 |
Description  |
|
|
CROSS-REFERENCE TO RELATED APPLICATION
This application is related to United States patent application Ser. No.
07/608,389, filed concurrently herewith, and entitled METHOD AND APPARATUS
FOR USING DOPPLER MODULATION PARAMETERS FOR ESTIMATION OF VIBRATION
AMPLITUDE, which is commonly assigned.
BACKGROUND OF THE INVENTION
The present invention relates generally to the use of low frequency
vibrations to investigate passive objects. More particularly, the present
invention relates to a method of and a system for using low frequency
vibrations to vibrate an object and then using high frequency wave energy
reflected by an object for analyzing the vibration of the object created
by the generated low frequency vibrations in order to examine the surface
or cross-section of the object and to produce an image of the examined
object. The pattern of vibrations produced in breast tissue and in other
organs, measured as a function of position within the tissue and as a
function of vibration frequency, are utilized to determine the elastic
constants of the tissue.
It is frequently useful to examine passive structures or objects by means
of the application of a swept frequency vibration or audio source.
Structures which are typically examined in that manner for flaws include
aircraft, ships, bridge trusses and other types of large structures. In
addition, soft tissue can also be examined in that manner.
In the examination of such structures, the vibration sources may be
temporarily mounted on exterior surfaces of, for example, ship hulls or
bridge trusses, or may be placed in the interior of the structures to be
examined, such as ship hulls or passenger aircraft. The interrogating wave
energy is then focused on a reference point on or within the object.
Such interrogating wave energy may vary from a laser, microwave or airborne
source, for use with aircraft, bridges, ships or other structures. Sonar
or ultrasound sources may be utilized for underwater inspection. The
vibration source of low frequency is swept over a broad frequency range so
as to excite eigenmodes with relatively high vibrational amplitudes. Once
an appropriate vibration frequency is found, the interrogating wave energy
can then be scanned over the surface or within a cross-section of the
interior for penetrating waves. The spot size or spatial resolution of the
scan, depends upon the wavelength of the vibration source used and the
particular apparatus used. For simply focused coherent sources, the spot
size will be equal to (1.2)(wavelength) (focal length)/(aperture radius).
Since sub-millimeter wavelengths can be achieved by the instant invention
using ultrasound and optical devices as interrogating wave sources,
millimeter scale, spatial resolution can be achieved utilizing the present
invention. Once an appropriate scan size or region of interest is
selected, an image is derived from point-by-point examination of the
Doppler shift of the reflection of the interrogating source back from the
object. Although different prior techniques have been described which
analyze vibrations using lasers or ultrasound, none of those techniques
make use of externally applied vibration and point-by-point scanning using
the method of the present invention, to generate a vibration image.
The method of the present invention is useful to generate an image whose
intensity or color is proportional the vibration amplitude calculated by
means of the inventive method, at each point on the object. That image can
be inspected for modal shapes and abnormally high or low vibration
amplitudes, and can also be compared with reference images obtained
earlier or from well characterized analogous structures. In addition, the
modal shapes at different frequencies can be analyzed to determine the
elastic constants of the material. The time required for analysis of each
spot utilizing the inventive method and system described herein is less
than three cycles of the vibration frequency when the frequency domain
estimator method is utilized and a fraction of a single vibration cycle if
the time domain estimator method is utilized. Thus, those methods can be
applied rapidly so as to permit real-time imaging. Because the methods are
also sensitive to vibration but are stable in the presence of noise,
vibrational amplitudes of less than 1/10 of a wavelength of the
interrogating wave energy can be easily detected utilizing the inventive
method and system.
One prior art approach to measuring amplitudes of vibration is shown in
U.S. Pat. No. 4,819,649, issued Apr. 11, 1989, to Rogers et al. That
patent is directed to a non-invasive vibration measurement system and
method for measuring the acoustically induced vibrations within a living
organism. That patent utilizes a continuous wave of high spectral purity
ultrasonic beams and utilizes two separate transducers, one for
transmitting and one for receiving the focused beams.
