On estimating the amplitude of harmonic vibration
from the Doppler spectrum of reflected signals
Sung-Rung Huang
Deœartment
ofElectricalEngineering,
University
ofRochester,
Rochester,
New York14•27
Robert M. Lerner
Deœartment
ofRadiology,University
ofRochester,
Rochester,
New York14•42
Kevin J. Parker
Rochester
Centerfor BiomedicalUltrasound,University
ofRochester,
Rochester,
New York14627
(Received8 May 1990;acceptedfor publication6 August 1990)
The Dopplerspectrumof echoesfrom a sinusoidally
vibratingscattererhasdiscretespectral
linesweightedby Besselfunctionsof the firstkind. Becausethe signaland spectrumare
complicated
functionsof the vibrationamplitude,a numberof differentapproaches
havebeen
tried in the pastto estimatethe vibrationamplitude,givena receivedsignal.Here, a new and
simplerelationship
betweenthe spread(or variance)of the Dopplerspectrumandthe
vibrationamplitudeis derived.A methodof estimatingthe vibrationamplitudeis proposed
basedon thisrelationanda noisecompensation
procedureis alsodemonstrated.
The
performance
of the estimators
is studiedthroughsimulations.
High accuracyis predicted
underpropersamplingconditions
evenwhenthe signal-to-noise
ratiois poor.Slightdeviations
from single-frequency
oscillation,aswould be causedby nonlinearor nonidealmediumor
sourceeffects,are found to have little contribution to the total estimation error.
PACS numbers: 43.60.Gk, 43.20.Fn, 43.30.Es
phones,and determinedthe vibrationamplitudeby fitting
the theoreticalspectrumwith an unweightedleast-squares
INTRODUCTION
The generalproblemof Dopplershiftsfrom objectswith
time-varyingvelocityin an inhomogeneous
or layeredmedium isquitecomplex.It is still a subjectof controversyinvolv-
inglinearandnonlinearderivations.
1,2However,whenthe
scatteringobjectis vibratingslowlysoasto producea w.avelength much larger than the geometricaldimensionsof the
scattereritself,the Doppler spectrumof the signalsreturning from sinusoidally
oscillatingstructuresis similarto that
of a pure-tone
frequency
modulation(FM) process.
3 This
spectrumis a Fourier serieswith spectrallineslying above
and belowthe carrierfrequency.The spacingbetweenspectral harmonicsis equalto the vibrationfrequency,and the
amplitudesof harmonicsaregivenby differentordersof Bes-
selfunctionsof the firstkind.3 A numberof applications
in
acoustics,optics,and radio haveled to researchon extracting the vibrationparametersfrom a measuredDopplerspectrum. Amplitude, phase,and frequencyof the oscillating
structureare the mosttypical parametersto be estimated.
Many techniqueshave beenproposedto estimatethe
vibrational
parameters.
Holenetal.4measured
thevibration
frequencyof oscillatingheart valvesby looking visuallyat
the spacingbetweenharmonicsin the ultrasoundDoppler
spectrogram.The vibrationamplitudeisestimatedby counting the number of significantharmonicsunder a certain
threshold.This procedureis relativelycoarsebut is related
to the observationin FM that the bandwidthis roughlyproportional to the modulationparameter,or amplitudeof os-
cillation.3 Taylor5'6studiedthe lasercalibrationof micro2702
J. Acoust.Soc.Am.88 (6), December1990
approximation.
Lerneret al.7 in their newtechnique
for
medicalimagingof elasticpropertiesof tissuecalled"sonoelasticityimaging,"havesuggested
the estimationof vibration amplitudeby calculatingthe ratio of the two largest
harmonics.
Jarzynskiet al.8 undertakea similarestimation
for precisionmeasurement
of the soundfieldswith laser
Dopplerby comparingthe ratioof carrierandfundamental
harmonics. Similar estimation had also been made by Cox
andRogers
9to studythevibrational
motionof auditoryorgansin fish.Observingthat the ratiosof the adjacentBessel
coefficients
increasemonotonically,
Yamakoshiet al.1ø
cameup with an estimatorby comparingthe relativemagnitude of the adjacentharmonicsin their studyof tissuemotion. They alsoderivethe vibrationphasefrom the fundamentalspectralcomponents
of twoquadraturechannels.
