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This is to certify that the

thesis entitled

ACOUSTIC ATTENUATION AND IMPEDANCE CHARACTERIZATION

BY BI-DIRECTIONAL IMPULSE RESPONSE TECHNIQUE
presented by

M.A. Anura P. Jayasumana

has been accepted towards fulfillment
of the requirements for

MASTERS degree in SCIENCE

66%?

Ma' professor

 

 

Date 12 th May 1982

 

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é //74<//

ACOUSTIC ATTENUATION AND IMPEDANCE CHARACTERIZATION
BY BI-DIRECTIONAL IMPULSE RESPONSE TECHNIQUE

By

M.A. Anura P. Jayasumana

A THESIS

Submitted to
Michigan State University
in partial fulfilment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Electrical Engineering
and System Science

1982

ABSTRACT

ACOUSTIC ATTENUATION AND IMPEDANCE CHARACTERIZATION
BY BI-DIRECTIONAL IMPULSE RESPONSE TECHNIQUE

By

M.A. Anura P. Jayasumana

This thesis presents an investigation into the use of
ultrasound reflection techniques for attenuation and impedance
imaging. For this purpose, the object to be imaged is modelled
in one of the following two forms. First the object is assumed
to consist of parallel homogeneous layers. Second, the object
is made of an inhomogeneous medium, which does not contain any
discontinuities in impedance. Expressions relating the impulse
response functions with the attenuation coefficient and impedance
have been derived. The theoretical development for a layered
structure has been verified by experiments. The results of the
experiments are presented. The method outlined in this thesis
yields the variation of the impedance and the attenuation
coefficient inside the object. To obtain this information, the
object has to be interrogated from opposite sides using an

ultrasonic pulse.

TO MOTHER AND FATHER

ACKNOWLEDGMENTS

The author extends his greatest gratitude to Dr. Bong Ho,
his thesis advisor, who has been a constant source of guidance
and support. Sincere appreciation is expressed to Dr. Ray
Nettleton for his review and suggestions to this thesis.
Appreciation is also extended to Dr. R. Zapp, a member of the
guidance committee, for his suggestions to this thesis. A very
special thanks is extended to Geetha Jayasumana for her help

and support.

TABLE OF CONTENTS

LIST OF TABLES

LIST OF FIGURES

I.

II.

III.

IV.

INTRODUCTION
1.1 Basic principles of ultrasound propagation
1.2 Previous work

THEORETICAL CONSIDERATIONS

2.1 Reflections from a layered structure

2.2 Reflections from an arbitrary impedance
profile

2.3 Improvement of the accuracy of the impulse
response function

ATTENUATION VARIATION FROM REFLECTED SIGNALS

3.1 Relationship of attenuation to impulse
response function of a layered object

3.2 Relationship of attenuation to impulse
response function of an inhomogeneous medium

EXPERIMENTAL PROCEDURE AND RESULTS

4.1 Data collection

4.2 Data processing

4.3 Experimental results

CONCLUSION

APPENDIX A

APPENDIX B

APPENDIX C

BIBLIOGRAPHY

iv?

Page

vi

31
36
42
42
48
68
72
74
76
78

TABLE

TABLE

TABLE
TABLE
TABLE

TABLE

TABLE

TABLE

TABLE

TABLE

TABLE

TABLE

TABLE

.10

LIST OF TABLES

Propagation velocities in some biological
materials

Characteristic impedance of some biological
materials

Biological attenuation coefficients

Results for test object 1 - using direct i.r.f.

Results for test object 1 - using restored
i.r.f.

Layer parameters for test object 1 from
direct measurements

Results for test object 2 using direct

i.r.f. and r1 = a1
Results for test object 2 using direct
i.r.f. and r6 = "b6

Results for test object 2
i.r.f. and r1 = a1

using restored

Results for test object 2 using restored

i.r.f. and r6 = -b6

Results for test object 2
i.r.f.

using direct

Results for test object 2 using restored

i.r.f.

Layer parameters for test object 2- from
direct measurements

Page

11
55

56

57

61

63

64

65

66

67

FIGURE
FIGURE
FIGURE

FIGURE
FIGURE

FIGURE

FIGURE
FIGURE
FIGURE
FIGURE

FIGURE
FIGURE
FIGURE

FIGURE
FIGURE
FIGURE
FIGURE
FIGURE

.3)

.5
00

00000000

##9##

CDNO‘Cfl-D

LIST OF FIGURES

Reflection and transmission of acoustic wave
A one dimensional layered structure

First and second order reflections from a
layered structure

A one dimensional inhomogeneous structure

Frequency spectrum of a typical incident
wave

Spectral extrapolation for impulse response
function improvement

Bi-directional ultrasonic interrogation
A layered structure
An inhomogeneous medium

Time relationships for bi-directional
ultrasonic interrogation

Incident waveform recording
Reflected waveform recording

Bi-directional impulse response functions
of a layered object

Test object 1

Incident waveform

Incident spectrum and the Hanning filter
Reflected waveform

Direct impulse response function of test

object 1 - before and after amplitude
detection

vi

17

19
20

26

29
32
32
37

37
43
43

43
49
50
51
52

53

FIGURE

FIGURE
FIGURE

FIGURE

FIGURE

4.9

4.10
4.11

4.12

A.1

Frequency restored impulse response
function of test object 1

Test object 2

Bi-directional direct impulse response
function of test object 2

Bi-directional frequency restored impulse
response functions of test object 2

Flow diagram of the amplitude detection
scheme

vii

54
57

59

60

77

CHAPTER I
INTRODUCTION

Use of ultrasound for biomedical imaging purposes has been under
investigation for more than three decades. Propagation of ultrasound
in biological material depends on the ultrasound propagation velocity,
the acoustic impedance, the attenuation in the tissue and the scatter-
ing of ultrasound by the inhomogeneties in the tissue structure. In
contrast, propagation of X-ray in tissue depends only on the density
of the tissue. Thus it appears that there is a considerable amount
of information contained in an ultrasonic wave propagated through a
tissue. This coupled with the noninvasive nature of ultrasound has
been its main attraction in biomedical applications.

In general, a biomedical ultrasonic imaging system will transmit
an acoustic pulse in the frequency range of 1-10 MHz through the
object under investigation and receive a waveform which is a result
of the transmitted wave interacting with the object. In transmission
techniques, the received signal is the deformed pulse resulting from
the incident pulse propagating through the object. In reflection
techniques the received signal is the waveform resulting from reflect-
ions of the incident pulse at the discontinuities inside the object.
Until recently, only the amplitude information of the received signal
was used for imaging. During the last decade, research has been
conducted on how to use the frequency and phase information of the

received signal for imaging and tissue characterization.

Ultrasonic impediography was a result of this. It allows the
determination of the acoustic impedance of the medium along the
path of propagation.

This thesis investigates a method which would allow extration
of information about the variation of acoustical attenuation in
addition to the variation of acoustical impedance along the path of
propagation. In certain physiological structures, knowledge of
attenuation may be more important than the knowledge of impedance.
For example, the impedance transition between gray and white brain
matter is only about 0.1%, whereas the corresponding change in
acoustical attenuation is two orders of magnitude largerls.

First, the basic properties of ultrasound propagation will be
briefly reviewed. Second, the basic principles of ultrasonic
impediography and range resolution improvement will be outlined.
Third, it will be shown that information sufficient to find out the
variation of attenuation along the path of propagation can be
extracted by transmitting two ultrasound signals from opposite sides
of the specimen. For this purpose, the object is modelled in one
of the following two ways. It can be modelled as consisting of
parallel layers of different impedancesl. Certain biological
materials however cannot be represented by distinct boundaries. In
such a case, the acoustic impedance is assumed to vary continuously
along the path of propagation. Any discontinuity has to be handled
by a combination of the two models. Experimental results are
presented to show the validity of the theoretical development. The

limitations and the means by which this method can be improved are

also discussed.

Next two sections of this chapter will review the basic proper-
ties of ultrasound propagation and present the state-of-the-art in

ultrasonic imaging and tissue characterization research.

1.1 BASIC PRINCIPLES OF ULTRASOUND PROPAGATION

The propagation of ultrasound in biological matter is due mainly
to the longitudinal waves, i.e. direction of propagation of the wave
is the same as the direction of vibration of particles. The propa-
gation due to shear waves can be neglected because the attenuation
coefficient for shear waves is extremely high6.

The wave motion in a lossless medium can be described by

 

Vzp - $3312— : 0 (1.1.1)
c at
and
c = :f = 'El— (1.1.2)
where
p = pressure
c = group velocity of the ultrasound wave
Ka = adiabatic bulk modulus of the material
k = adiabatic compressibility of the material
P = density of the material.

