Download zone inverse Doppler effect

Radio Science,Volume 33, Number3, Pages463-474, May-June 1998
Contribution to the problem of near-zone
Doppler effect
Yehuda
Ben-Shimol
inverse
and Dan Censor
Department of Electrical and Computer Engineering,Ben-Gurion University of the Negev
Beer-Sheva, Israel
Abstract. The existenceof an inverseDoppler effect in free spaceis again scrutinized,
followingsome papers predicting the existenceof such phenomenain the near zone of a
moving oscillatingthree-dimensionaldipole. In the presentpaper the wave scatteredfrom
a perfectly conductingthin cylinder moving in the presenceof a plane electromagnetic
wave is analyzed. The responseof such a cylinder may be consideredas due to either a
two-dimensionalmonopoleor a two-dimensionaldipole in accordancewith the polarization
of the incident wave. An intensivenumerical spectral estimation basedon the fundamental
definitionsof the terms "frequency,""spectrum," and "uncertainty" is performed on the
scatteredwave at various ranges. An inverseDoppler effect was not found for the twodimensionalcase. The same analysiswas applied to the caseof a moving three-dimensional
radiating dipole, reconfirmingpreviousresults which showedthe presenceof an inverse
Doppler effect in the near zone of the three-dimensionaldipole.
1.
Introduction
sionson inverseDopplereffectin freespace[Engheta
When an observerin free spaceis in motion rela- et al., 1980;Engheta,1990],whichanalyzethe elective to a monochromaticsource,the frequencymea- tromagneticfield of an oscillatingthree-dimensional
of a moving
sured by him/her will be higher than the source- (3-D) dipolein freespacein thepresence
observer.
However,
these
two
cases
are
different,
refrequency(blue shift) when approachingthe source
castingtheir different spectral contents. In the case
and lower (red shift) when receding. This effect is
media(e.g.,unmagnetized
coldplasma),
known as the Doppler effect [Doppler,1842] and of dispersive
has servedas a major subject for researchersdealing the discussionmay be restricted to a plane wave of
with variousapplicationsof this phenomenon(see certain frequency, and the Doppler effect is thereGill [1965]for a goodintroductionto the Doppler fore evident. On the other hand, the componentsof
effect). The term "inverseDoppler effect" is asso- the fieldswhich are radiated from the oscillating3-D
ciated with the possibility that an approachingob- dipole, as measuredby a moving observer,may be
serverwill recognizea red shift, and a recedingone, representedin the form of time-varying amplitude
a blue shift. Even if at a first glance such a phe- A(t) multiplied by an exponent of a time-varying
nomenonmay sound unrealistic, some publications phase4•(t), i.e.,
mention the possibility of an inverseDoppler effect
in movingdispersivemedia (for example,unmagne-
s(t) - A(t)ei'•(t)
(1)
tized coldplasma) [Frank,1943;Papas,1965;Ryd- In general,s(t) cannot be representedby a singlebeck,1960]wherethe inverseeffectmay occurunder frequencycomponent, and one must considerthe efspecial circumstances.There are also other discus- fect of the motion on both A(t) and •b(t). Therefore
the simple(and intuitive) definitionfor the Doppler
Copyright1998by the AmericanGeophysical
Union.
Paper number 98RS00033.
0048-6604
/ 98/ 98RS-00033$11.00
463
effect as usually given for plane wavesis no longer
appropriate, and new definitions must be specified.
The spectral analysis of signals which are representedby (1) may be called "time-frequencyanalysis" and already constitutes a fundamental research
464
BEN-SHIMOL
AND CENSOR:
NEAR
subjectfor many decades(see Cohen[1995]for a
summaryand referenceson this subject). The basic
objectiveof suchtime-frequencyanalysisis to devise
a function that will describethe energy densityof
a given signal in time and frequency. Analysis of
the propertiesof such densitiesis basedon analytic
signals,whichare in the form of (1), and their imaginary part is constructed from the Hilbert transform
ZONE INVERSE
DOPPLER
EFFECT
ciesis, and the mean frequency(i.e., instantaneous
frequency)no longerservesas a goodmeasurefor the
signal'sfrequency.
Common
motion
to all situations
is a "collision"
of "head-on"
between
observer
the observer and the
dipole, i.e., overlappingof position, thus generating
infinite field values. Furthermore, the analysisgiven
by Enghetaet al. [1980]and Engheta[1990]did not
considerthe properties of the amplitude which vary
complex representationof signalsand waves which rapidly in the near zone of the dipole. The effects
is common in electromagnetictheory is constructed of suchamplitude changeson the spectrum(or endifferently(i.e., quadraturerepresentation)
but may ergy density)do not appearin the instantaneousfreserve as a good approximation to the analytic repre- quency, which is computed from the phase alone.
In order to enhancethe analysis,the effect of amsentationfor band-limitedsignalsor, moregenerally,
for signalswhen most of the spectrum residesin the plitude changesmust be consideredtoo; thus parameters such as the instantaneous frequency must be
positive side of the frequency axis.
The purpose of the present work is to study the examinedcarefully with regard to the signal'sbandphenomenonof the near-zone Doppler effect in free width 2a, as done in the present work.
