Performance of short-time spectral parametric methods for reducing the variance of the Doppler...

7/29/2019 Performance of short-time spectral parametric methods for reducing the variance of the Doppler ultrasound mean

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Performance of short-time spectralparametric methods for reducing thevariance of the Doppler ultrasoundmean instantaneous frequencyestimation

H. Sava I L.-G. Dur and 1"2 G. Clou tier 1"31 Laboratory of Biomedical Engineering, IRCM, Montr6al, Qu6bec, Canada2 Departme nt of M edicine, University of Montreal, Montr6al, Qu6bec, Canada3 Departments of Physiology and Rad iology, University of Montreal, Montr6al, Qu6bec, Canada

A b s t r a c t - - T o achieve an accurate estimation of the instantaneous turbulent velocityf luctuat ions downstream of prosthet ic heart valves i n v i v o , the variability of thespec tra/m ethod used to measure the mean f requency shif t o f the Doppler signal ( i.e.the Doppler velocity) should be minimised. This paper investigates the performanceof various short-t ime spectra/ parametric methods such as the short-t ime Fouriertransform, autoregressive modelling based on two different approaches, autoregres-sive moving average mode/ring based on the Steig/itz-McBride method, and Prony'sspectra/method. A simulated Doppler signal was used to evaluate the performanceof the above ment ioned s pectra/me thods and Gaussian noise was added to obtain aset of signals with various signal-to-noise ratios. Two different paramete rs were usedto evaluate the performance of each method in terms of variability and accuratematc hing o f the theoretical Dopp ler mean instantaneous freque ncy variation w ithinthe cardiac cycle. Results show that autoregressive mode/ring outperforms the otherinvest igated spectra/ techniques for w indo w lengths varying between 1 and tOms.Am ong the autoregressive a lgor i thms implemented, i t is shown that the maxim umentropy method based on a block data processing technique gives the best resultsfor a signal-to-noise ratio o f 20 dB. However, at 10 and 0 dB, the L evins on-Du rbinalgor i thm surpasses the performance o f the maxim um entropy m ethod. I t isexpected that the intrinsic variance of the spectra/ methods can be an importantsource of error for the estimation of the turbulence intensity. The range of this errorvar ies f rom 0.38% to 24% depending on the parameters of the spectra/method andthe signal-to-noise ratio.Keywords--Dopp/er u l t rasound, Turbulence est imat ion, Spectra/analysis techniques,Heart valve diseases, M ean veloc ity estimationMed. Biol. En g. Comput., 1999, 37, 291-297

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1 Introduct ionTURBULENCE IS an irregula r edd ying moti on in whic h velo cityand pressure fluctuations occu r about their mean values. T heseperturbations are known to be irregular or random in time andspace, and may be composed of many smal l eddies or largervortices (STREETERand WYLIE, 1985). Turbulent flow motionsare usually decomposed into their mean and fluctuating parts.Formally, the presence of turbulence necessarily involvesthree-dimensional random velocities. However, in practice,the verification of the existence of random fluctuations in one

Correspondence should be addressed to Dr. G. Cloutier;email:, [email protected] received 27 Ma rch 1998 and in final form 21 January 19999 IFMBE:1999Medical & Biological Engineering & Computing

plane only is sufficient to confirm the presence of turbulence(STREETER and WYL IE, 1985). The intensity of the veloc ityfluctuations is an important parameter used to characteriseturbulence. This index is obtained by dividing the root-mean-square of the time-varying velocity fluctuations by the meanvelocity averaged in tim e and /or space (STREETER and WYLIE,1985; CLOUTIERet al. , 1996).To achieve an accurate estimation of the turbulence dow n-stream of prosthetic heart valves in vivo, a reliable methodshould first be established. In the past, the ho t-f il l anemom etrymethod was used to measure turbulence in pigs and humans(HASENKAM et al. , 1988; NYGAARD e t aL , 1992). However,this approach requires high technical skills and can beperform ed only during o pen heart surgery. In addition, calibra-tion problem s due to fibrin deposition on the heated sensor is amaj or drawback. Other techniques, such a s laser-Doppleranemometry and particle image velocimetry are limited to in