In one dimension, equation (1.1.1) becomes

2 2
at a x

The general solution to this equation is given by,

p = f(x-ct) + f'(x+ct) (1.1.4)

where f(x-ct) and f'(x+ct) are forward and backward travelling
waves respectively. Some basic properties of ultrasound propagation
will be considered now.
Acoustic velocity:

It has been found that the velocity of propagation, c, of
ultrasound in biological matter has only a_§mall_di§persigp, i.e.
it is independent of the frequency for most practical purposeszs.
The velocity in one kind of soft tissue is almost the same as that
in another. But the velocity in bones is much faster than that in

the tissue. The velocity is also a function of the temperature.

Propagation properties of some biological materials are given in

 

 

Table 1.1.
TABLE 1.1 Propagation velocities in some biological materials*.
Tissue Mean velocity (m/s)

Fat 1450

Human tissue, mean value 1540

Brain 1541

Liver 1549

Kideney 1561

Spleen 1566

Blood 1570

Muscle 1585
Skull-bone 4089

 

* From P.N.T.we11s25.

Characteristic impedance :

The characteristic impedance of a medium is defined as the ratio

of the pressure to the phase velocity associated with the propagation

of the acoustic waves. It is given by,

P k
l... = [T] = .122. , (1.1.5)
The characteristic impedance of the medium may be complexll. For
biological matter however, the imaginary part is negligiblezs.
Thus p = Zv (1.1.6)
where ' p = pressure
v = phase velocity or the velocity of particle vibration

Z

characteristic impedance of the medium.

Characteristic impedances of few biological material are given in

 

 

Table 1.2.
TABLE 1.2 Characteristic impedance of some biological materials*.
. 2 -5
Tissue Z (g/cm .s)x 10

Fat 1.38

Brain 1.58

Kidney 1-52

Human tissue, mean value 1.63

Spleen 1-54

Liver 1.65

Muscle 1.70
Skull-bone 7.80

* From P.N.T.Mell525.

Reflection and refraction at plane surfaces :

When an ultrasonic plane wave strikes the boundary between two
different media, it may be partially reflected as shown in Figure 1.1.
If the dimensions of the reflecting object are large compared to the
acoustic wave length, the geometrical laws of reflection apply. If
the wavelength is comparable with linear dimensions of the reflect-
ing object, these laws cease to apply and diffraction occurs.

Assume the case where the wavelength is small compared to the

dimensions of the reflector.

 

 

Medium 1 _ Medium 2
c1 , Z1 c2 , Z2
reflected ” transmitted
er 9t
9.
incident

FIGURE 1.1 Reflection and transmission of acoustic wave

By Snell's law,

 

6i = Br (1.1.7)
sin 8- C
l = 1
sin 9t ‘EE‘ (1.1.8)
where
c1 = acoustic velocity in medium 1
c2 = acoustic velocity in medium 2
9i = angle of incident wave
er = angle of reflected wave
at = angle of transmitted wave

The following boundary conditions apply.

1. The tntal pressure pn each side of the boundary must be eguaJ.
2. The pantiple velocity on each side of the boundary must be

pontinuous,
Hence,
pi+pr = pt (1.1.9)
Vi cos 9i - Vr COS 9r = Vt cos-,0t (1.1.10)
where
pi = acoustic pressure of the incident wave
pr = acoustic pressure of the reflected wave
pt = acoustic pressure of the transmitted wave
Vi = particle velocity of the incident wave
vr = particle velocity of the reflected wave
v = particle velocity of the transmitted wave.

From the definition of the acoustic impedance, equation (1.1.6),

 

 

pi = Z1 Vi (1.1.11)
pr = Z1 vr (1.1.12)
pt = 22 vt (1.1.13)
where
Z1 = acoustic impedance of the first medium
Z2 = acoustic impedance of the second medium.
From equations (1.1.7) - (1.1.13),
p Z cos 3. - Z COS 9
r = J; = 22 91+ 1 ‘5 (1.1.14)
pl ‘2 COS i 21 C059t
p 2 Z cos 0.
- _t = 2 1
‘t _ pi Z2 cos 9i + Z1 cos Bt (1’1'15)

The reflectivity

, r is defined as the ratio of pressure reflected

to the pressure incident and the transmissivity,'c as the ratio of

pressure transmitted to the pressure of the incident wave.

For normal incidence,

6i = Br

Thus from equations (1.1.14)

coefficient r is

and the transmissivity is

= St = 0 (1.1.16)
and (1.1.15), the reflection
Z - Z
= _____22 + 7.1 (1.1.17)
2 1
2 Z2
= 7-jfjr- (1.1.18)

The intensity reflection coefficient (“k and the intensity

transmission coefficient fat are given by,

 

 

2 2
r‘. = 4 =m:::::::t
pi 2 1' 1 t
2 2
_ pt . Z1 _ 4 Z1 Z2 cos 0,
[‘t _ 6T. 7—. - 2 (1.1.20)
1 2 (Z2 cos 6i + Z1 cos 6t)
From equations (1.1.17) and (1.1.18),
P P
t r
—— = +— . .
pi 1 pi (1 1 21)

Ultrasonic attenuation :

Attenuation of ultrasound in matter may be due to the combined
effect of several different mechanismszs. In certain situations
however, a certain mechanism may predominate. One of the mechanisms
is the result of deviation of the beam from a parallel beam.

Another is due to scattering by elastic discontinuities within the

 

medium. A discontinuity acts as a reflecting surface, the size of
which determines its effect as a scatterer. The scattered energy
will np_]onger propagate in the original direction, thus causing
attenuation. The attenuation due to absorption causes the acoustical
energy to be converted to some other form of energy, typically heat.
When an adiabatic stress and the resulting strain are not related

in a linear manner, a non-viscous mechanism occurs, in which the
dissipation of energy is proportional to the strain, rather than the
rate of change of strain. The loss of energy is constant for each
cycle, and hence the elastic hysterisis attenuation is proportional

to the‘fizcgnepcypof the ultrasound wave.

10

The attenuation in fluids mainly depends on thecyiscpsjty and
the.heat_conduction. The motion of particles is opposed by the
viscosity of the medium and this causes generation of heat. If the
heat conduction is significant, thermal energy may also move in
space. The time lag between the pressure and the density of the
medium is controlled by the time required for the viscous stress to
be equalized, or for the heat conduction to occur from high to low
pressure regions of ultrasonic field. This time lag accounts for
the energy absorptionzs.

Energy can exist in a medium in different forms such as mole-
cular vibrational energy, translational energy, lattice vibrational
energy and so on. In the presense of an ultrasonic wave, there can
be an increase in energy in one or more of these forms. This mecha-
nism, called_relaxation, can cauSe_aQSQrnLion—o£_acoustical.eoengy.
This form of attenuation is related to the frequency in a ncnljnear

manner.

No general theory of absorption seem to be possible for bio-

logical materialszs. But it is characterized experimentally by the
relation

I = I() exp( -20((f)d ) (1.1.22)
where

I = intensity of the acoustic beam after it has passed

through a distance d
Io = initial intensity of the acoustic beam
d = distance travelled through the medium

°<(f) attenuation coefficient characteristic of the medium.

11

The factor 2 in the exponent accounts for the fact that the intensity

is proportional to the square of the pressure. For biological material
21

in general ,

0((1-‘) = cxo f (1.1.23)
where

“0 = attenuation per unit distance at 1 MHz

f = frequency of the acoustic wave in MHz.

Attenuation coefficients of some biological material are given in

Table 1.3.

TABLE 1.3 Biological attenuation coefficients

 

 

Tissue Attn. coefficient Temp. Frequency
Np/cm.MHz 'C MHz
Liver, Human* .135 body 1.5
Liver, Human* .143 40 0.97
Liver, Porcine* .136 25 4.0
Spleen, Human* .058 18 1.6
Fat, Human* .069 37 1.0
Abdominal wall, Human* .074 - 1.0
Brain (average), Human@ .11 37 1.0
Cortical gray matter, Cat@ .08 37 1.0
Cortical white matter, Cat@ .14 37 1.0

 

* From comprehensive compilation of empherical ultrasonic properties

of Mammalian tissuelo.

@ From Biological Engineerin924.

12

1.2 PREVIOUS WORK

In recent years, a number of techniques have been developed to
extract information about biological objects from ultrasonic signals
which have interacted with the object. These techniques make use of
the frequency and phase information in addition to the amplitude
information available in the received signal.