In addition to the "instantaneous methods" it was
space on the basis of time-frequency analysis. The
decided
in the present work to examine the Doppler
caseof a radiating infinitesimal 3-D dipole has been
investigated
previouslyby Enghetaet al. [1980]and effectin the far and near zoneof the radiating dipole
Engheta[1990],who represented
the analyzedwave on the basis of short-time Fourier spectral analysis.
in the form of (1) and definedthe Doppler shift as The spectrogrammethod (which is basedon shortthe differencebetween the instantaneousfrequency time Fourieranalysis)usesboth amplitudeand phase
as measuredby the moving observerand the intrinsic in the calculation, thus preventingpotentially errofrequencyof 3-D dipole oscillations. Accordingto neous interpretations which are based on the phase
this definition, an inverse Doppler shift was found only.
The phenomenonof inverse Doppler effect in the
in the near zone of the 3-D dipole for some casesof
observer motion.
near zone of the 3-D elementary dipole is somewhat
unrealistic,becausethe effectsof the systemexciting
Usually, the instantaneousfrequency
the dipole and the associatedcircuitry are ignored.
aO(t)
This motivates the investigation of the near-zone
at
(2) Doppler
effect of scatteringby two-dimensional(2is usedas a generalizationof the term "frequency" D) monopoleand dipole where the excitation is exin cases where the amplitude is constant or varies ternal, for example, an incident plane wave. Accordslowly,i.e., caseswhich match the common intuition. ingly,the oscillating3-D dipolediscussed
by Engheta
Analysisof morecomplicatedcasesmust rely on en- et al. [1980]wasreplacedby a perfectlyconducting,
ergydistributions,and for mostof them the envelope thin cylinder moving in free spacein the presenceof
is concentrated in the vicinity of the instantaneous an exciting plane wave. The responseof the cylin• .........
•"• instantaneousfrequencyis not a local der may be taken to be due to a 2-D monopoleor
parameteras one may deducefrom (2); it is actually a 2-D dipole(in additionto a 2-D monopoleterm)
the averageof all frequenciesin a given time. Since for electric or magnetic polarization of the incident
the spreadof the frequenciesis determinedby the planewave,respectively.Van Bladel [1985]give a
behavior of the signal's amplitude, a measurement good summary of the spectral properties of the reof this spread may be defined as
flectedwavein the far zone. Michielsenet al. [1981]
of the realpart [Papoulis,
1977;Vakman,1968].The
and de Zutter [1982,1987]providegeneralanalyti(3)
cal calculationof the Fourier spectraof the scattered
and 2a is sometimes called the instantaneous band-
wave(for 3-D casesalso).
A fixed observerpositionedat the origin of the
width of the signal. The larger the instantaneous laboratory frame of referencemeasuresthe scattered
bandwidthgets,the widerthe spreadof the frequen- wave which contains information on the motion of the
BEN-SHIMOL
AND CENSOR:
NEAR
ZONE
INVERSE
DOPPLER
EFFECT
465
scatterer. The observer'spositionwas slightlydis- terer. An inverse Doppler effect has not been deplaced,to preventa collisionwith the movingcylin- tected in the near zoneof the movingcylinder.
der, and thusthe amplitudesof the electromagnetic
fieldsremain finite. However,the smallerthe offseta
gets(seeFigure6), the higherthe valueof the fields
are in the near zone.
2. The Three-Dimensional
Dipole
In this sectionwe will describesuccinctlythe anal-
While performinga spectralanalysis,the principle ysisreportedby Enghetaet al. [1980]and Engheta
of uncertaintyshouldbe bornein mind;that is, when [1990]. Only the parts whichare most essentialto
of this paper are included. Figure 1
dealingwith real situations,signalsof infinite time the discussion
describes
the
analyzed
configuration,in which an indurationare not available,and the spectralanaly-
finitesimal,oscillatingelectricdipole is positionedat
"uncertainty"in the contextof signalanalysis,which the origin of the laboratory frame of referenceand
shouldnot confused
with the principleof uncertainty lying along the z axis. An observeris movingwith
of quantummechanics.
The FouriertransformS(w) constant velocity in free spaceand measuresthe electromagneticfields which are radiated by the dipole
of the deterministic
time signals(t) is givenby
sis turns to spectral estimation. We use the term
for few cases of motion direction.
Let us denote the
laboratory frame of referenceby F and the frame
S(w)-
s(t)e-i•tdt
(4) of referencecomovingwith the observerby F•. The
and the powerspectrumPs(w) is definedby
-IS()l 2
(5)
and involvesvaluesof s(t) of all times. If s(t) representsthe measuredsignal, its exact spectral content
electromagneticfieldsof the oscillatingelectricdipole
in F are givenby equations(1) and (2) of Enghetaet
a/.[1980].(Note that there are some typographical
errorsin Engheta[1990].)