By virtue of its frequency domain processing, the device disclosed by
Rogers et al. cannot produce real-time imaging. In fact, the '649 patent
does not discuss imaging at all. All of the specific implementations
discussed in the '649 patent relate to frequency domain techniques which
are based upon the ratio of harmonic sidebands of the reflected signal,
which allows the intrusion of noise elements into the reflected sample
and, thus, into any analyzed signal.
The system and method disclosed in the '649 patent is also discussed in an
article written by the inventors which appeared in the Journal of
Vibration. Acoustics, Stress and Reliability in Design, entitled
"Automated Non-Invasive Motion Measurement of Auditory Organs in Fish
Using Ultrasound", Vol. 109, January 1987, pp. 55-59. That paper discloses
the use of external vibration to produce an FM Doppler shift and examines,
over long periods of time, the Doppler spectrum returning from a single
point. It is not a real-time system. The article assumes that, using very
small vibrations, the ratio of the carrier signal to the first harmonic is
indicative of the vibration amplitude. Like the '649 patent, the device of
Rogers and Cox disclosed in this paper is a non-scanning, slow, frequency
domain method which uses the ratio of harmonic sidebands and is restricted
to use with very low amplitudes.
Another approach used in the past is disclosed in an article entitled
"Imaging the Amplitude of Vibration Inside the Soft Tissues for Forced Low
Frequency Vibration", by Yamakoshi, Mori and Sato, published in the
Japanese Journal of Medical Ultrasonics, Vol. 16, No. 3, pp. 221-229
(1989). That article discusses an imaging system which can observe the
precise movements inside of soft tissues when an external vibration is
applied to those tissues. While the system described in that article does
scan and make vibration images, it uses frequency domain techniques which
are also based on the ratio of harmonic sidebands approach which are noise
sensitive. Furthermore, it is slow and not a practical real-time system
and is restricted to use with small amplitudes of vibration.
Yet another approach used in the prior art is that of Pierce and Berthelot,
as disclosed in Proceedings of the SPIE, Session EE. Engineering Acoustics
IV: "Laboratory and Measurement Microphone", "Absolute Calibration of
Acoustic Sensors Utilizing Electromagnetic Scattering from In Situ
Particulate Matter", Pierce and Yves (1988). That paper describes the same
FM Doppler spectrum utilized by Cox and Rogers and as described in their
article discussed above. However, Pierce et al. utilize laser methods to
measure the oscillation at a point. The methodology of Pierce et al. is a
non-scanning, slow, frequency domain technique which again uses the ratio
of harmonic sidebands and therefore suffers from the same noise
sensitivity problems as do the other prior systems discussed above.
All of the known techniques can be broadly classified as utilizing the same
approach to the estimation or determination of the vibrational parameters,
that is, using some ratio of spectral harmonic amplitudes. Thus, they all
suffer from the disadvantages of the ratio methods because they require
either intensive computation or larger lookup tables of theoretical Bessel
functions for comparison with the measured data. Further, ratio methods
work well only when the argument of the Bessel function is small, which
poses a severe limitation on the range of the estimation of the Doppler
spectrum.
As a practical matter, the performance of the ratio methods is highly
degraded since almost all Doppler spectra suffer from a poor
signal-to-noise ratio. Additionally, a sophisticated algorithm is required
to determine the best selection of the harmonic pair to be compared. The
present invention, on the other hand, utilizes a simple and noise-immune
method for vibration estimation or determination.
The present invention may also be utilized with soft tissue structures,
such as for breast imaging or the imaging of tumors. The criterion of
digital palpation for detecting such soft tissue abnormalities, namely
"stiffness" or "hardness" of a "lump", is not directly related to either
the ultrasound, x-ray or MRI appearance of a hard lesion. That is because
stiffness refers to solid mechanical properties measured at constant or
slowly varying force. However, ultrasound echogenicity relates to
inhomogeneties in structure, as measured using frequency pressure waves.