All
of thesetechniques
canbebroadlyclassified
in thesamecategory,or approachto estimationof the vibrationalparametersusingsomeratio of amplitudes.One of the disadvantagesof theratiomethodsisthattheyrequireeitherintensive
computationor large look-uptablesof theoreticalBessel
functionsfor comparisonwith the measureddata. Besides,
ratiomethodswork well onlywhenthe argumentof the Besselfunctionis small,which posesa severelimitation on the
rangeof estimation.Furthermore,in practice,the performanceof the ratio methodsis highlydegradedsincealmost
all Doppler spectrasufferfrom poor signal-to-noise
ratio.
Finally, a sophisticatedalgorithmis requiredto determine the bestselectionof the harmonicpair to be compared.
0001-4966/90/122702-11500.80
@ 1990Acoustical
Societyof America
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Therefore,this work presentsa simpleand noise-immune
algorithmfor vibration estimation.
The problemof vibrationamplitudeestimationis approachedthroughthe measurement
of the spectralspread
(or variance)of the Dopplerspectrum.The proposedestimationtechniquescan be implementedwithout difficulties.
For instance,in clinicalapplications
of Dopplerultrasound,
onecouldobtainthe necessary
parameterswith slightmodificationsof existinginstruments.Significantimprovement
on estimationaccuracycan be further achievedwith a noise
correctionalgorithm.The theoreticalderivationsof the estimationandresultsof simulationareshownin the following
section.The effectsof noise,sampling,and nonlinearityon
the estimator performanceare also demonstratedsubsequently.
Usingtrigonometricidentities,Eq. (7) canbe replaced
by theseries
3
s,(t) =,4 •
J. (/Y)cos[coot
+ n(coLt
+ q•)], (8)
where the modulationindex or the argumentof the Bessel
functions/3is directlyrelatedto the vibrationamplitudeof
the velocityor displacementfield as follows:
/3=
Aco.,
coL
=
2v.,coocos0
coLC0
=
2•'.,coo
cos0
= 4rr •
cos 0,
/•0
C0
(9)
whereAois the wavelengthassociatedwith the wave of frequencycooand propagationspeedCo.
Thus, giventhe Doppler spectrumas describedabove,
the estimationof thevibrationamplitudeisequivalentto the
estimationof the Besselargument/3. The exact spectral
I. THEORY
shapeof the Dopplersignaliscomplicatedanddependenton
A. Derivation of the Doppler spectrum
the parameter/3. Examplesof Doppler spectrafrom low,
Sincethe FM spectrumis well known,we brieflysum- medium,andhighvaluesof/3 andthe two quadraturecommarize results in this section and introduce our notation.
ponentsof the correspondingDoppler signalare given in
When a movingobjectis illuminatedwith an incidentlaser, Figs. 1 and 2, respectively.Given a measuredspectrum,it
radio,or acousticwave,the detectedbackscattered
signals canbeseenthat thebackwardestimationof/3isnot straightfromthat movingobjectwill demonstrate
a frequency
shift forward,evenin the noise-freecase.Experimentaltime-freknownastheDopplershift.If thescatterer
isoscillating
with quencydisplaydata from an ultrasoundB-scaninstrument
thevibrationvelocitymuchslowerthanthewavespeedand with Doppler capabilitiesare givenin Fig. 3. The vibration
the vibrationfrequencymuchlessthanthe cartier (incident frequencyis fixed, while the vibration amplitude,which is
wave) frequency,the spectrumof the detectedscattered proportional to the parameter/3, is increasedfrom left to
More sidebands
plusaliasingshowup
wavewill besimilarto that of a pure-toneFM process
since right, thendecreased.
when
the
parameter/3
increases.
As
shown
in Fig. 2, the two
the instantaneous
frequencyof the scatteredwaveshas a
quadrature
components
of
the
Doppler
signal
correspondDopplershiftproportionalto thevibrationvelocity.Assume
ing
to
those
in
Fig.
1
are
also
complex.
Thus
a
time
domain
that the transmittedor incidentsignalis
estimationapproachis not obvious.The estimationis even
st (t) = cos(coot),
( 1)
more difficultwhen the backgroundnoiseis mixed with the
andthe scatterers
arevibratingwith the form
signal,as shownin Fig. 4.
g(t) = •'msin(co,.torq),
(2)
v(t) =•(t) = v,,cos(co,.t
ørq),
(3)
where•e(t) is the displacement
of the vibration,v(t) is the
velocityof thevibration,co,.is thevibrationfrequency,
q is
the vibrationphase,g,, is the vibrationamplitudeof the
displacement
field,and v•, = co,.•e,,is the vibrationamplitude of the velocityfield.