Ultrasonic impediography is based on the time domajn characteri-
zation of the object in terms of an impglse_response_function. This

approach utilizes amplitude, frequency and phase information available

12-14

in the received signal. Jones has used the incident and

reflected signals to obtain the object impulse response function which
was then integrated to give the impedance variation along the path of
propagation. In addition Jones has also derived the relationships
between the impulse response function and the impedance variation for
different lossless structures in the presense and the absense of
multiple reflectionslz.

Accurate determination of the impulse response function plays a
critical role in impediography. The conventional approach is to
deconvolve the incident and reflected signals to obtain the impulse
response function. However, due to the fgg;;eebandw$d%h of the
transducer, only a smallwband of frequency components of the impulse
response function can be accurately determined. This limits the
range resolution as well as the accuracy in amplitude of the impulse

response function. SeyEral methods have been developed to overcome

this limitation. Papoulis, et al.23 describes a method in which the
reliable.freuenc¥_cnmponents of the impulse response function are
\\ ._.\

first‘used to makedan_initiafEestimate_9f_the_neak~1§lu££.and the

13

epochutimes. These values are then used to estimate the out of band

or ungg1iable_frequency_cpmppnents. This method is further discussed

in section 2.3.1 of this thesis.

KUC, et al.17 outline a figifiagLfilierinazapproach to this problem.
The Kalman filter processes the observed time samples to predict the
nezl_§§nmfl£LJuilue. The error between the predicted value and the

value actually observed is due either to the presense of noise or due
to a reflector.

Papoulis, et al.22 have demonstrated another method to improve

the range_gg§9;5;;95 of an impulse response function by spectral
extrapolation. This takes into account the fact that the impulse

 

response function has a f and hence in the frequency
domain it has a large bandw4zth. First they use only the reliable
section of the transfer function to obtain the impulse response
function. This would yield a signal having a large duration in the
time domain. It is truncated to the known impulse response function
duration, and this can now be used to estimate the other frequency
components of the transfer function. It has been shown that after a
number of iterations, the result approaches the actual impulse
response function, even in the presense of considerable noise. This
method is discussed in detail in section 2.3.2 of this thesis.

1-4

Beretsky, et al. have made use of the processing techniques

mentioned above in their research work. They have obtained the

impediograms of ngnbig1ggicgl_asflwell_as_biological.structunes with

censjderablez§ugcess-

Fourcade, et al.5 have used a pseudo— random§b1nary sequence as

the tcangmitted_signal. Cross correlating this incident signal with

14

the reflected signal will result in the impulse response function of
the system. The deconvolution in their system is achieved by
hardware implementation. As in other cases, the inherent_bandwidth
limitation of the transducer would deterjorate the resulting impulse
re;ppnse.£unction and a pracessing.technique—similar to the ones
described above will have to be used. Despite this drawback this
method has the advantage of a high qxgrgggflfigLQEQk:pgugg:ggt§p

- almost equal to unity. This feature is very useful in increasing
the range in in—vivo examinations, as there is a limit on the peak
power that can be applied without any harmful effect.

In all the impediographic methods listed above, the attenuation
due to the intervening material has been neglected. The_accunacy
of tmemmww The method
under investigation in this thesis not only takes into account the
attenuation in determining the impedance, but also allows the
determination of the attenuation itself.

Attenuation of ultrasound in biological tissue has been under
investigation for some time. The early researchers concentrated on
measuring the variation of the ultrasonic attenuation with frequency

in different tissue types8 and in understanding the mechanisms of

ultrasonic attenuation7. Kuc, et al.18 have qutlflmaiJLnMflinumejch
uses-a_§§u§§iaxL;uflsshto measure the attenuation coefficient slope

20

of soft tissue. Lizzi, et al. have demonstrated a method which

1
gives the attenuation characteristics of a homogeneousfltjssue as a

function of frequency.
The use of ultrasonic attenuation for imaging purposes began

9

very recently. Greanleaf, et al. have developed a clinical

15

imaging system with transmissive ultrasonic tomography. In this
system, ultrasonic pulses are transmitted through the breast in a
qggggg1=gigpe from many directions. The received signals are then
processed for arrival times and the change in amplitude. These
values are used in a convolution back projection reconstruction
algorithm to obtain estimates of the two dimensional distribution of
acoustic speed and attenuation within the scanned plane of the brest.

16 have used a phase insensitive detection method

Klepper, et al.
to find the attenuation distribution. They also used the acoustic
pulses transmitted through the sample from various directions to

construct an image using attenuation.

CHAPTER II
THEORETICAL CONSIDERATIONS

This chapter outlines the theoretical background necessary for
the proposed method of attenuation and impedance measurement. In_the
_£ir§t_§gctinn, a layered structure is assumed. This model results
in a straight forward determination of the attenuation and impedance
variations.(TThe_second_section—eonsidens an inhomogeneous model,
where the (Eoustieiimpedance is a continuous1functinnmo£_the.di§tange.
Any discontinuities would have to be handled by a combination of the
two models. Due to the nannou—bandwidxh of the ultrasonic transducer,
the impulse response function of the medium cannot be found
accurately. Two methods that can be used to improve the accuracy

of determining the impulse response function are described next.

2.1 REFLECTIONS FROM A LAYERED STRUCTURE

Although biological tissues are acoustically complex, it should
not necessarily be thought of as an inhomogeneous siurry of cellular
materiall. The microstructure of a tissue in general suggests that
it consists of a multilayered, qgnplangr, laminated structure in
both longitudinal and lateral extent. Lateral variation of the
acoustic boundaries can occur. The extent of the lateral area
covered by the ultrasonic beam can be reduced by focussing. Hence
the model like the one shown in Figure 2.1 can be used to represent

certain biological structures.

17

 

 

 

 

 

 

 

 

o 1 2 ... 1 1+1 ... N-1 N

20 21 22 ... 2, 2M zN_1 20

“o “1 0‘2 ... 0‘1 o‘1+1 “' '“N-1 0‘o

130 ‘31 132 131 1:1+1 1311-1 1:0
r1 r2 r3 r1 1+1 1+2 ' rN-1 IN

FIGURE 2.1 A one dimensional layered structure

Consider an object which consists of N+1 layers as shown in
39"
Figure 2.1. Tngqfiin§;.and the last layers are assumed to be made of
the_same_materia1, The following notation is used.

Zi = acoustic impedance of the ith layer
(Xi = attenuation coefficient of the ith layer in Np/s
i = time taken for the wave to propagate through the
ith layer
r. = reflection coefficient at the boundary between

Li511§t and jtflplayers
Note that the attenuation coefficient is given in Np/s. This can be
related to the attenuation in Np/m as
“(Np/s) = K (Np/m) x c (m/S) (2.1.1)

where c is the velocity of propagation of the ultrasonic wave.

18

The following assumptions are made.

1. Onlyfiglgétjand rd r r flections are considered. This can
be justified for most biological structures, where the change of
acoustic impedance from one layer to the next is IEULJgflngjggg;
If there is a 1argeocgaoggzin_ihe.impedance from one layer to

the next, as is the case/of a bone surrounded by soft tissue,

   

2. Each layer is homgggneous, i.e. there is no change in acoustical
properties within each layer.

3. The width of boundaries between the layers is small compared to
the acoustic wave length.

The response of the stnucture to an impulse can be derived as
follows. The impulse response consists offifiZfimpglses; each
corresponds to a reflection at one of the boundaries. Figure 2.2
shows the amplitude relationships at each boundary. The total

impulse response due to various layers can be expressed as

M
h(t) = Z} a, 5(t't1) (2.1.2)
where
1-1
t1 = 23 2 ‘tk (2.1.3)
k=o
1-1 2
a1, = exp(—Zuo‘to) 1*, 13:1 (1-rk ) exp(-zetktk) (2.1.4)

r. = Z1 ' Z1-1
‘ Z1 + Z1-1

(2.1.5)

19

mczuuzcum umgmxm_ w Eocm mcoepom—mwc gouge ucoomm cam umgwm

Ampwohpfbpoz. v98. Ame; v 2%:

 

term- 05-33.21-3ch-

     

 

     

Afipfiqupvpuuvaxm.Afiguavmg

Afipfixbpoxuvaxm. 2.7:

' s.

 

N.N maze:

”From-

cpnamuvaxm.fimagufivmc

Aopoamuvaxméc

AoPo81VaxmA

20

2.2 REFLECTIONS FROM AN ARBITRARY IMPEDANCE PROFILE
In this section the results of section 2.1 are extended for a

efi
structure with contin ' i ce. The iflhgmggenegys

medium can be assumed to consist of_azlarge_number_of;parallel.
hemogenqu§_1ayers with very éfigi? thickness as shown in Figure 2.3.

