Flow Velocity Profile via Time-domain Correlation Error Analysis And Computer Simulation

8/8/2019 Flow Velocity Profile via Time-domain Correlation Error Analysis And Computer Simulation

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Flow Velocity Profile via Time-Domain Correlation:Error Analysis and Computer SimulationS T E V E N G . FOSTER .P A U L M . E M B R E E . M E M H I : K , I I - . I . . ~ ; . , AN I )W I L L I A MD .O' B R IEN. J R . . F E L L ~ W . t x t

Abstract-An ultras onic flow veloci typrofi lemeasu remen tme th o demp lo y in g t ime-d o ma in co r re la t io n o f co n secu t i \ e ech o p a i r s h a s b eend ev e lo p ed . Th e t ime sh i f t b e tween a p a i r o f r an g e g a ted ech o es i s es -t ima ted by sea rch in g fo r th e sh i f t th a t r e su l t s in th e max im u m co r re -la t io n . Th is ech n iq u e s fu n d amen ta l ly d i f fe ren t h an h e au to co r re -la t io n tech n iq u es th a t d o n o t s ea rch fo r th e max imu m co r re la t io n . Th et ime sh i f t in d icate s th e d i s tan ce a g ro u p of s ca t te re r s h a s mo v e d f ro mwhich f low velocity s es t ima ted . The basis for he compu ter s imula -t io n s an d e r ro r an a ly se s o f th e sch eme in c lu d es a h an d p assed wh i teGaussian noise s ignal model for an echo from a scatter ing medium, thees t ima te o f f lo w v eloc it? f ro m h o th a s in g le sc a t te re r an d m u l t ip le sc a t -terers , and a derive d precis ion est im ation . The err or anal!


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FOSTER (/ < I / : F L O W VELOCITY P R O F I L E V I A T I M E - D O M A I N CORRELATIOU I 6 5

termined from the time-domain correlation technique forboth a single scatterer and multiple scatterers. In this sec-tion. the theoretical precision of he ime-domain corre-lation technique will also be evaluated.A. Signal Mode l ( , f a n E cho f r o m a Scattering Medium

The signal model for an ultrasonic echo from a scatter-ing medium w ith a large number of scatterers per wave-length (such as blood) is modeled as band passed whiteGaussian n oise (BPW GN ). Based upon Angelsens model[20]where there are a large number of scatterersn a smallvolu me , the reflection coefficient of each particle can bemodeled approximately as a Gaussian ndependent vari-able. Therefo re, the ultrasonic echoes are assum ed to beindependent and G aussian distributed. The received sig-nal is considered to be bandlimited since the exciting ul-trasonic pulse is bandlimited. There is also some bandlim-iting due to the frequency response of the receive amplifierand the nonze ro size of the scatterer, although these ef-fects are small compared tohe band-limiting of the trans-ducer.The signal at he receiving ransduc er consists of thesum o f all the individual echoes from each of the acousticparticles,delayed by the ound -trip imebetween hetransducer and particle. The mathem atical representationof the received electrical signal is given by

where A ( R ) is the andom eflectioncoefficient (zeromean, variance 0 2 ) . ( R , t - 2 1 R / c ) is the s ystem im-pulse response and c is the speed of so und in the medium.The system impulse response represents the echo from theparticle dV positioned at i? with a reflection coefficient ofunity. Since the system is assumed to be linear and band-limited, it is ch aracterized by a position dependent ultra-sonic impulse response.B. VelocityDetermination:SingleScatterer

The ssence ofhe ime-domain flow measurementtechnique, schematically show n in Fig. 1 , is to track scat-terers in a range cell within an ultrasonic beam and assessthe distance raveled by the ime shift between echoes.The change in range (from po sition 1 to 2 in Fig. 1) of asinglescatterersecho is found by acorrelationproce-dure. Th e range of the fi rst echo wavefo rm, w I I ) , esultsfrom an ultrasonic pulse transmitted at r = to . At a latertime to + T , a second ultrasonic pulse is transmitted andthe second echo waveform, W? ( t ) . s received, but shiftedin time by t , + T from the first ech o. In effec t, the time-axis origin is reset since the T term is ignored. The secondecho is shifted by t , from the first. This approach is car-ried-out throughout the paper.The correlation function of the two echoes is defined byR ( r ) = W , t )W?( + 7 ) d t. ( 2 )

Fig. I . Schematic epresentation of soundbeam elatlve o hedirectionof flow of scatterers. Here. he initial position 1 of Fcatterer is shifteddistance V T o position 2 . Angle 0 is me asurement (Doppler) angle andL is range from t ransducer.