The instantaneous
frequencyof the receivedor scattered waves will be shifted to
coo+ Acoa,
(5)
Aw,, = 2v,,Wocos0/Co,
(6)
wherecotis the instantaneous
frequencyof the scattered
waves,Aco
a istheDopplershift,Coisthepropagation
speed
of illuminatingwaveat frequencycoo,and 0 is the angle
betweenthe wavepropagationand the vibrationvectors.
Therefbre,the receivedor scattered
wavescanbe written as
(7)
sincetheinstantaneous
frequencyis,by definition,givenby
the time derivativeof the argumentof the carrier cosine
wave.
2703
Two Dopplerspectralparameters,spectralvariance(or
spectralspread)and meanDoppler frequency,are usually
defined as:
0'2o,
=
(co-- •)2S(co)dco
S(co)dco
(10)
and
(4)
Acoa
= Aco,,cos
(colt + •v),
s,(t) =A cos[coo
t + (Aco,,/coL)sin(co•t
+ qv)]
B. Vibration estimation from Doppler spectral spread
• =
coS(co)dco
S(co)dco
,
( 11)
wherec%istheDopplerspectralspread(a2•o
isthevariance
or secondmoment), • is the mean frequencyshift of the
Doppler spectrum (the first moment), and S(co) is the
Doppler power spectrum downshiftedto baseband.Note
that the mean Doppler frequencyshift • is not necessarily
zero sincethe Doppler signalis generallycomplexand the
Doppler spectrumS(co) can be asymmetric.
If the scattereris vibrating, the Doppler power spectrum that canbederivedfrom Eq. (8) in theprevioussection
is
S(co)-- 2•r •
J• (/3)6(co
-- nco•
),
(12)
n•--•-
J. Acoust. Soc. Am., Vol. 88, No. 6, December 1990
Huang eta/.' Amplitudeof harmonicvibration
2703
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#=3
o
o
'
0
(a)
I
16
'
'
'
I
'
32
'
'
I
'
'
4.8
I
'
Id•
p=3
32
i
0
64
Normalized
Frequency
16
#=•o
o
i
i
i
16
48
i
,
32
,
,
i
i
48
,
64
Normalized
Frequency
•
64
Normalized
Frequency
i
p=30
0
(e)
16
32
48
64-
Normalized
Frequency
/• = 30
ø I
I
16
32
48
64
Normalized
Frequency
0
(•)
16
•2
48
64
Normalized
Frequency
FIG. 1.Examples
of noise-free
Dopplerspectra
for low/5'= 3: (a) linearand(b) logscale;medium/5'
= 10:(c) linearand (d) logscale;high/5'= 30: (e)
linearand (f) logscale.Normalizedfold-overfrequency
flora= 64, normalizedsegment
lengthTFET= 4, for all cases.
2704
J. Acoust.$oc. Am., Vol. 88, No. 6, December 1990
Huang eta/.' Amplitudeof harmonicvibration
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-,-,
c-
0.0
0.2
0.4
0.6
0.8
1.0
Normolized Time
(-
fi= 10
o
0.0
0.2
(d)
o
o
0.4
0.6
0.8
1.0
Normalized Time
/•= 30
(-
o
E
o
(Y) o
I
•T
0.0
0.2
0.4
0.6
0.8
1.0
Norm01ized Time
lb)
o
0.0
0.2
(•)
/•= 10
-•c-
0.4
0.6
0.8
1.0
Normalized Time
/•=30
o
08
o.o
t•)
0.2
o.•
o.•
o.•
NormalizedTime
to
•
If)
o.o
0.2
o.•
NormalizedTime
FIG. 2. Examples
of twoquadrature
components
of noise-free
Dopplersignals
forlow/3= 3: (a) in-phase
and (b) quadrature-phase
component;
medium
/3= 10:(c) in-phase
and(d) quadrature-phase
component;
high/3-- 30:(e) in-phase
and(f) quadrature-phase
component.
Sincethefunction
isperiodic,
only onecycleis shownin the graphs.
2705
J. Acoust.Soc.Am.,Vol.88, No.6, December1990
Huangeta/.' Amplitude
of harmonic
vibration
2705
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(8)
fi=10
0.0
16.0
32.0
48.0
64.0
4•.0
64.0
NorrnolizedFrequency
(b)
ll•= 10
c:)
FIG. 3. Experimentalobservation
of Doppler spectrumfrom vibrating
structuresunderclinical B-scanultrasound.The lower portion is a time
( horizontal) -frequency(vertical) displayof data from the regionselected
by the cursorasshownon the upperportion.Vibrationfrequencyis held
fixed,whilevibrationamplitudeisincreased
andthendecreased
fromleftto
right.
wherethe powerspectrumhasbeendownshiftedto zero frequency,asby quadraturedetection.