In the first case,whichEnghetaet al. [1980]and
Engheta[1990]analyzed,the observermoveson the
equatorialplane of the dipole, along the y axis from
y - -c• to y - c• with constant velocity v - v•. It
is givenby (5) without usbeingableto associateeach
is evidentthat the observerapproachesthe dipolefor
spectralcomponentwith its rangein space(i.e., we
t• < 0 and recedesfrom the dipolefor t• > 0. Using
have here a completeuncertainty of range), so the
the transformationsof specialrelativity, we get
spectral contentsof the near and far zonesfrom both
approachingand recedingscattererswould be inseparable. In order to increaserange accuracy,the mea-
sured signal is sliced to overlappingsegments,and
a spectral analysis is performed separately on each
segment,usingthe short-time Fourier transform
sr½o)dto
z
to+T
The finite duration of the segmentedsignal,T, broadens the spectrum, and this broadeningis increased
as the duration of the segmentdiminishes. The implication of such broadening is an increase in the
uncertaintyof the spectralcomponents[Papoulis,
1977; Vakman,1968]. It is interestingto note that
the uncertainty principle applies also to the mea-
surementof the instantaneous
frequency [Papoulis,
1977; Vakman,1968]. In the presentcase,there is
no unique way of defining the optimal time segment
width, and hencea variety of width valueswas tested
during the simulation in order to detect an inverse
Doppler effect in the near zone of the moving scat-
Figure 1. An infinitesimaloscillatingelectricdipole
positionedat the origin of the laboratoryframe of
reference.
466
BEN-SHIMOL
AND
CENSOR:
NEAR
ZONE
INVERSE
DOPPLER
EFFECT
•m'•' (,B=0.01)
I Red
Shift
Ap.•roaching
E
and
•gfromDigole
--•--[iw(ik- ,-•)+ v (,ik
;p (•t,)•.
1+
4wvt'
,
exp[i (kv - w) t']
t' > 0
H•,- _r__
[-iw
(ik
+v-•)
-v i(%_•
+t'(vtl,)•.
_k2)]
exp[-i(kv+w)t']
<0
4wvt'
-1
-4
-2
0
2
4 kvt'
(S)
where p is the electric dipole moment, eo is the per- Figure3. Normalized
Doppler
shiftr•m,,,forH•, in
mittivity of free space,k - w/c, and c is the speed the equatorialplane of the dipole. Positiveand negative valuescorrespondto red and blue Doppler shifts,
of light in free space.
(Recomputed
after Engheta[1990]).
In order to representthe electromagneticfields in respectively.
F' in the form of (1) (whichis actually a quadrature
representation),one may write
w' = w[1-/•V]
exp(i•'e'0') exp(-iwt')
(9)
(10)
where
I d•'
exp(i•'m,•, ) exp(-iwt')
•]- kvdr'
(11)
,
•'•,0,,•,•,0, areusedfortheelectric
Themotion
affects
theamplitudes
E'•,, H,, and andthenotations
the phases•he,
' o,, •hm,•,whichmay be measured
by
the observerin F'. Neglectingthe effectof the motion
on the amplitudes(and the spectrum)Enghetaet al.
[1980]usedthe time derivativeof the phases
and%h'm,
•, fortheelectric
andmagnetic
fieldsofthe
and magneticfields,respectively.
Similarnotationis
usedlaterfor r/. From(10) it is understood
that r/
represents the normalized difference in the instanta-
neous frequencies between F and F' frames and is
given by
3-D dipole, respectively.
The instantaneousfrequencyin F' frame for low
velocities(i.e., 7 --- 1) is givenby
t
7]e'
O'--
2
2
t
2
[(kvt')• (l+•)--l]2qt-[kvt' (1+•)]•
t' <0
(12)
%'0' (? =0.0•)
IRedShifil
0.5
I [(kvt')2(i-/•)t'> 0
[(kvt
) (1-/•)-1]
+[k•t
(1-/•)]
[(kvt')2(i+$)-l]
--(1+•)
for the electric field and
-Approaching
Dip /
•m'qb'
--
t'>O
I [(kvt,)2(l_l•)+
[(kvt')2(1--•)+•]2+[kvt'(1--•)]2
_ [(kvt')2(1+l•)-l•]2-l•(1+l
•)
t' <0
[(kvt')2(1+ l•) _l•]2+[kvt' (1+ l•)]2
(13)
Plotsof r•e,0,andr•,,,, are givenin Figures2 and3,
respectively,showingan inverseDoppler effectin the
-0.5
-1
near zoneof the dipole,accordingto the definitions
givenin (11) and (10).
' J[
/Recidine
from
•ioole
The above difficulties, which arise from the defi-4
-2
0
2
4
Figure 2. NormalizedDopplershifttie,s, for E•, in
the equatorialplane of the dipole. Positiveand negative valuescorrespondto red and blue Doppler shifts,
respectively.
(Recomputed
after Engheta[1990]).
nition of the instantaneous
frequency,may lead us
to erroneous
physicalinterpretations
of the analyzed
phenomenon
(i.e., the Dopplershift in the present
case).If a phenomenon
suchasillustrated
by theexample,whichis describedin the appendix,doesexist
BEN-SHIMOL
AND CENSOR:
NEAR
ZONE
in the present case, the determination of the inverse
Doppler effect may be incorrect.
