IEEE TRANSACTIONS ON ULTRASONICS,FERROELECTRICS, AND FREQUENCY CONTROL VOL. 39, NO.2, MARCH 1992
187
Real and Complex Doppler Effects in Lossy Media
Dan Censor
Abstmct-The theory of the Doppler effect in the presence investigated. Of course it may be argued that as long as we are
of lossy media and moving scatterers is investigated. Essentially dealing with analytical complex functions, the continuation of
two kinds of phenmeoa emerge, which ue not disthguishabk
frequencies and propagation vectors into the complex domain
in the conventional a s e of lossless medip: 1) when the scatterers
o ~
motion involves a slowly is merely a formal step. On the other hand, the analysis and
move aniformly, or the ~ p t i of
varying velocity, it is shown tbat propagation in lossy media examples given below show that although the formal structure
involvcs complex Doppler effects, i.e, the spatial attenoation of of the problem is not much affected, certain modes of motion
the mediam is transformed into the temporal behavior of the create complex frequencies and are associated with spectral
received s&@,
YieMing complex Doppler Ssar freqaencies for a
broadening. This then leads to the legitimate question as
real exatation fmquency. This phenomeMM is closely dated to
to how the velocity can be gleaned from these signals. To
the qaestion of remote-sensingthe velocity by means of scatte*
of waves from moving objects. The compliations intrwloced by overcome this difficulty, it i,s proposed here that the received
the m i v e d complex fkequency signrrl am dsseossed. It is shown signal should be subjected to a temporal filtering process, thus
thatspeetrum b d e n i n g o~111ls,and that in certain cases, by counteracting the medium's effect and facilitating an efficient
jadieiopsly chosen temporal Mters the spectral degradation of spectral analysis.
the received signal am be mtemcted. The second class of
A natural question arising from this discussion is the folphenomena is associated with periodic and harmonic motion,
when thescatterer isvibratinga"d a h e d b t i o n . In this case lowing: Inasmuch as attenuation affects the received Doppler
tk incident wave is fnqaency modulated by the moving scatterer, signal, is it possible, knowing the velocity, to use Doppler
giviogrise to sidebandsat real freq~~~ncies.
It is shown that in this shifted signals for assessment of the parameters of the medium
use the spectral cantents of the'scattered signal is identical for at hand? This problem, again involving a transformation from
losskss and lossy media, provided the moving object remains in
the vidaity of some kat+. It is v e d that the tmosformation spatial to temporal variables, is considered below.
In principle, the theory presented here applies to any degree
f n the
~ spatial
~
to the temporal regime, a d by the Doppler
effect, faciktates &e assessment of the medium attenuation from of attenuation. In the limiting case of zero losses we are dealing
the attribates of the received Doppler shifted temporal @pd. with the (hyperbolic) wave equation. At the other extreme
Numerical simalatiom were performed to support and disphy stands the (parabolic) diffusion or heat conduction equation,
the aforementioned theoretical arguments.
governing highly lossy systems. It is interesting to see how the
Doppler effect behaves as we move from one extreme to the
[. INTRODUCTION
other. The following considerations apply to any wave system
HE DOPPLER EFFECT was announced [l]in 1843, (for in the presence of lossy media, e.g., acoustics, elastodynamics.
more detail on Doppler, his work and life, see [2]). Since As a concrete example, consider the electromagnetic case of
then the Doppler effect found many applications in practically waves in sourceless domains. The field equations are
all branches of science and engineering. The Doppler effect
V x E = -p&H
as a subject is dispersed throughout many disciplines and
in order to provide some linkage to the existing literature
the mention of some references is warranted. A general
introdum-on is given by [3]. The physical and mathematicaltheoretical aspects were emphasized by the relativistic Doppler
effect stated by Einstein [4] in his famous paper, see also In the conventional notation [ll],where at denotes the partial
Censor [5] and Van Blade1 [6]. Acoustical Doppler effect time derivative, and-constitutive equations D = EE,B = p H
found its application,in acoustical remote-sensing, especially and the Ohm "law" (actually another constitutive equation)
for medical Doppler ultrasound diagnostics, see for example j = aE are assumed, where for simplicity real, scalar e , p, 0,
Baker, Forster, and Daigle [7], Wells [8], and Fish [9]. It are chosen. Operating on (1) with V x and substituting from
is worth mentioning that in amustics the interaction of the the second equation, then using the identity (for Cartesian
classical Doppler effects and nonlinear harmonic generation components) [V x V X = V V -V2], we obtain the wave
equation
effects created some controversy [lo].
As far as the present author is aware, the Doppler phendmenon in the presence of lossy media was never adequately
T
9
Manuscript received March 25, 1991; revised June 10, 1991; accepted
October 11, 1991.
