Intrinsic spectral broadening (ISB) in ultrasound Doppler as a combination of transit time and local...

PII S0301-5629(00)00218-0

● Original Contribution

INTRINSIC SPECTRAL BROADENING (ISB) IN ULTRASOUND DOPPLERAS A COMBINATION OF TRANSIT TIME AND LOCAL GEOMETRICAL

BROADENING

GABRIELE GUIDI, CINZIA LICCIARDELLO and SIMONE FALTERI

Department of Electronics and Telecommunications, University of Florence, Florence, Italy

(Received25 November1999; in final form 6 March 2000)

Abstract—Doppler signals collected with a focused transducer are known to be affected by the so-called intrinsicspectral broadening (ISB). This article aims to point out how ISB is, in general, related to both the limited lateralextent of a focused beam (leading to a finite transit time), and the presence of several local insonation anglesaround the beam axis, due to focusing and diffraction effects (local geometrical broadening). The influence ofthese two elementary spectral contributions on the whole ISB is shown by considering the Doppler signal assimultaneously modulated in amplitude and frequency, and applying well-known relationships employed in thecommunication field. Such an analysis reveals that transit time and local geometrical broadening are twodifferent phenomena, whose simultaneous knowledge is necessary for correctly evaluating the overall ISB.Finally, thanks to a novel technique for separately measuring transit time and local geometrical broadeningeffects on transducers with markedly different focusing properties, more than 1000 experimental acquisitionsshow how a proper combination of such measured contributions gives an accurate ISB estimation, confirming thetheoretical expectations. © 2000 World Federation for Ultrasound in Medicine & Biology.

Key Words:Ultrasound Doppler, Transit time, Geometrical broadening, Intrinsic spectral broadening, Dopplerartefacts, Vector Doppler.

INTRODUCTION

In clinical practice, the Doppler signal is commonlyinterpreted as a histogram that allows association witheach frequency component and, therefore, at each ve-locity, a spectral amplitude proportional to the numberof red cells travelling at that particular velocity. Ac-cording to this vision, the spectral behavior reproducesthe statistical distribution of blood cell velocities inthe sample volume (SV) under examination at the timeof analysis.

Unfortunately, to have such equivalence, the acous-tic wave used to measure velocity from the Doppler shiftshould be a plane wave, but this is not practically pos-sible for several reasons, such as the finite dimension oftransducers employed in real equipment, and the focus-ing of ultrasound (US) beams necessary to concentratethe acoustic energy in a limited region of space.

As a matter of fact, those characteristics involve abroadening of the Doppler spectrum, which is intrinsi-

cally due to the actual mechanism of delivering andreceiving acoustic energy and, for this reason, is calledintrinsic spectral broadening or ISB (Evans et al. 1989).Such spectral broadening is combined with other sourcesof broadening, such as the presence of a velocity gradientwithin the sample volume, or the contribution of accel-eration (signal nonstationarity); therefore, the Dopplerspectrum deviates from its supposed histogram function-ality and the related diagnostic indexes (which are nor-mally evaluated without taking into account these as-pects) may be strongly influenced.

ISB has been studied as a cause of artefacts, but alsoas a possible source of information complementary tothat attainable with conventional Doppler techniques.For a line flow, it has been demonstrated that the angledependence of Doppler bandwidth differs from that ofDoppler frequency; a proper integration of this comple-mentary information, in principle, might eliminate theneed to know the insonation angle to have absolutevelocity information from Doppler signals. Studies onDoppler bandwidth for exploiting its complementary ve-locity information were started in the last 3 decades, withmajor contributions by Newhouse et al. (1976, 1977,

Address correspondence to: Gabriele Guidi, Department of Elec-tronics and Telecommunications, University of Florence, Via S. Marta,3 - 50139 Florence, Italy. E-mail: [email protected]

Ultrasound in Med. & Biol., Vol. 26, No. 5, pp. 853–862, 2000Copyright © 2000 World Federation for Ultrasound in Medicine & Biology

Printed in the USA. All rights reserved0301-5629/00/$–see front matter

853

1980, 1987) and Newhouse and Reid (1991), who intro-duced a set of theorems and properties of the Dopplerbandwidth collectively known as “transverse Dopplertheory,” with the purpose of extracting velocity informa-tion from Doppler bandwidth rather than from Dopplershift.

