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Ultrasonic Doppler velocimeter using cross-bispectral analysis
ULTRASONIC IMAGING 1, 144-153
(1979)
ULTRASONIC DOPPLER VELCCIMETER USING CROSS-BISPECTRAL ANALYSIS Takuso
Satol
The Graduate School at Nagatsuta Tokyo Institute of Technology 4259 Nagatsuta, Midori-ku, Yokohama-shi,
Japan
and Osami Sasaki Faculty of Engineering Niigata University Gakkocho 1, Nagaoka-shi,
Japan
The cross-bispectral characteristics of three simultaneouslydetected signals are used to realize a new ultrasonic Doppler vector velocimeter. This system has a special feature in that it is not affected by additive Gaussian noise. The principle, construction and some experimental results are shown. Key words:
Bispectral analysis; ratio; three-dimensional;
Doppler effect; ultrasonic;
signal-to-noise velocity.
Introduction The ultrasonic Doppler velocimeter is a powerful means for the measurement of the velocity of an object in an optically-opaque medium, such as the flow of blood in human body, without physical contact with the object [l]. In most applications, however, only the velocity component in one direction has been measured. In order to measure the direction of a velocity vector, at least three components of the velocity vector must be measured. A direct solution is given by arranging three independent conventional velocimeters and combining the results for the estimation of the velocity vector. In this method, however, information about the movement of the object at an instant in time is not measured directly, but only through the combination of independent measurements of velocity components. In other words, in this method, the intrinsic detailed relations which are included in the simultaneously-detected signals are not fully utilized. For this purpose, we consider in this paper that if we detect signals simultaneously with multiple detectors, then any movement of the object must be reflected in the simultaneously-detected signals in a specific way according to the geometry of the detecting system. Hence, if we could detect and use the specific relations between simultaneously-detected signals, instead of independent *Author
to whom correspondence
should
0161-7346/79/020144-10$02.00/O Copyright All rights
@ 1979 bv Acudemic Press, Inc. in my form resen*ed.
qf reproduction
144
be addressed.
ULTRASONIC DOPPLER VELCCIMETER
measurements of three components, a more effective measuring system which extracts even very faint phenomena in the observing volume would be realized. We adopted the cross-bispectrum as the specific relation between the three simultaneously detected signals. The bispectrum B,(wl,w2) of a random signal X(t) with zero mean is defined as the relation among three frequency components of wl, ~2 and Al+ w2 in the signal 12-41. This lation
statistical parameter is related to the function Rx(Tl,~2), which is defined by
Rx(Tl,T2)
corre-
= E~X(t)X(t+~l)X(t+'2))
11)
where E(X) indicates the expectation Fourier transform as follows: Bx(wl,w2)
third-order
of X, by the two-dimensional d-r 1d-r 2.
= llRx(rl,72)e-'w1'1-'w2T2
Hence the bispectrum is related to the three random variables X(t), The bispectrum has a significant value if and X(t+'rl) and X(t+T2). only if there is a relationship between the three random variables. That is, it has a significant value when there is a significant relationship among frequency components ~1, w2 and wl+w2 in the frequency domain. The third frequency ~3 is determined from wl and w2 by 03 = wl"*'2 due to the stationarity of the random signal [2,3] That is, if a random signal is stationary, we have a non-zero value only for the combination of frequencies which satisfy the relation w1+w2 = w3. This is the definition of the bispectrum of a single random signal. Now, let us extend the concept for the case where three random signals are treated. If we replace X(t), X(t+Tl), and X(t+T2) by and X3(t+T2), respectively, then the cross-bispecxl(t) , X2(t+Tl) trum among Xl(t), X2(t) and X3(t) is defined as follows: Bx~~x~,x~(~~,W~)
= ~~Rxl,X2,x3(~1,~2)-jW171-jW2T2
where
d7 1 dT 2 , (3)
Rx~'~~,x~(T~J~)
= E(Xl(t)X2(t+rl)X3(t+r2)}
.