Tbe author is with the Department of Electrical and Computer Engineering,
Ben Gwion University of the Negev, Beer Sheya, 84105, Israel
IEEE Log Number 9105557.
where denotes one of the Cartesian components of the field
and clearly 0 = 0, E = 0 correspond to the cases of the wave
equation, diffusion equation, respectively. Assuming a plane
wave solution of the form eik'r-iwt, where the propagation
0885-3010/92$03.00 Q 1992 IEEE
-
I.
.
188
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vector k and the (angulay) frequency w, may be complex, (2)
now becomes
which is tantamount to a four-dimensional Fourier transformation of (2). A similar procedure can be applied to other
physical models, e.g., acoustics, or waves in a fluid plasma
model [12]. In general, lossy media will include time and space
derivatives of even and odd degree, corresponding to factors
of i in the transform, hence in general the dispersion equation,
obtained by a Fourier transform of the wave equation, relating
the propagation vector k to the frequency w, can be written
in the form
F(k,w)= 0
(4)
and will be complex. These conclusions apply as well to other
wave fields, e.g., acoustics and elastodynamics.
The Doppler effect is introduced through a simple case
based on propagation of plane waves in the presence of a
uniformly moving reflecting boundary. This example reveals
the presence of complex Doppler shifted frequencies in the
scattered signal. We then consider the Doppler effect due to arbitrary motion, and as an example focus on the vibrating plane
reflector at normal incidence. Two kinds of approximations
are effected, depending on the characteristics of the motion.
It is shown that for the case of a slowly varying velocity we
obtain the results of the complex Doppler effect once again.
On the other hand, if the amplitude of the vibratory motion
is small, then another approximation is adequate, yielding real
frequency sidebands. An additional example is provided by
the rippling plane, which introduces scattered waves in new
directions.
A discussion follows, in which the physical reasons for the
two kinds of Doppler effect in lossy media are explained.
Measurement of Doppler effects is facilitated by a spectral
analysis of the time domain signals. As long as we deal with
real Doppler effects this scheme is straightforward, although
the estimation of the velocity is sometimes complicated due to
the effects introduced by the apertures and flow fields involved
[7l-[9],[13],[14]. In the case of lossy media and for the range
of parameters where the complex Doppler effect is significant,
the usual spectral analysis yields degraded spectra. This is
manifested by the broadening of the spectra caused by the
migration of the real frequencies into the complex domain. The
broadening is analyzed for simple cases, and remedial steps are
proposed, in terms of special temporal filters. It is also shown
that these filters cause a windowing effect on the spectrum:
The filter, while counteracting the spectral broadening and
increasing the resolution at one frequency range, may cause
additional broadening and increased fuzziness in other spectral
regions.
Finally we briefly address the question of estimating the
medium’s attenuation from the properties of the Doppler
signals.
11. THE DOPPLEREFFECTFOR PLANE WAVES
AND UNIFORM MOTION
A simple case for investigating the question of the Doppler
effect in lossy media is presented by the normal reflection of
a plane wave from a plane interface in uniform motion. We
assume a plane wave
cp - ezkz-awt
(5)
1-
launched in the z direction. The propagation vector k and
frequency w satisfy the relevant dispersion equation (4). Inasmuch as a lossy medium is assumed, the branch of the complex
function is chosen, which displays attenuation in the direction
of propagation. In an active medium, such as inside a laser
medium, an exponentially growing wave might be physically
feasible. In the electromagnetic case ‘Pi corresponds to the
relevant Cartesian component of the E or H fields.
It is now assumed that the wave is reflected by a boundary,
uniformly moving according to
(6)
z ( t ) = ut.
At the boundary we have to satisfy conditions prescribed
by the physical model. The boundary conditions for the
electromagnetic case can be stated exactly only for the case of
uniform motion. This involves a relativistic transformation to
the frame of reference in which the boundary is at rest, application of the conventional boundary conditions for objects at
rest and application of an inverse Lorentz transformation back
to the “Laboratory” system of reference. This has been amply
discussed and implemented by Einstein [4], and others [5], [6].
The exact nature of the boundary conditions is not the main
theme in the present discussion, and the particular conditions
chosen do not affect the general conclusions reached here.