 

Y‘O r1 r2 r3 . . . Y‘k_1 Y'k rk+1 . . . Y‘N_1

 

 

 

 

 

 

 

 

 

 

 

to t1 t2 t3 . . . tk-l tk tk+1 . . . tN-l

FIGURE 2.3 A one dimensional inhomogeneous structure

The impulse response function of the structure shown in Figure
2.3 can be written using equation (2.1.2)-(2.1.5) as
N
h(t) = Z: a. 5(t-t.) (2.2.1)
1=0 ‘ ‘
For the k th and (k-1) st layers,

h(tk)

ak (2.2.2)

h( (2.2.3)

tk-l) ak-1

21

 

 

Therefore,
h(tk) ak
h t = .____ (2.2.4)
( ) a
k- k-1
Substituting for ak and ak_1 from equation (2.1.4),
k']. 2 m
- . - It.
h(tk) - rk jEE (1 rJ ) exp( 2 J J)
— k-Z 2
h(tk_1) rk_1 3:2 (l-rj ) exp(-ngtfi)

2
rk (l'rkq? exp(‘2‘xk-1tk-1)
rk-1

(2.2.5)

 

But from equation (2.1.3),

215 = t

k-1 ‘ t

k (2.2.6)

k-1
Hence the ratio of the amplitudes of the impulse response function

at t = tk and t = tk-l becomes

“(tk- 5 rk-l

 

h(tk) rk (1-rk_12) exp(-0<k_1 (tk-tk_1))

which can be expressed as

Ln h(tk) - Ln h(tk_1)

_ 2
- Ln rk - Ln rk_1 + Ln(1-rk_1 )

+ Ln(exp(-°< (t (2.2.7)

k-l k'tk-1))
From equation (2.1.5), the ratio of acoustic impedances of the two

layers is given by,

 

 

22

Hence,

Ln Zk - Ln Zk-l

H
r
3
/\
PA
+
1
X'
\J
l
r.
3
r\
p;
I
1
7?
‘4

1‘” Ln 2 - Ln 2

rk-o 0 k k-1

U
DO
x

X’

Also,

lim 2

rk" o Ln(1-rk

) =-rk

From equations (2.2.7) - (2.2.9),

Ln(%(Ln Z - Ln Z

Ln h(tk) - Ln h(tk_1) k k_1))

’ Lh(%(Ln Zk-l ' Ln Zk'2))

+ %(Ln Z - Ln Z )2

k-1 k-2

+ Ln(eXp(’ “k-1(tk-tk-1)))

= Ln(Ln Zk - Ln Zk-l)

- Ln(Ln Zk-l - Ln Zk-Z)

+ %(Ln 2 - Ln 2 2

k k-l)

+ LH(eXp(" uk'1(tk-tk-1)))

Let t

(2.2.8)

(2.2.9)

23

 

 

 

 

Ln h(tk) - Ln h(tk- At) At = Ln Ln Zk - Ln Zk-l
zst ' zst
- Ln Ln Zk-l - Ln Zk-2
.At
'Ln Z - Ln Z 2
+ %(At)2 [ k t k I}
A
+ Ln(exp( k_1(tk tk 1)))
For an inhomogeneous medium,
tk " tk-l = At "'" 0
Hence as lim 9
Cit-o o
d _ 9.. L
8‘1? Ln 11(1) Ft .At - Ln dt Ln 2‘15:t _ Ln dt Ln 2 Ft
k k k-l
2 2
At d
+ I" [313' L" Z1t=tk]
+ Ln(exp(-°‘k_1tk))

Adding and substracting Ln(exp(—'%k_2tk_1)) from the right side or
equation (2.2.10),

d _ d.
E Ln h(t) t___tk .At — Ln(exp(-0kk_1tk. dt Ln Z1t=tk ))

d
.. Ln(exp(—0tk_2tk_1. (TE Ln Z1t=tk_1 ))

- Ln(exp(—( 0‘ k-1'“ k-2)tk-1))

2 2
At (1 1
+ ‘4 [dt L” Z1t=tk] (2'2‘1 )

24

Using the identity,

Ln(exp x) = x
and letting
0( _o( = A0(
k-1 k-2 ’
L = 1 a d
dt Ln h(t) t=tk A—E [Ln(exp(- k_1tk. 3? Ln 4W.”
d
- Ln(exp(-“k_2tk_1.aE-Ln Z1t=tk_1))]
nwc Tot d 2
+ SE tk-1 + ‘71" [Eff L" Z1t=tk]
As lim ,
At-OO
d _ d d da..
at L" h(t) - dt' Ln [exp(-Ni).aE-Ln Z J + dt' t (2.2.12)

Note that 04 is a function of t.

Integrating equation (2.2.12) with respect to t yields

Ln h(t) Ln [exp(-“fig? Ln Z] + 113%? -dt + c

-°‘t+Ln [£3an Z] +mt- (“At +c

d
[.11-(TE

d
Ln [C dth Z . exp(- [and]

an + Ln exp(-fkdt) +c

where C is the integration constant.
The impulse response function is thus given by

h(t) = c %Ln 2 . exp(-Imdt) (2.2.13)

 

The coefficienE§E§tan be determined by considering a lossless

medium, i.e. a medium with 04:0, The relationship between h(t) and

25

Z for such a medium is given by13,
h(t) 16%an 2 (2.2.14)
Therefore,
C = % (2.2.15)
The complete impulse response is thus given by,
h(t) = a exp(-jd;(t).dt) ag-Ln Z(t) (2.2.16)
0
where
at Z(t) = impedance_encountened—by—the—u4trasonic—pu4seaat
4EHMLJ;
“(13) = ade—wmwmmfim

ultrasonic pulse at_time t;~

2.3 IMPROVEMENT OF THE ACCURACY OF THE IMPULSE RESPONSE FUNCTION
The acoustic impulse response function of an object can be

obtained by deconvolving the incident and the reflected signals. It

has been shown that a narrow bang sounoe.is necessary to maintain a

good lateral resolutionz. Practically, even in cases where lateral

resolution is not of main concern, the incident spectrum is limited

by the transducer bandwidth. The conventional approach for determi-

nation of the impulse response function consists of the following

steps.

1. Obtain the Fourier transforms X(f) and Y(f) of the incident and
reflected signals x(t) and y(t) respectively.

2. The transfer function H(f) is given by

H(f) = $3 (2.3.1)

26

3. The impulse response function h(t) is obtained by determining the

inverse Fourier transform of H(f).w(f), where H(f) is a window‘?

function in the frequency domain, whigh_eItenoatee_raoidly_eot§ige
the fregugongxflMLJdLl(£).
h (t) = F'1( H(f). 11(1) ) (2.3.2)

 

-h ---—..-..-

N

i
I
I
I
1
1:1

FIGURE 2.4 Frequency spectrum of a typical incident wave

The frequency spectrum of a typical incident waveform is shown
in Figure 2.4. Due to the presense of noise, the transfer function
computed using equation (2.3.1) is nej_neliable_at—the—£ehM¥§+4#L¢he
spectrum—X(f), 0uiside_the.frequencymband (fi.f2). In the time domain
this will result in an ambiguity in the location and amplitude of
the returned echoes, due to noise and interference from overlapping
echoes. This ambiguity can be reduced by restoring the frequency
components of the impulse response function h(t), that are outside
the significant frequency band of the incident spectrum. Two tech-

niques that can be used for this purpose are described below.

 

 

27

2.3.1 FREQUENCY RESTORED IMPULSE RESPONSE FUNCTION23
This method is suitable for improvement of the impulse response
function of a layered structure. Due to the presense of noise, the

calculated transfer function will have a value Ho(f), which is un-

reliable outside the main frequency band (f1,f2). Th2_£§§nliing
ewmmmulflnmwby a—suiiablwndow
function—H(f) that attenuates rapidly outside this band.
Hw(f) = Ho(f).w(f) (2.3.3)
where
110(1) = M1 (2.3.4)
X(f)

The inverse Fourier transform of Hw(f), hw(t), is a modulated wave-

form and the locations and the amplitudes 24+ i=1,2,... k of its

 

peaks determined by a demodulation scheme give a first order estimate

of the interface locations and the corresponding reflection coeffi-
cients. This estimate can be improved by the following procedure.
1. Form the frequency spectrum

13(1) = 2, a1. exp(—j27i 111.) (2.3.5)
1

2. Form the new estimation of the transfer function
H1(f) = Ho(f).N(f) + (1 - N(f)).P(f) (2.3.6)

3. The inverse Fourier transform h1(t) of H1(f) is the new estimate

of the impulse response function.

