl EEG ndwidth (Nizy et l., 25). Hence, their removl requires the usge of other filtering techniques, rther thn low-pss filters (Allen et l., 2). Insted of using n verge templte, El-Ttr nd Fokpu (211) nd Ferreir et l. (213) proposed the usge of n rtefct wveform estimte sed upon set of sinusoidl wveforms, for implementtion of the grdient rtefct modelling nd susequent EEG restortion. In ddition to estimting the rtefct wveform, the sum-of-sinusoids model cn e used to predict the vriility of the grdient rtefct wveform over the time (Ferreir et l., 213). The sum of sinusoids is widely employed for representtion nd nlysis/synthesis of speech nd udio signls, which ressumed s periodic signls. Such signl representtion hs proven to e effective ecuse of its widespred pplicility nd simplicity of implementtion (Brillinger, 1987; Prsd et l., 28; McAuly nd Qutieri, 1986). This pper presents simple nd ccurte pproch for modelling of signl frequency components of interest nd rtefcts in physiologicl signls y the sum of set of sinusoids. Our method constitutes modifiction from the pproch proposed y Ferreir et l. (213) for removl of underlying grdient rtefcts from the EEG signl ndwidth. According to our methodology, ech signl frequency component is pproximted seprtely s sinusoid, whose mplitude prmeter is estimted y using the mgnitude of the Discrete Fourier Trnsform (DFT). The phse of the DFT is used for estimtion of the sinusoid phse s well, s descried in the section Methods. The proposed method shows to perform n effective modelling of ECG signl components s well s stisfctory removl of underlying grdient rtefcts from the EEG signl y the sutrction of the estimted sum-ofsinusoids model, s shown in the section Results. Methods Signl Frequency Components Modelling The Discrete Fourier Trnsform (DFT) cn e used to decompose finite periodic discrete-time sequence, yi, into its k complex sinusoids or exponentil components. In the opposite wy, the Inverse Discrete Fourier Trnsform (IDFT) llows recovering yi from those resulting components. In the literture, some pproches hve een descried in order to compute the DFT nd the IDFT efficiently, known collectively s Fst Fourier Trnsform (FFT) lgorithms (Prokis nd Mnolkis, 1996).Eqs. (1) nd (2) constitute one representtion of such FFT lgorithms (Oppenheim, 1999): N (i 1)(k 1) Y k = y i ω N, (1) i=1 N y i = 1 N Y (i 1)(k 1) kω N, (2) k=1 where Yk is the Discrete Fourier Trnsform of yi; N is j2π N the length of yi; ndω N = e. Ech frequency component of yi cn e lso fitted or pproximted y the following sinusoid wveform (Brillinger, 1987; Ferreir et l., 213): s k,i = A k cos 2πf k t i + φ k, (3) where the mplitude Âk is estimted y using the DFT mgnitude of the k-th frequency component nd the signl length ssocited with yi: A K = 2 Y k N. (4) The time ti is computed s: t i = iδt, (5) where Δt corresponds to the inverse of the smpling frequency fs. The frequency fk, ssumed known, is provided y Eq. (6): f k = k f s N. (6) In order to mtch fk with the corresponding frequency component within yi, N ws set s multiple of the rtio fs / fk. Estimtion of the sinusoidl phse is performed y clculting the DFT phse of the k-th frequency component: Consistency Anlysis φ k = rctn I(Y k). (7) R(Y k ) Our methodology ws implemented in MATLAB environment. For nlysis of consistency, Eqs. (1) (7) were pplied for estimting single sinusoid (si) in Gussin white noise (ni): y i = s i + n i. (8) Zero men Gussin white noise ws mixed with 74