Therefore it will be a good strategy to choose the simplest
conditions and leave the discussion of the physical relevance
for specific problems, as they arise in various branches of
physics. Thus it will be assumed here that the field must vanish
on the boundary surface. In the context of mechanical waves
(e.g., acoustics), this amounts to @ standing for the velocity,
pressure field, for rigid, soft bodies, respectively. To satisfy
this condition, the total field @ must include a scattered wave
a, and therefore we choose
9 = aZ+ a, = ezkz--ZWt
+
~~-k,z-zw,t
(7)
and the boundary condition is satisfied on imposing A = -1
for the amplitude of cpP and provided at the boundary the
exponents be equal, that is
kvt
- wt = k,vt - a r t
(8)
must be satisfied for the constant parameters k,, w,. Substituting the dispersion equation (4) yields
k~ - w ( k ) = -k,v
- W , = -k,v
-~ ( k , )
(9)
which in principle can be solved for k,, and then through
( 4 ) w, is finally obtained. The aformentioned argument is
based on the linear wave equation, which is adequate for the
description of the Doppler effect, at least to the first order in
CENSOR REAL AND COMPLEX DOPPLER EFFEClS IN LOSSY MEDIA
189
the Mach number v/c. In most cases, including acoustics, only
the first order Doppler effects are of interest. In cases where
the nonlinear effects are of the same order of magnitude as the
Doppler effect, the linear wave equation is inadequate even for
deriving first order approximations. In order to discuss higher
Mach number approximations of the Doppler effect, additional
terms in the fundamental field equations must be retained
[lo], (151. Below, this point is also considered in connection
with periodic motion. To the first order in the velocity (9) is
rewritten as
w,=w(k)
( ));;:
1--
=w
(
1--
wp:;w))
(lo)
which is recognized as the “classical” Doppler effect, but note
that the phase velocity Vph in a dispersive medium is w (or
k) dependent, not merely a constant, and might be complex
in general. To derive IC,, substitute w,. from (10) into the
dispersion equation (4). A common mistake is to assume that
k, = k. This is the so-called quasi-static approximation [5].
Obviously this is inconsistent because u n l i e k,, w,., the pair
k, wr does not satisfy the dispersion equation (4) even to the
first order in the velocity. Inasmuch as (10) is accurate only
to the first order in the velocity, in terms involving v we are
justified in writing w ( k ) or wph(k) in terms of k, w, rather
than in terms of IC,., w, which already contain effects of first
order in the velocity.
Insofar as the theory is concerned, the result (10) is the
“classical” Doppler effect which one would have expected,
and therefore it is not very interesting. However, the fact that
presently ‘uph hence also w r , are in general complete has far
reaching practical implications. It means that a simple spectral
analysis of the received signal
e-‘”’t
(11)
will not correspond to a line spectrum, i.e., a &impulse at
the frequency w,, because Fourier analysis deals with real
frequencies only! Instead, as shown below, the spectrum will
be broadened.
111. TfE DOPPLER EFFECT DUETO ARBITRARY MOTION
AND PLANEREFLECTORS
Scattering by arbitrarily moving objects has been previously
discussed for lossless media [6], [16)-[18]. Closed solutions
are available only for a few cases. Inasmuch as first order
velocity effects are sought, perturbation schemes can be used
to derive the leading velocity effects [16], [17]. The problem
is stated here in a similar manner. The excitation 9;is given
in (5). For a plane scatterer the total field is now represented
as a sum of the incident wave and a superposition (integral)
of scattered plane waves:
ip
= ai+ ipr -- eikz--iwt
g ( y ) dv
(12)
where the contour C is undetermined at this stage. Subject to
the aforementioned boundary conditions, (6) and (12) prescribe
at the boundary
eikz(t)-iwt
+J,e--i(kuz(t)+vt)g(y)dv = 0.
The problem now reduces to the question of finding the
weighting function g ( v ) and the contour C ( v ) ,i.e., the adequate values of the frequencies v, which satisfy (13). For
uniform motion the solution is trivial and reduces to (7).
In more general cases g ( v ) has to be determined and its
substitution into the integral (12) in turn determines the
scattered wave.
As an example, consider the case of harmonic motion of
the scatterer. As a word of caution, especially in the regime
of acoustical waves, it must be pointed out that this problem
generated a lot of controversy as to the validity of the analysis
in terms of the linear wave equation, and the need to include
nonlinear effects in the analysis [lo], [19]-[25]. Finally even
the severe critics [25] conceded that for certain ranges of the
parameters, the Doppler mechanism can dominate. One of the
criteria proposed [Z] was that the Doppler effect dominates
when the dimensions of the scatterer are small compared to the
acoustical wavelength excited by the its motion. This criterion
can be met in our discussion if R, the (angular) frequency of
the mechanical harmonic motion, is small compared to w . In
any case, this is not the central theme of the present discussion.
N o kinds of limiting cases will be considered. If the
equation of motion of the reflecting plane is taken as
where R, and t, the time of observation, are sufficiently small,
such that sinRt can be expanded about the small argument.
For simplicity, consider the leading term sinRt M Rt only.
This yields (10) with 11 replaced by En. Inasmuch as the
problem has been linearized, with v = 50 as a constant
velocity, in this case the Doppler shifted scattered wave will
possess a complex frequency. This statement holds also if a
few higher power terms in Rt are retained.
A different Doppler effect is apparent when this expansion
is not valid and the equation of motion of the reflecting plane
is chosen as
Vmax
z ( t )= R sinRt
implying
&@
dt
= ,,U
cos Rt
where ,,w
is the maximum velocity. Substituting (15) into
(13) involves exponentials with sinusouidal functions in the
exponents and can be recast in terms of series of Bessel
functions [17]. In order to deal with the first order effects,
let us define a small perturbation parameter = kw,,/R.