_ -1
h1(t) - F ( H1(f)) (2.3.7)

4. Detect the peak amplitudes a, and their positions ti and
repeat steps 1-4. After two or three iterations, a considerable

improvement can be expected.

 

28

2.3.2 IMPROVEMENT OF THE IMPULSE RESPONSE FUNCTION BY SPECTRAL
EXTRAPOLATIONZZ

This method extrapolates the frequency components of H(f)
outside the reliable frequency band (f1,f2) by using the frequency
componets inside the reliable band. This technique is applicable for
impulse response functions of both layered and homogeneous structures
The reflected signal has a finite time duration which depends on the
thickness of the object under test. This fact is made use of in
order to extrapolate the spectrum. The procedure is as follows.
1. Calculate the transfer function H(f) using equation (2.3.1.).

(See Figure 2.5a)

2. Let
H(f) f1<|f|<f2

N1(f) = (2.3.8)
0 otherwise

(See Figure 2.5b)
3. Inverse Fourier transform N1(f) to obtain w1(t)
111(1) = F‘1 111(1) (2.3.9)
(See Figure 2.5c)

4. The duration of the impulse response is assumed to be T. T
depends on the thickness of the object. The new estimate of the
impulse response function is

{w1(t) o < t < 1

otherwise

h1(t) (2.3.10)

(See Figure 2.5d)
5. Compute the Fourier transform H1(f) of h1(t) and form the spectrum

H(f) = w (f) f < |f|<f
"o ”2(f) : -.-. 1 1 2 (2.3.11)
' H1(f) otherwise

 

29

 

 

 

 

 

 

 

 

 

FIGURE 2.5

Spectral extrapolation for impulse response
function improvement.

8" H(f)
: 3 E i L
-12 -f1 f1 12 f
b. 111(1) u
i i 1 ;
42 -11 f1 12 f
C.
w1(t) A
f
d.
. (I
h1(t)
VT '1
e.
W2”) )1
/\L\/' 1
/'. i . f\ :

3O

N2(f) is shown in Figure 2.5e.
6. Repeat steps 3,4,5 until the required improvement is achieved.
Next chapter makes use of the theory developed in sections 2.1
and 2.2 to devise a method to find the variation of attenuation

along the path of propagation of an ultrasound signal.

CHAPTER 111
ATTENUATION VARIATION FROM REFLECTED SIGNALS

This chapter explains how the information about the variation
of attenuation inside an object can be obtained by using the
incident and reflected signals. As two reflected signals, obtained
by transmitting a wave from either side of the object (see Figure 3.1)
are required, this method is apglicable only for objects with aC§E§E£>
lggufli;odong.the_be§gggxis. ’However, the range can be increased by
increasing the amount of incident signal power.

The first section assumes the object to be a layered structure.
An inhomogeneous_object, where the acoustical impedance is a
continuous function of the distanee_j§_eonsidered_next. The impulse
response of the object when interrogated from the opposite sides

are given by h1(t) and h2(t).

3.1 RELATIONSHIP OF ATTENUATION T0 IMPULSE RESPONSE FUNCTION
OF A LAYERED OBJECT

Consider a layered structure with N+1 layers - N interfaces -
as shown in Figure 3.2. The_LiggLzand.1gggggggggézggg=gggomgg=go
b3:£§§:§§@§:¥§$§£iglg which normally is the case because the object
is immersed in water. Using the results of section 2.1, h1(t) and
h2(t) can be written as
N

h1(t) = a18(t-t1.) (3.1.1)

1 1

31

32

 

 

 

 

x(t) x(t')
_’ «IQ——
y1(t) _ Test object y2(t')
‘— -——'
FIGURE 3.1 Bi-directional ultrasonic interrogation.

 

 

 

 

1 2 3 4
A
2o Z1 Z2 Z3
0(0 “1 012 0(3
T"o 1:1 1‘2 1‘3
1‘1 r2 r3 r4
FIGURE 3.2

i-1

1-1

i-1

i-1

 

1-1

A layered structure.

 

 

1+1 1+2

1+1
“1+1
1:1°+1

 

1+1 “1+2

N-l

N-I
N-1

‘h-1

 

N-1

 

 

33

where,
i
t1. = Z 2tk (3.1.2)
k=0
1"]. 2
ai = exp(-Zflbtb). r. k=1 (1-rk ) exp(-Zfiitk)
(3.1.3)
Z. - Z.
1 1-1
vs. :2 -— (3.1.4)
‘ Z1 + Z1-1
Similarly,
N
h2(t) = Z, b.8(t-t'.) (3.1.5)
j=1 .1 .1
where,
N
t'. = Z 211k (3.1.6)
J k=j
N 2
. = - - N512 . . - - aft
bJ exp( 2 J J) rJ kl};1 (1 rk ) exp( 2 k k)
(3.1.7)

The minus sign accounts for the fact that the reflection coefficient
changes its sign when the incident wave is from the opposite side.
T23 values a, and b}(i,j4i_1g2ee.JNLuauLJMLJznuLtfixzmdflnojixmLJme
imguTge.nesponse_£unctions_hTLtJ_and h2(t).?

Using equation (3.1.3), the ratio of ai to a1.+1 is

i-
exp(-20cot0). r1. T11 (1-rk2) exp(-2&k’Ck)

 

 

a1 = kfl
a. 1
1+1 exp(-Zaoto)- r1+1 "IT (l-rkz) 6Xp(-2°‘ktk)
k=1
r1
= _ (3.1.8)

 

ri+1(1-ri2) exp(-Zthi)

 

34

Similarly, from equation (3.1.7)

 

 

2
exp(-2¢{Ci)- ri(1'ri+1 )

N
TT 2
.. exp(-Zu-tO). Y'. ‘ (1’1‘ ) exp(-2": )
bi = 1 1 1 k=i+1 k k k
bi+1 ( 2“ ‘t ) r INI (1-r 2) ( ZKft )

 

r1+1

Using equations (3.1.8) and (3.1.9),

 

 

 

2 2
a1 b1 = r1 (1'r1+12)
a. 5. 2
1+1 1+1 (1-r1 ) r1+1
Define R, as
2
+6 . ”1
'51 1-r.2 :9
#4..
Then
a, bi = Ri
a1+1 b1+1 R1+1
or
R = R al1+1 b1+1
i+1 i ‘ “SET—EE—T'
r. S. RAH
1+1 1+1 Ri+1 + 1
where
__ {+1 a1+1
Si '-
‘1 a1+1

(3.1.9)
(3.1.10)
rln‘TL
)2,-R. 5 (3.1.11)
0+RL)V- ;.
r34 1,,:—.
L \’T v
\/z
e:
Nit: t ( Hei
.///
///
/' (3.1.12)
/
f /
lfi L“!
(3.1.13)
(3.1.14)

35

Dividing equation (3.1.8) by equation (3.1.9),

 

a1 b1+1 = r1 ri+1
a. 5. 2 2
1+1 1 ri+1(1-r1. ) exp(-2°(i‘Ci) - r1.(1-r1.+1 ) exp(-2‘91)

which can be written as,

a. b

«I = 1 1+1 _ 2 _ 2

Hence the attenuation constant is given by,

a. b.
_ _l_. 1 1+1 2 2
“1 ‘ 41'L"1—1T <1-r.-><1-n-.1> mm

The following algorithm can be used to find the reflection coeffi-
cient at each interface and the attenuation and impedance in each

layer. The value of 3% is assumed to be known.

1. From equations (3.1.3) and (3.1.7)

r exp(2u6to) a1 (3.1.16)

1

rN -exp(2¢%1%) bN (3.1.17)
El’bN and Tbcan be read from imgglse response functions.

2. Use equations (3.1.11) through (3.1.14) with i=1 to find Ri’ R1

+1
and ri+1 respectively.
”2 < )
R. = ————-——— 3.1.11
1 1 _ r 2

a. .
Ri+1 = R, .[TEFFTéFtél (3.1.12)

 

36

R. *5
rm = 3+1 #T (3.1.13)
1 1+1
where,
+1 if a. > 0
51+1 = ”1 (3.1.14)
-1 if a. < (3
1+1

3. Use equation (3.1.15) to find the attenuation constant “3,

 

. b.
_ 1 , a1 1+1 2 2
“'1 " 747?- . Ln [ET—F.— . (l-Y‘i ) (1-Y‘i+1 4(3.1.15)
1 1+1 1
4. From equation (3.1.4),
Z. 1 + r. Z.
1 _ 1 1-1
7. _ 1 _ r. , _7__ (3.1.18)
0 1 o

This equation can be used to find the normalized impedance of
the i in layer.
5. Repeat steps 2,3 and 4 until all the impedances, attenuation

coefficients and reflection coefficients are found.