This statistical parameter represents the relationship between frequency components of w1 in Xl(t), w2 in X2(t) and wl+W2 = W3 parameter in X3(t). Thus, it can be used as the statistical describing the three simultaneously-detected signals Xl(t), X2(t) and X3(t). The special feature of this statistical parameter may be explained most dramatically in connection with the common additive Gaussian noises components. The usual disturbing noise in Doppler velocineter systems nay be considered to result from the addition of many independent random factors, such as small independent turbulences, perturbations and thermal noise. The observed noise may be considered as the addition of these noise components; they tend to be Gaussian as the central limit theorem asserts [2].
14.5
SAT0 AND SASAKI
It is well known that the bispectrum of any Gaussian random signal vanishes completely. In other words, there is no relationship between the three frequency components ~1, "2 and wl + ~2 in Gaussian random signals. On the other hand, in the present system, the three simultaneously-detected signals have, of course, specific relationships between them, since they result from the movement of a specific object in the observed volume. if there is common additive Thus, it becomes very clear that, Gaussian background noise in the three detected signals, crossbispectral analysis of the detected signals will eliminate them completely, while maintaining the information of the velocity of the object under measurement in the form of the bispectral characteristics. Of course, as we are using three detectors, the velocity information in the form of a vector in three-dimensional space is reflected in the bispectral region in a specific way. Since the statistical relationship among the three it is also clear that the noise independently employed, each detector can be eliminated. Thus, a Doppler velocimeter which is ing noise can be realized. Its construction results are given in the following sections.
Principle
and Construction
of the
signals is introduced
not affected by surroundand some experimental
System
The construction of the system based upon the bispectral approach is shown schematically in figure 1. Ultrasonic waves are transmitted from the transducer focused on the observation region. They may be focused by using a transducer array or concave transducers and the depth resolution may be improved by using pulsed waves or M-sequence modulation 151 as in conventional ultrasonic Doppler velocimeters, so that a small sampling volume is obtained. The ultrasonic waves scattered by the moving object in the sampling volume are detected by three detectors. The three signals are heterodyned and their cross-bispectrum calculated in a minicomputer. The results are interpreted to yield the value of the velocity vector. The relationship between the frequencies of the detected signals ri(t) fDi
velocity vector and the are as shown in figure
When the velocity vector is V(vx,vy,vz), (i = 1,2,3) includes the corresponding (i = 1,2,3) which are given as follows.
fD,=
$ (vxsinacosfil
fD, = hl{
fD3=
vxsincl'cosB
i {vxsinacos6
the detected signals Doppler frequencies
+ vysinclsin~
- vs(cosa
+ v sincl'sinB Y
- v (cosa' z
+ vysinclsinB
146
Doppler 2.
- vz(coscx
+ l)} + 1))
+ l)}
(5)
in
ULTRASONIC DOPPLER VELOCIMETER
s(t) 1
-0mr *;r):
amplifier rdt)
83 ultrasonic sign01 enerator
V(VX’VY,V
tronsmitter
receiver
rz(t)
2 \’
A’..
velocity vector output uv,, vy I
Fig.
Schematic
1.
construction
of the bispectral
ultrasonic
Doppler
velocimeter.
T:Transmitter
Fig.
0:Object
2.
R:Receiver
Geometrical
That is, they are determined ity vectors of the velocimeters
arrangement.
by the relations between the sensitivand the velocity of the object.
If we use the fixed condition sincl.sinB' = sina' .sin@ = 1 sina.sinB, then the Doppler frequencies are as follows: T
fDl rffl = fD2 = I 1fD3
Now,
detected
b
CI
V
a2 b
~2
V
a3 2b cl
V
al
let us show what by the cross-bispectral
class
X
Y
=
[Al
WI
.
(7)
z
of velocity vectors can be analysis of the detected signals.
147
SAT0 AND SASAKI
A
s
FfD, as ?z k
0
3
(
$I = tan-l
a, + a2 - a3
-
cg
04
)
Fig.
3.
Special
plane
Fig.
4.