X-ray absorbtion is related to the density and the presence of high atomic
number elements such as calcium. In magnetic resonance imaging, the image
brightness is related to the proton density and spin-spin and spin-lattice
relaxation processes. Thus, no in vivo modality is available which
directly assesses the stiffness of a region of tissue. The instant
invention directly assesses some mechanical properties of tissue and the
presence of stiff inhomogeneties can be detected using low frequency
vibration and the disclosed novel imaging and analysis techniques.
SUMMARY AND OBJECTS OF THE INVENTION
In view of the foregoing, it should be apparent that there still exists a
need in the art for a method of and system for providing for the remote
inspection of structures or soft tissues using an externally applied
vibration source and which provides for the generation in real-time of an
analysis of the returned signal from the object under investigation for
Doppler shift properties. It is, therefore, a primary object of this
invention to provide a method of and system for remotely inspecting
structures and soft tissues using externally applied vibration and Doppler
shift measurement which is characterized by providing an analysis on a
real-time basis of the return signal of an interrogating wave for Doppler
shift properties.
More particularly, it is an object of this invention to provide a remote
inspection system as aforementioned which provides for an analysis of the
Doppler shifted interrogation signal in order to derive the local
vibration amplitude of the reflector or object under investigation as
either a gray scale or color image.
Still more particularly, it is an object of this invention to provide a
system for remotely inspecting objects in which the image which represents
the analyzed Doppler shifted signal reflected from the object is generated
by scanning the object with wave energy such as laser beams, microwaves,
sonar, or ultrasound.
Another object of the present invention is to provide a system for the
remote inspection of structures which is not restricted to small
amplitudes and is noise insensitive.
A further object of the present invention is to provide a system for the
remote inspection of objects which utilizes a time domain method which
permits the real-time imaging of the vibration amplitudes produced by
reflection of the object over a region of interest from as little as three
samples of quadrature components in the time domain.
A still further object of the present invention is to provide a system for
the remote inspection of objects in which a time domain estimator method
is utilized but which provides a system which is not restricted to small
vibration amplitudes and is not a ratio of harmonic sidebands estimators.
Thus, the system is very flexible and noise insensitive.
Briefly described, these and other objects of the invention are
accomplished in accordance with its system aspects by using a low
frequency vibration source to force the oscillation of an object under
investigation and a pulsed ultrasound imaging system is utilized to detect
the spatial distribution of vibration amplitude of the object in
real-time. Due to the different mechanical properties of the abnormal
regions of the object from those of normal regions, vibration patterns of
the object at different vibration frequencies are used to determine the
abnormalities. A vibration source generates a source of vibrations which
is fed to a pulsed or coherent ultrasound imaging device as well as being
transmitted onto the object under investigation.
The reflected Doppler shifted waveform generated by reflecting the pulsed
ultrasound waves off of the vibrating object under investigation is
received by the pulsed ultrasound imaging device which computes the
vibration amplitude and frequency of the object in order to form a
vibration image. The computations use two alternative methods, either on a
frequency domain estimator basis or on a time domain estimator basis.
Those methods may or may not include a noise removal process, depending
upon the working environment.
The results of calculations of vibration amplitude and phase from some
target position are stored in a conventional scan converter which produces
an image of the estimates as a function of position over the entire scan
plane. Also, synchronization between the estimator systems and the
vibration source may or may not be employed, depending upon the actual
vibration estimation method utilized.
With these and other objects, advantages and features of the invention that
may become hereinafter apparent, the nature of the invention may be more
clearly understood by reference to the following detailed description of
the invention, the appended claims and to the several drawings attached
herein.
BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 is a block diagram showing the system of the present invention;
FIG. 2 is a block diagram showing the apparatus of the present invention;
FIG. 3 is a block diagram showing the system for noise correction for the
frequency domain estimation method embodiment of the present invention;
FIG. 4 is a block diagram showing the system for the time domain estimation
method embodiment of the present invention;
FIG. 5 is a schematic block diagram showing the phase and co-phase signal
estimation system for the time domain estimation system of FIG. 4;
FIG. 6 is schematic block diagram of additional phase processing circuitry
which may be utilized in connection with the time domain processing
circuitry of FIG. 4;
FIG. 7 is a schematic block diagram of successive-phase estimator circuitry
which may be utilized in connection with the phase and co-phase estimation
circuitry of FIG. 4;
FIG. 8 is a schematic block diagram of successive co-phase estimator
circuitry which may be utilized in connection with the phase and co-phase
estimation circuitry of FIG. 4;
FIG. 9 is a schematic block diagram of bi-phase estimator circuitry which
may be utilized with the phase and co-phase estimation circuitry of FIG.