3.2 RELATIONSHIP OF ATTENUATION TO IMPULSE RESPONSE FUNCTION
OF AN INHOMOGENEOUS MEDIUM
In this case, the medium is assumed to have a continuous
variation of impedance along the path of propagation as shown in
Figure 3.3.

The following notation is used.

x(t) incident waveform

y1(t)

reflected waveform when the object is

interrogated from left

 

x(t)

 

 

 

 

 

 

 

 

 

 

 

y](t) , Test object
FIGURE 3.3 An inhomogeneous medium.
t/2
A
(t) '(t')
Z(t) Z'(t')
(iv, T/2

FIGURE 3.4

 

Time relationships for bi-directional ultrasonic
interrogation.

y2(t)

h1(t)

Z(t)

Z'(t')

°‘(t)

«(1')

38

reflected waveform when the object is interrogated
from right

impulse response function of the medium when
interrogated from left,

i.e. y1(t) = x(t) * h1(t)

impulse response function of the medium when
interrogated from right,

i.e. y2(t) = x(t) * h2(t)

impedance encountered by the acoustic signal,
when it has travelled a time t/2 from the left
end

impedance encountered by the acoustic signal,
when it has travelled a time t'/2 from the right
end

attenuation coefficient (Np/s) encountered by
the acoustic signal,when it has travelled a time
t/2 from the left end

attenuation coefficient (Np/s) encountered by
the acoustic signal, when it has travelled a time
t'/2 from the right end

total round trip time of the ultrasound signal

- i.e. time taken for the signal to return after

being reflected at the far end.

From Figure 3.4 1t 1S seen that h](t) and h2(t ), where t = T-t,

corresponds to ref ' fr m he ° ’ m.

Hence,

Z(t)

(x(t)

From equation (2.2.

111(1)

h2(t')

Now, the aim is to

z'(1-+)‘/,
1/
o“(T-t)

16),

t

39

1:. exp(-I =o((t).dt) .

O

t
1/2 exp(-0f

o<'(t).dt) .

DID.

t

d

E1

Ln Z(t)

Ln Z'(t')

(3.2.1)
(3.2.2)

(3.2.3)

(3.2.4)

fiqg:tE§:5itenu5fion>and thqfifipéflgncepof a thin

,_sogment of the body as shown in Figure 3.4. From equation (3.2.3)

with t=t and t=t

1

Similarly,
h2(‘52 )
h2 t1 )

Also
t + t' =

and

29

 

 

 

 

 

t
13 exp(-f 1 O‘(t).dt) . %‘E L” Z(t) =t1
0
t
15 exp(-d! 2 “(12).dt) . 3? Ln Z(t) =t2
. d
t —L Z(t)
exp(-f1 x(t).dt) . 3t n t 1
(3.2.5)
d
t1 —. L Z'(t') 1.!
exp(- 20(‘(t).dt) . 3t n It 4:?
ti F Ln Z'(t')lt.=ti
(3.2.6)
1 (3.2.7)
1 (3.2.8)

40

Using equations (3.2.1), (3.2.2), (3.2.7) and (3.2.8),

 

 

3? Ln Z(t) [ t=t1 - 9%?' Ln Z'(t') t|=ti (3.2.9)
QE-Ln Z(t) 1:12 = €131) Ln Z'(t') t.=té (3.2.10)
12 ti
f «(1) dt -.-. I ot'(t).dt (3.2.11)
t
1

Multiplying equations (3.2.5) and (3.2.6),
1111,) 12(1))
h1(t2) h2(ti)

 

t!
exp( [111 «(1). dt ).exp(jt_[ 2 .11-(1)31)
t2 1

6"L" Z(t)|t= -t dt' L" 2 (t )11' =té

' d
dt Ln Z(1")It-"t2-CIT' Ln Z' (t' )|t' 't1

Using equations (3.2.9) and (3.2.10),

 

 

 

 

h 11(t ) h 2(t') t1
= exp(-2 f 0((t).dt )
h12(t ) h 2(t1) t2
i.e. ,
1 h t h 1'.
1f 0‘(t).dt = 1 Ln 1(2) 2(1) (3.2.12)
2 h1(t1) h2(t§)

From equations (3.2.3) and (3.2.4),

. _ , 1 ti .
111(11) 112(11) - 4. exp(- ot(t).dt).exp(-l[ 1x (t).dt)
. gE-Ln 2(1)|t=t1 . :7 Ln 2' (1 )‘t. ‘ti

 

41

Using equations (3.2.1) and (3.2.9),

. t1 T
h1(t1) 112(11) 1. exp(-6f «(t).dt) . exp(-t] «(t).dt)

1
d 2
. (-1) 3‘1? Ln Z(t) I t=t1

T d 2
-144 exp(-J “(t).dt) . (E L" Z(t) \ t=t

1

This reduces to

d

1
exp(% .j «(t).dt) . [-4 h1(t).h2(T-t)] ’2
0

By integrating with respect to time,

T -%
Ln g 5' [exp(-J ck (t).dt]

1

t /2
.J [—4 h1(t).h2(T-t)] .dt

 

 

 

(3.2.13)
Let
’X‘ A'3F H = exp(-IIT «(t).dtL (3.2.14)
0
where(§;:ib the total attenuation of the medium,
From equations (3.2.13) and (3.2.14),
t »
Z(t) -P 2
Ln = (A ) 2 -4 h (t).h (T-t) .dt
Z(O) T or [ 1 2
(3.2.15)

Equation (3.2.12) can be used to find the attenuation variation along
the medium. The result can be used to determine AT from equation:

(3.2.14). Finally, equation (3.2.15) can be usgg__to find the vari-
ation o£_jmpedance along.the.path_o£_propagation.

 

CHAPTER IV
EXPERIMENTAL PROCEDURE AND RESULTS

The theoretical development in section 3.1 was verified using
two models. Section 4.1 outlines the procedure used to obtain the
data -the incident and reflected wave forms. The method of analysing
the data is given in section 4.2. Next, the results obtained for the
two models are compared with the values obtained by direct measure-

ments.

4.1 DATA COLLECTION
The systems used for sample recording of incident and reflected

signals are shown in Figures 4.1 and 4.2. To calculate the attenua-
tion and impedance variation in the test object, the incident signal
waveform and the reflected signal waveforms, when the ultrasound
signal is applied on either side of the object, are required. The
waveforms werg_mgngallymsamglgdfig§jng the delayed sweep feature of
the Tektronix 465 gathode Bay Qscilloscope. A sampling rate of igflfl;
was chosen, as the significant frequency components of the incident
signal were below 5 MHz. The experimental procedure is as follows.
1. Record the incident signal waveform.

This was done by observing the reflections at the air - water

interface as shown in Figure 4.1. As the acoustic impedance of

25

water is about 3700 times that of gin , total reflection with a

phase change of.;§g;_was_a§§gmad. The attenuation of ultrasound

signal in water was neglected.

 

42

43

 

 

Panametrics

5050
Pulser/
Receiver

 

 

water

 

 

 

 

 

 

 

 

 

Tektronix
465
CRO

 

 

FIGURE 4.1

 

 

Panametrics
5050
Puiser/
Receiver

Incident waveform recording

 

 

 

 

 

 

Tektronix
465
CRO

 

 

Figure 4.2

A
in Test object S3

 

 

 

Reflected waveform recording

44

2. Record the reflected signal when interrogated from side A of the
object.
This was done using the system shown in Figure 4.2.

3. Record the reflected signal when interrogated from side B of the
object.

Same method as in 2 above.

4.2 DATA PROCESSING
The following procedure was used to obtain the impedance and the

attenuation variation of the object.
1. Estimate the acoustic impulse response function of the object when

it is interrogated from either side of the object.

This was done using the following two methods.

a. Direct Impulse Response Function

The reflection function of the object,H(f) is the ratio of the

Discrete Fourier Transforms of the reflected and incident signals.