Cross-bispectrum II.
n determined
by the
corresponding
V(V Then the relation
x
al + a2 -a3 ,v , CP Y Doppler frequencies fD + fD = fD . 1
2
0 s Frequency
,
4, f D,
arrangement.
geometrical
to velocity
First, let us consider a class of velocity in a specific plane n which is determined by the arrangement of the system as shown in figure 3. are given by
P
____m-- --
vector
vectors
in plane
included
geometrical
These
velocities
vx) .
(8)
of the detected
signals
have the
3
There exists an important relationship between three frequency components fl, fp and f3, which satisfy the equation fD, f fD2 = it is known generally as the bispectrum of a stationary random fD3: signal. Hence as far as the velocity vector in the plane n is concerned, it is clear that the Doppler frequencies can be detected through bispectral analysis of the detected signals as stationary random signals. solution matrix
The velocity of a set manipulation.
vector components can be obtained directly as the of linear equations given in eq. (7) by means of For example, v is given by x -c2(fD1
v
x
-
fD,)
.
= cl
(al+a2-a3)
-
c2
(2al-a31
(9)
Hence, for instance, if only one velocity vector exists, as shown in figure 4, the corresponding cross-bispectrum has a single peak in the fDr and fD2 plane with the combination of the third frequency fDs = fD1 + fD2- That is, roughly speaking, the direction of OP corresponds to the direction of the velocity vector, and the length of OP represents its magnitude. In this way, the velocity vector in the plane n can be measured through cross-bispectrum analysis. For the observation of the velocity vector which is not included in the plane II as shown in figure 5, two methods can be considered. The first one is the mechanical rotation of the whole
148
ULTRASONIC DOPPLER VELOCIMETER
Fig.
5.
Measurement
of the velocity
vector
outside
system, so that the observable plane n coincides one. In this case, the same signal processing The second method is the equivalent plane by changing the signal processing, cal arrangement of the system unchanged. ing relations are obtained. +d,f
dlf
D2
n.
with the required can be used.
rotation of the observation while keeping the geometriIn this case, the follow-
=f
Dl
plane
(10) D3
where a3 + r(c1 - a2 + r(c1
-2a2
dl =
2al - a3 + rc1 al - a2 + r(c1 -
d2 =
and the velocity v
x
=
components
v
Y z
=
-
a2c)fD1
b(cl(al
cmmvA c
c2)
+
(11)
,
are
c(fD, - fD1) + a2 - a31 -
cl(al
(c2A v
- 2~2) - cp)
+
al
given
by
c2(2al
(-ClA
f
alc)fD2
+ a2 - a31 - c2(2al x
a.2)
- as))
(12)
'
where dial
- d2a2
c = Cl - dlcl
- d2c2
A =
a3
-
In this case, the definition frequency combinations which system, W3' In our measuring the movement of object in the ship between them still exists, frequency space being different random signals. where
(13)
of cross-bispectrum is extended for do not satisfy the relation wl + w however, these components result f?om observing volume; hence the relationin spite of the analyzing region in from that in the case of stationary
As for stationary common random signals, the relation ~1 + ~2 = w3 is not satisfied;
149
we are now observing hence, they donot
SAT0 AND SASAKI
, l-
(Hz) 60 t
l-
b-
06
L
V
L
L
40 Frequency 50 fo,
.
07
60 0-k)
Fig.
6.
Cross-bispectrum
for
Fig,
7.
Cross-bispectra for different directions
the
0 d '0
velocity
20
40
Frequency
vector
the velocity vectors in plane n.
f D,
parallel with
60(HZ)
to y-axis. three
take significant value. The non-stationary signals which happen to system, if satisfy the relation of ~1, w2 and w3 in our observation they are Gaussian random signals, also vanish in the process of the cross-bispectral analysis. The other noise components which enter independently into each detector can be eliminated in the same way as in the previous case. Experimental
Results
A practical system based on these principles was constructed. Three MHz ultrasonic waves were used in water. An acrylic disk with random thickness was rotated in various orientations to serve as a convenient moving object. Other a = 30°, a1
= 0.90,
parameters ~1' = 14.48O, a2
=
0.25,
were
as follows:
f3 = 60", a3 = 0.50,
B' = 25.66', a = 0.43,
The detected signals were A/D converted and were processed in a minicomputer.