The physical interpretation given in the literature toISB is twofold:1. If a single flow line crosses a focused US beam, the

related ISB can be seen as caused by finite transit timeunder the beam (transit-time broadening). Accordingto this point of view, the signal amplitude is weightedby the beam and its spectrum has a bandwidth in-versely proportional to the transit time (Newhouse etal. 1976).

2. On the other hand, if the ultrasonic field is decom-posed in a set of plane waves along different direc-tions, the ISB can be attributed to the presence ofseveral insonation angles (geometrical broadening)(Newhouse et al. 1977; Bascom et al. 1986).

The study of Doppler spectral broadening within theframework of transverse Doppler led to some theorems,one of which states the equivalence between geometricaland transit-time broadening (Newhouse et al. 1980), as aconsequence of a similar property found in the laserDoppler field (Edwards et al. 1971).

Such equivalence can be qualitatively explained byconsidering what happens in the focus of a focusedtransducer. If we decompose the active element intoseveral elementary spherical wave sources, it is knownthat, in the focal zone, the interference between differentwave contributions produces an acoustic field concen-trated around the beam axis, with the wave fronts parallelto each other, as happens in a plane wave. This is whysome authors in this case talk about “plane wave condi-tions” (Ata and Fish 1991). Although, in those condi-tions, if we imagine a flow-line crossing such field, therelated Doppler spectrum is not a line; it is insteadbroadened by the windowing effect originated by thelimited beamwidth in that point.

Indicating asW the aperture width andF the focallength of a focused transducer, it is well-known that adirect relationship holds between the beamwidth in thefocus of a transducer and theW/F ratio (Hunt et al.1983). Thanks to such relationship, Newhouse et al.(1980) indicate the relative Doppler spectrum bandwidth,BD/fD, due to the transit time in the focus, as a functionof W/F:

BD

fD>

W

Ftanu (1)

whereu is the insonation angle, taken as shown in Fig.1a. The same equation, later confirmed with diffraction

theory (Censor et al. 1988), was intuitively interpreted asdue to the extreme directions under which the flow lineis insonated, and its contribution indicated as a uniquesource of spectral broadening, with the term “geometri-cal broadening.”

This approach has been demonstrated useful in anumber of applications as, for example:● for verifying the invariance with velocity of the rela-

tive Doppler bandwidthBD/fD with a thread phantom(Tortoli et al. 1992);

● to confirm such invariance with flow phantom insteady conditions (Cloutier et al. 1993);

● for explaining the invariance of the Doppler band-width with range cell size above a critical beam-to–flow angle (Tortoli et al. 1993);

● for evaluating the peak frequency correction with asingle element transducer (Tortoli et al. 1995), andwith an array (Winkler and Wu 1995);

● for extending such corrections in case of wide andnarrow beams (Willink and Evans 1996);

● for in vivo blood velocity evaluation from Dopplerbandwidth (Guidi et al. 1997); and

● for predicting maximum frequency and spectral widthin steered Doppler beam systems (Hoskins et al.1999).

Although this approach is interesting for the unifyingvision of the problem and the consequent simplification,some limits hold:1. The statement “geometrical broadening and transit-

time broadening are one” (Newhouse et al. 1980) maysound a bit confusing. It tends to “mask out” theactual physical phenomenon responsible for the cre-ation of the spectral broadening that is due to asimultaneous modulation in amplitude and frequencyof the transducer excitation signal. This is particularlyevident for a strongly focused transducer. Because thetwo broadening causes are claimed to be the same,one could think to evaluate the Doppler spectrumbandwidth in a location far from the focus as thetransit-time reciprocal. But, as observed experimen-tally by Newhouse et al. (1977), it does not corre-spond to experimental results. The above-mentionedstatement could be better understood as: “geometricalbroadening and transit-time broadening in the focusare one”;