For thisparticularBesselspectrum,the meanfrequency
• is zerosinceJ_, (/3) = ( -- 1) "J, (/3) and the powerspectrum is thereforesymmetricaboutzerofrequency.Thus, the
spectralspreadcanbe calculatedfrom the zeroth and second
momentsof the spectrumdefinedas
mo= •
O
r2,,(/3),
(13)
m2=
(noL) J • (/3),
(14)
Z
wheremk is the k th momentof the Doppler spectrum.
However, the zeroth moment is the total energyof the
signaland is equalto unity. One can easilyshowthis using
the followingmathematicalidentity:
eiasin
0= •
Jn(•) ei"O.
( 15)
Squaringtheaboveequationandreplacing0 with -- 0 in one
term, we have
1 • eilysinOeil•sin( - o)
Z
m=
2706
--c•
Z
n=
Jm(•)Jn(•) ei(m
- n)O.
--oo
J. Acoust.Soc. Am., Vol. 88, No. 6, December 1990
ß
.
0.0
.
115.0
32.0
NormolizedFrequency
FIG. 4. Exampleof noisyDoppler spectrumwith/g = 10, signal-to-noise
ratio SNR = 20 dB, normalizedfold-overfrequencyfto•a = 64, and normalizedsegmentlengthTFFT --4, (a) linearscale,(b) log scale.
Integratingthe aboveequationover one period (0 = 0 to
2rr), and usingorthogonalityof exponentialfunctionsfor
m•//,
1-- •
j2.(/3)__mo,
(17)
a result that has been noted in the literature. •
The secondmoment can be derived in the sameway by
takingthe firstderivativeof the Eq. (15) with respectto 0,
(16)
/• COS
Oe
i/3sin
0__ Z
n•
nJ, (•)einO.
Huang ot a/.' Amplitudeof harmonicvibration
(18)
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Squaringfrom the aboveequationasbefore,
j 2,,
(,8)6(0
- nco
r)+N(co
))dco
• 2COS
20eifisinOeifi
sin
( - o)
=
•
m•
•
--oo
mnJ,•
(/•)J,(/•)ei('•-
X
(19)
J•,(l•)6(co
-- ncoL
) + N(CO)
do
= (mzs + m2.•v)/(mo,s+ mo,•v)
Then, integratingover oneperiodagain,
•--
•
n•
= [o•o,s
+ (1/SNR)o•,,•v]/[1+ (1/SNR)],
2
2
2 -- •
2
nJ•(/5')=m2.
(20)
In general,all momentsof the Besselspectrumcan be
calculatedfrom derivativesof Eq. (15) by properdifferentiation,squaring,
andintegration.Low-ordermomentsof the
Besselspectrumare givenin Table I.
Thesemomentsof the Besselspectrumcan alsobe calculatedfrom the generatingfunctionof Besselfunction:
/(•,Z)•
•
zngn(•)=e(B/2)[z-•l/z)l
(21)
which can be found in a standard handbook on bessel func-
tions.11By takingthe 1stthroughthek th derivativeof the
generatingfunctionwith respectto z and substituting
z= 1
into the resultingexpressions,
all momentsof the Bessel
spectrum,asfunctionsof theparameter/5',
canthenbecalculated from lower-ordermomentsby squaringand simplealgebraicmanipulation.
From thispointof view,the Besselspectrumis actually
a one-parameterfunction.Therefore,the secondmoment
servesasa goodestimatorof the spectrum.
One can estimatethe vibrationfrom the Doppler spectral spreadas
oao_,
= (m• -- m•2)/mo = [m• -- (•) • ]/mo,
(22)
and for this caseof the Doppler spectrum,• = O,thus
or
/5'= x/•(rro•/cot).
(24)
This indicatesthat the amplitudeparameter/5'can be
estimatedfrom the standarddeviationof the power spectrum. This straightforwardresulthas,apparentlynot been
previouslyderivedfor the caseof FM broadcastor Doppler
spectrumfrom vibratingobjects.
(25)
where• is the meanDoppler shift of the noisysignalgiven
by
• =
co
J• (/5')6(0 -- ncot.