The two echoes are identical except for the secondbeing time delayed from the first, and expressed aswz(r)= W, ( t - t , )

Hence ,R ( r ) = wl( t ) \ t j I ( t - t , + 7 ) t = R11(7 -

where R , (7- , ) is the autocorrelation function of I ( tan dmaximalwhen heargumentof heautocorrelationfunction is zer o, .e . , when r = t , . Themagnitude ofchange in range of the second echo with respect to that ofthe first echo is found at the position wh ere the correlatiofunction of the two echoes is mJx imal.In Fig. I , the axial component of the scatterers veloc-ity along the beam axis is V,.,. Th e axial distance that thescat terer has m oved i n time T is designated as AL. whereA L = V,.,T = VT co s ( e ) . At time t = to an ultrasonicpulse is transmitted, and at time t = to + r I the echo fromthe scatterer at position 1 is received. At time t = to + Ta second pulse is transmitted and the echo from the scat-terer at position 2 arrives at ime t = to + T + t? . Thepropagation ime for he echo from position 2 is t2 - tgreater han hat for he echo returned from posit ion 1.This is the ime shift between the two echoes from h isonescatterer, hat s, t,, = t2 - r l , where ts is positivewhen the scatterer is moving away from the transducer (asin Fig. l ) and negative when the scatterer is moving o-wards the transducer. The axial flow velocity compon entis (assuming VAcos ( e )


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I I I I I I IA B C

I I I I I I I I I I I I I I I I I J I I ~8. 1. 2 3. 4. S. 6. 7. B. 9. 10. 11. 12. 13. 14. I . 16. 17. IS . 19 29 21

I1 1 I I I-. 4 - . 2 8 . 0 . 2 .+ -. 4 -. 2 8 . 0 . 2 . 4 - .4 - . 2 0 .0 .2 , 4

Since the vessel is orientated at an angle 0 ( the measure-ment angle) the resultmustbescaled by l /cos (e) , toyield he speed of the scatterer as (assuming V co s ( 8 )


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adex'@mixture, used as a blood m imicking substance. Theexperitne ntal set-up is described in [ 141. Th efront wallsignal is near 4.5 ps and the back wall ech o is near 18 vs.The center sections of the signals appear to move to theright slightly while sections near the walls do not mo ve.Three 0.8-ps range-gated sections of the hree signalsar edenoted by the symbols A . B. an d C in th euppercurvesofFig. 2. Theseare used to llustrate he ime-domain correlation method in the figure's lower portion.The discrete time correlation oefficient between two rect-angularly windowed. zero mean signal sections. e l H ) an de, n is given by

:L - I

p ( s ) =\ -

( 8 )where .S is th eshiftbetweensignals samples) , r is therange t o beginning of section to be measured (samples) .p ( S ) is the correlation co efficient between echoes ( - Ip ( s ) I ), and N is thewindow ength (sam ples) . Forthe sample correlation functions shown in Fig. 2. the win-dow engthwas 40 samples and thesampling ate ( , f , )was S0 M H z so that 40 samples correspond o a round-trip ength of 4 X at 5 M H z . The physical ength of th ecell is on e-half this value. that is. 2 X . since range is de-termined by round-trip travel time (h er e 0.8 ps) and the"range cell length" is therefore c N , / 2 f i or 0.62 mm ( f o rc = 1540 m / s ) . P re v ious w o r k [21] . [ 2 2 ] ndicated hatthis window length is a good com prom ise between rangeresolution ndmeasurementprecision (asdiscussed i nSection 111-B of this pa pe r),

The three colu mn s of correlation coefficient functionsin Fig.2 epresentcorrelations at ranges A , B. an d C .The top row represents correlations of signal 1 with signal2 , the middle row of signal 2 with signal 3 and the bottomrow of signal I with signal 3 . As the examples show, themaximum correlat ion of the center section (ran ge B ) is apositive time shift, corresponding to a flow velocity awayfrom the transducer. The correlation peak of signal 1 withsignal 3 is at a shift twice that of signal 1 with signal 2 orsignal 2 with signal 3 . The two correlat ion examples nearthe tube walls ( ranges A an d C ) show less shift, as ex-pected for the case of fully developed laminar Row.Because the signal is sampled at discrete times, the cor-relation function can be calculated only at discrete im edelays and themax imum is likely o fall between wopoints, limiting the accuracy of the location of the corre-lation pe ak. The method used to estimate he maximumpoint of the correlation function is to find the maxim umdiscrete correlation and ts wo neighboring points, fi t aparabola to the points, and determine the maximum of the

f ( x 1t

I I + xx0-1 x. xo+ l

Fig. 3 . Graph~cal reprcsentat ionh a t shows t h a t three points o f correlationtunetion near its muxirnum ca n he approximated by parabola.

parabola 1221.1231. Th is is llustrated in Fi g. 3 . A com-puter evaluation of he variance of this estim ate (jitter)versus samp les per wavelength (or sampling ratelcenterfreque ncy ) is shown graphically in Fig. 4 or 3 system Q's(ratio of the center frequen cy of the u ltrasonic pulse o its3-dBbandw idth). and it is quitesmall (for ten samplesper wavelength) when compared to the precision of thetime domain correlation method [ 2 2 ] .Th e magnitude of the velocity of the scatterer at a par-ticular range is given by

where S,,,,,s the t ime shift with maximum correlation.Since the time-domain correlation method compares twsuccessive choes that have assedhrough the ameintervening tissue. frequency dependent attenuation of thetissue path will not affect themean flow velocitymea-sureme nt. T hus, the average time shift determined by timedomain correlation is unbiased by the frequency responseof the tissue.