2. The practical use of eqn (1) is affected by somelimitations:● The most widely used approach to measure Dop-

pler spectrum bandwidths is to set a thresholdgf (afraction of the amplitude spectrum peak value),revealing the maximum and minimum frequencycomponents corresponding to such level. The mea-sured bandwidth, evaluated as the difference be-tween maximum and minimum frequency, gives a

854 Ultrasound in Medicine and Biology Volume 26, Number 5, 2000

result near to the theoretical one ifgf ,, 1. In realapplications, the presence of various forms ofnoise, together with the speckle-like nature ofDoppler signals, may produce a noise floor thatneeds to be cut out by increasing the threshold level(e.g., gf 5 0.5 @26 dB). The bandwidth evaluatedwith such increased threshold represents, therefore,only an approximation of the theoretical one, theapproximation worsening as the threshold ap-proaches unity. Moreover, once defined a specificthreshold, the underestimation error changes withthe spectral shape; the more it is peaked, the largeris the underestimation.

● The beamwidth,w, must satisfy the conditionw ,, W, which could be not satisfied in general(for example, in one of the experiments reportedbelow W/w56).

3. Such lack of contact with the physical reality may bea disadvantage when the broadening for sample vol-umes with small axial lengths compared with thebeamwidth has to be predicted. For insonation anglesfar from 90°, so that the transit time is due to the SVlength, rather than the beamwidth, a mixture of mod-ulations is present, and to take into account only the

transit time was shown to give good prediction ofexperimental bandwidth vs. angle results only in thefocus (Mc Ardle and Newhouse 1995), where nofrequency modulation is present.

This is why other authors have interpreted the “geomet-rical broadening” as the frequency deviation due to thevarying orientation of the wave vector along the flow linecrossing the beam (Fish 1986).

Looking at the problem from this point of view, wecan imagine that a single point target moving at constantvelocity v and crossing an acoustic beam, generates aDoppler signal that, if detected from a pulsed-wave (PW)flowmeter employing a sample volume long enough tonot influence the transit time, can be expressed in ageneral complex form:

s~t! 5 A~t! z ejFD~t!, (2)

whereA(t) is influenced by the amplitude field, andFD(t)by the phase field distribution. The latter term can beseen as composed of two parts:

FD~t! 5 F# D 1 DFD~t! (3)

Fig. 1. Travel of a scatterer into an acoustic field. (a) Schematic illustration of simultaneous actions due to transit timeand phase bending; (b) effect of such actions on the corresponding Doppler signal.

ISB in ultrasound Doppler● G. GUIDI et al. 855

where the first term,

F# D 5 2pf#D (4)

is related to the prevailing Doppler shiftf#D, which wouldbe available if the transducer would generate an idealplane wave directed along the transducer axis.DFD(t)isthe contribution due to the difference between the phasefield of a plane wave and the actual phase distribution ofthe focused beam. As shown in Fig. 1a, it is originated bythe target crossing a curved phase front; by decomposingthe actual field in local plane waves, it originates instan-taneous frequency shifts proportional to the slightlyvarying Doppler angles along its path across the USbeam (Guidi and Falteri 1999).

In terms of communications theory, the signal de-scribed by eqn (2) can be seen as modulated, both infrequency and amplitude, with a carrier frequency rep-resented by the average Doppler shift,

f#D 5F# D

2p. (5)

Although the geometrical broadening mentioned by Ne-whouse et al. (1980) is a “global” parameter, related tothe focused transducer as a whole, in the second case, thegeometrical broadening is a “local” feature, due to thefield phase distribution along the scatterer trajectory,which obviously varies from one zone to another of thesame beam.

Giving to “geometrical broadening” this secondmeaning, the equivalence between transit time and localgeometrical broadening does not hold anymore. Even ifthis might be seen as a disadvantage, in this way, wehave the possibility of studying the intrinsic broadeningin its fundamental components and how these compo-nents are influenced by complex field distributions.