) + N(co) dco
X
j2 (/5')6(0- not.) + N(co) do
(26)
ink,s andink,•varethek th momentof signalandnoiseabout
thecorresponding
meanfrequencies
respectively,
rro•,s
isthe
Doppler spectralspreadof vibration only, rro•,•vis the
Dopplerspectralspreadof noiseonly,andSNR isthe signalto-noiseratio givenby
SNR =
j 2 (,8)6 (co-- not-) do
X
N(co)dco
=mo,s/mo,•v= 1/mo,s.
(27)
As long asthe noiseis stationary,the momentsof noise
power spectrumcan be estimatedwhen the vibrationis removed or halted. Once the noise moments have been esti-
mated, the noise-freevibrational Doppler spectralspread
can then be estimatedfrom the noisysignalas
cr2o,,s
= cr2•,
[ 1 + (1/SNR)] -- (1/SNR)a2o,,•v
.
(28)
In someapplications,the vibrationis inherentand cannot be controlledexternally.The noisecompensation,
in this
case,can be doneby estimatingthe signal-to-noiseratio as
well as the Doppler spectralspreadof the noisefrom the
finite bandwidthwhite noiseassumptionas follows:
mo,•v=
Nodo = 2NOB,
(29)
co2N
odo = 3 3
'"2NoB
B
'
(30)
--B
in2,N:
o;o,,•v
= (2NOB
3/3)/2NOB= B 2/3,
(31)
whereNoisthe powerspectraldensityof thewhitenoise,and
B is the one-sided bandwidth
of the white noise.
C. Noise correction algorithm
In practicalsituations,noisepresentsproblemsin parameterestimation.The Dopplersignalstendto be 30-50 dB
lower than the carrier in many applications,therefore,the
signal-to-noiseratio for Doppler signalis usuallypoor. Additive, stationary,and uncorrelatednoisecan be removed
from the Dopplerspectralspreadvibrationestimator.If stationary uncorrelatednoisewith power spectrumN(co)•is
added in the received backscatteredsignal, the noisy
Doppler spectralspreadrro•can be written as
2707
J. Acoust.Soc. Am., Vol. 88, No. 6, December1990
TABLE I. Low-ordermomentsof the Besselspectrum.
All odd moments
0th moment
2ndmoment
4thmoment
6thmoment
0
1
«/•2
«/•2_•_38•4
«/•2_•½•4 _••_6•6
Huangeta/.' Amplitudeof harmonicvibration
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Even when the noiseis not white, the noisecompensation is still possibleaslongasnoisepowerandnoisespectral
spreadcan be estimateda priori by statisticaltechniques.
II. RESULTS
AND DISCUSSIONS
SimulatedDoppler spectraare obtainedby taking fast
Fourier transform (FFT) of finite segmentsthat are composedof the two quadraturecomponents
of Eq. (7). Figures
5-9
are
families
of
the
estimation
errors
[ (/•-/•)//•]
X 100% asfunctionsof variousparametersin
the estimationprocess,where/• is the estimatedvibration
parameterand/• is the true parameter.
A. Effects
of noise
The estimationerrorsare plottedasfunctionsof signalto-noiseratio from 0-40 dB in Fig. 5 (a). White Gaussian
noise was added independentlyinto the two quadrature
components.
The normalizedsamplingfrequency(defined
assamplingfrequencydividedby thevibrationfrequency)is
128. Or equivalently,the normalizedfold-overfrequency
(definedas fold-overfrequency,or aliasingfrequency,or
half-sampling
frequencydividedby thevibrationfrequency)
is 64. A high samplingrate wasusedto reducethe aliasing
Noise-Dee
o
og
•
;,, •.--.-/•:•o
\
• k'v••
"%•• "•"--...
.................
2;100
Normalized Fold Over Frequency
o
0.0
1•.0
(a)
20.0
30.0
40.0
Signal-to-Noise
Ratio
SNR = 40 dB
o
o
o /•=1
,
o.
I•
o.o
(b)
'
I
•0.0
'
i
20.0
i
30.0
FIG. 5. Plot of estimationerrorsin percentageagainstsignal-to-noise
ratio
from 0-40 dB at normalizedfold-overfrequencYfro.d
= 64 and normalized
segmentlength TFrr -- 4, (a) withoutnoisecorrection(theoreticalprediction and resultsfrom one simulation), (b) with noisecorrection(average
overfivesimulations).Note theexpandedscalein (b) showingdrasticerror
reductionachievedby the noisereductionprocedure.