3 Signl Processing Reserch Vol. 2 Iss. 4, Decemer single sinusoids possessing different frequencies fk (.25, 3, 1, 5, 2, nd 8 Hz).The SNR of the resulting (oserved) signl ws vried from -1 to 6 db. At ech SNR, 1 trils were crried out in order to estimte the percent error for the sinusoid mplitude nd phse estimtes θ k : error = 1 θ k,q θ k θ k %, (9) where θ k,q is the verge of the prmeter estimte, tking into ccount the 1 performed trils. Also the men squre error (MSE), ρmse, chrcteristics were estimted y Eq. (1): 1 ϱ MSE = 1 log 1 1 θ k,q θ k 2, (1) q=1 In order to ssess the performnce of the sinusoidl phse nd mplitude estimtion, the Crmer-Ro Bound (CRB) ws lso clculted using Eqs. (11) nd (12) (Prokis nd Mnolkis, 1996; Ppoulis nd Pilli, 22; Ky, 1993): vr A k 2 σ2 N, (11) vr φ k 2 σ2 N A k 2, (12) where σ 2 corresponds to the vrince of the Gussin noise. vlues of frequency fk is shown, considering different SNR levels nd sinusoid length N = It cn e oserved tht s the SNR increses, the estimted error for the mplitude nd phse estimtion tends to %. TABLE 1 ESTIMATED PERCENT ERROR ACCORDING TO EQ. (9), FOR DIFFERENT VALUES OF FREQUENCY fk, SNR, AND 1 TRIALS SNR (db) Percent error (%) θk fk(hz) Ak φk Ak φk Ak φk Ak φk Ak φk Ak φk Ak φk Ak φk Chrcteristic of the MSE for the mplitude estimtion Signl (.u.) Originl signl Oserved signl -1log 1 (MSE) CRB Our method SNR (db) 1 Signl (.u.) -1 1 Chrcteristic of the MSE for the phse estimtion Signl (.u.) c Estimted signl -1log 1 (MSE) FIG. 1 APPLICATION OF EQS. (1) (7) FOR ESTIMATING A SINGLE SINUSOID MIXED WITH WHITE GAUSSIAN NOISE: (A) ORIGINAL SINUSOID (Ak = 1; φk = 1 rd); (B) OBSERVED (MIXED) SIGNAL (SNR = 1 db); (C) ESTIMATED SINUSOID (A k =.9995.u. AND φ k = 1.4 rd) Fig. 1 shows the single sinusoid of Fig. 1 (Ak = 1.u.; fk = 1 Hz; φk = 1 rd; N = 8192) mixed with Gussin white noise, SNR = 1 db. Δt ws set t 1/248 Hz. In Fig. 1c, the sinusoid estimted y using the pproch of Eqs. (1) (7) is depicted. The corresponding percent error is equl to -.492% nd.44%, respectively, for Âk nd φ k. In Tle 1, the percent error for different 4 3 CRB Our method SNR (db) FIG. 2 CHARACTERISTICS OF THE MSE FOR THE AMPLITUDE AND PHASE ESTIMATIONS (N = 8192; fk =.25 Hz; 1 TRIALS). SUCH CHARACTERISTIS ATTAIN THE CRB EVEN UNDER LOW SNR, DEMONSTRATING THE ACCURACY ESTIMATION OF THE PROPOSED MODELLING APPROCH Fig. 2 depicts the MSE chrcteristics for the mplitude nd phse prmeters. It cn e noticed tht those chrcteristics closely ttin the respective CRB estimtes, even under low SNR. Also, the frequency 75

4 Signl Processing Reserch Vol. 2 Iss. 4, Decemer 213 dependency for mplitude nd phse estimtions cn e neglected (Tle 1). Such chrcteristics, ssocited with the low percent error for the mplitude nd phse prmeters, show the ccurcy of using Eqs. (1) (7) for modelling nd extrction of single sinusoidl frequency components (Ky, 1993). Therefore, they do indicte the fesiility of using the proposed method s filter nk pproch for modelling nd extrction of signl frequency components (Goodwin, 28; McAuly nd Qutieri, 1986). Thus, we hve pplied Eqs. (1) (7) for modelling lrger ndwidths of rel ECG excerpts s well s rising grdient rtefcts in EEG signls. Such signls were collected for reserch focused on epilepsy nd post-trumtic stress disorder (Vn Liempt et l., 211; Ferreir et l., 213). The ECG/EEG (EpG) frequency components of interest were modelled s sum of sinusoid components, s follows (Ferreir et l., 213): K EpG i = s k,i, (13) k=1 Different vlues of signl length, N, were chosen in order to evlute the influence of the signl length over the otined results. The influence of the totl numer of K modelled frequency components ws nlysed during the methodology ppliction s well. Results