In spite of the mentioneddifficulties,in many cases
the instantaneousfrequency is consideredas an in-
INVERSE
DOPPLER
EFFECT
467
lOO
lO
stantaneousparameter, i.e., the frequencyof the signal at a given time t is determined locally with no
concernto the signal's properties in the past or the
future. However, the definition of the instantaneous
frequencyis associatedwith analytic signals,which,
in turn, are calculatedby usingthe signalvaluesfor
all times. Actually, the instantaneousfrequencyis
o.1
O.Ol
-3
-2
-1
o
1
2
3
the averagefrequencyat a giventime [Cohen,1995],
and the spreadof signal'senergyaroundthis average Figure 4. Instantaneousquality factor Q for the
(3-D) dipole.
frequency(i.e., instantaneous
bandwidth)will deter- electricfield of the three-dimensional
The inverseeffect residesbetweeneach pair of bold
mine whether the use of the averageis appropriate
points and is acceptedonly for large Q values.
in the case at hand. In other words, one must con-
sider the effect of the motion on the amplitude. The
ratio between the instantaneousfrequency and the
instantaneousbandwidth may be called an "instantaneousquality factor" (similar to the definitionof
the quality factor in filter theory) Q, i.e.,
for the magneticfield. LargeQ factors(i.e., Q > > 1)
indicate narrow bandwidth, and the use of instantaneousfrequencyas the dominant frequencyin F'
frame is valid. In contrast, small Q values (i.e.,
1) indicatethat the bandwidthis wide and the
instantaneousfrequency cannot represent the meaUsing(7) and (8), the normalizedinstantaneous
band- sured signal as a dominant frequency. Figures 4 and
width (2a) is given by
5 showthe behavior of Q versuskvt' for different values of the observer'svelocity. It seemsthat for the
electric field, the use of the instantaneousfrequency
is valid for most of the range, showing that an inkv
dt •
verseeffect doesexist. As for the magnetic field, the
inverse effect which was reported by Engheta et al.
Q- 2a'
(kvt')
(•t')
(14) Q < <
[1980]and Engheta[1990],happensverynearto the
[(kvt')2(l+B)-l]2+[(kvt')(l+B)] 2
fort • <0
[(k•t,)•(1-•)-l?+[(•t,)(1-•)]
dipole'slocation where Q valuesare relatively small,
or, in other words, the inverse effect is acceptable
only for very small/• values(i.e.,/• (0.01).
In many cases(i.e., for signalsof relativelylargeQ
factors) the instantaneousfrequencycoincideswith
•
fort'
>0
fortheelectric
field
and
(15)the
of the maximum
of the distribution
ver-
sus time, and therefore if one determines the frequencyof the signal as the frequencywhere the max-
1
imal powerspectrum(or distribution)occurs,the re-
dt'
_ • ((•+•)(•')•-•)•+•[(•-•)(•')•+•]_
(•t,) (•t)[((•+•)(•t ) -•) +[(•t )(•+•)]']
for t' • 0
(kvt')
location
[(1-B)(kvt,)2-B]2+[(kvt,)(1-B)] 2
for t' • 0
(16)
sultswill likely be identical. Another fact which must
be remembered is that in reality the observer may
only measurereal signalswhich do not come in the
form A(t)cos•b(t). Therefore, in order to calculate
the dominant frequency,one must calculate an analytic complexsignal (by usingthe Hilbert transform
of the real part as the imaginary part of the complex
signal)from which the phasemay be determinedor,
alternatively, use spectral techniques.
468
BEN-SHIMOL
AND
CENSOR:
NEAR
INVERSE
DOPPLER
2o',.
LO-3
of the instantaneouscylinder'spositionin Fo).
i n-S
the transformationsof specialrelativity. In Figure 6
the y axes of ro and r• coincideat t• = to = 0; and
The calculation
-3
-2
-1
0
1
2
3
a is the minimal
Figure 5. Instantaneousquality factor Q for the
magnetic field of the 3-D dipole. The inverseeffect
residesbetweenthe bold point and the vertical axis
and is acceptedonly for large Q values.
3.
EFFECT
of reference. The incident plane wave may be electrically polarized or magnetically polarized parallel
to the cylinder's axis. A fixed observer,positioned
at the origin of the laboratory frame of referenceFo,
measuresthe wave scattered by the cylinder. This
scattered wave is analyzed in order to determine the
Dopplereffectasa functionof time (i.e., asa function
LO-•
_
ZONE
The
Two-Dimensional
Case:
The
Moving Cylinder
of the scattered
wave is based on
distance between the observer and
the cylinder axis. The incident wave Uinco is plane
and may be written in Fo as
Uinco- foeikø'rø-iwøtø
(17)
where fo definesthe polarization of the incident wave
and has a magnitude of one, ko definesthe direction
of propagation,and wois the angularfrequency.
The resulting scatteredwave measuredby an ob-
The inverse Doppler effect which takes place in
the near zone of the 3-D radiating dipole raisesthe server in Fo when expressedin terms of Fo is a comquestionwhether sucha phenomenonalso occursfor plicated expressioninvolving Fo time and spacecoother type of radiators or scatterersin free space. ordinatesro, to, in mixed form. The representation
The following description and analysis covers the
two-dimensional
case.
is simpler in form when the scattered wave in Fo is
expressedin terms of time and spacecoordinatesr•,
Figure 6 describesthe configurationof the follow- t•, of r• [Censor,1967,1969,1972]
ing theoretical experiment: A circular cylinder with
constitutive parameters Pi, ei, moves with constant
=
velocityv in the presenceof a plane electromagnetic
wavein freespace(e0,P0). Here we denotethe frame
of referencecomovingwith the cylinder by F• in or- where the scattering coefficientsin Fo are
der to avoid confusionbetween "primed" physical
quantities(that is, measuredin the comovingframe
+
+
of reference)and the derivativesof other physical
quantities which appear later in the laboratory frame and an(a•) may be approximatedby
Yo
Ho
P--Xo
Figure 6. A movingcylinderin the presenceof an incidentplanewave.