after
A = 0.5mm cl
= -3.73,
heterodyne
c2 = -3.93 detection
First, the velocity vector along the y-axis was measured. Of course, it is included in the plane II. The observed crossbispectrum is shown in figure 6. One peak can be seen at (48 Hz, 48 Hz). This peak corresponds to the velocity vector V with vx = as easily derived from the relation shown Vz = 0 and v y = 109 n&s, before. As for the resolution of the observed velocity, the data length for the measurement of the cross-bispectrum was 1 second and the width of the bispectral window for smoothing of the bispectrum was set to 7 Hz, resulting in a frequency resolution of about 7 Hz.
150
ULTRASONIC DOPPLER VELOCIMETER
(mmk)
----a
true
--,
measured
velocity
(Hz)”
vector
velocity
vector
f
75
20 .9 y’ 10 bfff!i- --
-
_____---e=2
0 80 -10
IS
e=-5” ----a_____-
100:
‘o.75 ( :
120 vvww
,
Fig.
8.
Measured velocity vectors in plane
vectors II.
Fig.
9.
Cross-bispectrum
for
Fig.
10.
a velocity
Power spectra of three and without corrupting
The velocity were also observed are shown in figure
for
three
different
vector
independently noise.
velocity
outside
plane
measured
signals
n.
with
vectors of several directions in the plane n The results by rotating the object in the plane. have peaks at the appropriate 7. The bispectra
151
SAT0 AND SASAKI
----a
true
-measured
Fig.
11.
Cross-bispectrum
Fig.
12.
Velocity
positions.
figure ities.
The
8.
These
vector
velocities
results
of signals derived
to
vector velocity
by additive
from observed
corresponding
are
corrupted
velocity
vector
noise.
cross-bispectrum.
the
in good agreement
results
with
are
the
true
shown
in
veloc-
Next, to measure the velocity vector outside the plane II, the moving object was rotated so that the new plane was at about 10 degrees to the plane n. The cross-bispectrum is shown in figure 9. In this case the cross-bispectrum had one peak atfD1 = 42 Hz and the velocity vector from these fD2 = 34 Hz. We were able to derive results, and the result showed good agreement with the true velocity vector. Finally, to show the ability of additive noise, the received with Gaussian-like noise.
of the system to eliminate the effect signals were corrupted intentionally
The power spectra of three original and three disturbed signals are shown in figure 10 by the dotted and solid lines, respectively. In this case, the velocity vector was in the y-axis direction. If the velocity vector were to be derived from these power spectra, as in the conventional three-dimensional Doppler velocimeter, it is clear that we could not distinguish one real velocity from the many ghost velocities because of many peaks in fd,, due to the Gaussian noise. The cross-bispectrum obtained with our new method is shown in figure 11. In this case, one peak is observed. That is, the effect of corrrupting noise is eliminated by the application of the bispectral analysis. The velocity vector obtained is shown in figure 12. This result is also in good agreement with the true velocity. Conclusion Doppler
The effectiveness of the new three-dimensional velocimeter has been shown theoretically
152
ultrasonic and experimentally.
ULTRASONIC DOPPLER VELOCIMETER
The applications of the method for measurement of flow characteristics in practical Situations, such as the flow of blood in human of simplified hardware, are under study. body r and construction Acknowledgment The authors reviewers of this
would like to express their sincere Journal for their kind suggestions
thanks to the and advice.
References
[ll
P. N. T., Wells, (Academic Press,
r21
Harris, B., Spectral New York, 1967).
r31
Time Series Data Analysis Brinllinger, D. R., -~(Holt, Rinehart and Winston, New York, 1975).
r41
Sato, T. and Sasaki, K., Am. 9, 404-408 (1977). -
IIs1
Peterson, W. W., Error Correcting Tokyo, 1967) Chapt. 7.
Physical Princia of -- Ultrasonics London and New York, 1969). Analysis
of Time Series, --
Bispectral
153
Holography, Codes
(Univ.