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v i t r o evaluat ion and t rans lucent f luids . Recent ly, Dopplerul t rasound has been used to quant i ta t ively evaluate the turbu-lence p roduced downs t ream of na t ive and pros the t i c hea r tvalves i n v i v o (NYGAARD e t a l . , 1994a, 1994b).The accurate evaluat ion of turbulence is important , espe-cia l ly in the case of mechanical pros thet ic heart valves as thecharacteris tics of the f low through the va lve have b eenassociated with bloo d cel l t rauma (JON ES, 1995).Downs t ream of obs t ruc ted m echanica l p ros the t i c hea r tvalves , high turbulence intens it ies are expected because o fthe perturbing influence of the leafle ts on the f low dynamics ,and the p resence o f m uch h ighe r R eynolds num bers in th i sregion of the cardiovascular sys tem. Physiological ly, thecharacterisa t ion of turbulence is of re levance because of i tsassociat ion with red blood cel l lys is and thrombus formation(SCHMID-SCH()NBEINet a L , 1981; WURZINGER an d SCHMID-SCHONBEIN, 1987). Fu rtherm ore, turbu lence is also associatedwi th the l ack o f p res sure recovery obse rved downs t ream o fmechanical heart valves (VANDERVOORT et a L , 1995;NIEDERBERGER e t a L , 1996).In a recent s tudy by our group (CLOUTIER e t a l . , 1996),b lood f low tu rbu lence a t s eve ra l pos i t ions downs t ream of a86% area reduct ion s tenosis was characterised us ing param-eters extracted from the Doppler spectrum. In that s tudy,turbulence was quantif ied by computing the re la t ive temporalDoppler veloci ty f luctuat ions and the Doppler backscat teredpower us ing au toregres sive m ode l l ing o f 2 m s Dopple r s ignalsegments. In C lout ier ' s s tudy, a l though the re la t ive changes inturbulence intens i ty were correct ly mapped, unexpected highva lues were m easured com pared to ho t - f i lm anem om et ry .Among the reasons explaining these high values , the variancein t ime associated with the es t imator of the ins tantaneousDop pler freq uency shi t~ ma y have contributed s ignif icantly.The accuracy o f indices extracted from the Dop pler s ignal ma ybe affected by the bias and va riance of the es t imator, thecompatibi l i ty of the spectra l m etho d with the s ignal, the s ignal-to-noise ra t io (SNR), and by parameters such as data window,m ode l o rde r o r sm ooth ing window type .Although several s tudies have been carried out on theDoppler s ignal to inves t igate the performance of differentspectral analysis meth ods (KALUZYNSrd, 1987; TALHAMI andKITNEY , 1988; VAITKUS e t a L , 1988; SCHLrNDWEIN andEVANS, 1990; AHN and PARK, 1991; DAVID e t a L , 1991;FORSBERG, 1991; ZEIRA e t a l . , 1994; GUO e t a L , 1994;WILLINK and EVANS, 1995; KEETON e t a l . , 1997), little efforthas been m ade to s tudy these m e thods in the con tex t o fturbulence veloci ty f luctuat ion measurements . The object iveof this pap er is to inves t igate the perform ance of variousspectra l es t imation techniques in terms of their specif icvariance and bias wh en app lied to the es timation of theDopple r m ean ins tan taneous f requency , which i s used toextract the f low veloc i ty as a func t ion of time. The im pact ofvarious parameters such as the model order, window length andSNR is inves t igated for the fol lowing techniques : short- t imeFourier t ransform (STFT), autoregress ive modell ing (AR)based on two different approaches , autoregress ive movingaverage m ode l l ing (ARMA) based on the S te ig l i t z -McBridemethod, and Pr ony ' s spectra l metho d. The variance and bias o feach es t imator were assessed us ing a s imulated Do ppler s ignalrepresent ing an aort ic f low waveform.