11. ERROR NALYSISA. Precis ian of Time-Domain Corre la t ion Method

Thederivation of the precisionof he ime-domainmethod is essentially identical to the derivation of the timearrival of a radar signal (241, (251. The der ivation's over-all assumption is the signal level is large compared to thenoise leve l . The der ivation also assum es that the windowlength is large compared to the ultrasound wavelength andth ecorrelationpeakca n be estimated romawidebandsignal with additive white noise 12 l ] . The validity of theseassumptions is addressed later (see Section 111).It is also assumed that the additive white noise present

in the two received signals causes a small deviation (muchless han awavelength) in the ime-delayestimate (F).Here the noise is small comp ared to echoes and the echoesare h ighly correlated at the rue ime delay ( T ~ ) ) . hus ,the variance of ? about T( ) is 1211

where N , , is the white noise power spectral density of eachecho ( W / H z ) , is the R M S bandwidthof he eceived

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I68

1 o

F 0.842WAW>

0.6YSL0

a-lg 0 .4CT2L0U

0.27L

0.0NUMBER OF SAMPLES PER WAVELENGTH

Fig . 4 . Graphical representation of'jittcr (variance ofrhih estimate) rewit-ing Iron1 the parabolic ti t as functlon o f numher o f \ample\ per a a v c -length for three system Q'\.

e c h o e s ( r a d / s ) , p is the maxim um correlation coefficientof the echoes without noise and E is the average energyof each angegatedecho ( J ). Thu s, he precision oftheime-domainorrelationelocitystimate1 JvAR(.i)/~,] s

whereSNR is the ignal-to-noise atio, d E / N u . F orasampledsignalwhere the noisespectraldensity is con-stantith frequency, the S N R is given byJN V A R ( s ) / V A R ( n ) ,where IV is the number of samplesin the range gate, VAR (S) s th e signal variance and VAR( U ) s the noise variance in the range gate .

As indicated by ( 1 l ) . the precision of the time shift es-timate is inversely related to the product of the SNR andR M S ban dw idth. The precision value for each time shiftimproves by an orde r of magnitude when the SNR is in-creased from I O dB to 30 dB. The precis ion of the imeshiftestimatecan be improved by decreasing hepulserepetition requency hatwould esult in a argermeantime shift for he same axial flow velocity. How ever, asmore scatterers enter and leave the beam during the timebetween ech oes, the maxim um correlation of the consec-

utive echoes decreases . Thus, the prccis ion of th e velocitestim ate will decrea se s tllc timebetween echoes increases . On the otherhand. i f the imebetweenechoesdecreases t o ;I very smallvalue. the point ofmaximumcorrelation and the correspondingvelocityestimate wivary randomly due to the random noise in the backscatered signal.Fig . 5 shows an experimental result of the correlationcoefficient asa function o f ' lateral eparationdistance.From these measurcme nts and the theoretical result forcylindrical beam 1221, a ncarly linear approximation (asuming he correlation function is independent of 8 ) appears valid. Wh en the ateralscparationdistance is onhalf of the heam width. hecorrelationcoefficie nt is rduced from 1 . O t o about 0 . 3 . The inear curve in Fig .can be expressed as

P ( t i ) = I - I . 2 t r ( 1where is the fraction of the eamw idthhe catterershave moved perpendicular to the beam (lateral separatioand is represented i n terms of the scatterers ' velocity,timebetween nitiation of the transmitted pulses, T , th3-dB heam width , BW. at the measurem ent range and thmeasurement angle. 8. a s

L ' T s in (8)L1 = BW ' ( 1and is valid over theapproximate ange 0 < . 0.6estimated from Fig. 6. Because the highly focused trans-ducerbeamwidth is only 3.3 wavelengths ( F = 2 ) anthe window length is only 4 h (round tr ip) . the small vlocity gradient at the center o t the parabolic protile shounot affect this measurement.Fig . 6 show s the precision (see ( I 1 ) ) of the time shifestimate for difl'erent tmnsduccr measurem ent anglcs anthe inear correlation coefficient function o f ( 12 ) . Thevalueswere ypical for theexperimentalnleasurementswith he S -MHz transducer used in flow measuremenwith an SNR of 20 dB [ 141. ( 2 ] . For each measuremenangle. the precision value dec rease s t o a minimum value( 3 . 7 7 0 a t 4 8 2 n s f o r 3 O C ; 6 . 5 ~t 2 7 8 n s f o r 4 5 " : 1 1 . 2at 160 ns for 60" an d 23.2% at 74 ns for 75 " ) and thebegins to increase due t o the decorrelation of the echoe

In sum ma ry, the heoreticalprecision o f ' the ime-domain correlation method depen ds on the measurement angle, the system R M S bandwidth. the mea n time shift be-tween echoes. and the S NR .B. Errors Associcttrd b t t i f l 1 Corrrl~lriotlM r t l l o d

In this discussion he erm "range cell" is definedathe vollumeof scatterers contained within he window edsegment of the echo. Since the echoes from the scatterinmediumcannot be entirelycontained in th ecorrelationwindow. there arc several sources of error associated with ecorrelationmethod.Theyare: I ) a windowedwaveform do es not necessarily have its m aximum correlationat the true shift: 2 ) the range cell contains sca tterers that

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FOSTER c f nl : F L O W VELOCITYPROFILEVIA T I M E - D O M A I N CORRELATION

I0 0 125 0250 0375 05W 0625 0750LATERALSEPARATION (BW)

F ig . 5 . Measured cor re lat ion coef f ic ient (sol id ine) and i t s l inear approx-imation dashe d ine)verws ateralseparationdistance cxprcssed ;ISfraction of beamwidth perpendicular o beam axis) . Measurement wasmade at the center o f 23-h diameter tube in whi ch constant laminar f lowwas established. S MHz. h ighly focused ransducer ( F = 2 ) h c am w i dt hwas 2. 3 A , cor re lat ion ength was 4 A. and measurement angle ( B ) was15".