Diagnosis,
(John
Wiley,
and Theory, J. - Acoust. ~ Tokyo
Press,
-Sot
(1979)
ULTRASONIC DOPPLER VELCCIMETER USING CROSS-BISPECTRAL ANALYSIS Takuso
Satol
The Graduate School at Nagatsuta Tokyo Institute of Technology 4259 Nagatsuta, Midori-ku, Yokohama-shi,
Japan
and Osami Sasaki Faculty of Engineering Niigata University Gakkocho 1, Nagaoka-shi,
Japan
The cross-bispectral characteristics of three simultaneouslydetected signals are used to realize a new ultrasonic Doppler vector velocimeter. This system has a special feature in that it is not affected by additive Gaussian noise. The principle, construction and some experimental results are shown. Key words:
Bispectral analysis; ratio; three-dimensional;
Doppler effect; ultrasonic;
signal-to-noise velocity.
Introduction The ultrasonic Doppler velocimeter is a powerful means for the measurement of the velocity of an object in an optically-opaque medium, such as the flow of blood in human body, without physical contact with the object [l]. In most applications, however, only the velocity component in one direction has been measured. In order to measure the direction of a velocity vector, at least three components of the velocity vector must be measured. A direct solution is given by arranging three independent conventional velocimeters and combining the results for the estimation of the velocity vector. In this method, however, information about the movement of the object at an instant in time is not measured directly, but only through the combination of independent measurements of velocity components. In other words, in this method, the intrinsic detailed relations which are included in the simultaneously-detected signals are not fully utilized. For this purpose, we consider in this paper that if we detect signals simultaneously with multiple detectors, then any movement of the object must be reflected in the simultaneously-detected signals in a specific way according to the geometry of the detecting system. Hence, if we could detect and use the specific relations between simultaneously-detected signals, instead of independent *Author
to whom correspondence
should
0161-7346/79/020144-10$02.00/O Copyright All rights
@ 1979 bv Acudemic Press, Inc. in my form resen*ed.
qf reproduction
144
be addressed.
ULTRASONIC DOPPLER VELCCIMETER
measurements of three components, a more effective measuring system which extracts even very faint phenomena in the observing volume would be realized. We adopted the cross-bispectrum as the specific relation between the three simultaneously detected signals. The bispectrum B,(wl,w2) of a random signal X(t) with zero mean is defined as the relation among three frequency components of wl, ~2 and Al+ w2 in the signal 12-41. This lation
statistical parameter is related to the function Rx(Tl,~2), which is defined by
Rx(Tl,T2)
corre-
= E~X(t)X(t+~l)X(t+'2))
11)
where E(X) indicates the expectation Fourier transform as follows: Bx(wl,w2)
third-order
of X, by the two-dimensional d-r 1d-r 2.
= llRx(rl,72)e-'w1'1-'w2T2
Hence the bispectrum is related to the three random variables X(t), The bispectrum has a significant value if and X(t+'rl) and X(t+T2). only if there is a relationship between the three random variables. That is, it has a significant value when there is a significant relationship among frequency components ~1, w2 and wl+w2 in the frequency domain. The third frequency ~3 is determined from wl and w2 by 03 = wl"*'2 due to the stationarity of the random signal [2,3] That is, if a random signal is stationary, we have a non-zero value only for the combination of frequencies which satisfy the relation w1+w2 = w3. This is the definition of the bispectrum of a single random signal. Now, let us extend the concept for the case where three random signals are treated. If we replace X(t), X(t+Tl), and X(t+T2) by and X3(t+T2), respectively, then the cross-bispecxl(t) , X2(t+Tl) trum among Xl(t), X2(t) and X3(t) is defined as follows: Bx~~x~,x~(~~,W~)
= ~~Rxl,X2,x3(~1,~2)-jW171-jW2T2
where
d7 1 dT 2 , (3)
Rx~'~~,x~(T~J~)
= E(Xl(t)X2(t+rl)X3(t+r2)}
.