Of course, since IC is in general complex for the present case
of lossy media, also E must be complex, in order that the
displacement in (14) be real. Approximating
it becomes clear that the reflected wave must involve frequencies
(13)
W,W-
=w-R,w+=w+R
.
(18)
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with corresponding propagation vectors k - , k+ determined by
the dispersion equation (4). Consequently, to the first order in
E, the scattered wave 'P, is given by
ip
- -e-tkz-iwt
r -
+ E[e-ik+z-iw+t
- e--rk-z-iw-t
3
(19)
and substitution of (15) into (19) and evaluation of (13) verifies
that (19) satisfies the boundary conditions to within
the first
power in E. Note that the expression in the parentheses in (19)
is already multiplied by E, hence to the first order in E the
expression in parentheses is evaluated to order eo. Rewriting
E in the form
Once again E = kv,,/R
is the small Rerturbation parameter.
Instead of (12) we must allow now a superposition of waves
having different directions of propagation, but it turns out that
the frequencies of the leading terms are once again given by
(18). Using approximations of the kind shown in (17), the
incident wave (5) at the boundary is now recast as
(Ky-Rt)--Ewt
= e--rwt
~ ( yt), = <sin (Ky - Rt)
(21)
i.e., the rippling takes place at a frequency R and mechanical
wavelength 27r/K, at an amplitude <.For distance y and time
t such that the argument of (21) is small, one may expand the
sin function as a power series. Keeping the leading term only,
the incident wave (5) becomes at the boundary
- -etkEKy-~k<Rt-~wt
t -
(22)
indicating both a spatial shift (aberration) and a temporal
(Doppler) shift simultaneously. Consequently the reflected
wave is chosen as
0r -- - e - t k , = ~ + ~ k v y ~ - t w r t
(23)
where k,,, krY,w, satisfy the dispersion equation (4) relevant
to the medium, and at the boundary 'P,
'P, = 0 must be
satisfied. To the first order w, is given by (10) with 'U = @2,
k,, = 2 k < K , and k,, is derived from the dispersion equation.
Using these values in (23), it is then verified that to the first
order in the small parameters the boundary condition is indeed
satisfied. Inasmuch as krY, w, are once again complex, this
solution falls within the category of the complex Doppler
effect.
For the other kind of the Doppler effect, corresponding to
(15), we now write
+
x(t) = -sin (Ky - Rt)
%ax
R
and corresponding to (16) we now have
d4t) - --V,,cos(Ky
dt
- 0t).
+
[1+iEsin ( K y - Rt)]e-*wt
erKy-w+t
2(
- e--rKy-w-t
)
(26)
and therefore the scattered wave is chosen in the form
-e - z k z - z w t
and comparison to (10) reveals that the condition of small
first-order effect in the Mach number is coupled in (20)
with the ratio w/R: If this ratio is large, then the relative
must be reduced accordingly, otherwise the
velocity v,/wpuph
approximation (17) becomes inapplicable and must be replaced
by a different method, e.g., the full l3essel function series
expansion [16], [17]. The striking finding here is that (19),
unlike (10) or the approximation based on (14), contains the
real frequencies given in (18).
More complicated modes of motion may be analyzed by
these methods: Consider for example a rippling plane interface
on which a mechanical wave propagates in the y-direction
according to
%
E
- Ee-ak+,z+xKy-~w+t
+
~~
--2
k- ,2-1 K y -aw
-t
(27)
Substituting (24) in (27), using the same approximations that
lead to (26), and adding up (26), (27) shows that (27) indeed
satisfies 'Pa 'P, = 0 at the boundary. The values of k+z, k - ,
are determined by the relevant frequency and K (which defines
the component of the propagation vector in the y direction)
from the dispersion equation (4). Once again we obtain here
the real Doppler effect involving the real sideband frequencies,
as in (18). In the present example the Doppler effect involves
also modified propagation vectors.
+
IV. DISCUSSION
Boundary value problems in the presence of time dependent
boundaries provide the mathematical modeling for the phenomena described previously, constituting a generalization of
the celebrated Doppler effect. In (12) a mathematical statement
of the total field in terms of a superposition of plane waves
is presented. Substitution of the time dependent boundary
condition yields (13). This applies to the special case of onedimensional motion along the x-axis only. The formalism can
be generalized to arbitrary motion, various boundary conditions, and even vector waves. However, systematic solutions
of forms such as (12) do not exist. Simple cases have been
considered previously, for which the solutions, subject to
certain perturbation type approximations, are easy to construct.