2 S p e c t r a l a n a l y s is m e t h o d sTradit ional ly, the t ime -freq uen cy analys is of the Doppler

s ignal is performed us ing the STFT, in which the s ignal isdivided into small sequentia l or overlapping segments and thefas t Fourier t ransform (FFT) appl ied to each segment .292

How ever , the jo in t t im e-f requ ency re so lu t ion o f the S TF T i sinherent ly l imited due to the un certa inty principle , which s tatesthat the analys is resolut ion in t ime and frequency cannot bearbi t rarily small b ecause the prod uct o f these resolut ions islower bounded (ZEIRA e t a l . , 1994).In an a t tempt to a l levia te the inheren t l imitat ions of the FFTapproach , o the r a l t e rna t ive t im e-f requ ency es t im a t ion proce -dures have been prop osed. Th ese a l ternat ives fa l l into threecategories: (a) short- time parame tric methods; (b) bi l ineart im e-f requen cy m e thods ; and (c ) w ave le t t rans form s . W i th ineach o f these ca tegor ie s one can im plem ent d i f fe ren t m e thodsin accordance with the specif ic parameters of interes t and thesignal characteristics. Th is pape r focuse s only on short-timep a r a m e t r i c m ethods which a s sum e a gene ra ting m ode l fo r theprocess . This parametric appro ach to spectra l es t imation can bedivided into three s teps . In s tep o ne, an appropria te parametrict ime-series model is se lected to represent the bas ic character-ist ics of the experim ental signal . In s tep two, an es t imate o f theparameters o f the mo del is com puted. In the f inal s tep, thees t imated parameters are inserted into the theoret ical powerspectra l dens i ty express ion approp ria te for that model . B ecauseof their autocorrela t ion extens ion property, short- t ime para-m e t r i c m e thods can use shor te r windows than the S TF Twi thout unduly com prom is ing the i r f requency re so lu t ion .However, short- t ime parametric methods are sens i t ive to poormo del f i t and non-s ta t ionari ty within the short s ignal segm ents(KAY and MARPLE Jr ., 1981). As a l l the im plemented para-metric methods used in this paper are well es tabl ished in thel itera ture , only a b rief descript ion of these m ethods is presentedbe low.A R a n d A R M A m o d e l l i n g : The gene ra l ARMA m ode l , a l soknow n as po le -ze r o m ode l l ing , a s sum es tha t a tim e se r ies x [ n ]can be m od e l led a s the ou tpu t o f a f i lt e r con ta in ingp po les andq zeros :

P qx [ n ] = - y ~ a [ i ] x [ n - i ] + ~ , b [ i ] w [ n - i ] ( I )i=1 i=1

exci ted by a zero-mean, uni t -variance, uncorrela ted randomdata sequence ( i.e. norm alised whi te noise) w [ n ] which is takento be un observable . In terms o f sys tem identif ication, theARMA m ode l i s cha rac te r i s ed by a ra t iona l sys tem func t ionof the fo rm :q b [ i ] z - iB ( z ) _ i =o (2 )H ( z ) - - - - pA ( z ) ~ a [ i l z_ i

i= 0where A ( z ) a n d B ( z ) represen t the z - t rans form of the au to-regress ive (AR) and m o ving-ave rage (MA ) branch of theARM A m ode l wi th a [0] = 1 . Hav ing e s t im a ted the sys temparameters a [ i ] a n d b [ i ] , one can then e s t im a te the powerdensi ty funct ion as :

~ S ( f k) = 1 1 3 (e xp ( - -2 f i r f k ) )1 2 (3 )[ f 4 ( e x p ( - - Z f i Z f k ) ) l 2

where the hat notat ion, , represen ts an es t imator operatorra ther than the real value, a n d f k i s the f requency cor re spondingto the spectra l index k.A R m ode l l ing can be cons ide red a s a spec ia l case o f theAR MA process . I t is one o f the m os t wide ly used approachesdue to i ts assumption that the s ignal comprises a se t ofresonances or s inusoids , which is an appropria te model forma ny practica l signals (KAY and M ARPLE, Jr., 1981), includin gthe Dopp ler ul t rasound s ignal . Different a lgori thms wereproposed to es t imate the autoregress ive process parameters .