The purpose of this paper is, first, to show how theDoppler spectrum bandwidth can be evaluated as thecomposition of transit time and local geometrical broad-ening at a chosen threshold.

Second, thanks to an original experimental method,transit time and local geometrical broadening are sepa-rately evaluated in different zones of two differentlyfocused transducers. Their combinations, according tothe previously presented equation, are then comparedwith the Doppler bandwidth directly calculated on theDoppler signal. Such global bandwidth, due to probefocusing, is addressed throughout the text as ISB.

A signal-processing approachConsidering the general form of the Doppler signal

reported in eqn (2) in case of Gaussian amplitude mod-

ulation function and parabolic phase distribution, thehypothesis is not far from the actual behavior for severalreal beams; the resulting signal is a Gaussian-weightedlinear chirp, as the one shown in Fig. 1b.

The spectrum of such a signal was evaluatedanalytically, both in the radar field (Cook and Bern-feld 1967; Skolnik 1970) and in the more generalsignal processing area (Papoulis 1977; Wilhjelm1993).

The spectrum bandwidthBD, evaluated at a thresh-old gf, can be related to some signal parameters, evalu-ated in the time domain at a thresholdgt. With referenceto Fig. 1, indicating byDt 5 Ds/v the transit timecorresponding to a thresholdgt 5 g and with Df 5f(k1) 2 f(k2) the frequency spread due to phase fieldcurvature along the scatterer trajectory at the samethreshold, the bandwidth of such signal evaluated atgf 5g is given by:

BD 54

p Îln2~g!

Dt2 1p2

16Df 2, (6)

as demonstrated analytically in the Appendix.This equation, independently evaluated within this

work, was found to be a generalization (i.e., valid for ageneric threshold) of a similar one obtained by Fish(1986) starting from Skolnik’s results (1970), which waslater used for estimating the influence of the phase fielddistribution on the Doppler bandwidth through a numer-ical simulation (Ata and Fish 1991). It could be easilydemonstrated that the latter equation is equivalent to eqn(6) for g 5 1/e.

If we indicate the term 1/Dt as the transit-timebroadening and the termDf as the local geometricalbroadening, eqn (6) demonstrates that, in general, ISBresults from a combination of the two contributions 1/Dt(transit-time broadening) andDf (local geometricalbroadening) that cannot, therefore, be considered eitherequal to each other, or equal to the global spectrumbroadening.

Furthermore, considering different zones along thebeam, the two terms exhibit opposite behaviors. Forexample, at the focus, the phase modulation is absent,because the phase fronts are parallel, so thatDf > 0 andthe transit time is minimal, exerting the strongest influ-ence on ISB. On the other hand, away from the focus, thebeam enlarges and the transit time grows, reducing itsbandwidth contribution; phase fronts, having a greatercurvature, give rise to a frequency modulation that pro-duces the most significant broadening contribution to thewhole BD.


METHODS

The need for experimental data to confirm this pointof view about the genesis of ISB led to develop differentmethods for simultaneously measuring both the transittime, Dt, and the local geometrical broadening,Df, for areal transducer.

The experimental data reported in the followingsection were supplied by a method, recently published byGuidi and Falteri (1999). It is based on a modified threadphantom equipped with a thin nylon thread, where sometight knots smaller than the US wavelength are used aspoint-like scatterers moving at constant velocity alongstraight trajectories (Fig. 2).

The transient signals produced by such targets wereacquired in its “in phase” and “in quadrature” compo-nents (IQ), and processed off-line with a MATLAB-based software procedure capable of automatically iden-tifying each knot. The user can set a proper absolutethreshold level, usually215 dB below the maximumamplitude of the converted signals, to discard the weak-est transients whose peak levels do not reach the thresh-old, for which the S/N ratio is insufficient for furtherprocessing. More details about such processing are re-ported in the cited article.