4;
FIG. 10 is schematic block diagram of 1-shift cross-phase estimator
circuitry which may be utilized with the phase and co-phase estimation
circuitry of FIG. 4;
FIG. 11 is schematic block diagram of the phase-ratio estimator circuitry
for vibration frequency estimation for use as part of the circuitry of
FIG. 4;
FIG. 12 is a schematic block diagram of the auto-correlation estimator
circuitry which may be utilized in conjunction with the circuitry of FIG.
4;
FIG. 13A is a schematic block diagram illustrating how a conventional
velocity estimator can be utilized to provide an improved approximate
estimation of vibration amplitude signal;
FIG. 13B is a schematic block diagram showing how a conventional variance
estimator can be utilized to produce an improved approximate estimation of
vibration amplitude signal;
FIG. 14 is a diagram showing the geometry of a cylinder which is utilized
in discussing the present method of breast imaging using modal vibration
analysis; and
FIG. 15 is a block diagram showing a method of imaging breast tissue
according to the present invention.
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT
Referring now to detail in the drawings wherein like parts are designated
by like reference numerals throughout, there is illustrated in FIG. 1 in
block diagram form the overall system of the present invention. A low
frequency (approximately 1-1,000 Hertz) vibration source 100 applies a
signal to the object under investigation 102 in order to cause the object
under investigation 102 to begin vibrating.
An ultrasound transmitter and receiver 110 is used to generate ultrasonic
radiation which is aimed at a point on or in the object under
investigation. The waves reflected from the object under investigation are
received by the receiver portion of the ultrasound transceiver 110 and
supplied as an input to the pulse ultrasound imaging device 104. The
vibration source 100 is connected to supply the same signal which is used
to vibrate the object under investigation 102 to the pulse ultrasound
imaging device 104. The pulse ultrasound imaging device 104 may be an
ACUSON 128 Color Doppler Imager available from ACUSON, Mountain View,
Calif.
The pulse ultrasound imaging device 104 receives the vibration signal
generated by the vibration source 100 as well as the ultrasound reflected
signals from the object under investigation and utilizes the two signals
to produce the received signals 106. The received signals are input into a
vibration parameter calculator 108, which may preferably be specialized
real-time circuits as shown in FIGS. 3-12. Alternatively, it may be a
computer, such as a Sun 3/50 workstation networked to a Sun disk server.
Alternatively, a personal computer having real-time digital signal
processing characteristics could be utilized in place of the Sun computer.
The vibration parameter calculator 108 generates the vibration images 112
which may be displayed by means of a color or monochrome CRT, laser or
other printer or other display device 113.
The pulse ultrasound imaging device 104 is used to detect the spatial
distribution of vibration amplitude in real-time. The spatial distribution
of the vibration amplitude is generated by means of the vibration source
100, which may preferably be a 5" diameter woofer loudspeaker available
from the Radio Shack Division of Tandy Corporation, Dallas, Tex., and
others, causing the object under investigation 102 to vibrate, and the
ultrasound transceiver 110, which may be a 3.5 MHz or 5 MHz array
available from ACUSON.
Due to the different mechanical properties of the abnormal regions of the
object under investigation 102 from those of its normal regions, vibration
patterns of the object 102 at different vibration frequencies are used to
reveal the abnormalities.
It should be understood that the vibration source 100 may use sound or any
type of electromagnetic radiation which causes the object under
investigation to vibrate. In addition, the ultrasound transmitter and
receiver 110 may alternatively utilize any type of coherent, pulsed or
continuous electromagnetic radiation that can be detected by the
appropriate imaging device 104, for example, light waves, microwaves, etc.