H(f) = {(3 (4.2.1)
As the spectrum of H(f) is Hfltflliflhle_QutSide_the_signifiggn;

f299!§fl§¥_hand_nf_XL£), a Hanning filter H(f) was used to filter

out the unreliable frequency components of H(f).

m = W (4.2.2)

where

 

2
cos "(I‘m fo-Af< f <fo+ 21f
2Af

H(f) (4.2.3)

0 otherwise

- I O O
fart-5f < f < f +Af 15 the reliablLinequenmam—QLXLf—L The

 

45

inverse transform h'(t) of H'(f) is a filtered version of the
impulse response function. The algorithm used for this method
is given in Appendix A. The impulse response function is
estimated using a software implemented amplitude detection
scheme (Appendix C).

b. Frequency Restored Impulse Response Function
Section 2.3.1 outlines a method by which the impulse response
function can be improved by estimating the frequency components
outside the significant frequency band of the incident waveform.
This method was used to find an improved impulse response
function. The method of implementation is given in Appendix B.
Amplitude detection scheme given in Appendix C was then used
to find the echo amplitudes.

2. Obtain the variation of impedance and attenuation using the impulse
response functions. This is done by using equations (3.1.11)
through (3.1.17). Assuming the attenuation of ultrasound in
water to be negligible, equations (3.1.11) through (3.1.17) can

be written as follows.

r1 = a1 (4.2.4)
rN = 'bN (4.2.5)
For i = 1,2, ...N,
R. = ———Y:-‘—2——2— (4.2.6)
1 1 - r1
Ri+1 = R1. fig—ill?)— (4.2.7)

i+1

where

i+1

exp(-Zai'ci)

46

R %
___ 5, __i:1_ (4.2.8)
1+1 1 + Ri+1
+1 if a. >' 0
= . “'1 (4.2.9)
-1 1f ai+1 < 0
a. b. ‘5
= 31—1—1? - (1 ' riz). (1 " ri+12) (4'2'10)
1+ i

I a. b.
= 1 1 1+1 2 2
R— Ln[a—1—+-1—b—i . (l-Y‘i )(1-Y‘i+1 )] (4.2.11)

_ '| 1-1 (
- . 4.2.12)

 

 

notation is used (See Figure 4.3)

number of interfaces

= reflection coefficient at the i th interface

= magnitude of the reflected impulse due to the
i th interface, when interrogated from side A

= magnitude of the reflected impulse due to the
i th interface, when interrogated from side B

= attenuation coefficient (Np/s) in the layer
bounded by i th and i+1 st interfaces

= propagation time in the layer between i th and
i+1 st interfaces

= acoustic impedance of the layer between i th and

i+1 st interfaces

4Z

 

 

 

 

1)
hlm
a1
/\?3 JN‘N
1\/ \Vi \/’ g:
a '.
a2 N'Z aN-1
h (t) 4
2
bN-l
bN—Z
/\"2
b b1
3
b
FIGURE 4.3 Bi-directional impulse response functions of a

layered object.

48

exp(-20911.) = loss factor - WM
propagates forward and backward in the layer

bounded by i th and i+1 st interfaces.

 

Equations (4.2.6) through (4.2.12) and either the equation (4.2.4)
or equation (4.2.5) are sufficient to find the variation of attenuation
and impedance along the path of propagation. Hence it is possible to
find the variation of attenuation in two ways - using equations
(4.2.4) and (4.2.6) through (4.2.12) or equations (4.2.5) through
(4.2.12). The results obtained using both sets of equations are

given in section 4.3.

4.3 EXPERIMENTAL RESULTS

Two models were investigated using the method outlined in section
4.2. The first object to be tested was a symmetrical structure of
three layers. The second was a five layered structure. The results
obtained are also compared with the values obtained by direct measure-

ment of parameters.

4.3.1 A THREE LAYERED STRUCTURE

The first model to be used in the experiment is shown in Figure
4.4. It consisted of three layers - a machine oil layer separated
from water by two acrylic cast (plexi glass) layers. The incident
signal waveform, its frequency spectrum and the Hanning filter
function used in the calculation are shown in Figures 4.5 and 4.6.

As_this is a figmmetricallstrugtuge, it was interrogated only from

one side. The corresponding reflected signal is shown in Figure 4.7.

 

 

 

49

 

 

 

 

 

 

 

 

water a.c. oil a.c. water

 

 

 

 

 

 

 

FIGURE 4.4 Test object 1.

The direct impulse response function calculated using the Hanning
window of Figure 4.6, before and after amplitude detection are
given in Figure 4.8.

The impulse response function obtained using the frequency
restored method is given in Figure 4.9. For this purpose, frequency
components up to 7.27 MHz were estimated. The Hanning filter
function shown in Figure 4.6 was used to weight the original and
estimated frequency components. The algorithm used for this
purpose is given in Appendix B. One iteration was made. The results
obtained using each method are given in Tables 4.1 and 4.2. Table

4.3 gives the results obtained by direct measurement of parameters.

50

.ELowm>mz pcmcwocH

m.¢ mmszu

 

 

51

N12 o~\m

.cmurw» mcwccm: mga vcm Escuomqm ucmumucH

 

m.¢ mmszu

AmVZm
Acvx

 

 

 

 

.Egowm>m3 empompmmm

n...» mag:

1P

 

 

 

 

 

 

 

E»

53

.cowuompmu

 

 

 

 

wasprQEm cmpmm vcm mcocmn H pumwno “mop mo :owpucsm wmcoammc mmpzaew pomcmo w.¢ mmzuHu
. a mmfi.o-nma
m.‘p moo o-" w\/ NN om m m H
I m n 4 v 0 ll llllll 4 .4 . n n 0
ammo.oumm
NmH.ouHm :
g“
mL u \#
4N kmé 3:. ..... :>?>>»>= ><>>>< - L

 

 

 

A
__r

Apvc

H.o

any;

54

.H pommno pmmw mo cowpucsw mmcoammc mmpzasw cocopmmc xocmzcmgm

oaafi.o-nmm

q.

 

)

m.¢ mmstm

 

mmmo.onma

omo~.ou

Hm

 

3v;

55

TABLE 4.1 Results for test object 1 - using direct i.r.f.

a. Reflection coefficients

 

 

Interface Reflection coefficient
Water / Acrylic cast 0.1820
Acrylic cast / Oil - 0.1590
Oil / Acrylic cast 0.1590
Acrylic cast / Water - 0.1820

 

b. Layer parameters

 

 

Layer Normalized Loss factor Attenuation
impedance coefficient (Np/s)
Water 1.00 1.0000 0.0000
Acrylic cast 1.45 0.8976 0.0448
Oil 1.05 0.5189 0.0198
Acrylic cast 1.45 0.8976 0.0448

Water 1.00 1.0000 0.0000

 

 

56

TABLE 4.2 Results for test object 1 - using restored i.r.f.

a. Reflection coefficients

 

 

Interface Reflection coefficient
Water / Acrylic cast 0.1690
Acrylic cast / Oil - 0.1774
Oil / Acrylic cast 0.1774

Acrylic cast / Oil - 0.1690

 

b. Layer parameters

 

 

Layer Normalized Loss factor Attenuation
impedance coefficient (Np/s)
Water 1.00 1.0000 0.0000
Acrylic cast 1.41 0.8589 0.0633
Oil 0.98 0.5739 0.0168
Acrylic cast 1.41 0.8589 0.0633

Water 1.00 1.0000 0.0000

 

TABLE 4.3

measurements.

Layer parameters for test object 1 from direct

 

Material

Normalized impedance*

Attenuation
coefficient (Np/s)

 

Acrylic cast

Oil

0.13 - 0.18
0.0145

 

* Normalized with respect to impedance of water.

 

 

 

water

a. 0

milk

oil

a.c. water

 

 

 

 

 

 

 

FIGURE 4.10 Test object 2.

 

 

 

 

 

 

58

4.3.2 A FIVE LAYERED ASYMMETRICAL STRUCTURE

The model shown in Figure 4.10 was the second object to be used
in the experiment. It consisted of five layers - three layers of
acrylic cast, a layer of milk and a layer of machine oil. The oil
used here was different from the one used in the test object 1.

Test object 2 was interrogated using ultrasonic pulses from
sides A and B. The corresponding direct and frequency restored
impulse response functions are given in Figures 4.11 and 4.12

respectively. The Hanning window used in this experiment is given
by.

2
cos 1‘ 1:351:18.” 1.28< f< 4.42

H(f)

0 otherwise
where f is the frequency in MHz.

In the frequency restoring method of calculating the impulse
response function,frequency components up to 7.85 MHz were estimated.
The impulse response function obtained by each method was used to
calculate the attenuation and impedance variations in the two ways
outlined in section 4.2. The corresponding results are tabulated in
Tables 4.4 through 4.7.