(18)
BEN-SHIMOL
AND CENSOR:
NEAR
ZONE
INVERSE
DOPPLER
EFFECT
469
Scattered
si•.alforE polarization
-'
'
13:0.01'
0.5
0
-0.5
-0
;
-1
,
kvt
Figure 7. The wave scatteredfrom a thin cylinder as measuredby an observerin the
laboratory frame of reference.The imaginary part is representedby a dashedline.
i7•
ao
m
-
21n(2/6k•d)
5 = 1.781...
the frequency may be separated from the effect on
the amplitude, since the transformed waves remain
plane. For nonplanar waves, the given definition of
an(a•)
• -e-i'm'in•r(k•d/2)2n
(n!)
2 n- 1,2,3,...Doppler effect is still useful if the given nonplanar
(20)
for electrical polarization and
ao
•
-i•r (k•d/2)2
an(a•)
• e-ina•i•r
(k•d/2)2n
(n!)
2 n- 1,2,3,...
(21)
for magneticpolarization.For electricalpolarization
the monopoleterm a0 is dominant,as seenin (20).
For magneticpolarization,a0 anda• are of the same
order, and thus the monopoleand dipole terms are
dominant,as can be seenfrom (21). Here
are the anglessubtendedby the propagationvectors
k0, k•, respectively,and the velocityv in F0,
respectively,as describedin Figure 6.
In the presentpaper,a symmetricalDopplereffect
in the far zoneswas chosen,prescribingthat the incidentwavepropagatesin a directionperpendicular
waveis recastas a superposition(integral) of plane
waves [Censor,1991;Stratton,1941]. The transformation of such nonplanar wavesfrom one inertial
frame to another may be derived by direct transformation of each componentplane wave included in the
givennonplanarwave spectralrepresentation.In the
present case the scattered wave in r• and in F0 are
not planar, as may be seenfrom (18), and the effect of the motion on the phase cannot be separated
from the effect of the motion on the time-dependent
amplitude.
The presence
of a frequencycomponent(i.e., a sine
wave of a given frequency)in a given time signal
is identified by a high peak in its power spectrum
(equation(5)) at that frequency.This propertywill
serveto identify the dominantfrequencycomponent
of the scatteredwave. Sincean analytic expression
of the short-timeFouriertransform(6) of the scattered
wave (18) is not available,the powerspectral
to the velocity(i.e., k. v = 0). ThereforeCto= •r/2.
density
must be estimatedwith a numericalspectral
Hence,for the incidentwave,a transverseDopplerefestimator.
Spectral estimatorsmay be divided into
fect is discussed.For cto- •r/2, the scatteredwave,
two
main
categories:
(1) nonparametric
(or classical)
for both electricand magneticpolarization,is illustrated in Figure 7.
4. Spectral Analysis of the Doppler
Effect
The relativisticDopplereffect,as statedin the be-
spectral estimators which are based on the discrete
Fouriertransformalgorithm(DFT) and (2) parametric (or modern) spectralestimatorswhich are based
on somemodels describingthe behavior of the analyzed signal or its spectrum.
In the present simulation it was decided to use
ginningof this paper, is derivedfrom the Lorentz a DFT based algorithm and its efficientimplementransformationswhen the electromagneticfields are tation, the fast Fourier transform(FFT) algorithm
planewaves.In this casethe effectof the motionon [Cooleyand Tukey,1965;Ifeachorand Jarvis,1993;
470
BEN-SHIMOL
AND CENSOR: NEAR ZONE INVERSE
Kay, 1988; Marple, 1987]. An extensivediscussion
DOPPLER
EFFECT
1
of spectral analysisand estimation is given by Kay
[1988]and Marple[1987].
0.8
The useof a numericalcomputationforcesus to set
specificvaluesfor eachparameterwhichtakespart in
the analyzedconfiguration.The cylindermotion has
been simulatedsothat the cylinder traversesa trajectory parallel to the x axis from large negativex values to large positivex valuesmovingwith constant
velocity. The cylinder is locatedbelow the x axis of
•
0.4
0.2
0
the laboratoryframe of referenceFo (seeFigure6) so
it doesnot coincide(i.e., collide)with the observer,
and infinite
field values are thus avoided.
The inci-
dent plane wave is chosenat I GHz frequency,and
the velocity of the cylinder is much lower than the
speedof light, althoughthe relativisticformalismis
retained for all cases(variouscaseswere tested by
us, but only a few resultsare presentedin the figures
0.6
-1.4
-1.2
-1
-0.8
-0.6
-0.4
[GHz]
Figure 9. Normalized PSD of the wave scattered
from a moving dipole (H polarization). The time
signal coversthe near and far zones.
given here). Consequently,the Doppler frequency pies causepoorer frequencyresolution. The FFT alshift in the far zone is much lower than the carrier
frequency,and thus the resolutioncapability of the
spectralestimatormust be consideredin orderto get
resultswith sufficientaccuracy.For the chosenveloc-
ity, say3 x 10• m/s (i.e.,• - 0.01),the Dopplershift
in the far zone is of the order of megahertz,and may
be even ]essin the near zone. For carrier frequencyin
gorithm is designedfor optimal DFT computation,
i.e., for N discrete frequency values. In order to
make such computations practical, the location of
the peak of the power spectrum is obtained in two
steps. An initial estimation of the peak location is
computedwith a shortFFT algorithm(i.e., low number of discretefrequencyvalues). This initial esti-
the RF range(1 GHZ in the presentexample),a vast mation is used later to define the limits where this
numberof samples
(2 x 10s samples
for a resolution peak occurssolely (i.e., without sidelobes). Later,
of 10 Hz with a samplingfrequencyof 2 GHz) has an optimization algorithm searchesfor the location
to be includedin eachanalysisin order to detect the
frequencychangewith sufficientaccuracy,thus rendering such computationsimpractical. Fewer sam-
I
1
I
of the peak using the DTFT (discretetime Fourier
transform)algorithm (i.e., the frequencyvaluesare
continuous). The use of the initial FFT estimation
prevents the detection of sidelobesand reducesthe
number of DFT computations which are neededfor
the requiredfrequencyresolution(1 Hz in the present
paper).