It is, however, necessary that both the vibration source 100 and the
transmitter 110 utilize coherent radiation which can be readily detected
by the imaging device 104.
As disclosed in FIG. 1, the present invention provides for the remote
analysis and imaging using, for example, radar, microwaves, sonar,
ultrasound, lasers, or any other type of electromagnetic radiation of
vibrated structures such as airplanes, bridges, ship hulls or tissue, to
detect flaws and cracks as revealed by the abnormal vibration amplitudes
generated by such flaws. The stress concentrations which occur around such
cracks and flaws are detected by using the Doppler techniques described
herein.
The image is generated by scanning the vibrating object under investigation
102 with coherent, pulsed or continuous, focused or unfocussed wave energy
such as laser, microwaves, sonar or ultrasound, and analyzing the return
signal for Doppler shift properties. By using either the frequency domain
estimation method shown and described in connection with FIG. 3 herein or
the time domain estimation method shown and described in connection with
FIGS. 4-12 herein, the Doppler shifted signal is analyzed 106 to derive
the local vibration amplitude of the reflector or object under
investigation 102. The result, when combined with information regarding
the position of the interrogated object 114 is vibration images 112, which
can be displayed as either a gray scale or a color image.
In general, the method of the present invention for the examination of a
passive structure requires the application of a swept frequency vibration
or audio source, such as the vibration source 100. In examining a
passenger aircraft, for example, that may be accomplished by placing a
bank of loud speakers in the interior. Vibration sources may also be
temporarily mounted on exterior surfaces, such as the wings, hulls of a
ship or bridge trusses. The interrogating wave energy generated, for
example, by the ultrasound transceiver 110, is then focused on a reference
point or within the object. The vibration source 100 is swept over a broad
frequency range so as to excite eigenmodes of the object under
investigation 102 with relatively high vibrational amplitudes.
Upon finding an appropriate vibration frequency, the interrogating wave
energy is scanned over the surface or within a cross-section of the
interior of the object under investigation 102 for penetrating waves. The
time required for analysis of each spot is less than three cycles of the
vibration frequency when the vibration parameter calculator 108 is
utilizing the disclosed frequency domain estimator method and a fraction
of a single vibration cycle where the vibration parameter calculator 108
is utilizing the time domain estimator method disclosed herein.
The spot size, or spatial resolution of the scan produced by the
transceiver 110, depends upon the wavelength used and the particular
apparatus. For simply focused coherent sources, it has been found that the
spot size will be equal to (1.2)(wavelength)(focal length)/(aperture
radius). Since the present invention achieves the use of sub-millimeter
wavelengths using ultrasound and optical devices, millimeter scale spatial
resolution is achieved. Also, vibrational amplitudes of less than 1/10th
of one wavelength of interrogating wave energy can be easily detected
utilizing the estimation methods in the vibration parameter calculator
108.
Upon selecting a scan size or region of interest, an image is derived from
a point-by-point examination of the Doppler shift in the wave energy
reflected by the vibrating object under investigation 102. The image
intensity or color is proportional to the vibration amplitude calculated
by means of the estimation method which will be described later herein, at
each point of the object 102. That image can be inspected for modal shapes
as well as abnormally high or low vibration amplitudes, and can also be
compared with reference images obtained earlier or from well characterized
analogous structures.
Referring now to FIG. 2, there is shown a block diagram of the apparatus of
the present invention. As shown in that figure, the object under
investigation 102 is vibrated by a vibration transducer 214 which is
powered by an amplifier 216. The amplifier 216 is driven by a vibration
function generator or vibration source 100 whose frequency and signal
shape are adjustable. After the vibration of the object under
investigation 102 has been accomplished, it is scanned by means of an
ultrasound transmitter/receiver 110 which is controlled by a transmit
signal generator 200 and a transmit/receive switch 202.