Fig. 6. Theoretical precision ( ( I )) ersus t ime hhift and measurement an-gle.Beamdecorrelation s assumed tobe inear. S N R i s 20 d B ( I O ) ,R M S bandwidth is 2 . 5 M H z ( 15.7 M r / s ) a n d b e am w id th i\0.6 m n l .

move at different velocities: 3) an actual range cell willcontain ultrasonic energy from scatterers outside the rangecell due to finite bandwidth effects; and 4) as the scatterermoves through the beam, the backscattered ultrasonic sig-nal is modulated by the field pattern of the transd ucer.I ) Error Associated with W indowing: The process ofwindowing may shift hepointof maximumcorrelationaway rom he rueshift if the echoes are dentical. T oevaluate his error sources, the second echo will be as-

sumed to be identical to the first with no shift. Hence thetrue shift between the two echoes m ust be zero. The es-timated shift. .i, is defined by (7 ) . A computer simulationhas been used to deter min e the bias and variance of theestimate i and showed hat .iwas unbiased (zero mean )for all rectangular correlation windows [22].The term"jitter" in Fi g. 7 is defined as he standarddeviation (square root of the variance) of the estimate .i.The signalmodelused in th esimulation is band-p assedwhite Gaussian noise (BPW GN) and the system Q is de-

fined to be the ratio of the center frequency of the ultra-sonic pulse to its 3 dB bandwidth. A six-pole Chebyshevresponse is used (with1-dB ipple) in order to simulateth e transfer function of the transducer and receiver elec-tronics. Th e window ength is shown in wavelengths ofthe center frequency of the ultrasonic pulse.The j i t ter decreases as the window length increases be-cause he correlation more closely approximates he au-tocorrelation function (which has zero jitter in estimatingof the true shift). The bottom curve of Fig. 7 simulates aBPW GN w ith an infinite Q (purely sinusoidal). Note thatthe minima and maxima occur in the same positions for aQ of 32 as hoseof he infiniteBPWGN case.This isexpectedsince , with a Q of 32,wavefo rm will closelyresemble a slowly modu lated sinusoid. As the value of Qdecreases further the minima and maxima lose their iden-tity as comp ared o hose of ahigher Q system. Fig. 8shows the jitter versus Q for various correlation lengthswhere the correlation engthswerechosenas heworst


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3.0

2 . 5

WINDOW LENGTH (WAVELEN GTH )F ig . 7 . Graphical cpresentation of jltt er (\t and ard deviation o t estimatei ) s function o f window length for tour \ystem Q . \ .

2 OOl 75 -

- 1 50 -fP-o 1 2 5 -mz

100-

Ob 1'0 210 310 4'0 : $0 ; 810SYSTEM Q

Fig. 8. Graphical rcpresentation of jlt ter a\ function o t s y s t c n l Q lor clghtdifferent window lengths. From top- to-bottom the windo- length5 arc1 .2 5 , 2 . 25 . 3 . 2 5 . 4 . 25 . 5 . 2 5 . 6 . 2 5 . 7 . 2 5 . an d 8 . 2 5 h .

case lengths for the pure sinusoid case ( t z + 0.25 wave-lengths where n is an intege r) .2) Error Associuted with Scatterers Moving a t D l f e r -en t VelocitiesWithin the Rcrrlge Cell: Thecattererswithin a range cell will be mo ving at different velocitie ssince hevelocity profile across hevessel isnot neces-sarily uniform . Thu s, onlypart of the range cell of thefirst signal will line up with the correspo nding part of thesecond s ignal. The other parts of thesignal of the firstecho will not be perfectly aligned with their correspon d-ing parts in the second echo since these parts com e fromscatterers that are mov ing with different velocity. A com-puter simulation was used to estimate the jitter due o dif-ferent flow rates in the ange ell and the esults regraphically represented in Fig . 9 where jitter is plotted as

4510- 35 -

0 1 0 2.0 3.0 4.0 5.0 6.0 7.0WINDOWLENGTH NWAVELENGTHS

Fig. 9. Gr a p h ic a l reprercntation 0 1 jitter ( \ t a n d a d devia t ion o f th c emate i ) \ functwn o t u i n d i ~ength o i eleven \cattCrer q m r a t i o n dtances ( I n u a \ c l e n g t h ~ ) l o rI sy\ tem Q o l 2.

a function of window length for various scatterer separation distances.Separation is detined ( i n wavelengths) as the distanthat the fastest scatterer gains ov er the hlowest scatterin the time nterval between he nitiation of successiechoes (221. The separation is a one-w ay separation. Fthis simulation. the scatterers at the edge o f the range cenearest to the transducer are assumed t o have the slowesvelocity and the scatterers at the farthest edge are assumto have the fastest velocity. The scatterers betMjeen thetw oedgesareassumed to havevelocities hat ncreaselinearly with depth into the range cell.Because of the results shown in Figs . 7 an d 8 , the chaacteristics shown in Fig . 9 should be similar for other stem Q's (he r e Q = 2 ) . Th e effect of the f ini te correlatiwindow ength is shown in the bottom curve (zero sepration): i f the correlatio n wind ow effect were not presenthen this curve would lie on the horizontal axis. Separation can be determinedanalytically or ful ly developelaminar flow from the axial velocity profile, V i ( ) ,