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In this paper, two of them are implemented; the maximumentropy method (M EM) in which the autoregressive parametersare found via a block solution of the Yule-Walker equations,and linear prediction coding based on the Levinson-Durbinalgorithm (LAR). Bo th of these algorithms are described indetail by MARPLE (1987).The ARMA Steiglitz-McBride method, which is an iterativealgorithm for computing the model parameters of eqn. 3, wasimplemented. Th e full description o f this method with itsadvantages and drawbacks can be found elsewhere(STEIGLITZ and MCBPdDE, 1965; JOO et aL, 1983). Due tothe nature of the Doppler signal, the MA model order of theSteiglitz -McB ride alg orithm was set to zero (i.e. b[0] = 1 andb[i] = 0 f or i S 0). Th is is justified b y the fact that thesimulated signal is a sum of sinusoidal components whichshow only peaks in the corresponding power spectrum.Prony 's method: This method seeks to fit an exponentiallydecaying sinusoid model to the data, in contrast to AR andAR MA methods that seek to fit a random mod el to the second-order data statistics. The mo del assum ed in Pron y's method is aset of p exponentials of arbitrary amplitude, phase, frequency,and dam ping factor (MARPLE, 1987). The discrete-time fu nc-tion is described by:

px [ n l = ~ h [ m l z [ m ] "m=l n = 0 , 1 , 2 . . . . . N - 1 (4)where

him] = A[m] exp(jqSim])an d

z[m] = exp[(d[m] + j2r~f[m])At]In eqn. 4, A[m] is the am plitude, qS[m] is the phas e in radians,d[m] is the damping factor, f [m ] is the frequency in hertz, andAt represents the sample interval in seconds.Although Prony's method normally terminates with thecomputation of the above mentioned parameters, i t is possibleto calculate the spectrum base d on the following form ula (KAYand MARPLE Jr., 1981):

'~[A] = IS[.fk][ 2 (5)where

pX[fk] = ~ A[m] exp(j~b[m])m= l2d[m]x [d[m] 2 + (2re[/"k - - f [ m]] ) 2 ]

3 Simulated Doppler signalTo co mpare the perfo rmanc e of the different spectraltechniques, a synthesised signal is required which can beused as a reference signal. In this paper, a Doppler signalsimilar to one obtained downstream of an aortic heart valve issimulated. Fig. 1 shows Doppler flow waveform s recordeddownstream of a St-Jude aortic heart valve installed in a flowphantom (DURAND et al., 1998). The Doppler signal wassynthesised using an approach similar to that described by

MO and COBBOLD (1986, 1989) to ma tch the spe ctral char-acteristics of Fig. I. The metho d is briefly summarised belowas it constitutes the basis for the spectral algorithm compar-Medical & Biological Engineering & Computing

Fig. 1 Experimenta l short-time Fourier freque ncy distribution ofDoppler signals recorded downstream (four diameters) of aSt-Jude m echanica l aortic heart valve. The 23 mm valve wasinstalled in a flo w phan tom described in DURAND et aL(1998), and measurem ents w ere perfor med at a mean flo wrate o f 4.51/min

isons. The bidirectional Dopple r signal was simulated as a sumof random phase sinusoids covering the frequency range[O,fmax ], as described by Guo et al. (1994).

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Ms(t) = ~ a m exp[j(2rtfmt + ~bm) (6a)m= lw h e r e

an d

M = 2fm~xTa m = x / Z S( t , f m ) A f y m

fm = (m - 0 . 5 ) A f

293

where fl = 1.67 Hz and fmax is in kilohertz. In the aboveequation, all frequencies belo w 500 Hz are remov ed to simulatethe highpass wall fi l ter used during clinical exams.Although both in-phase and quadrature compone nts of theDopp ler signals were generated, o nly a portion o f the in-phasecardiac cycle signal was used in this study to investigate theperformance of the different spectral estimation techniques.Thi s cho ice is justified by the f act that the statistical character-istics of the in-phase an d quad rature signals are the same. Inaddition, by analy sing the signal onl y within a predefined timeinterval, the computation time of the overall process issubstantially reduced. As the maximum turbulence occursaround the maximum systolic pressure (SHEN, 1961;GERRARD, 1971), this p art of the flow signa l was selected.Fig. 2 shows the normalised theoretical t ime-fre quenc y distri-bution o f the simulated Do ppler signal. This bias and variance

{ . 5 s i n ( 2 r ~ f ~ t ) + l 0 ~ < t ~ < 0 . 3 sfmax = 0 0.3 S < t ~< 1 S (6b)