For the “surviving” transients, for example the oneshown in Fig. 3a, the instant corresponding to the peakamplitude is taken as the time origin, identifying thetarget passage from the beam axis. In this time-amplitudereference, two cross-points between the Gaussian-likeamplitude and a relative threshold,g, identify the in-stantst1 and t2, as shown in Fig. 3b. In this case, thethreshold was chosen to be 6 dB below the maximum tohave good S/N conditions, so thatg 5 0.5.

The parameters involved in eqn (6) were then eval-uated as:

Dt 5 t2 2 t1 (7)

Df 5 f~t2! 2 f~t1!. (8)

For calculatingDf, the approach was to derive numeri-cally the phase of the complex signal by estimating phasedifferences between samples divided for the time lag.The frequency behavior betweent1 and t2 was thenapproximated as linear and the trend estimated with alinear regression on the instantaneous frequency sampleslaying in such time range. In this way, the high varianceof the frequency estimates due to the numerical deriva-tion was greatly reduced (see Fig. 3c).

EXPERIMENTAL RESULTS AND DISCUSSION

The purpose of the experiments was to analyze theDoppler spectrum bandwidth in different frequency andamplitude modulation conditions, comparing the band-width evaluated directly on the spectrum of acquiredsignals, with the bandwidth evaluated as the compositionof the two elementary componentsDt andDf.

Two single-element 8-MHz focused transducerswere employed in the experiments. As summarized inTable 1, the two transducers, indicated hereafter as A1

and B2, have greatly different focusing properties.Thanks to the system described in the previous

section, the Doppler transients associated with the knotpassage through the beam, have been characterized interms of transit time and frequency modulation for bothtransducers.

Because the frequency and amplitude modulationpresent on the Doppler signal can vary significantly withdistance, the point targets were moved along trajectoriesintercepting the beam axis at a distance ranging from18.2 to 36.6 mm for transducer A in 92 steps 200mmlong, and from 12 to 80 mm for transducer B in 136 steps500 mm long. Five transients were acquired at eachdistance, so that 460 transients were analyzed for trans-ducer A and 680 for transducer B.

The smaller distance range explored for A is due toits strong focusing, which results in an insufficient S/Nratio out of the focal zone, while transducer B, for whichthe focusing is weaker, allows a good response even farfrom the transducer face.

In both cases, the beam axis-to–trajectory angle was45° and the target velocity was 18 cm/s.

1 Manufactured by ESAOTE Spa, Florence, 50127, Italy.2 Manufactured by Krautkramer Branson Inc. (formerly KB

Aerotech), Lewistown, PA 17044, USA.

Fig. 2. Mechanical device for producing a sequence of fivepoint-scatterers transients, produced by the 140-mm innerknots, with leading and trailing pulses generated by the bigger

knots.


In Fig. 4, a set of results related to the more stronglyfocused transducer are shown. In the upper part, the twoelementary contributions evaluated on time-domain sig-nals are shown. Figure 4a shows the amount of frequencymodulation,Df, measured at26 dB for each point target

(gray dots) and the average over up to five values (l).The frequency modulation is clearly larger in the nearand in the far field, but it is nearly absent immediatelyafter the focus (range 22–24 mm). Figure 4b showssingle and average transit time contributions, 1/Dt, at thesame threshold. Its behavior is opposite to the previousone: large bandwidths in the focal zone (20–24 mm)decrease for distances out of this range. Figure 4c showsthe composition of the averageDf (26 dB) andDt (26dB) according to eqn (6). Finally, Fig. 4d reports thespectral bandwidth calculated directly in the frequencydomain with three different thresholds, over the spectrumobtained by ensemble-averaging the spectra of each tran-sient. The triangles represent the26 dB bandwidths, thatshould be the same as those in Fig. 4c.

Fig. 3. Typical Doppler transient generated by the passage of a point-scatterer through the investigated beam. (a) IQsignals; (b) amplitude; (c) frequency. Setting a relative threshold on the amplitude, two instants are selected and the

corresponding transit time,Dt, and frequency variation,Df, have been measured.