This statistical parameter represents the relationship between frequency components of w1 in Xl(t), w2 in X2(t) and wl+W2 = W3 parameter in X3(t). Thus, it can be used as the statistical describing the three simultaneously-detected signals Xl(t), X2(t) and X3(t). The special feature of this statistical parameter may be explained most dramatically in connection with the common additive Gaussian noises components. The usual disturbing noise in Doppler velocineter systems nay be considered to result from the addition of many independent random factors, such as small independent turbulences, perturbations and thermal noise. The observed noise may be considered as the addition of these noise components; they tend to be Gaussian as the central limit theorem asserts [2].
14.5
SAT0 AND SASAKI
It is well known that the bispectrum of any Gaussian random signal vanishes completely. In other words, there is no relationship between the three frequency components ~1, "2 and wl + ~2 in Gaussian random signals. On the other hand, in the present system, the three simultaneously-detected signals have, of course, specific relationships between them, since they result from the movement of a specific object in the observed volume. if there is common additive Thus, it becomes very clear that, Gaussian background noise in the three detected signals, crossbispectral analysis of the detected signals will eliminate them completely, while maintaining the information of the velocity of the object under measurement in the form of the bispectral characteristics. Of course, as we are using three detectors, the velocity information in the form of a vector in three-dimensional space is reflected in the bispectral region in a specific way. Since the statistical relationship among the three it is also clear that the noise independently employed, each detector can be eliminated. Thus, a Doppler velocimeter which is ing noise can be realized. Its construction results are given in the following sections.
Principle
and Construction
of the
signals is introduced
not affected by surroundand some experimental
System
The construction of the system based upon the bispectral approach is shown schematically in figure 1. Ultrasonic waves are transmitted from the transducer focused on the observation region. They may be focused by using a transducer array or concave transducers and the depth resolution may be improved by using pulsed waves or M-sequence modulation 151 as in conventional ultrasonic Doppler velocimeters, so that a small sampling volume is obtained. The ultrasonic waves scattered by the moving object in the sampling volume are detected by three detectors. The three signals are heterodyned and their cross-bispectrum calculated in a minicomputer. The results are interpreted to yield the value of the velocity vector. The relationship between the frequencies of the detected signals ri(t) fDi
velocity vector and the are as shown in figure
When the velocity vector is V(vx,vy,vz), (i = 1,2,3) includes the corresponding (i = 1,2,3) which are given as follows.
fD,=
$ (vxsinacosfil
fD, = hl{
fD3=
vxsincl'cosB
i {vxsinacos6
the detected signals Doppler frequencies
+ vysinclsin~
- vs(cosa
+ v sincl'sinB Y
- v (cosa' z
+ vysinclsinB
146
Doppler 2.
- vz(coscx
+ l)} + 1))
+ l)}
(5)
in
ULTRASONIC DOPPLER VELOCIMETER
s(t) 1
-0mr *;r):
amplifier rdt)
83 ultrasonic sign01 enerator
V(VX’VY,V
tronsmitter
receiver
rz(t)
2 \’
A’..
velocity vector output uv,, vy I
Fig.
Schematic
1.
construction
of the bispectral
ultrasonic
Doppler
velocimeter.
T:Transmitter
Fig.
0:Object
2.
R:Receiver
Geometrical
That is, they are determined ity vectors of the velocimeters
arrangement.
by the relations between the sensitivand the velocity of the object.
If we use the fixed condition sincl.sinB' = sina' .sin@ = 1 sina.sinB, then the Doppler frequencies are as follows: T
fDl rffl = fD2 = I 1fD3
Now,
detected
b
CI
V
a2 b
~2
V
a3 2b cl
V
al
let us show what by the cross-bispectral
class
X
Y
=
[Al
WI
.
(7)
z
of velocity vectors can be analysis of the detected signals.
147
SAT0 AND SASAKI
A
s
FfD, as ?z k
0
3
(
$I = tan-l
a, + a2 - a3
-
cg
04
)
Fig.
3.
Special
plane
Fig.
4.