As mentioned earlier, there has to be some balance betweenthe requirementsof an exact frequency
determination(i.e., analysisof longsignalduration)
'
I I]=0.01
0.8
•0.6
and the determination
of the exact time
when this
frequencyoccurs(i.e., analysisof short signalduration). In the extreme casewhere the signalfrom all
rangesis used to calculate the spectrum, the power
spectral density (PSD) containstwo large peaks at
frequenciesbelongingto the far zones(and which
0.4
0.2
0
-10.5
-10
-9.5
-9
are predicted by the relativistic transformations for
plane waves),but there is no way to determinewhen
Figure 8. Normalized
powerspectraldensity(PSD) these frequenciesappear in the original signal (see
of the wave scatteredfrom a movingmonopole(E Figures 8 and 9 for electric and magnetic polarizapolarization).The time signalcoversthe near and tion, respectively). Furthermore, the analyzed sig[ GHz ]
far zones.
nal contains information
from the far and near zones
BEN-SHIMOL
AND
CENSOR:
NEAR
which cannot be resolvedfrom the power spectrum.
In order to decreasetime uncertainty and enable a
spectralanalysisat "specificlocations"of the cylinder, a sliding rectangular time window was introduced, i.e., the scattered wave was divided into overlapping segments. For each such segmentthe PSD
is calculatedusing samplesfrom that segment,and
the frequencyin which the maximal power value appearedwastaken to be the Dopplershift belongingto
the center of the current segment. The window size
(i.e., the time extent of eachsegment)has beenvaried from small valuescoveringabout one cycleof the
measuredsignal in the far zone to the extreme case
where a singlewindow coveredalmost the full signal,
in order to get the optimal window which yields the
best frequency-locationinformation.
Summarizingthe descriptionabove,the numerical
simulationand analysisis composedof the following
steps:
1. The scattered wave, as measuredby the observerat the origin of F0, is simulatedusingequation
ZONE
INVERSE
DOPPLER
EFFECT
471
7.5
[•=0.01
,
Nindow
size
!
,,-•
2.5
2Cycles
, .........
"i
i .
20 Cycles
.....
t
40Cycles
I
-2.5
-5
-7.5
ß
.,
-40
i
-20
kvt
0
20
40
Figure 11. NormalizedDopplershift for the moving
2-D dipole. The incidentwaveis magneticallypolarized, and the D FT-based algorithm is used for the
computations.Eachline representsa differentdifferent window width which is measured in numbers of
cyclesof the measuredsignalin the far zone(solid
line, two cycles;dashedline, 20 cycles;dash-dotted
line, 40 cycles). In general,narrow windowsfollow
better rapid amplitude changesin the near zone.
(18) andthe scatteringcoefficients
givenin equations
(20), (21) (for electricand magneticpolarization,respectively),and (19). The time rangecoversthe near Doppler shift is definedas the frequencyin whichthe
and far zonesof the observer.Examplesof suchsig- maximal value of the power spectrum occurs.
3. A more accurate computation of the Doppler
nalsare shownin Figure 7 for both electricand magfrequency(usingthe abovedefinition)is obtainedby
netic polarization.
2. The scatteredwaveis dividedinto overlapping an optimization algorithm. The algorithm is based
segments,and eachis zero paddedand processedby on a minimum finding algorithm aroundthe Doppler
an FFT algorithm. From the FFT transform a power frequencywhich is determined by the previousstep.
4. The frequencyof the peak is assignedto the
spectrum is calculated. The initial estimation of the
center of the time segment
5.
__._,,
•
Discussion
•3=0.01
Some results of the computed Doppler shift are
presented
in Figures 10 and 11 for a moving 2-D
-•,,
0.5
monopole and a moving 2-D dipole, respectively. In
the far zone, the computed Doppler shift fits the theoretical value which may be obtained analytically
•
....... 20Cycles
•
[
.....
40
Cycles
by substituting the asymptotic expressionof Han-0.5
t,'x
kel functionfor large argumentsin (18). In the near
zone,the transitionfrom the to < 0 range (cylinder
-1
approaching)to the to > 0 range(cylinderreceding)
is not abrupt; that is, the Doppler frequency shift
-1.5
-6
-4
-2
0
2
4
6
changesgradually, but the changeis not monotonic.
Figure 10. Normalized Doppler shift for the mov- The fast change of the amplitudes of the scattered
ing 2-D monopole. The discreteFourier transform waves in the near zone of the cylinder is shown in
(DFT) basedalgorithmis usedfor the computations. Figure 7 and appearsas a high frequency-component
In general,narrow windowsfollow better rapid am- in the PSD. This may explain the differencebetween
plitude changesin the near zone.