2708
J. Acoust.Sac. Am., Vol. 88, No. 6, December 1990
#=5
'
/•=•o
4-
__/•=
x
20
2,1½
40.0
Signal-to-Noise
Ratio
(dB)
•
o
lb)
NormalizedFoldOver Frequency
FIG. 6. Plot of estimationerrorswithout noisecorrectionagainstnormalizedfold-overfrequencywith TFrr = 4, (a) noise-free,(b) signal-to-noise
ratio SNR = 40 dB. As samplingfrequencyincreases,underestimations
originatedfrom aliasingdecreasein all cases.In (b), noisespectralspread
causeslarge errors at high samplingratessincewidebandwhite noiseis
employedin simulations.
Huang eta/.' Amplitudeof harmonicvibration
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ageestimationerrorsover five simulations.The estimation
errors are swingingback and forth around zero as the parameter/3 changes.This indicatesthat the estimatoris unbiased.The maximum estimationerror for poor signal-to-
..............
/5'=5
....... /5'=10
•.-/•
= :•o
--- -/• = 50
noise ratio at 0 dB is still within 4%. The estimation error of
high/3 is muchlessthan that of low/3, asexplainedearlier.
B. Effects of sampling
The aliasingeffectcan be seriouswhen the fold-over
frequencyor samplingfrequencyis not proper.Figure 6 (a)
shows the noise-free estimation
o.
0.0
1.0
' 2.0
' ' 3.0
' ' 4.0
' ' 5.0
' ' 6.0
' ' 7.0
' ' 8.0
' ' 9.0
' '
Normolized
Segment
Length
10.0
FIG. 7. Plot of noise-freeestimationerrors againstnormalized segment
length TFrr -- 4. The finite-lengtheffectproducesoverestimation.
When
thesegment
lengthisanintegralmultipleof vibrationperiod,theestimation
error dropsto a minimum.
error at normalized
fold-
over frequencyexpressed
as a powerof 2. The effectof the
impropersegmentation
isminimalheresincethe normalized
total lengthisfour, aswill beexplainedlater. The estimation
errorsblow up at the point when the parameter/3is larger
thana threshold.
Thissampling
criterioncanbeexpressed
as
X ----2ffold•/2timaxiL,
(32)
wheref• is the samplingfrequency,froldis the fold-overfrequency,andfL is the vibrationfrequencyin Hz.
It is interestingto notethat the aliasingalwaysresultsin
underestimation.The reasonis that harmonicshigher than
fold-overfrequencyare foldedback into lower frequency
contentsandthe co
2 or n2 term associated
with the higher
error resultingfrom the finiterepresentation
of the infinite
spectrum.The normalizedsegmentlength (definedas segmentlengthdividedbythevibrationperiod)is4. Two setsof
curvesare shownin Fig. 5 (a). The rapidly fluctuatingone
comesfrom a singlesimulation,whilethe smoothoneis the
theoreticalpredictionof the deviationof the estimation
withoutanynoisecorrectionusingEq. (28). The simulation
resultsagreewith the theoreticalprediction.Theseresults
showthat thesignal-to-noise
ratio mustbehigherthanabout
•30 dB to achieveacceptableaccuracyof estimationunless
noisecorrectionprocedureis performed.
From Eq. (25), the performance
of the vibrationestimator without noisecorrectionwill be degradedwhen the
signal-to-noise
ratio is low or the spectralspreadof noise
only is largecomparedto that of vibrationonly. When the
parameter/3is small,the spectralspreadof the Besselspectrum is narrow sincethe bandwidth (or spectralspread) is
proportionalto theparameter/3from Eq. (23). In thiscase
of low/3, the spectralspreadof noiseis comparableor even
largerthan that of vibrationonly, thusthe performanceis
highlydegradedby the additionof the noise.When the parameter/3is large as in widebandFM, the performanceof
the estimatoris fairly goodaslong as/3is still smallerthan
normalizedfold-overfrequency.The overestimation
is due
to the subtractionin Eq. (28).
Figure 5(b) is a plot of the estimationerror with the
noisecorrectionalgorithm.The vibrationwas removedto
estimatethe momentsof the noise-onlyDoppler spectrum,
andthentheparameter/3is estimatedfrom the noisysignal
byEq. (28). Samplingconditions
arethesameasthatin Fig.
5 (a). Note the expandedverticalscaleisFig. 5 (b). Obviously, dramaticerror reductionis achievedthroughthe noise
correctionprocedure.The resultsof Fig. 5(b) are the aver2709
J. Acoust. Soc. Am., Vol. 88, No. 6, December 1990
harmonicsin calculatingthe secondmomentis smallerthan
what it shouldbe in the theoreticalexpressions(10), (14),
and (23).