Cross-bispectrum II.
n determined
by the
corresponding
V(V Then the relation
x
al + a2 -a3 ,v , CP Y Doppler frequencies fD + fD = fD . 1
2
0 s Frequency
,
4, f D,
arrangement.
geometrical
to velocity
First, let us consider a class of velocity in a specific plane n which is determined by the arrangement of the system as shown in figure 3. are given by
P
____m-- --
vector
vectors
in plane
included
geometrical
These
velocities
vx) .
(8)
of the detected
signals
have the
3
There exists an important relationship between three frequency components fl, fp and f3, which satisfy the equation fD, f fD2 = it is known generally as the bispectrum of a stationary random fD3: signal. Hence as far as the velocity vector in the plane n is concerned, it is clear that the Doppler frequencies can be detected through bispectral analysis of the detected signals as stationary random signals. solution matrix
The velocity of a set manipulation.
vector components can be obtained directly as the of linear equations given in eq. (7) by means of For example, v is given by x -c2(fD1
v
x
-
fD,)
.
= cl
(al+a2-a3)
-
c2
(2al-a31
(9)
Hence, for instance, if only one velocity vector exists, as shown in figure 4, the corresponding cross-bispectrum has a single peak in the fDr and fD2 plane with the combination of the third frequency fDs = fD1 + fD2- That is, roughly speaking, the direction of OP corresponds to the direction of the velocity vector, and the length of OP represents its magnitude. In this way, the velocity vector in the plane n can be measured through cross-bispectrum analysis. For the observation of the velocity vector which is not included in the plane II as shown in figure 5, two methods can be considered. The first one is the mechanical rotation of the whole
148
ULTRASONIC DOPPLER VELOCIMETER
Fig.
5.
Measurement
of the velocity
vector
outside
system, so that the observable plane n coincides one. In this case, the same signal processing The second method is the equivalent plane by changing the signal processing, cal arrangement of the system unchanged. ing relations are obtained. +d,f
dlf
D2
n.
with the required can be used.
rotation of the observation while keeping the geometriIn this case, the follow-
=f
Dl
plane
(10) D3
where a3 + r(c1 - a2 + r(c1
-2a2
dl =
2al - a3 + rc1 al - a2 + r(c1 -
d2 =
and the velocity v
x
=
components
v
Y z
=
-
a2c)fD1
b(cl(al
cmmvA c
c2)
+
(11)
,
are
c(fD, - fD1) + a2 - a31 -
cl(al
(c2A v
- 2~2) - cp)
+
al
given
by
c2(2al
(-ClA
f
alc)fD2
+ a2 - a31 - c2(2al x
a.2)
- as))
(12)
'
where dial
- d2a2
c = Cl - dlcl
- d2c2
A =
a3
-
In this case, the definition frequency combinations which system, W3' In our measuring the movement of object in the ship between them still exists, frequency space being different random signals. where
(13)
of cross-bispectrum is extended for do not satisfy the relation wl + w however, these components result f?om observing volume; hence the relationin spite of the analyzing region in from that in the case of stationary
As for stationary common random signals, the relation ~1 + ~2 = w3 is not satisfied;
149
we are now observing hence, they donot
SAT0 AND SASAKI
, l-
(Hz) 60 t
l-
b-
06
L
V
L
L
40 Frequency 50 fo,
.
07
60 0-k)
Fig.
6.
Cross-bispectrum
for
Fig,
7.
Cross-bispectra for different directions
the
0 d '0
velocity
20
40
Frequency
vector
the velocity vectors in plane n.
f D,
parallel with
60(HZ)
to y-axis. three
take significant value. The non-stationary signals which happen to system, if satisfy the relation of ~1, w2 and w3 in our observation they are Gaussian random signals, also vanish in the process of the cross-bispectral analysis. The other noise components which enter independently into each detector can be eliminated in the same way as in the previous case. Experimental
Results
A practical system based on these principles was constructed. Three MHz ultrasonic waves were used in water. An acrylic disk with random thickness was rotated in various orientations to serve as a convenient moving object. Other a = 30°, a1
= 0.90,
parameters ~1' = 14.48O, a2
=
0.25,
were
as follows:
f3 = 60", a3 = 0.50,
B' = 25.66', a = 0.43,
The detected signals were A/D converted and were processed in a minicomputer.