Although very simple, these cases shed light on the general
behavior of such solutions. It has been shown that for constant,
or slowly varying velocity, and provided the time of observation can be properly limited, the scattered Doppler signal
involves complex frequencies. This complex Doppler effect
can be explained in terms of the following primitive argument:
When the object moves toward the receiver, the scattering
takes place from locations gradually closer to the receiver. As
time passes the signal traverses a shorter round trip distance
through the attenuating medium, and therefore at the receiver
the signal grows in amplitude. This time modulated amplitude
of the received signal corresponds to a complex frequency
with a negative imaginary component. The opposite happens
when the scatterer moves away from the source, resulting in
a complex frequency having a positive imaginary component.
Admittedly, this explanation is very primitive and is based
on the "quasi-static approximation," which suffers from an
intrinsic inconsistency [5]. Note that (12), (13) are not based
on such a primitive notion. If a solution of (12) is available, it
CENSOR REAL AND COMPLEX DOPPLER EFFECIS IN LOSSY MEDIA
191
will correctly take into account the kinematics and dynamics
of the underlying physical model.
Using the same argumentation, it is possible to explain why
the assumption that 6 = Icw,,/R
is a small parameter (cf.
(15), (16)) leads to real frequencies in the Doppler shifted
signal. In this case the traverse of the scatterer in terms of
wavelengths of the exciting signal is small. The scattered
signal originates from a limited region in space whose distance
relative. to the receiver does not appreciably vary.
In a realistic situation it is expected that the two effects
will be displayed simultaneously: Vibratory motion will produce sidebands, but as the span of the motion increases, the
frequencies will gradually leave the real axis and move into
the complex plane. Computer simulations of such combined
effects are given below.
v.
1 ei(w-u)t
e iwt d t = --
1
T
T
.
=-sin(w-v)~
n
(28)
and for real v and sufficiently long pulses, the spectrum
becomes increasingly sharper, approaching in the limit a line
spectrum, i.e., a S-impulse function situated at w = v:
2n
27ri(w-v)
-T
lim 1
2X
.
e
iwt
-T
dt = S(w - v).
+'W
(29)
The case of a complex frequency v = a + ip is different. For
this case the transformation (28) yields
1
2lrz'(LJ-v)
-
- eJre--r(d--a)r
z
27ri(&d- v )
e3~et(d--a)r
-/
-
ePT
Jm
27r
(31)
where the asterisk denotes the complex conjugate. Clearly the
spectrum described by (31) is different from (29). As long as
T is sufficiently large for (31) to hold, the shape, excepting
the amplitude, of this spectrum is independent of the value T
chosen, because epTis a constant. The shape of the spectrum is
symmetrical about w = a, where at the central value it attains
the maximum amplitude, and falls off as Iw - a ( increases. The
amplitude (31) falls off to 1/& of its peak for w satisfying
OW = (W - a ( = p
(32)
where Aw is half the bandwidth, hence the relative bandwidth
of this spectrum can be written as
SPECTRAL CONSIDERATIONS
The detection of the Doppler frequency shift is performed by
subjecting the received time signal to a Fourier transformation.
Usually this is done digitally, by properly sampling the time
signal and subjecting the data to a fast Fourier transform
(FFT) algorithm. The result consists of complex data which are
usually displayed as real and imaginary, or modulus and phase
parts of the complex spectrum. In the case of Doppler spectra
due to stochastic ensembles of particles, encountered for
example in medical pulsed Doppler systems [7]-[9], [13], [14],
the significant indicator is the modulus, usually referred to as
the amplitude spectrum, or its square-the power spectrum.
In order to keep the following argument as simple as possible, we consider, as previously, the simplest case of a plane
wave. This is achieved by modeling the sourceheceiver and
scatterer as point sources, i.e., small compared to wavelength
and situated many wavelengths apart. At the receiver we
assume a time signal e-iut, existing between times -T to +T
and vanishing elsewhere, where v is in general a complex
frequency. This adequately describes the fact that we are
dealing with pulses. The Fourier transform will be written in
the form
JTe - i u t
spectrum is given by
e3Tet(d-O)T
27Ti(u: - v )
(30)
where for positive a and T finite but sufficiently large the
decreasing term involving e-DT is neglected, compared to
the exponentially growing term. The corresponding amplitude
2Aw - 2 p
-a
a
(33)
In real situations we deal with a whole spectrum of Doppler
shifted frequencies arriving at the receiver. This might be
the result of dealing with a collection of scatterers moving
according to some spatially distributed velocity field. Even
when only one scatterer is involved, or if we have a collection
of scatterers all moving at the same velocity, the receiver
might have a finite aperture, as for example in the case of
ultrasound transducers. In such a case we deal with a finite
width spectrum (rather than a single line as in (29)) from the
start. The additional spread introduced by the lossy media,
acting on each spectral component according to (31), with the
attenuation determined by the geometry of the problem and
the medium involved, further obliterates the information we
try to glean from the received signal.