Figure6 (b) isthesamesimulationwith 40 dB signal-tonoise ratio. Interestingly,as the fold-over frequencyincreases, the estimation errors increase as a second-order
polynomial.This resultsfrom the property of the white
Gaussiannoiseusedin simulation.Sincethe spectralspread
of a uniformspectrumisproportionalto the secondpowerof
the bandwidth,asshownearlierin Eq. (31 ), the estimation
error increases
approximatelyasa second-order
polynomial
in normalizedfold-overfrequencywhenno filteringprocess
is involved.Therefore,filtering shouldbe taken if the sampling frequencyis increasedto avoid the aliasingerror. In
practice,filterscan be designedto optimizethe estimation
accordingto the actualspectralmomentsof signalandnoise.
Figure7 is the plot of noise-freeestimationerror against
normalizedtotal samplingsegmentlength Tvrr. To exclude
the effectof aliasingerror,the normalizedfold-overfrequency is setto 64. The estimationerror is minimal wheneverthe
normalizedtotal samplinglengthis equalto an integer.The
largeerror of nonintegralnormalizedtotal samplinglength
is due to the sharpdiscontinuityof the time domainsignal
from the inherentperiodicitywhenFFT analysisisexploited
without windowing.This indicatesthat windowingor synchronizationis requiredin practicalanalysis.Note that the
finite samplinglength alwayscausesoverestimation.This
happensas expectedsincethe sharpdiscontinuityin time
domaingeneratessignificantsidebandsthat can subsequently increasethe spectralspread.
C. Effects of nonlinearity
If the vibration is not perfectlysinusoidaldue to some
medium or vibration sourcenonlinearity, the Doppler specHuang otaL' Amplitudeof harmonicvibration
2709
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• = 10, Nonlinearity = 10%
/• = 10, Nonlinearity = 2 %
(c)
•.o
•.o
•4.0
0.0
16.0
•.o
•.o
NormalizedFrequency
NormalizedFrequency
• = 10, Nonlineari•y = 2 %
• = 10, Nonlinearity = 10 %
(d)
I' 0.0
.......
16.0
•.0
,•.0
6,.0
,•.o
0.0
NormalizedFrequency
•.o
NormolizedFrequency
•.o
FIG. 8. Examplesof noise-freeDoppler spectrumfor 2% and 10% nonlinearitywith fundamentalvibrationamplitude/3= 10, normalizedfold-over
frequencyffo•a
= 64 andnormalizedsegmentlengthTFvr = 4. Note that thepeaksof thespectrashiftfrom thecaseof no nonlinearity(Fig. 1). The relative
ratiosof harmonicsalsovary asa functionofnonlinearity,vibrationamplitude,and fold-overfrequency:(a) 2% nonlinearity,linearscale,(b) 2% nonlinearity, log scale,(c) 10% nonlinearity,linearscale,(d) 10% nonlinearity,log scale.
tral shapewill deviatefrom the one parameterBesselspectrum. Assumingthe vibrationis periodicand can therefore
be represented
by a Fourier series,both the ratio of harmonics and the spectralspreadwill be differentfrom that of a
pure sinusoidalvibration. Assumingthat only the fundamental and secondharmonicsare significantin the vibration, the returnedsignalnow canbe written as
Sr (t) = A COS
[ COot
-t-/• sin(CO
Lt -t-• 1)
-t-/•2sin(2COL
t -t-•%)],
J. Acoust. Soc. Am., Vol. 88, No. 6, December 1990
An expressionfor the Fourier seriesexpansionof the
abovesignalcanbe obtainedfrom the analysisof multitone
FM 3withslightmodification
asfollows:
$r(t) =
•
•
J• ([3)J.([32)
Xcos[COo
t + m(CO•t+ qo•)+ n(2CO•t
+ q02)
].
(34)
(33)
where/3and/32are the modulationindicesof the fundamen2710
tal and secondharmonicsof the vibration, respectively.
The nonlinearity is definedas
Huang eta/.' Amplitudeof harmonicvibration
2710
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error comesfrom the nonlinearity.If the nonlinearityis removed, estimator without noise correction gives accurate
estimationwithin 0.01%, while the accuracyis increasedto
0.0001%
after noise correction.
If widebandwhitenoiseis addedbut no filteringprocess
is used (same parameters,exceptffold = 64 fL ), then the
estimation error for estimator without noise correction rises
to 30.4%, while the error remains as small as 1.75% after
,
,
,
5.0
10.0
Nonlineority
(%)
FIG. 9. Plot of noise-freeestimationerrorsagainstnonlinearity.The errors
are almost the samefor all valuesof/•' from 0.1-50. The errors due to nonlinearity are lessthan 2% when nonlinearityis lessthan 10%.