V , ( , . ) = V,,,a,[ - ( 2 r : i W ' I ( 1where V,,,.,, is the maximum axial velocity in wavelengthpe r u n i t t ime. r is theaxial distancevariable in wavlengths and N h is the vessel diameter in wavelengths (seFig . 10). Thechange i n distance, S ( r ) , xperienced bthe scatterers at the end o f the range cell compared to thoat thebeginning o f the angecell in t ime, T , the ime



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F I ~ .O. Geometry used in derivation of jitter resulting from error a w -ciated with scatterers movlng at difere nt veloc ities within range csll.

intervalbetween he nitiation of thepulses is approxi-matelyS ( r ) = T [ V h ( r+ d r / 2 ) - V A ( r- d r / 2 ) ] ( 1 4 a )

ddr= T - V h ( r ) r = -4( TV,, , , , /NX) ( 2 r / N X ) d r

( 14 b 1where dr is herangecell engthprojectedonr-axis inwaveleng ths. The worst-case sep aration occu rs at the ves-sel walls, where

S ( N A / 2 ) = -4 -V,,, d r , (15)NAEquation (14b) can be formulated in terms of experi-

mentallymeasured quantities.Thedistance thatamid-streamcatterermoves in t ime T i s H ow eve r,hesystem measures directly he ncrease in round-trip pathlength due to this movem ent, that is, S = 2T V,,, cos 0 ,where S is defined as this axial separation distance. Thevalue dr can be related to the range cell axial len gth , d r '(determined by the window length) where dr = dr ' sin 0 .Therefore,S ( r ) = - 2 ( s / N h ) ( 2 r / N h ) tan 0 d r ' . ( 1 6 )

For a 6.9 6 mm diameter vessel ( N h = 24 h ) nclined at45 " with respect to the transducer, an ultrasonic pulsewitha center frequency of 5 M H z , Q at 4, a measured S of 0 .7X. and a window length ( d r ' ) of 8 X (physical length ofrange cell is4 X becau se of round-trip tim e) the calculatedworst-caseseparation (a t vessel wall, that is, at r = 12h )would be 0.47 X. The j i t ter associatedwith the velocityestimateusingan S X window ength sabout 6% of awavelength (from Fig. 9). Norm alizing he itter by theaxial velocity (i .e .. divid ing the jitter by the axial flow,s = 0.7) yields a value of 8 % .

Th is is roughly he amount of norma lized jitter foundexperimentally for this size vessel with this amount of .Yfor a window ength of 6 .5 X. The rangecell will beslightly larger than a 6 .5 h window length because the Qis not zero . Th e cell will be roughly l .5 X longer due toth e Q, bringing the total window ength to 8 X, which iswhat was used previously. Th e error becom es smaller asthe range cell nears the center of the vessel.3) Error Associared \t?ithTime Duration c$ th e SysttJrn

Impulse Response: Thesystem mpu lse response s hewaveform at the output of the system due o the ultrasonicpulse generatedan ddetected by thesystem) reflectingfrom a single scatterer. The echo from a medium whichconsists of many scatterers is the convolution of the sys-tem impulse espon se with the density unction of th escatterers. T he bandlimited frequency response of the sys-tem will cause the received signal at a particular range tobe the result of the convolution of the system impulse re-sponsewith heenergy reflecting rom sc atterers.Thelonger the temporal duration of the impulse response, thegreater will be the fraction of energy at a particular rangearising romenergy romearlierscatterers in the rangecell (and before the range cell). The velocity es timate willbe shifted towards the velocities of those earlier scatter-ers. naddition o hevelocityestimationbeingbiasedtowards early scatterers, the estimate will also lose rangeresolution. The mathem atical representation of this effectis assumed to be 1" V ( t )1h(r - ) 1 2 d t- m

V ( 7 ) = (17)! h ( t ) I Z t0

where T represents the location of range cell, V ( ) is thevelocity profile that the p ulse passes through, h ( t ) s th esystem impulse response (assumed causal), and V ( 7 ) isthe estimate of velocity at range. For good range resolu-t ion, h ( 2 ) must have the lowest possible temporal dura-tion.4 ) Error Assnciuted with Intensity Profile Across th eSound Beutr~:As a scatterer passes through a sound beamthe intensity of the scattered ultrasoun d will be directlyproportional to the intensityof the sound beam at the scat-terer locatio n. Hen ce, the intensity of a scatterer will in -crease as the scatterer nears the axis of the beam and willdecrease as the scatterer mov es away from the axis . Th isamplitude modulation effect is known as beamwidth mod-ulation. With many scatterers in the beam , som e will beapproaching the beam axis and so me will be m oving aw ayfrom the beam axis, causing two successive echoes to ookless similar . The j i t ter of the est imate i due to this errorsource is shown in Fig. 11. Here jitter (standard deviationexpressed as a percentage of the w avelength) is presentedas a function of the fractional amount of beamwidth thatthe scatterer moves for various windo w lengths. The lat-



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17112-

10

00' 0 . k 0 ;o 015 0 ;o 0s oo 0 . kPERPENDICULARDISTANCE (BW)