S( t , f m ) is the time-varying theoretical power spectral densityfunction defined over the range [0,fmax], Af = f m a x / M is anincremental frequency domain step, Ym is a z-squared randomvariable with two degrees of freedom, and ~bm is a randomphase uniform ly distributed ov er [0, 2n]. T he sinusoidal ampli-tudes a m are Rayleigh distributed, and the num ber o f sinusoidsM is determined by the signal duration T.In this study, a D oppler signal o f 1 s was simulatedcontaining positive frequencies during the systole and nullvalues during the diastole. The m axim um frequency was seta t 4.5kHz and the sampling frequency at 20kHz. Themaximum frequency waveform of the s imulated s ignal wasdefined a s :


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of the spectral methods to estimate the mean instantaneousfrequency were evaluated within the time window includedbetween 120 and 180ms.To inves t igate the performance of each method, SN R of20dB, 10dB and 0dB were used. The noise levels were notselected to mimic different turbulence intensities but to test thestability of the spectral techniques to estimate the instantaneousmean frequency. Laminar flow is considered in these simula-tions. I f a spectral method cannot p rovide a low variance o f themean frequency when no turbulence is present, i t is very likelythat this method will perform poorly in situations wherefrequency fluctuations (turbulence) exist. For each of thespecified SNR, 20 different realisations were generated toevaluate the variance of the estimate for each method. Thedesired SNR were obtained by adding white Gaussian noise tothe simulated D oppler signal (ZEIRA et aL, 1994).

4 Assessment of the performance of spectralestimatorsThe perform ance of each technique relies on the accuracy ofthe assessment of the mean frequen cy parameter. The instanta-neous mean frequency waveform was estimated according tothe following equation:

N- I

?mean[n]_ k=0 (7)N- I ^Z sin, k]k=0

where n and k correspond to the discrete time and frequencyvariables, respectively. The parameter S represents the powerof the signal computed from the different algorithms, an dfk isthe frequency corresponding to the spectral index k. The fmeanwas estimated for each of the 20 realisations and for each SNR.Then, the mean and standard deviation o f this parameter wereevaluated. In order to q uantify the variability of each estimate,a variance coefficient (i.e. Var-CoejO defined as:Var - Co ef = 1 ~_, std(?mean[rl]) (8 )U~ ,[),.,o.[n]]

was calculated for each method. In eqn. 8, s t d ( ) . . . . [n])represents the standard deviation of a specific methodtowards estimation o f the insta!ataneous mean frequency ateach data sample position n, #[f,,ea,[n]] is the mean value ofthe estimated instantaneous mean frequency of that methodobtained by averaging the corresponding 20 mean frequencyvalues of the Doppler signal, and N is the numb er of spectral

lines in the time window centred around the peak systole.Although expressed differently, the Var-Coef is equivalent tothe relative turbulent velocity fluctuation index defined inCLOUTIERet al. (1996).In addition, the normalised mean-square-error (NMSE)between the estimated and theoretical mean frequency wasmeasured.N M S E = 1 n=l-~(#[fmean[n]]f~mean[n-fmean[n])2 (9)

Of course , the theoret ical mean frequency waveform can beestimated from the theoretical power spectrum density of thesimulated ~ignal using eqn. 7. However, bearing in mind thatthe theoretical power spectral density function underlying theDop pler signa l is given b y (MO and COBBOLD, 1989):S( t , f ) = A ( f m ~ _f )2 exp[_B(fmax _f )2 ] (10)

one can estimate the theoretical instantaneous mean frequencywaveform analytically by using,fm~ , ,~ fm~O

fmean = I 2I fmax + ~-- ~ fmax< 0where fmax, A, and B are time-varying parameters repre-senting, respectively, the max im um frequency of the Dopp lerspectrum, the power scaling factor, and the bandwidth factor.In this study, A and B had values obtained by the proceduresdescribed in Guo et al. (1994).