Table 1. Parameter summary of the two transducers used inthe experiments

Transducer A(strongly focused)

Transducer B(weakly focused)

Focal depth (mm) 20.75 3026 dB beamwidth at the

focus (mm) 0.45 1Aperture diameter (mm) 9.4 6


By observing the bandwidth vs. distance behavior inFig. 4d (i.e., an estimation of ISB), it is clear that itdiffers from transit time and the frequency modulationcontributions shown in the upper two Figs. 4a and b,confirming that ISB is not equal either to the transit time,or to the frequency modulation broadening. On the otherhand, it is evident that their combination according toeqn (6), shown in Fig. 4c, gives a picture that, althoughnot identical, is very similar to the actual bandwidth vs.depth plotted in Fig. 4d. Apart from a considerablevariance in the measurement ofDt and Df, differencescan be ascribed to the not-exactly–Gaussian amplitudemodulation of the beam, by the crossing at 45° of thebeam axis, which involves a slight asymmetry in theweighting function, and for the frequency modulationthat, in general, may be not exactly linear.

This experimental result seems to confirm that, al-

though not exactly represented by eqn (6), in principle,ISB is originated by a proper combination of transit-timeand frequency modulation broadening.

Figure 5 reports similar results for transducer B,whose weak focusing involves a different weight of thetwo elementary contributionsDt andDf. Figure 5a showsa large frequency modulation before the focus (range10–30 mm), no modulation immediately after the focus(35–40 mm), and a moderate (almost constant) modula-tion for longer distances. As for the previous transducer,the transit-time broadening (Fig. 5b) is weak near thetransducer face, maximum at the focus, and decreasing atgreater distances.

Both the spectral bandwidth obtained as compositionof Dt andDf according to eqn (6) (Fig. 5c), and the band-width calculated directly with a26 dB threshold on thetransients’ spectra (Fig. 5d), give a similar decreasing trend,

Fig. 4. Transducer A. (a) Frequency modulation at26 dB vs. range; (b) 1/transit time at26 dB vs. range; (c)composition of the two frequency and amplitude modulation contributions according to eqn (3) withg 5 0.5; (d)

spectral bandwidth measured directly from the ensemble averaged spectra of signals from point targets.


which shows how ISB is, in this case, determined mainly bythe frequency modulation involved in the complex phasefield distribution in the near field, and, from the focus tolonger distances, its main contributor is transit-time becausealmost no frequency modulation is present in that zone. Forthis reason, the high phase measurement uncertainty in thenear field, clearly visible in Fig. 5a, is “reflected” on thebandwidth calculated as composition ofDt andDf (Fig. 5c),which differs from the bandwidth evaluated on the spec-trum for short transducer-to–target distances (Fig. 5d, range0–30 mm). On the contrary, for larger distances, a betterprecision in the measurement of both termsDf and 1/Dt wasobtained, as shown by the small scattering of instantaneousvalues with respect to the average values visible in the twoupper figures in the 30–80 mm range. This allowed us toobtain a “composition” bandwidth (Fig. 5c), characterized

in such range by a much cleaner trend, near to the behaviorof Fig 5d.

By comparing the results of Figs. 4 and 5, it ispossible to see that, in the former case, the highly vary-ing Dt andDf tend to cancel each other, giving an almostconstant ISB at every transducer-to–target range, and, inthe latter, the two elementary contributions impose theirbehavior on ISB in different beam zones. In both cases,the results seem to confirm that ISB is determined by acombination ofDt andDf.

CONCLUSION

The study in this paper identified the origin of ISBas a combination of transit-time broadening and localgeometrical broadening.

Fig. 5. Transducer B. (a) Frequency modulation at26 dB vs. range; (b) 1/transit time at26 db vs. range; (c)Composition of the two frequency and amplitude modulation contributions according to eqn (3) withg 5 0.5; (d)

spectral bandwidth measured directly from the ensemble average spectra of signals from point targets.


The theoretical analysis was based on well-knownsignal-processing approaches, used in several fields ofengineering and telecommunications.