Fig. I l . Graphical epresentation o f beamhidthmodulation itter ( s t a n -dard deviation of th eestimate i)s function of distance hescatterermoves across the beamw idth (normalized to th e 3 d B beamwidth ) fo rthree window lengths o f 2 . 4. an d 8 X w i t h sy\tern Q o f 2 .

era1 distribution of the sound beam is assumed to have aGaussian profile. For the case of no scatterer move ment,the jitter reduces to the jitter associated with the windowlengtheffect (see Fig. 9). The error associated with thebeam profile dw arfs the correlation effect for good SNR .In sum mary, the error from the beamwidth modulationof the scatterers represents the greatest source of error forvelocity estimation where all of the scatterers i n the beamaremoving t the samevelocity. When he cattererswithin he angecellaremoving at differentvelocities,the itter caused by a small velocity gradient can easilyexceed the other error sources.

I V . VERIFICATIONY C O M P U T E RIMULAT IOKA detailed computer s imulation was performed in orderto evaluate the effects of the system quantities on the es-timation of flow velocity at various points across a vessel[22]. Within the beam width the intensity is nonzero; out-side the beamwidth no sound is assumed to exist. No noisewas added to the simulation and no averaging of velocity

results was perfo rme d. All distance measurements are inwavelengths relative to the center frequenc y of the ultra-sonic pulse.Fig . 12(a) shows one example of the compu ter s imu-lation esultsof hevelocityprofilesdetermined by thetime-domain correlation method. The topm ost curve is theactualparabolicvelocityprofile hatanunbiasedultra-sonic flow meter would measure. The other top curve isthe ime-domain correlation method estimate of he ve-locity profile. Both velocity profile curv es are norm alize d

to the midstreamvalueof heactualvelocitywith hemaximum of he actual flow at 100%. The s tandard de-viation curve is the jitter of the estimate normalize d bdividing by the maximum ac tu d axial flow. The precisiocurve ( i n percent) is a me asure of the precision of the velocity estim ation algorith m and co nsists of the jitter ( standard deviation of the velocity estimate) d ivided by the estimateof hemeanvelocityexpressed asapercentage.Theprecisioncurve is not norma lized.When he valuexceeds l o o % , t is truncate d to 100%.The same graph-ical format is used in Fig s. 12(b ), (c) . and (d) for whicthe respective vessel diamete r are 24, 12, and 6 h (foudifferent measurement angles are shown in each of Figs12(b),12(c),and12(d).The standard deviation curves of Figs. 12(b)-(d) nd i-cate the total jitter from all sources discussed in SectioIll- B. The primary jitter errors are from beamwidth mo dulation and from velocity gradients across the range cell.Since hese hree igureshave he same size range celthe elocity radient crossheange ell hould bsmallest in the argestvessel. Thecontribution o midstream jitter from velocity gradients is least in the largesvessel (24 X ) at he mallestviewingangle (45" ) . Asmaller viewing angle will result i n a longer path througthe vessel minimizing furth er he effect of velocity gradients. In Fig. 12(b) (upper right) the midstream jitter i3 . 7 % .The perpendic ular positional change (S , ) of a scatterewith respec t to the beam axis is related to the axial posi-tional change ( S ) of the scatterer by the angent of hemeasurement angle o r S, = S . tan ( 8 ) . The perpendiculapositional change at the center J f the vessel s 0.5 . ta(45") or0.5 avelength. The beamwidth normalized perpendicular positional chan ge is 0 . 5 / 2 = 0.25. This corresponds to ajit t er of 3 . 7% (see Fig. 1 l ) . Hence , the midstream value of jitter for 8 = 45" in Fig . 12(b) is largela result of beamwidth modulation. Th e midstream valueof jitter for the smaller vesselsnd larger angles are largthan the 3 . 7% baseline value for beamwidth modulation.indicating that the jitter introduced by velocity gradientsis increasing.The itter from velocity gradients can be observed iFigs. 12(b )-(d) at the edges of the vessel where the effectof beamw idth modulation are minimal. For example, thebeamwidthmodulationofFig.12(b) is largestatmid-stream (since he scatterers are moving he fastest here)with a value of 3 . 7 % . Al l other jitter above this baselinis a result of velocity gradie nts, and near the edges of thvessel the majority of the .jitter is caused by the large ve-locity gradients.I n Figs. 12(c) and ( d) the ji t ter is greater than the 3 .7 %baseline set by beam width modulation because the vesseldiame ters are sma ller and have the sam e axial flow velocity and range cell size as the 24 wavelength diameter vesse l in Fig .12(b).hence, the velocitygradientsmustbegreater .Table I examines thenterdependencies of windowlength, beamw idth, vessel diameter, and viewing angle.



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l 7 3

~ o a o r

0 7 3 -

' 0 6 I t

too

'0 6 12

too

10

' 0 3 6 9

The values of these quantities are approximately the sam eas the experimental quantities [ 141, ( 2 ] , [26] and henceallow he results of the simulation o be compared withtheory. Numbers w ithout parentheses in Table I are heestimates of the actual flow at m idstream divided by theactual lowatmidstreammultiplied by 100 (percentagebias) and the numbers in parentheses are the precisions ofthe velocity estimation (see ( 1 ) ) .