Fig. 3 illustrtes ppliction of Eqs. (1) (7) for modelling nd extrction of frequency components in the ECG signl. In Fig. 3, n exemplry ECG excerpt (originl ECG signl: N = 8192; fs = 248 Hz) nd the respective modelled signl resulting from the sum of totl K = 1 sinusoid components (estimted ECG signl) re depicted. The vlue K = 1 ws clculted using Eq. (6), for fk rnging from to 25 Hz (fk = 25 Hz), in ccordnce with the clinicl frequency rnge of the electrocrdiogrm (Olson, 21). Clerly, it cn e oserved tht the proposed pproch effectively chieves to model the considered ECG ndwidth. For the signls of Fig. 3, the crosscorreltion coefficient is equl to.999. The difference etween the originl nd estimted ECG depicted in Fig. 3, therefore, corresponds to the non-modelled high-frequency ctivity ove 25 Hz. Also it is noticed tht incresing the numer of K modelled components, the MSE decreses (Fig. 4). The sme ehviour is noted when N is incresed, considering specific fk (Fig. 4). In this cse, however, the MSE stilizes round certin vlue (6 µv 2 ) fter the length of the signl reches certin numer of smples, s expected x 14 4 Originl nd estimted ECG signls Originl ECG signl Estimted ECG signl MSE (µv 2 ) Numer of modelled frequency components, K Difference etween originl nd estimted ECG signls MSE (µv 2 ) 15 1 f K = 25 Hz Signl length, N FIG. 3 (A) APPLICATION OF EQS. (1) (7) FOR MODELLING THE FREQUENCY COMPONENTS OF AN ECG EXCERPT AND THE RESPECTIVE ESTIMATED SIGNAL. K = 1 MODELLED FREQUENCY COMPONENTS OR SINUSOIDS (fk RANGING FROM TO 25 Hz) WERE TAKEN INTO ACCOUNT; (B) DIFFERENCE BETWEEN ORIGINAL AND ESTIMATED SIGNAL FIG. 4 MSE VARIATION FOR DIFFERENT VALUES OF: (A) k (RANGING FROM 1 TO 5; N = 8192); AND (B) N (fk = 25 Hz) Figs. 5 nd 6 depict the usge of the proposed method for crrying out the grdient rtefct wveform modelling nd susequent removl from the EEG signl, in ccordnce with Ferreir et l. (213). Fig. 5 depicts four representtive EEG excerpts (N = 496; fs = 76

5 Signl Processing Reserch Vol. 2 Iss. 4, Decemer Hz), electrode positions P3, CP1, FC5, nd F6 with grdient rtefct interference (verge round 15 μvpk-pk), recorded from three different sujects In Fig. 5, the respective low-pss filtered signls re depicted. As cn e oserved, considerle signl distortion is still oserved in these signls ecuse of the underlying rtefct interference (Nizy et l., 25). 2 Suject 1 - Electrode position P Suject 1 - Electrode position P Suject 1 - Electrode position P Suject 1 - Electrode position P FIG. 5 (A) EEG EXCERPTS WITH GRADIENT ARTEFACT INTERFERENCE; (B) EEG SIGNALS OF (A) AFTER APPLICATION OF A LOW-PASS FILTER: A CONSIDERABLE DISTORTION IS STILL OBSERVED IN SUCH SIGNALS BECAUSE THE UNDERLYING ARTEFACT COMPONENTS WHICH REMAIN IN THE EEG BANDWIDTH FIG. 6 (A) MODELLED UNDERLYING GRADIENT ARTEFACT OF THE SIGNALS OF FIG. 5B USING THE PROPOSED METHOD; (B) RESTORED EEG SIGNALS AFTER SUBTRACTION OF THE ARTEFACT WAVEFORM MODELS OF (A) FROM THE SIGNALS OF FIG. 5B Hence, we used Eqs. (1) (7) nd (13) to model nd sutrct the frequency components corresponding to the underlying rtefcts. According to Nizy et l. (25), these frequency components occurs in 77