The experiments conducted on a customized threadphantom allowed confirmation of the theoretical predictionsfor two differently focused transducers, producing defi-nitely different transit-time and geometrical broadening.

The main conclusion is, therefore, that it is notcorrect to confuse transit-time and local geometricalbroadening, when dealing with Doppler bandwidth forany purpose.

In particular, if ISB artefact cancellation in Dopplermeasurements is proposed (as on some recent commer-cial equipment), the importance of a beam characteriza-tion using knowledge of both transit-time and local geo-metrical broadening should be recognized. In otherwords, to properly characterize a beam for Doppler ap-plications, it is not sufficient to obtain amplitude beam-plots of a particular probe, giving information only ontransit times at different locations. Only with the simul-taneous knowledge of amplitude and phase field distri-butions can the ISB behavior of a beam be fully charac-terized.

Acknowledgement—The authors thank Piero Tortoli for several ex-tremely useful discussions.

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APPENDIX

The bandwidth equation reported in the text as eqn(6) can be found starting from the general expression ofa Gaussian-weighted linear chirp signal:

s~t! 5 e2~a2jb!t2, (A1)

where a and b are two real positive constants.This simplified expression is equivalent to the sig-

nal represented in eqn (2), considered in base-band in-stead of modulated in phase and amplitude around acarrier frequency . Hence, any consideration about thespectrum shape and, therefore, about the spectral band-width made on the signal, eqn (A1), holds true also forthe signal of eqn (2).

The spectrum of the signal represented in eqn (A1),given by:

^@s~t!# 5 S~ f ! 5 E2`

`

e2~a2jb!t2e2j2pftdt, (A2)

is found to be (Papoulis 1977):


S~ f ! 5 Î jp

b 1 jae

p2f2

a21b2 ~a1jb! (A3)

To find the spectral bandwidth with respect to a thresholdg times lower than the amplitude peak, in place of thisspectrum its normalized amplitude can be considered:

uS~ f !umaxuS~ f !u

5 eap2f2

a21b2. (A4)

The points at which this function falls to a valueg , 1identify a maximum and a minimum frequency whosedifference gives the spectral bandwidth. These two fre-quencies can be easily found from:

e2

ap2f 2

a21b2 5 g (A5)

2ap2f 2

a2 1 b2 5 ln g (A6)

f1 5 21

p Î2Sa 1b2

aD ln g;

f2 5 11

p Î2Sa 1b2

aD ln g.

(A7)

So the bandwidth is:

B~g! 5 f2 2 f1 52

p Î2Sa 1b2

aD ln g, (A8)

which is a real value becausea andb are positive, andg , 1.

To correlate this expression with the bandwidthcontribution given by amplitude and frequency modula-tion, respectively, the same threshold is applied to thetime-domain signal amplitude, obtaining two time in-stants given by:

e2at2 5 g (A9)

t1 5 2Î2ln g

a; t2 5 1Î2

ln g

a. (A10)

The difference between them gives the transit time:

Dt~g! 5 t2 2 t1 5 2Î2ln g

a. (A11)

The corresponding contribution of frequency modulationis calculated as the frequency variation between the twotimes t1 and t2. Because the signal phase variation is:

f~t! 5 bt2, (A12)

the corresponding frequency variation can be found as:

f~t! 51

2p

tf~t! 5

b

pt, (A13)

and the frequency modulation contribution as:

Df~g! 5 f~t2! 2 f~t1! 52b

p Î2ln g

a. (A14)

Hence, from eqns (A11) and (A14), respectively, thefollowing equations can be derived:

a ln~g! 5 24 ln2 g

Dt2 (A15)

b2

aln~g! 5 2

p2Df2

4. (A16)

Finally, by substituting them in eqn (A8), an expressionof B, as function ofg, Dt andDf, can be written:

B 54

p Îln2 g

Dt21

p2

16Df2. (A17)


Intrinsic spectral broadening (ISB) in ultrasound Doppler as a combination of transit time and local...

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