5 Results and discussionFig. 3 depicts the analyticalfmea . values for the portion o f thecardiac cycle investigated in this paper (between 120 and180 ms). In all the abo ve m ention ed parametric methods, the

model order (m) investigated was varied from 8 to 40. Lowermodel orders were not used as the t ime-varying number ofsinusoids (i .e. M) used to generate the Doppler signal at eachmoment in time for the selected part of the cardiac cycle isbetween six and nine, as determined by eqns. 6a and 6b. On theother hand, higher model orders are not recommended for theapplication in hand (SCHLINDWEINan d EVANS,1990; FISHet al., 1997) due to the nature of the m ean frequen cy parameter.A Hanning window was used for each window length as it hasbeen show n that this type of win dow is more suitable when theinstantaneous mean frequency is the parameter under investi-gation (WANG and FISH, 1996). Th e w indow length was rang edfrom 1-10ms as short windows are required to improve the

Fig. 2

294Theoretical time-frequency distribution o f the simulated Doppler signal. The segment on which the spectral techniques were tested isbetween 120 and 180ms

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Fig. 3 Theoretical time-varying mean freque ncy or the seIected pa rtof the simulated signal

t ime resolution of turbulence estimators, and longer windowsimprove the frequency resolution but emphasise the change innon-stationary signal characteristics (CARDOSO t al., 1996). AW - 1 point overlapping window was applied from onesegment to the next, where W is the numb er of samplepoints in each window length.Fig. 4 shows the perform ance of the parametric methodsinvestigated in this paper for the case corresponding toSN R= 0 dB. T he characteristics o f the corresponding graphsfor other SNR ratios are very similar to those shown in Fig. 4,apart from the fact that the performance of each method isbetter when the SNR is high. From Fig. 4, it is clear that thewindow length is the paramete r that has the strongest effect onthe performance of the time-frequency distribution of al lmethods. With the exception of the ARM A method , themodel order seems to have a minor impact on the Var -Coe fan d N M S E . This is due to the fact that, for the purpose of thisstudy, we w ere interested only in the estimation of theinstantaneous m ean frequ ency and not in the spectral resolutionor accurate estimation of frequency components within thecorresponding spectrum. In this sense, results presented in thispaper agree with previous wo rks which have sug gested that themodel order is not such a crucial parameter when estimatingthe instantaneous mea n frequ ency of the Doppler signal(SCHLrNDWEN and EVANS, 1990). It see ms that a mo delorder in the range 8 -12 would give the best results. Byincreasing the window length the performance improvessubstantially for each method in terms of the Var-Coef andNMSE.Fig. 5 depicts the performance of each method for a modelorder of 10 and that of the STFT using different windo wlengths and SNR s. O nly the results of the Var-Coef arepresented since it is this index that has to be minimised toobtain a reliable estimator of the mean frequency. Ho wever, i tmust be ment ioned that the behaviour of the N M S E is verysimilar to that o f the V a r - C o e f .Results presented in this figure show that algorithms basedon the AR model outperform all the others, and the ARMAmethod gives the worst results. This can be attributed to thefact that true AR algorithms reflect more closely the nature ofthe simulated signal than the other methods. For the iterativeSteiglitz-McBride ARMA method, although it was modified toan AR model, i t does not seem to be well adapted for theDoppler signal. Among the two AR algorithms, the MEMseems to perform better than the LAR method at 20 dB SNRs.However, for lower SNRs (SNR = 10 and 0 dB) the perfor-mance of LAR is superior. These results can be explained bythe fact that AR modelling performs better under a GaussianMedical & Biological Engineering & Computing

Fig. 4 Performance of various short-time spectral parametric meth-ods for SNR= 0 dB an d varying w indow length and modelorder

random process which is almost always the case for theDoppler signal (FONTArNE et al., 1997). It can be observedfrom the results of Fig. 5 that the performan ce of all themethods improves or reaches a plateau as the window lengthincreases. This suggests that a w indow length of at least 10 msmust be used when the estimation of large-scale turbulence isthe primary concern under investigation. Howeve r, there mightbe other cases when a shorter window is needed to track downfaster fluctuations of the Doppler signal attributed to small-scale turbulent m otion s (TRITTON, 1988). In these circu m-stances, our work suggests that use of the LAP, is the bestoption amo ng the short-time spectral parametric methods testedwhen the signal is corrupted with noise.