6 Signl Processing Reserch Vol. 2 Iss. 4, Decemer 213 frequency ins, whose fundmentl re multiples of the inverse of the echo-plnr imging slice time (ST). Thus, Eq. (6) is chnged to: f k = k f s ST, (14) nd the frequency components which rise into the region of ±1 Hz round the fundmentl frequency were then modelled (Ferreir et l., 213; Nizy et l., 25). Fig. 6 shows the resulting modelled underlying rtefct for the EEG excerpts of Fig. 5. In turn, Fig. 6 depicts the restored EEG fter sutrction of the sumof-sinusoids models of Fig. 6 from the signls of Fig. 5. Finlly, in Fig. 7, the wveforms otined y verging the rtefct wveforms shown in Fig. 6 (segmented t ech slice time), nd estimted y the rtefct verge sutrction (AAS) method (Allen et l., 2) re depicted for comprison purposes. The respective slice time tken into ccount for verging is equl to ms (= 155 smples)(ferreir et l., 212). The verge cross-correltion clculted for these wveforms is equl to.996, considering ll nlysed EEG excerpts (Ferreir et l., 213). 2 1 Suject 1 - Electrode position P Smple Smple Smple 1-1 Our method Averge rtefct templte Smple FIG. 7 AVERAGE ARTEFACT WAVEFORMS OBTAINED BY AVERAGING THE WAVEFORMS DEPICTED IN FIG. 6A (RED TRACE), AND OBTAINED BY THE AAS METHOD (BLUE TRACE). THESE WAVEFORMS ARE QUITE SIMILAR, SHOWING THAT THE ARTEFACT VARIABILITY OVER THE TIME IS PREDICTED BY THE SUM-OF-SINUSOID MODELS DEPICTED IN FIG. 6A As cn e oserved in Figs. 6 nd 7, the proposed method predicts the vriility of the rtefct wveform over the time, unlike the verge templte estimted y using the AAS method (Ferreir et l., 213; Allen et l., 2). This chrcteristic could e useful for investigting nd understnding the nture nd morphology of such interference s well s estimting more ccurte rtefct wveform or templte, in order to improve the EEG restortion (Ferreir et l., 213; Allen et l., 2; Koskinen nd Vrtiinen, 29). As still oserved y Ferreir et l. (213), incresing of the length of the EEG excerpt results in pproximting the rtefct wveform represented y the sum-of-sinusoids model to the verge wveform templte estimted y the AAS method. Discussion As discussed y Brillinger (1987) nd Prsd et l. (28), the sum-of-sinusoids model cn e used to provide some very importnt solutions nd rel life pplictions in different res. For instnce, in udio nd speech processing, modelling sed upon the sum of sinusoids hs een widely employed ecuse of its simplicity nd flexiility of implementtion (Goodwin, 28; El-Ttr & Fokpu, 211). In this wy, y comining sum-of-sinusoids modelling with Discrete Fourier Trnsform properties, we hve mde use of the pproch descried in Eqs. (1) (7) to otin n nlyticl representtion of frequency components in physiologicl signls. As cn e oserved in Figs. 3, 6, nd 7, frequency components of the ECG nd grdient rtefcts in EEG excerpts cn e ccurtely modelled using the proposed methodology. For the ECG signls shown in Fig. 3, the cross-correltion etween the originl nd estimted signls is equl to.999 (verge equl to.996), indicting strong frequency ctivity for the modelled ndwidth ( 25 Hz). Figs. 3 nd 4 confirm these results. As depicted in Fig. 4, low MSE is oserved etween originl nd estimted signls (t 6 µv 2 ), for signl length higher thn 8192 smples. In the sme wy, s shown in Figs. 5, 6, nd 7, the proposed pproch chieves n ccurte representtion of the underlying grdient rtefct wveform (Ferreir et l., 213). Therey, such n rtefct cn e stisfctorily removed from the EEG signl (Fig. 6). Moreover, Fig. 7 revels tht the rtefct wveforms depicted in Fig. 6, verged t ech echo-plnr imging slice time, re quite similr to the rtefct templtes estimted y the estlished AAS methodology (Allen et l., 2; Ferreir et l., 213). The verge cross-correltion clculted etween these verge rtefct wveforms is equl to 78