Increasing the viewing angle results in a decrease in theprecision of th evelocityestimation.This trend is fol-lowed for all co mb ination s of the other three quantitiesexcept he 6-X vesselwith a BW = 4 fo r both windowlengths. The reason for this exception is due to the esti-mate of the midstream flow is decreasing faster than thevariance of the estimate. Hence . the precision (the ratioof the woaforementionedquantities) will increase. n-



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24 I 2 6BW = 4 LE N = 4 0 = 4 5 "0 = 5 5 ;

6 = 6 5 "'' 6 = 75"

" LE N = 8 H = 45 '6 = 5 5 "6 = 65"

B W = 2 LE N = 4 H = 4 5 "r e = 75 "e = 5 5 0

Y X . O i 2 , Y l Y 3 . 1 ( 6 . 6Y7.Yi-l 7 1 Y 2 . h ( 7 . h9 8 . 3 1 5 . 5 ) Y3 .0 i 8.498.31 1 0 . 1 1 Y3.2 ( 17.9 7 . 6 ( 2 . 2 ) 01.81.5.09 7 . 6 i 2 . 9 ) 9 1 . 1 ( 5 . 89 7 . 6 i 4 . 1 ) Y0.9 (6 .59 7 . 5 ( 7 . 2 1 9O.XiR.IY8.8r 6 . 3 ) 9 7 . 0 i 7 . 099 . I 8 . 5 ) Y6.4( 8.698 .51 13.3 ) 96. I i 14 .

79 .978 477 X) 7 7 . 676 n73.97 0 x6 9 . 6

YY .68 8 . 2) 87 .7

"Results of a computer simulation of the cstim at~ on l mld\tream H o u(the number without parenthesis) and the preci\ion i n the ve lo c i t ) est ima-tion (the number In parenthes is). The quantities varied are: he ve\se l di -ameter. the window length ( L E N ) . he beamNldth of the transducer i B W j .and the viewing angle ( 8 ) . The f i xed quan tities are velocity profile ( p a r a -b o li c) , s y st em Q ( 4 ) and maximum axia l separation (.\ = 0 . S X ) .

creasing he window ength resul ts in more precision aspredicted in Fig. 7 . Reducing hevesseldiameterde-creases heprecisionbecause he velocitygradientsar ehigher i n the smaller vessels for identical range cell sizes.A larger beamw idth has a greater precision in the largertubeswhere hevelocitygradientsare mallacross herange cell. The smaller thewindow ength he essbiasthere s in the velocity estim ate. Bias is the deviation ofth estimatero m at theessel center flow. T heshorter window length allows less local averaging of thevelocity rofile. Th e ma llerhe vessel iameterhegreater the bias. Here the size of the range cell remainedcons tant, but due onarrowing hevelocityprofile, therange cell must now contain a larger number of scatterersof differingvelocities,hence.biasing he esultdown-ward . Bias gets worse for ncreasing viewing angle. es-pecially for the smaller vessels. The larger angles reducethe axial contribution to the echo and increase the radialcontribution (by the cosine of the viewing angle), hence,increasing hevelocitygradients i n the angecell therangecelldimensions are ndependent of ang le but de-pendent on beamwidth, window ength, and mpulse re-sponse).Large eamwidths ave reater ias.Thebeamwidth will smooth out velocity nformation, hence,detail in the rang e cell will be lost.

V . CONCLUSIONSome important observations can be drawn frotlr t a r -comp uter simulation results. The bias is directly relatedto the size of the range cell (a smaller range cell will av -erage over a smaller portion of the velocity profile). Sev-eral quantities control the precision: l ) A narrower beam-width will result in improved recision: 2 ) A longer

window ength will result in a better precision in the etimate of th eactual hift; 3 ) A largevelocitygradientacross the range cell will result in poorer precision. Quantities I ) an d 2 ) must be traded off against 3 ) .These results suggest an accurate one dimensio nal vlocity profile acro ss hevesselcan beobtained. A higdegree of precision between measured and rue velocityexists over most of the vessel.A C K N O W L E D G M E Y T

Theauthorsacknow ledge hc useful comments romRobert W . Gill, Ilmar A . Hein. Robert Skidmore, V eijoSuorsa, Gregg E. Trahey, Peter N . T . Wel l s and the excellent reviewers are also very much appreciated.R E F E R E N C E S

[ I ] P. M . Enlbrce and W . I) . O'Brien, Jr., "PulsedDoppler a c c u ra\sersnlcnt d u e t o frequency-dependent t tenuation an d Rayles c a t t e r ~ n g rror sources." IEEI , ' 7rcul.s. R i o m c 4 . E t ~ g . . o l . 3 7 ~opp . 377-326. Mar. I Y Y O .[ 2 ] R . W . ill,"Measurement of blood Row bl ultraaound:Accuracand wurces ol er[-or." U/rrcr.\o~tr~ded. B ; / ] / . v o l . I I , pp. 625-631985.131 C. P . Je thwa. M . Kaveh. G . R . Cooper , a n d F . Saggio. "Blood f lmeasurenwnt< u \ l n g ultra\onic random signal Doppler s)stem." f ETrtrm. Sonic5 U/ f rmou . . v o l . SU-22 . pp . l-It. a n . 1975.