In many systems, e.g., instrumentation for medical ultrasound diagnostics, the signal processing at the receiver starts
by downshifting the received spectrum by the carrier frequency
(so called RF, i.e., radio frequency, even if the context is
ultrasound) W O . This operation, when applied to signals as in
(28)-(30), amounts to taking a = 0 in the pertinent expression
for the spectrum. Obviously the definition (33) of the relative
bandwidth becomes inapplicable. Usually the systems are
equipped with special detectors capable of determining the
sign of w - a,thus facilitating the distinction between motion
toward and away from the radiatinglreceiving antenna or
transducer. Clearly in such circumstances the broadening effect
of the spectrum due to medium losses is even a more salient.
In order to avoid this broadening effect, or at least reduce it
as much as possible, it is suggested here that a filtering process
be applied to the received signal. Rewriting (10) in the form
wr = w - 2kv = w - 21cv - 22yv
(34)
where k = K + iy is the complex propagation vector and
the ratio y / of~ its imaginary to real parts is a measure of
the losses encountered in the propagation process. Neglecting
phase factors, the received time signal due to (34) is now
e-aw,t
- e-2yvye-iwt+i2~vt
(35)
I92
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clearly showing the Doppler frequency shifting due to the
velocity v, and the exponential attenuation resulting from the
motion of the scatterer away from the transmitterheceiver
location. Downshifting the received signal by the carrier
frequency, as explained previously, amounts to putting w = 0
in (39, but note that by doing so the information regarding the
sign of w -2nv is lost unless special measures are incorporated
in the instrumentation. Presently we must also make sure
that the sign of attenuation, amplification exponent involving
y, when the scatterer moves away, toward the transducer,
respectively, is correctly taken into account. If the received
signal (35) is multiplied by a factor eayUt,the time dependent
exponential modulation will be eliminated and the spectrum
will show a sharp line at frequency CY.This scheme, as
attractive as it seems to be, is not always possible: Inasmuch
as the velocity and the loss factor are not always known, the
suggested time domain filtering will have to be done on a
trial and error basis, and therefore a real-time implementation
of this idea is probably unfeasible. The problem is further
complicated when more than one scatterer is present, each
moving according to its individual velocity. In this case a
windowing effect will appear, by which the improvement of
the spectral region relevant to the Doppler effect of one particle
will cause additional degradation of the spectra associated with
the motion of other particles. Finally, when the transducer
or antenna are more complicated, a more detailed analysis
is necessary, taking into account the aspect angles of various
parts of the aperture with respect to the motion. A few relevant
simulations are presented below.
VI. REMOTE-SENSING OF MEDIUMLOSSES BY MEANS
OF THE COMPLEX DOPPLER
EFFECT
It is clear from (34), (35) that the inverse problem can also
be formulated: If a scatterer is given whose velocity is known,
then by measuring the return signal one can measure the
parameter 7.This is a somewhat oversimplified statement of
the problem, but the principle involved is clear. The advantage
of such a set up is obvious. One might measure the medium’s
losses by performing a one way forward propagation experiment or a measuring a backscattered signal. By comparison
of the amplitude at two points, and knowing the distance
between these two locations, the losses incurred may be
measured. However, the measurement of the Doppler signal
(35) is simpler in many respects. We do not have to know
the locations, and we do not have to repeat the measurement
to derive dependable average values. In the present case we
can exploit time segments of the Doppler signal to estimate
the losses from the exponential decay or growth of the signal.
The price we have to pay is the need to know the velocity.
This is yet a raw idea and needs more research. Theoretical
limitations must be assessed and corresponding simulations
are needed to support the conclusions.
VII. NUMERICAL
SIMULATIONS
The following numerical simulations are based on the quasistationary formalism. As stated previously [ 5 ] , this formalism
is inconsistent even to the first order in the Mach number v/c.
However, if all we are interested in are frequencies (and not
propagation vectors), and with due care, the formalism may be
used for simulating first-order Doppler effects. An important
aspect of the following simulations involves the sampling rate.
It will be noticed that we are dealing with carrier frequencies
in the MHz band, e.g., fo = w0/27r = 3 MHz, and the
number of samples in the hundreds, e.g., 512, over a time
span of fractions of seconds, e.g., 0.1. s. This appears to be
an inadequate sampling rate, far below the required Nyquist
frequency, which is at least twice the highest frequency we
attempt to recover. In order to avoid the necessity of high
sampling rate, the time signal is first downshifted by W O .
This is implemented, for example, in medical pulsed Doppler
ultrasound instrumentation. In the hardware implementation it
is also necessary to use a dual channel quadrature arrangement
that distinguishes between positive and negative signs of the
frequency difference. More detail is available in the literature
[7]-[9]. The software implementation of this idea is trivial
and merely involves the suppression of the factor e-awot.
Once this is performed, the present algorithm applies only
to the computation of the absolute value of the spectrum, i.e.,
the amplitude spectrum. Finally, it is noted that the graphs
displayed below zoom onto the interesting results and therefore
do not contain all the available time signal and FlT data.