Nonlinearity = N2 • (/3://3) X 100%.
(35)
Figure 8 a-d showsthe spectraof 2% and 10% nonlinearityfor the caseof/3 = 10.The spectralshapesand peaks
are differentfrom that of pure tonevibration (Fig. 1). Surprisingly, this doesnot affect the Doppler spectralspread
much.Estimationfor noise-free,smallaliasing,andproperly
sampledsignalhasbeenperformedas shownin Fig. 9. The
resultsshowthat nonlinearitylessthan 10% contributesless
than 2% estimationerror. In comparison,all conventional
estimatorsthat utilizeamplituderatioswouldbedramatically affectedby 10% nonlinearity.This can be appreciatedby
comparing the patterns of peak amplitudes in Fig. 1(c)
againstthosein Fig. 8 (c).
D. Combined
effects
A casethat is typicalin "sonoelasticity
imaging,
"7'12
with the followingparameters,was performedto showthe
combined error:
Co= 1.5X 10• cm/s,
fo = Wo/2rr= 7.5 MHz (i.e., ,go= 0.2 mm),
0 = 100,
f• = co•/2•r = 200 Hz,
/3 = 10 (i.e., •m = 0.16 mm, Vm-- 20.3 cm/s),
SNR = 20 dB,
noisecorrection.The large error is due to the large wideband-noisespectralspreadas predictedin Eq. (31 ). This
suggests
that either noisecorrectionor filteringis necessary
for noise reduction when the samplingfrequencyis increasedto reducethe aliasingerror in practicalimplementation. If filtering is applied, the effectson signal spectral
spreadmust be taken into accountduring the filter-design
phase,sincehigherharmonicsof the signalsare removedas
well. It shouldbe notedthat the performanceof the estimator with noisecorrectioncanbeimprovedby increasingsampling frequencywithout filtering.
III. CONCLUSION
We haveanalyzedthe signalreflectedfrom a sinusoidally vibrating object. An estimator of vibration amplitude
based on the derived relation between Doppler spectral
spreadand vibrationamplitudehasbeenproposed.A noise
correctionalgorithm is also proposed,which can improve
the estimation accuracy dramatically. Simulations show
good results within 4% error given signal-to-noiseratios
rangingfrom 0-40 dB. Adequatesamplingfrequency,asgiven in the text, must be satisfiedaccordingto the expected
maximum vibration amplitude. Proper filtering can reduce
the spectralspreadof noiseand then reducethe estimation
error. Windowingor synchronizationisrequiredin practical
implementationto reducethe finite length effect.The proposedestimatorsurvivesthroughvibrationnonlinearityless
than 10%, whereas other known estimators that make use of
the harmonicratio of amplitudesdo not tolerateslightnonlinearities.Overall, the proposedestimatoris robustin the
presenceof noisecomparedto earlier estimators.The estimation can be obtainedvia simplecalculationsfrom the parametersin someexistingDoppler instruments,especially
thoseavailablein clinical Doppler ultrasound.To displaya
realtime vibration image, existing faster time domain
Doppler spectralspreadestimators(e.g., Refs. 13, 14) can
be appliedwith modifications(e.g., changingscanningpatterns and samplingfrequencyto reduce,theeffectof vibration phase)and/or synchronization(with low-frequencyvibration) to display the vibration amplitude. The results
shouldbe relevant to echocardiology,sonoelasticityimaging, lasercalibrationof soundfieldsand vibration,and other
radio, radar, and sonarapplications.
f•old= 12.8fL = 2560 Hz (i.e., fs = 5120 Hz),
TFFT-- 10 TL = 50 ms (i.e., LFrr ----256) l,
N2= 10%,
ACKNOWLEDGMENTS
error without noise correction is 3.39%, while the estima-
The supportandencouragement
of Dr. R. Gramiakand
Dr. J. Holen are gratefullyacknowledged.
This work was
supportedby the Departmentof ElectricalEngineering,
tion is reduced to 1.97% after noise removal. Most of the
University of Rochester.
where T•r is the segmentlengthfor FFT analysisin ms,
and L •r is the numberof samplesin T•r. The estimation
2711
J. Acoust. Soc. Am., Vol. 88, No. 6, December 1990
Huang eta/.' Amplitudeof harmonicvibration
2711
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