Figure 10 (monopole)and Figure 11 (dipole). The
•
[ -- 2Cycles
•'• ""-
kv!
,
,
472
BEN-SHIMOL
AND
CENSOR:
1.5
NEAR
. .
0.5
-0.5
-1
-1.5
-3
-2
-1
0
1
2
3
Figure 12. NormalizedDopplershift (relativeto
w/•)versuskvt forthe moving2-D monopole.
The instantaneous
frequencyis computednumerically.The
increasein frequencyin the nearzonefollowsthe fast
amplitude change.
relative amplitude changefor the dipole caseis faster
than in the monopole casewhich is supported by the
resultsshownin Figures 10 and 11, i.e., the increase
of the Doppler shift in the near zone of the cylinder.
Close examination of Figures 10 and 11 showsthat
near the origin, for 0 < kvt << 1, the Dopplershift
is still positive. This shouldnot be interpretedas an
inverseDoppler effect sinceit is causedby the finite
width of the slicing window. As the width of the
ZONE
INVERSE
DOPPLER
EFFECT
The effectis more pronouncedfor the 2-D dipolecase
(i.e., magnetic polarization). In addition, a numerical computationof the instantaneousfrequencyfor
the 2-D caseis presentedin Figures 12 and 13. The
results agree with Figures 10 and 11. The numerical
computationof the instantaneousfrequencyis much
faster and more efficientthan the DFT-based algorithm but is more sensitiveto the presenceof noise,
which usually occurs in real situations. Moreover,
the computationsof differencesas an approximation
to the analytical derivatives may be inaccurate for
fast changes(which appear in the near zone), and
one has to sample the signal with a very high sampling frequency,which may lead to other numerical
problemssuch as accuracy and round-off errors.
Varioussetsof simulationparameterssuchascylinder velocity,segmentwidth, samplingfrequency,etc.,
were used in order to discover an inverse Doppler
shift in the near zone of the 2-D cylinder. Such an
effect was not found.
The DFT-based algorithm was applied to the mea-
suredfieldsof the 3-D movingdipole. Sincea singularity occursin (7) and (8) for t' = 0, the trajectory
of the moving observerwas slightly displacedabove
the y axis and parallel to it, so collisionwas avoided.
The results are given in Figures 14 and 15 and are
in agreementwith the resultsgivenby Enghetaet al.
[1980]and Engheta[1990],exceptfor a very narrow
slicingwindowincreases,
moreinformationfromthe regionnear the origin, wherethe rapid changeof sigpast (relative to the window'smidpoint) is involved nal values is characterizedby very high frequencies
with the computation, thus slowingthe responseto
the change in the Doppler shift at positive times.
where the peaks of the PSDs occur and are selected
as the dominant frequencies.
1.5
7.5
2.5
-2.5
-5
.5
-7.5
kvt
,
,
Figure 13. Normalized Doppler shift (relative to
co//)versuskvt for the moving2-D dipole. The instantaneousfrequencyis computednumerically.The
increasein frequencyin the near zonefollowsthe fast
amplitude change.
-3
-2
-1
0
1
2
3
4
Figure14. Normalized
Doppler
shift(relative
to
co•) computedwith the DFT-basedalgorithm.The
inverseDoppler shift appears in the near zone. Close
to the origin, the effectof the fast amplitudechange
is dominant. The extent of the high frequencylags
in proportion to the width of the slicingwindow.
BEN-SHIMOL
AND
CENSOR:
NEAR
ZONE
INVERSE
DOPPLER
EFFECT
473
applied, and none of them show the existence of an
inverse Doppler effect for the 2-D case.
1.5
For the three dimensionalcase,the analysisand
1
resultswhichwere givenby Enghetaet al. [1980]
and Engheta[1990]are rechecked
with respectto the
0.5
instantaneous bandwidth
o
in the near zone. This ex-
amination supports the phenomenon of the inverse
-0.5
Dopplereffectfor the three-dimensionaldipolein free
-1
space, excluding a very narrow zone where the fast
change in the amplitude is not reflected in the in-
-1.5
-2
-4
-2
0
2
4
6
Figure 15. NormalizedDoppler shift (relative to
w•3)for the magneticfield of the 3-D dipole as computed with the D FT-based algorithm. The inverse
Doppler shift which appearsin the near zone is very
closeto the origin, and thus may be invalid. Close
to the origin, the effectof the fast amplitude change
is dominant. The extent of the high frequencylags
in proportion to the width of the slicing window.
6. Summary
stantaneousfrequency. Analyzing the fields of the
three-dimensional
dipole with the D FT-basedalgorithm supportsthe existenceof the inverseDoppler
effectphenomenonand showsthat in a very narrow
regionthe fast changeof the amplitude is dominant.