The transmitter/receive switch 202 functions to control the mode of
operation of the ultrasound transmitter/receiver 110, that is, to place it
either in the transmit or receive mode. The transmit/receive switch 202
causes the ultrasound transmitter/receiver 110 to be in the transmit mode
while it is receiving a transmit signal from the transmit signal generator
202. If it is not receiving a signal from the transmit signal generator
202, the transmit/receive switch 202 causes the ultrasound
transmitter/receiver 110 to be in the receive mode and it functions to
transmit the Doppler echo signal received by the ultrasound
transmitter/receiver 110 to the received signal amplifier 204.
The output from the received signal amplifier 204 is fed to both a detector
206 and to the beta estimators 220, alternative embodiments of which are
shown in more detail in FIGS. 4-12. The output from the detector 206 is
fed to a logrithmic amplifier 208 whose output is in turn fed to the
storage and scan converter 210. The output from the beta estimators
circuitry 220 is likewise fed to the storage and scan converters circuitry
210. The output from the storage and scan converters 210 is the vibration
images which are displayed on the display 113.
A timing and position control circuit 212 produces a timing and position
control signal which is used to control the vibration function generator
100. A synchronization signal generator 218 receives the output signal
from the vibration function generator 100 and generates a synchronization
signal S which is fed to both the beta estimator circuitry 220 and the
storage and scan converters circuitry 210, as well as the timing and
position control circuitry 212. The timing and position control circuitry
is also connected to control the transit signal generator 200, the
transmit/receive switch 202, the received signal amplifier 204, the
detector 206, the logrithmic amp 208, the storage and scan converter
circuitry 212 and the beta estimator circuitry 220.
A conventional ultrasound B-scan color Doppler imaging instrument, such as
that manufactured by ACUSON described above, includes the equivalent of
the transmit signal generator 200, the transmit/receive switch 202, the
received signal amplifier 204, the detector 206, the logrithmic amplifier
208, the storage and scan converter circuitry 210, the display 113 and the
timing and position control circuitry 212.
Such a conventional ultrasound imaging instrument may be utilized with the
following modifications. A vibration transducer 214 is added. A
synchronization signal generator 218 generates a synchronization signal
based upon the output from the vibration function generator 100 which is
fed to the beta estimator circuitry 220 and the storage and scan converter
circuitry 210. The synchronization signal consists of information relating
to the phase and frequency characteristics of the vibration function
generator 100. Those characteristics are used in the calculations
performed by the beta estimator circuitry 220 in order to produce the beta
estimators. The estimate of beta is stored, together with the conventional
ultrasound received signal amplitude in the storage and scan converters
circuits 210, which outputs the information collected over the entire scan
plane as a video signal of the vibration images to the display 113.
It should be noted that the output from the storage and scan converter
circuitry 210 can be two separate images, such as a conventional
ultrasound image and a separate image of the variations in beta over the
scan plane. Or, the output from the storage and scan converter circuitry
210 may be a signal image with color overlay that can be used to display
the beta (vibration) information overlaid on conventional gray scale
ultrasound images.
FIG. 3 shows the operation of the frequency domain estimation method of the
present invention which may be used in conjunction with the vibration
parameter calculator 108 or beta estimator circuitry 220 to affect the
remote analyzing and imaging of the object under investigation 102. The
operation of the frequency domain estimation method is based upon the fact
that the Doppler spectrum of echoes of a sinusoidally vibrating scatterer
(object under investigation 102) contains discrete spectral lines weighted
by Bessel functions of the first kind. The frequency domain estimation
method is based upon the use of a new and simple relationship between the
spread or variance of the Doppler spectrum of the echoes and the vibration
amplitude of the scatterer.
The frequency domain estimation method for use with the remote inspection
and analyzing system produces high accuracy even under conditions when the
signal-to-noise ratio of the Doppler spectrum return from the scatterer is
poor. In addition, deviations caused by slight non-linearities of the
vibration of the scatterer have been found to have little contribution to
the total estimation error.