6 ConclusionsSimulated Doppler signals have been used to compare theperformance of a number of parametric spectral analysismethods regarding the va riance o f the instantaneous meanfrequency estimation in terms of the window length, modelorder and signal-to-noise ratio. Results show that overall ,autoregressive modelling based on the LAR method givesbetter performance than the other methods for signal-to-noiseratios between 0 and 10 dB, Howev er, for a higher SNR of20 dB, the performance of ME M and LAR methods is compar-able, with the MEM having the upper hand. Results also

showed that the length of the window is the most importantfactor in terms of accurate estimation of the mean frequency ofthe Doppler signal. It is suggested that a duration of at least1999, Vol. 37 295


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. t 0

N t2o

2 4 1 0 d B S N R20

t2 . . . .

0 2 4 6 8 1 0window ength, ms

ST F T method

LAR method5

MEM method

PFIONY methodt 5 B SNR12 ~ d96

ARMAmethod12

0 2 4 6 8 1 0window ength, ms

2.01.81.01.8

1.20.90.6 . . . .1.50.90.80.3 . . . .

3.02.52.0

0 2 4 6 8 1 0window ength, msFig . 5 Per formance o f t he ST FT and var ious spect ra l paramet r i cmethods fo r di f ferent SNR s and window lengths. For para-metr ic methods , a model order of 10 was selected

1 0 m s i s r e q u i r e d t o s i g n i f i c a n t l y r e d u c e t h e v a r i a n c e o f t h es p e c t r a l e s t i m a t o r o n t h e o v e r a l l r e s u l t s . H o w e v e r , s i n c e i t i si m p o r t a n t t o h a v e g o o d t i m e r e s o l u t i o n t o t r a c k ra p i d t u r b u l e n tm o t i o n s , i t c o u l d b e o f i n t e r e s t t o i n v e s t i g a t e a l s o t h e p e r f o r -m a n c e o f j o i n t t i m e - f r e q u e n c y d i s t r ib u t i o n m e t h o d s o r w a v e l e tt r a n s fo r m s . A s f o r t h e o r d e r o f t h e m o d e l , i t h a s b e e n s h o w nt h a t i t d o e s n o t p l a y a s i g n i f i c a n t r o l e i n t h e v a r i a n c e o f th ee s t i m a t o r o r t h e N M S E p a r a m e t e r .A c k n o w l e d g m e n t s - - T h e a u t h o r s w o u l d l ik e t o t h a n k t h e N a t u r a lS c ie n c es a n d E n g i n e e r in g R e s e a r c h C o u n c i l o f C a n a d a f o r s u p -p o r t i n g t h is p r o j e c t ( g r a n t # O G P I N 3 35 ), a n d M r s L o u i s A l l a r da n d D a m i e n G a r c i a f o r p r o v i d i n g t e c h n i c a l a s s i s t a n c e . T h e f i r s ta u t h o r ( H . S . ) w a s s u p p o r t e d b y a r e s e a r c h f e l l o w s h ip f r o m t h eH e a r t a n d S t r o k e F o u n d a t i o n o f C a n a d a .

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Author's biographyGuY CLOUTIER obtained his Phi) in BiomedicalEngineering from the l~cole Polytechnique ofMontreal in 1990. Between 1990 and 1992, hepursued his post-doctoral training with Dr. K.Kirk Shung at the Laboratory of Medical Ultra-sonics, Bioengineering Program, PennState Uni-versity. Dr. Cloutier is currently Director of theLaboratory of Biorheology and ExperimentalUltrasonography at the Research Center of theUniversity of Montreal Hospital, Visiting Scientist at the Laboratory ofBiomedical Engineering of the Clinical Research Institute of Montreal,

and Associate Professor of Research at the Institute of BiomedicalEngineering, Departments of Physiology and Radiology, University ofMontreal. His research interests include the characterisation of redblood cell aggregation dynamics with ultrasound, the 3D morphologicand haemodynamic assessment of lower limb arterial stenoses withDoppler ultrasound, and the quantification of the effect of blood flowturbulence and red cell haemolysis on the dysfunction of prostheticheart valves. Dr. Cloutier is recipient of a research scholarship fromthe Fonds de la Recherche en Sant6 du Qurbec.

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