after
A = 0.5mm cl
= -3.73,
heterodyne
c2 = -3.93 detection
First, the velocity vector along the y-axis was measured. Of course, it is included in the plane II. The observed crossbispectrum is shown in figure 6. One peak can be seen at (48 Hz, 48 Hz). This peak corresponds to the velocity vector V with vx = as easily derived from the relation shown Vz = 0 and v y = 109 n&s, before. As for the resolution of the observed velocity, the data length for the measurement of the cross-bispectrum was 1 second and the width of the bispectral window for smoothing of the bispectrum was set to 7 Hz, resulting in a frequency resolution of about 7 Hz.
150
ULTRASONIC DOPPLER VELOCIMETER
(mmk)
----a
true
--,
measured
velocity
(Hz)”
vector
velocity
vector
f
75
20 .9 y’ 10 bfff!i- --
-
_____---e=2
0 80 -10
IS
e=-5” ----a_____-
100:
‘o.75 ( :
120 vvww
,
Fig.
8.
Measured velocity vectors in plane
vectors II.
Fig.
9.
Cross-bispectrum
for
Fig.
10.
a velocity
Power spectra of three and without corrupting
The velocity were also observed are shown in figure
for
three
different
vector
independently noise.
velocity
outside
plane
measured
signals
n.
with
vectors of several directions in the plane n The results by rotating the object in the plane. have peaks at the appropriate 7. The bispectra
151
SAT0 AND SASAKI
----a
true
-measured
Fig.
11.
Cross-bispectrum
Fig.
12.
Velocity
positions.
figure ities.
The
8.
These
vector
velocities
results
of signals derived
to
vector velocity
by additive
from observed
corresponding
are
corrupted
velocity
vector
noise.
cross-bispectrum.
the
in good agreement
results
with
are
the
true
shown
in
veloc-
Next, to measure the velocity vector outside the plane II, the moving object was rotated so that the new plane was at about 10 degrees to the plane n. The cross-bispectrum is shown in figure 9. In this case the cross-bispectrum had one peak atfD1 = 42 Hz and the velocity vector from these fD2 = 34 Hz. We were able to derive results, and the result showed good agreement with the true velocity vector. Finally, to show the ability of additive noise, the received with Gaussian-like noise.
of the system to eliminate the effect signals were corrupted intentionally
The power spectra of three original and three disturbed signals are shown in figure 10 by the dotted and solid lines, respectively. In this case, the velocity vector was in the y-axis direction. If the velocity vector were to be derived from these power spectra, as in the conventional three-dimensional Doppler velocimeter, it is clear that we could not distinguish one real velocity from the many ghost velocities because of many peaks in fd,, due to the Gaussian noise. The cross-bispectrum obtained with our new method is shown in figure 11. In this case, one peak is observed. That is, the effect of corrrupting noise is eliminated by the application of the bispectral analysis. The velocity vector obtained is shown in figure 12. This result is also in good agreement with the true velocity. Conclusion Doppler
The effectiveness of the new three-dimensional velocimeter has been shown theoretically
152
ultrasonic and experimentally.
ULTRASONIC DOPPLER VELOCIMETER
The applications of the method for measurement of flow characteristics in practical Situations, such as the flow of blood in human of simplified hardware, are under study. body r and construction Acknowledgment The authors reviewers of this
would like to express their sincere Journal for their kind suggestions
thanks to the and advice.
References
[ll
P. N. T., Wells, (Academic Press,
r21
Harris, B., Spectral New York, 1967).
r31
Time Series Data Analysis Brinllinger, D. R., -~(Holt, Rinehart and Winston, New York, 1975).
r41
Sato, T. and Sasaki, K., Am. 9, 404-408 (1977). -
IIs1
Peterson, W. W., Error Correcting Tokyo, 1967) Chapt. 7.
Physical Princia of -- Ultrasonics London and New York, 1969). Analysis
of Time Series, --
Bispectral
153
Holography, Codes
(Univ.
Diagnosis,
(John
Wiley,
and Theory, J. - Acoust. ~ Tokyo
Press,
-Sot