7 Signl Processing Reserch Vol. 2 Iss. 4, Decemer s well. This fct demonstrtes tht the rtefct vriility over the time is predicted y the sum-ofsinusoid models (Ferreir et l., 213). Therefore, such fetures llow the usge of the proposed sum-of-sinusoids modelling pproch for n effective representtion nd extrction of frequency components in ECG signls nd grdient rtefcts in EEG signls. In future work, such method shll e pplied nd ssessed for other types of iologicl signls nd other types of rtefct. Conclusions In this work, we hve mde use of filter technique sed upon the sum of set of sinusoidl wveforms to model nd extrct signl frequency components in physiologicl signls. According to our modelling pproch, ech frequency component is pproximted s sinusoid, whose mplitude nd phse re estimted y mking use of the Discrete Fourier Trnsform mgnitude nd phse prmeters. The proposed methodology shows to chieve n ccurte representtion of signl components in ECG signls nd underlying grdient rtefcts in EEG signls. The sutrction of the sum-of-sinusoids model still llows stisfctory removl of the underlying grdient rtefct from the EEG signl. Therefore, such chrcteristics enle its use s n effective pproch for extrction nd filtering of frequency components of interest in iologicl signls. ACKNOWLEDGMENT We re grteful to Sski vn Liempt, M. D., nd Col. Eric Vermetten, M. D., Ph.D. from the University Medicl Center /Centrl Militry Hospitl, Utrecht, for providing the dt presented in this work. This work hs een mde possile y grnt from the Europen Union nd Ersmus Mundus EBW II Project, nd y grnt from CNPq Science without Borders Progrm. REFERENCES Allen, P., Polizzi, G., Krkow, K., Fish, D., Lemieux, L. Identifiction of EEG events in the MR scnner: the prolem of pulse rtefct nd method for its sutrction. NeuroImge8 (1998): Allen, P., Josephs, O., Turner, R. A method for removing imging rtifct from continuous EEG recorded during functionl MRI. NeuroImge 12 (2): Brillinger, D. Fitting cosines: Some procedures nd some physicl exmples. In Applied proility, stochstic process nd smpling theory, edited y I.McNeill nd J. Umphrey, Dordrecht: Reidel, El-Ttr, A., Fokpu, O. Modeling MR induced rtifcts contminting electrophysiologicl signls recorded during MRI. 33 rd Annul Interntionl Conference of the IEEE EMBS, Boston, Msschusetts, August 3 Septemer 3, 211, Proceedings: , 211. Ferreir, J., Cluitmns, P., Arts, R.M. Grdient rtefct correction in the EEG signl recorded within the fmri scnner. 5 th Interntionl Conference on Bio-inspired Systems nd Signl Processing, BIOSIGNALS 212, Vilmour, Portugl, Ferury 1-4, 212, Proceedings: , 212. Ferreir, J., Cluitmns, P., Arts, R.M. Grdient rtefct modelling using set of sinusoidl wveforms for EEG correction during continuous fmri. Signl Processing Reserch 2 (213): Goodwin, M. The STFT, sinusoidl models, nd speech modifiction. In Spring hndook of speech processing, edited y J. Benesty, M. Sondhi, Y. Hung, New York: Springer, 28. Ky, S. Fundmentls of sttisticl signl processing: estimtion theory. Upper Sddle River: Prentice Hll, Koskinen, M., Vrtiinen, N. Removl of imging rtifcts in EEG during simultneous EEG/fMRI recording: reconstruction of high-precision rtifct templte. NeuroImge 46 (29): McAuly, R., Qutieri, T. Speech nlysis/synthesis sed on sinusoidl representtion. IEEE Trnsctions on Acoustics, Speech, nd Signl Processing 34 (1986): Nizy, R., Beckmnn, C., Innetti, G., Brdy, J., Smith, S. Removl of FMRI environment rtifcts from EEG dt using optiml sis sets. NeuroImge 28 (25): Olson, W. Bsic concepts of medicl instrumenttion. In Medicl instrumenttion: ppliction nd design, edited y J. Wester, th ed. New York: Wiley, 21. Oppenheim, A., Schfer, R. Discrete-Time Signl Processing. 2 th ed. New Jersey: Prentice-Hll, Ppoulis, A.; Pilli, S. Proility, rndom vriles, nd 79

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