Since the scattering object 102 is caused to vibrate slowly by means of the
vibration source 100 so as to produce a wavelength much larger than the
geometrical dimensions of the scatterer itself, the Doppler spectrum of
the signals returning from essentially sinusoidally oscillating structures
is similar to that of a pure tone frequency modulation (FM) signal. That
spectrum is a Fourier series having spectral lines lying above and below
the carrier frequency. The space in between the spectral harmonics is
equal to the vibration frequency and the amplitudes of harmonics are given
by different orders of Bessel functions of the first kind. Typically, it
is desirable to determine the amplitude, phase and frequency of the
oscillating structure.
The FM spectrum is well known and, thus, its use in connection with
determining a Doppler shift of a moving object will be discussed only
briefly. When a moving object is illuminated with an incident laser, radio
or acoustic wave, the detected back scattered signals from that moving
object contain a frequency shift known as Doppler shift. If the scatterer
is oscillating with a vibration velocity much slower than the wave speed
and of a vibration frequency much less than the carrier or incident wave
frequency, the spectrum of the detected scattered wave will be similar to
that of a pure tone FM process since the instantaneous frequency of the
scattered waves has a Doppler shift proportional to the vibration
velocity. The Doppler spectral moments of the reflected signal are usually
defined as:
##EQU1##
where .sigma..sub..omega. is the Doppler spectral spread and thus
.sigma..sub..omega..sup.2 is the variance or second moment, .omega. is the
mean frequency shift of the Doppler spectrum (i.e., the first moment), and
S(.omega.) is the Doppler power spectrum.
Since the object under investigation 102 is vibrating, the Doppler power
spectrum can thus be written as:
##EQU2##
where W.sub.L is the low frequency vibration frequency and where the power
spectrum is down shifted to a zero frequency, using, for example,
quadrature detection. For this particular Bessel spectrum, the mean
frequency w is 0 and the power spectrum is therefore symmetric about a 0
frequency. The zeroth moment only has been noted by Watson in his Treatise
on the Theory of Bessel Functions, Chapter 2, pp. 14-15, 31, The MacMillan
Company, New York, N.Y. (1945) as:
##EQU3##
The second moment can be derived in a similar way to obtain the result of:
##EQU4##
In general, all moments of the Bessel spectrum can be calculated from the
generating function of the Bessel function given by Watson on page 14,
equation (1) of his Treatise. By taking the first through kth derivative
of the generating function with respect to z and substituting z=1 into the
resulting expressions, all moments of the Bessel spectrum as functions of
the parameters can then be calculated from the lower order moments by
squaring and simple algebraic manipulation.
Approached from that point of view, the Bessel spectrum becomes a one
parameter function. Therefore, the second moment is a good estimator of
the spectrum itself. Thus, the vibration amplitude can be estimated from
the Doppler spectral spread as:
##EQU5##
The above new algorithm indicates that the amplitude parameter beta can be
estimated from the standard deviation of the power spectrum.
Even utilizing the above equation, noise still presents a problem in
providing a correct parameter estimation. For example, the Doppler signals
tend to be 30-50 dB lower than the carrier in many applications. Thus, the
signal-to-noise ratio for Doppler signals is usually poor. Therefore, it
is necessary to remove stationary and uncorrelated noise from the Doppler
spectral spread vibration estimator method result. The signal-to-noise
ratio SNR can thus be represented by the
##EQU6##
where m.sub.O,N is the zeroth moment of the noise.
As long as the noise is stationary, the moments of the noise power spectrum
can be estimated or determined when the vibration is removed or halted.
Once the noise moments have been estimated, the noise-free vibrational
Doppler spectral spread can then be determined from the noisy signal as:
##EQU7##
Even in some applications in which the vibration is inherent and cannot be
controlled externally, the noise compensation can be performed by
estimating the signal-to-noise ratio as well as the Doppler spectral
spread of the noise from the finite band with white noise. Even if the
noise is not white, the noise compensation can still be provided as long
as the noise power and noise spectral spread can be estimated by
statistical techniques.
The method for applying the frequency domain estimation system is shown in
FIG. 3. First, the external vibration source 100 is turned off at step
300. Then, the spectral moments estimation for noise is performed at step
302. This is done using convention | | |