In Fig. 1 the broadening of the Doppler spectrum is investigated for the case of a simple isotropic point scatterer
uniformly moving in the radial direction with respect to a
point source. A linear acoustic medium is considered, with real
source frequency of fo = w0/27r = 3 MHz. The propagation
constant k = n for the lossless case is real and is taken as 12
006, i.e., for this limiting case we have a real phase velocity
of 1570 m/s. For the lossy case k = n iy is complex,
and the Doppler effect is investigated for various values of
the loss ratio y/n. The motion is in the radial direction
according to s ( t ) = 50 vt, where v = 0.2m/s, 10 = 0.06m,
and the time is taken from t = 0 to t = 0.1 s, sampled
at 51? points. This is also the number of the points used
in the F l T algorithm. The downshifted amplitude spectrum,
centered about zero frequency, is displayed for various values
of y/n. The broadening effect due to attenuation is evident: for
Y / K . = 0 the spectrum is almost a line spectrum as predicted
by (29). In view of the discrete nature of the simulation scheme
and the finite number of samples, also because our algorithm
contains a 1/r inverse distance factor, the spike displayed in
Fig. 1 is considered to be a satisfactory approximation. As
increases the spectrum becomes broader, as predicted by
(30H31). It should be noted that the losses considered here
are enormous, and these values are chosen to emphasize the
salient characteristics of the problem. In any real set up such
values would surpass the dynamic range of the instrumentation
under consideration.
It has been argued previously, that for vibratory, or harmonic
motion as expressed by (15) and the discussion following
it, there should be a negligible effect on the spectrum as
long as the amplitude of this motion is small, such that the
attenuation does not vary significantly along the path. In
such a case we expect the frequencies to remain real, i.e.,
upon subjecting the time signal to the FFT algorithm, extra
+
+
CENSOR REAL AND COMPLEX DOPPLER EFFECTS IN LOSSY MEDIA
1 93
1
I
0.8
0.6
.s
s
U
0.4
P
0.2
0
-m
-1500
-1OOo
-500
Frequency (Hz)
0
so0
Fig. 1. Spectrum broadening due to complex Doppler effect. For larger loss
factor -,/ti the spectrum is broader.
spectrum broadening is not expected. To test this aspect of the
problem the following parameters have been chosen: Similar
to (14), harmonic motion of the scatterer is assumed, with
the mean distance from the point source taken as zo = 0.1
m, 5d.Z mm, and R=lOlr. The time was taken between
t=O and t=l s, sampled at 256 points, which is also the
number of data used in the FFT algorithm. As in the previous
example, fo = w0/27r = 3 MHz, k = n for the lossless case
equals 12006, corresponding to a phase velocity of 1570 m/s.
From the point of view of frequency modulation, the present
situation corresponds to a modulation index [26] of value 6.
It should be noted that variation of the ratio 7/n in this case
amounts to modulation of the harmonic part of the time signal
by a factor e-r'oCOSRt,which changes the time signal, and
because n is kept constant and 7ln is varied-can also be
viewed as a change in the value of the modulation index.
Consequently, the amplitude of any sideband.is not conserved,
although its location along the frequency axis is unaffected. In
Fig. 2 the spectrum for the lossless case -y/n=O is depicted. The
dotted line shows the envelope of the sidebands' amplitude.
Fig. 3 displays the case yln=l. The envelope is changed but
the overall width of the spectrum is essentially unaffected.
Only when the losses become very high, as in Fig. 4, where
the case $n=5 is displayed, the broadening of the spectrum
becomes appreciable. The facts that the width of the sidebands
and the spacing between them are practically unaffected are
displayed in Fig. 5 . Here the results for ~IIE=Oand 7/n=5
are superimposed and the graph zooms in on the fust few
sidebands.
In order to test the ideas of temporal filtering explained
previously, (34), (35) and the following discussion, additional
simulations have been performed. Two particles moving simultaneously are considered. Both start at time t = 0 at a
distance 0".
from the transmitter/receiver, which is agah a
point radiator, and move until t = 0.02 s. One particle moves
toward the source with a velocity w1 = -0.2 m/s, the other
is moving away according to 712 = 0.4 m/s. The loss factor
chosen here is -y/n = 0.02, which is quite low compared
to the previous examples. In the presence of much higher
values the exponentially decreasing signal due to the particle
moving away from the source is completely swamped by the
-75
-50
-25
0
Frrsucncy(Hz)
25
50
75
Fig. 2. Spectrum due to harmonic motion. Lossless case. See text for a
detailed description of parameters.
1
0.8
P
0.6
.s
0.4
!
f
P
0.2
0
-75
-50
-25
0
fimuency OW
25
50
Fig. 3. Spectrum due to harmonic motion. Loss factor ? / K
75
= 1.
1
0.8
1
0.6
i!