[4 ] B Angel \en. "Instantaneous requency.mean requency, and vaance o f mean frequ ency e\timato r for ultrasonic blood velocity Doppler s ~ g n a l s . " / LE Trurl.\. Bioruc.6. ,,g.. bel. BME-28, pp . 73741. 1981.151 B . .Angelhen a n d K . Kri\tollersen."Diwrete imeestlrnation o fmean Doppler frequency In ultrasonic blood velocity measurements.I E E E Trot!.$.E i o r n c~ / .E f i X r . . v o l . BME-30 , pp . 207-214. 1983.161 D . W . Baker. "Pulsed ultrasonic Doppler blood f l o w sens ing," f ETrot ! . \ . Sot~ic..\U//ra\ott . .vol. SU-17. pp . 170-IX5. 1970.171 P. A. Gran dcha mp. "A novel pulsed directionalDopplervelocimetcr. hephase-detection protilometcr." in Proc.. 2m / Eur . Corr:.rrcrtour,t/ in Mrd . . pp . 137-143. 1975.[ R ] M .Brande\tinl."Application o f the phasedetectionprinciple it ranscutaneous velocity protile meter." in Pro ( . . Zt7tl Eur. Corfg .rrtr.\ouud i n M o d . , pp . 144-152, 1975.191 -, "TopRow-.A digital Full-range Dop pler v e l o c i t y meter." / E

[ I O ] L, Hatl e a n d B. Angelsen. h p p / c r C'/rrcl.\our?t/ it1 Crirt!io~og!.:Ph.i c w / Pr rw l p / < , c r d C/irlicu/ lpp / ic u / ion . P h ~ l a d e l p h ~ a . P A :ea aF e b ~ g e r .1982.

[ I I ] P. Atkinson and J . P . Wo o d co ck . D o [ J / J / c ~/ / r ~ r \ o u ~ r de / I f s U seClitlical ,~c , [~ . \rf rc , r~1c,~r/ .ewYork:AcademicPress. 1982.1121 C. Kasai, K. N a m e k a w a . . A . Koyano. and R . Omoto , "Rea l - t~me w/ t % E Trcrus. Sonic\ L'/rrtr,\orz., v o l . SU-32, pp . 458-464, 1985.dimen\~ona l l ood R o w i m a g ~ n g s ~ n g n autocorrelation technique

[ 131 M . I . Skolnlk . R t h r H ~ d h o ~ i . ew York:McGraw-Hill . 197[ 141 P. M . Emhreean d W . D O'Brien. Jr . . "Volumetricblood flowt ime dorna~n orrelation:Experimentalverification," / E E E Trtr

U/rr~~.so/f.t , r ror l rc . . Fr-cq. Cou/r. . v o l . 37 . n o . 3 . pp . 176-189. M1990.[ 151 D. Dotti, E.Gatti . V . Svel to. A. Ugpc. a nd P. Vidal i , "Blood R

measurement by ultrawund correlation techniques," Erlcrgitr R i u curc', v o l . 2 3 . pp . 571-575, Nov . 1976.[l61 M . Bassini. D . Dotti . E. Gatti ,P.Pizzolati , nd V . S v e l t o . ".ultrasonicnon-inva\ivcblood lowmeterbased o n cross-correlatitechniques," U/ rru ,y m. / ! I / . P ro< . . 1979.1171 M . Bassini. E . Gatti. T. Longo, G . Martinis. P. Pignoli. and P. Piz-zolati , "111 1, ivo recording o f blood velocityprofiles andstudiesrirro o f protilc alterations induced by known \tenoacs," T u t r s Hrth r . J . , vol. 9. pp . 185-194, June 1982.

[ IX] 0 . Bonnefousand P . Pe qu e . "T imedomain ormulation of pulsDoppler ultrawund an dbloodbelocityestimation by cross correlt ~ o n . " Ulfrerson. / r u c r , q i u g . v o l . 6 . p p . 7 5 - 8 5 , 1986.

TrClt7.S. s O ! ? ; C S u/fi'O,\~JH.,ol. su - 25~p . 287-293. 1978.


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j 19) 0 . Bonnel 'ous . P. esque. an d X Bernard. " A new \ e loc i t }e \ t imnto r

Steven G . Foster, photograph a n d hiographq n o t ~ t a l l a h l e t timc o f puh-lication

Paul h l . Emhree ~ S ' X O - ? r l ' X l S'XI-h1 '82-S '83-lvl'8.5) ua5 horn in Rcad lng . PA In 1950. He re -celved th e B . S . degree i n electricalengineer ingfrom Lehlgh Unlverrirq. Bethlehem. P A . In 198 .and he M . S . an d P h . D degree \ in electrical e n -ginecrinp from Unlver\it) ol ! l l~no i> . Ur hana . [ L .In 1Y82 and 1986. re\pectivcly.From I981 t o 1981 he ua \ a member ul . tech-nlcal s t a t 1 a t Bell T elephonc Ldboratoriss. K c a d -ing, PA . u h c r c h e v.ork.cd In the field 01' a n a lo gfilter dehlgn o f CMOS integratedcIrcuIt\. A t t e rcompleting the P h . D . in the field 01 nedlca l u l t r a w u n d . he p ined Ph i l lp rUltrasound. Santa Ana. C A . In 1986. His current interes t \ i n \ d \ c he ma l -

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Flow Velocity Profile via Time-domain Correlation Error Analysis And Computer Simulation

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