Appendix:
A Simple Example
The behaviorof "instantaneous
frequency"(i.e.,
the time derivativeof the phase) does not always
serve as a generalization of the term "frequency,"
asillustratedby the followingexample[after Cohen,
1995]' Take a time signal
f (t)
The existenceof the inverseDoppler effectin free
spacehas been scrutinized,in view of somepapers
reportingthe effectin the vicinity of a movingradi-
-- 81(t) 4. 82(t)
-__ Aleiw•t4. zx2c
A _iw=t
(A1)
= A(t)eJ•ø(t)
ating 3-D dipole. The difference between the terms
"frequency"
and "instantaneous
frequency"andtheir
relation to the physicalmeaningof the Doppler effect were carefullyconsidered.A theoreticalexperiment consistingof a movingperfectlyconducting
thin cylinder in the presenceof an exciting plane
wave was discussed. The responseof the cylinder
may be taken to be due to a 2-D monopoleor a
2-D dipole for electric or magnetic polarization of
the incident plane wave, respectively,therefore enabling us to compare our results with those of the 3-
where the amplitudes A1, A= are taken to be real
constantsand Wl, w= are positive angular frequen-
cies. Sincew•, w= are taken to be positive,s(t) is
analytic. The powerspectrumof s(t) consistsof two
delta functionsat (angular) frequencieswl and w=,
i.e.,
i0
:
',!
:,,'
',,r
•, :
•,•
D casewhichwerereportedby Enghetaet al. [1980].
The spectral analysis is based on the discrete time
Fouriertransform(DTFT) and its fast implementation (FFT). The measuredsignal was divided into
overlappingsegmentsin order to reducethe time uncertainty of the Doppler shift. A classicalspectral
estimatorwas applied to eachsuchsegment,and the
Doppler shift was defined as the differencebetween
the location of the highest peak of the PSD and the
frequencyof the exciting plane wave. The time instance of this Doppler shift was set at the center
of the time segment. Intensive numerical analysis
involving various sets of simulation parameterswas
-lO
-20
-30
:'i
il
]i
ii
i
'• I ............
•_--•.2,•2=• ß
::i
:;!
'!i
(o,
=10rad/sec
1
cøz
=20rad/sec
1
-40
Figure 16.
The instantaneous
frequency
of
Aleiw't4-A2e
iw2t.Thedashed
linesshows
thepos-
sibilityof negative
frequency
values
foran analytic
signal.
474
BEN-SHIMOL
AND
CENSOR:
S(w) = A• 6(w- w•) + A26(w- w2)
NEAR
(A2)
The amplitudeA(t) and the phaseqo(t)are givenby
ZONE
INVERSE
DOPPLER
EFFECT
de Zutter, D., The dyadic Green'sfunction for the Fourier
spectra of the fields from harmonic sourcesin uniform
motion, Electromagnetics,2, 221-237, 1982.
de Zutter, D., Green'sfunctionfor the Fourierspectraof
n(t)- v/A•
2+A22
+2A•A2
cos(w•
- w2)t(A3)
the fields from two-dimensional
sources or scatterers in
uniform motion, Radio Sci., 22, 1197-1203, 1987.
Doppler,
C. J., •lberdasfarbige
LichtderDoppelsterne
and
und einiger anderer Gestirne des Himmels, Abh. K.
BShmischenGes. Wiss., 2, 467-482, 1842.
Engheta, N., An overview of the theory of the near zone
inverseDoppler effect, in Recent Advancesin Electromagnetic Theory, edited by H. N. Kritikos and D. L.
respectively.The instantaneousfrequencyis calcuJaggard,pp. 56-73, Springer-Verlag,New York, 1990.
lated as the time derivative of the phase, i.e.,
Engheta, N., A. R. Mickelson,and C. H. Papas, On the
near-zone inverse Doppler effect, IEEE Trans. Antendqo(t) I
1
A• - A•i
nas Propag.,AP-œ8, 512-522, 1980.
Frank, I. M., Doppler effect in a refractive medium, J.
(^s)
Phys. USSR, 2, 49-67, 1943.
Figure 16 illustrate the instantaneousfrequencyof Gill, T. P., The Doppler Effect, Academic, San Diego,
Calif., 1965.
the signals(t) for two cases.The followingsimple
exampleillustrates somedifficultiesarisingfrom the Ifeachor, E. C., and B. W. Jervis, Digital Signal Processing: A Practical Approach,Addison-Wesley,Reading,
definitionof the instantaneousfrequency:
qo(t)
- arctan
A1c0s031t
q+A2cosw2t
03(t)
-- dt ---• (031
q-032)
q-• (031
-032)A•(t
)
Mass., 1993.
1. The instantaneousfrequencyis not necessarily
Kay, S. M., Modern SpectralEstimation: Theory and Aponeof the distinctfrequencies
in the spectrum(put
plication, Prentice Hall, Englewood Cliffs, N.J., 1988.
A1 = A2 in (A5), for example).
Marple, S. L., Jr., Digital SpectralAnalysis with Applica2. The instantaneousfrequencymay be continutions, Prentice Hall, Englewood Cliffs, N.J., 1987.
ous, ranging over an infinite number of values even Michielsen, B. L., G. C. Herman, A. T. de Hoop, and
D. de Zutter, Three-dimensionalrelativistic scattering
for a signalhaving a spectrumconsistingof few dis-
tinct frequencies.
of electromagneticobject in uniform translational mo-
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tion, McGraw-Hill, New York, 1965.
for negativefrequencies).
Papoulis, A., Signal Analysis, McGraw-Hill, New York,
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Rydbeck, D. E. H., The Doppler effect in dispersive,inho-
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Y. Ben-Shimol and D. Censor, Department of
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University of the Negev, Box 653, Beer-Sheva,
84105, Israel.
(e-mail:[email protected]];
[email protected])
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(ReceivedMarch 3, 1997; revisedDecember5, 1997;
acceptedJanuary5, 1998.)