.sx
0.4
a
0.2
0
-75
-50
-25
0
25
50
75
Frrsuency (Hz)
Fig. 4. Spectrum due to harmonic motion. Loss factor - y / ~ = 5.
exponentially increasing signal due to the particle approaching
the source. The number of sampling points is 256 for each
particle, and as before k = 12006 is the real part of k. In Fig.
6 the two individual time signals (real part) are displayed, in
Fig. 7 the sum is displayed. The ordinate is chosen arbitrarily.
The? graphs dearly show how the approaching particle's time
signal, because of its higher energy, masks the signal produced
by the retreating particte. The time filtering is performed by
multiplying the time sequence of Fig. 7 by the exponential
e2Ywt, where v = v1 = -0.2 m/s. The resulting time signal
is displayed in Fig.8. The elimination of the exponential
194
IEEETRANSACTIONS ON ULTRASONICS, FERROELECTRICS, AND FREQUENCYCONTROL, VOL. 3 ~ NO.
, 2, MARCH 1992
I
I
I
I
I
I
0.6
B
56 0.2
5
;. -0.2
I
2
-0.6
-1
-15
-10
0
-5
10
5
15
i
0
Fnsucncy (Hz)
0.004
0.008
0.016
0.012
0.02
Tune (sec)
Fig. 5. Zoom on the central parts for loss factors 7 / K = 0 and 7 /ti = 5. It
is emphasized that the width and the spacing of the sidebands is unaffected.
Fig. 8. Time signal (real part) after exponential temporal filtering.
1
I
I
I
I
I
I
I
I
I
1500
2000
0.8
0.6
0.4
0.2
0
0.02
1
,
0.004
,
,
I
I
i
0.008
0.012
T i (sec)
1
0.016
i
0
-2CW -1500 -1000
-500
0
500
Fraluency (Hz)
1000
Fig. 9. Amplitude spectrum corresponding to the (complex) time signal
associated with Fig. 8.
Individual (real part of) time signal due to two
oppositely moving particles.
Fig. 6.
- 1 ,
0
0.016
0.008
0.012
T i (sec)
0.004
I
0.02
Fig. 7. Combined (sum of real parts of) time signal due to two
oppositely moving particles.
growth contributes to the sharpening of the spectral peak
corresponding to the approaching particle. At the same time
the retreating particle is further attenuated by the additional
exponential e2yut and therefore its spectrum is even more
degraded. The amplitude spectra before and after filtering are
displayed in Fig. 9.
VIII. CONCLUSION
A preliminary investigation of the effect of propagation
losses on Doppler signals is reported. Simplified theoretical
models are employed, backed up by numerical simulations.
The sjmplest way of looking at the problem is to realize
that signals from various distances are differently affected
by the medium loss parameter. This gives rise to exponentially increasing or decreasing time signals. Such signals
may contain different energies, and therefore weak signals,
due to retreating particles will be masked by higher energy
signals present. When performing a spectral analysis, the
weak signals might disappear if the losses are sufficiently
high. Even if this problem does not exist, the losses and the
associated exponentially modulated time signals will cause
spectral broadening. This decreases resolution and should be
considered as spectrum degradation.
Sometimes the adverse results can be compensated by
temporal filtering, i.e., by judiciously multiplying the time
signal by an exponential signal with an opposite sign exponent.
The point of view relating the effects to distances traversed
by propagating waves is useful also for understanding the
spectra due to periodically moving scatterers. If the scatterer
vibrates abour some point in space, with small changes in the
losses incurred during a cycle, then the spectrum will again be
real, i.e., the broadening of the spectral lines will be negligible.
More theoretical work and simulations, maybe also experiments, are needed in order to better understand the effects
of losses on multiple particle systems and arbitrary apertureslantennas involved.
CENSOR REAL AND COMPLEX DOPPLER EFFECTS IN LOSSY MEDIA
ACKNOWLEDGMENT
The author is grateful to members of the Department of
Electrial a d Qmputer Engineering, Ben Gurion University
of the Negev, Beer Sheva, Israel: Mr. E. Zonnenxheh for
his useful comments, and particularly for his help in deriving
the computational results, and Dr. D. Wulich for his helpful
comments regarding signal analysis.
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Dan Censor received the B.Sc., M.Sc., and D.Sc.
degrees in 1962, 1963, and 1967, respectively, all
from the Technion-Israel Institute of Technology,
Haifa.
He is a Professor of Electrical Engineering in
the Department of Electrical and Computer Engineering, Ben Gurion University of the Negev, Beer
Sheva, Israel. His current interests are in the areas of
relativistic electrodynamics, rays, and wave propagation and scattering systems, e.g., electromagnetic
and acoustical, linear and nonlinear, lossless and
absorptive, and Doppler effects in the presence of moving media and moving
objects.
Dr.Censor is a member of the URSI Israel National Committee and of the
MIT Electromagnetic Academy.