It is seen that when equation (4.2.4) is used, the accuracy
of the results become less for the layers closer to the B end.
Similarly, the accuracy of the results for layers near the A end
decreases if rN =-bN - equation (4.2.5) - is used. This is to be
expected since the signal to noise ratio is less for the reflections

from the far end of the object. Hence the two sets of results can

be combined to give more accurate values by using the parameters of

 

59

.N pomnno ummp we mcowpocam mmcoammc mmrzasw pumgwu chowuumcwu-vm

HH.¢ mmstm

 

 

 

 

 

 

 

. --m
. --H ohfio.o-nen omNH o - a
oomo o - n mHNo.o-um
m.. .< $111--.. mm/ 4? mm ....... NH)? 1 NM nullL.
ae~o.on~n
OHNH.onQD
wamfi.o-umm
m; p mama. -uoa
may 7 PE? 8 - a... ----.i a . a
.8/\ e;(\ _ . . . .4, ----..uul,
mhfio.ou . memo.ou a
. -m
Ammo o- . eHNH.-uHm

= AHVN;

 

1 E:

60

.N pummno pmmp mo mcowuocsm mmcoamw; mmraasw umgoummc hocmzumcm Pacovpomcwu-wm

 

 

 

 

. H . a
m;_pemmo o-- n omo.o-mn moeo 0.. a
(7..., <41 ..... >>.
mNNO.O-Nn
... $851.3
w .
.1.“ - a3 . m y... a
. a -1. ./\ .,1111l n “
Rad-m. mo¢o.o-:< m<

emmo.on m

NH.¢ mmzwmm

 

 

 

 

mNNH.o-ma
------- a 1 will... o
«HRH.o-cg N
13;
1
memfl.o-Na
NH ma
...... . 4 1 -1-- o
«HRH.o-Hm H
A3;

 

61

TABLE 4.4

a. Reflection coefficients

 

Interface

Reflection coefficient

 

Water / Acrylic cast 1 0.171
Acrylic cast 1 / Milk - 0.185
Milk / Acrylic cast 2 0.101
Acrylic cast 2 / Oil 0.072
Oil / Acrylic cast 3 0.139
Acrylic cast 3 / Water - 0.232

 

b. Layer parameters

Results for test object 2 - using direct i.r.f. and

 

 

Layer Normalized Loss factor Attenuation
impedance coefficient (Np/s)
Water 1.00 1.0000 0.0000
Acrylic cast 1 1.41 0.9185 0.0708
Milk 0.97 0.6173 0.0201
Acrylic cast 2 1.19 0.9160 0.0730
Oil 1.38 0.2679 0.1090
Acrylic cast 3 1.82 1.0000 0.0000
Water 1.13 1.0000 0.0000

——r

62

TABLE 4.5 Results for test object 2 - using direct i.r.f. and

a. Reflection coefficients

 

Interface Reflection coefficient

 

 

 

 

Water / Acrylic cast 3 0.171

Acrylic cast 3 / Oil - 0.102

Oil / Acrylic cast 2 - 0.052

Acrylic cast 2 / Milk - 0.073

Milk / Acrylic cast 1 0.136

Acrylic cast 1 / Water - 0.125

b. Layer parameters

Layer Normalized Loss factor Attenuation
impedance coefficient (Np/s)

Water 1.00 1.0000 0.0000
Acrylic cast 3 1.41 1.0000 0.0000
Oil 1.15 0.2664 0.1100
Acrylic cast 2 1.04 0.9127 0.0760
Milk 0.89 0.6108 0.0206
Acrylic cast 1 1.18 0.9047 0.0830
Water 0.91 1.0000 0.0000

 

TABLE 4.6

a. Reflection coefficients

63

 

Interface

Reflection coefficient

 

Water / Acrylic cast 1
Acrylic cast 1 / Milk
Milk / Acrylic cast 2
Acrylic cast 2 / Oil
Oil / Acrylic cast 3

Acrylic cast 3 / Water

0.171
- 0.147
0.111
0.107
0.118
- 0.192

 

b. Layer paramerers

Results for test object 2 - using restored i.r.f. and

 

 

Layer Normalized Loss factor Attenuation
impedance coefficient (Np/s)
Water 1.00 1.0000 0.0000
Acrylic cast 1 1.41 1.0000 0.0000
Milk 1.05 0.4843 0.0302
Acrylic cast 2 1.31 0.7630 0.2250
Oil 1.63 0.4053 0.0752
Acrylic cast 3 2.06 1.0000 0.0000
Water 1.40 1.0000 0.0000

 

64

TABLE 4.7

a. Reflection coefficients

 

Interface

Reflection coefficient

 

Water / Acrylic cast 3
Acrylic cast 3 / Oil
Oil / Acrylic cast 2
Acrylic cast 2 / Milk
Milk / Acrylic cast 1

Acrylic cast 1 / Water

0.171
0.106
0.096
0.098
0.131
0.153

 

b. Layer parameters

Results for test object 2 - using restored i.r.f. and

 

 

Layer Normalized Loss factor Attenuation
impedance coefficient (Np/s)
Water 1.00 1.0000 0.0000
Acrylic cast 3 1.41 1.0000 0.0000
Oil 1.14 0.4042 0.0754
Acrylic cast 2 0.94 0.7611 0.2220
Milk 0.77 0.4826 0.0304
Acrylic cast 1 1.01 1.0000 0.0000
Water 0.74 1.0000 0.0000

 

65

TABLE 4.8

a. Reflection coefficients

Results for test object 2 - using direct i.r.f.

 

Interface

Reflection coefficient

 

 

 

 

Water / Acrylic cast 1 0.171

Acrylic cast 1 / Milk - 0.185

Milk / Acrylic cast 2 0.101

Oil / Acrylic cast 2 - 0.052

Acrylic cast 3 / Oil - 0.102

Water / Acrylic cast 3 0.171

b. Layer parameters

Layer Normalized Loss factor Attenuation
impedance coefficient (Np/s)

Water 1.00 1.0000 0.0000
Acrylic cast 1 1.41 0.9185 0.0708
Milk 0.97 0.6173 0.0201
Acrylic cast 2 1.19 0.9127 0.0760
Oil 1.15 0.2664 0.1100
Acrylic cast 3 1.41 1.0000 0.0000
Water 1.00 1.0000 0.0000

 

56

TABLE 4.9

a. Reflection coefficients

Results for test object 2 - using restored i.r.f.

 

Interface

Reflection coefficient

 

 

 

 

Water / Acrylic cast 1 0.171

Acrylic cast 1 / Milk - 0.147

Milk / Acrylic cast 2 0.111

Oil / Acrylic cast 2 - 0.096

Acrylic cast 3 / Oil - 0.106 ‘

Water / Acrylic cast 3 0.171

b. Layer parameters

Layer Normalized Loss factor Attenuation

impedance coefficient (Np/s)

Water 1.00 1.0000 0.0000

Acrylic cast 1 1.41 1.0000 0.0000

Milk 1.05 0.4843 0.0302

Acrylic cast 2 1.31 0.7630 0.2250

Oil 1.14 0.4042 0.0754

Acrylic cast 3 1.41 1.0000 0.0000

Water 1.00 1.0000 0.0000

 

67

the layers closer to side A from the set of results obtained by

assuming r = a and the parameters of the layers closer to side B

1 1
from the set of results obtained by assuming rN = —bN . These results
are given in Tables 4.8 and 4.9. The parameters obtained by direct

measurements are given in Table 4.10.

TABLE 4.10 Layer parameters for test object 2 from direct

 

 

measurements.
Material Normalized impedance* Attenuation
coefficient (Np/s)
Acrylic cast 1,2,3 1.50 0.1300 - 0.1800
Oil 1.03 0.0350 - 0.0440
Milk 0.98 0.0147 - 0.0217

 

* Normalized with respect to impedance of water.

CHAPTER v
CONCLUSION

The conventional ultrasound imaging systems make use of the ampli-
tude information of the reflected or transmitted signals. Recent
research has resulted in impediography techniques, where the impe-
dance of the medium, derived from the reflected signal, rather than
the amplitude of the returned echoes is used as the imaging parameter.
The impediographic methods make use of the frequency, phase and ampli-
tude information available in the reflected signal. However, there are
.some_cases, such a§=nggigg=giutge=§ggine_where the attenuation

coefficient would be a better imaging parameter than the impedance.
Research has begun very recently on the use of the ultrasound attenu-
ation coefficient as tbe_imaging_narameten1 The aim of this study was
to develop such a method. None of the research, that we came across
in this area use reflection techniques. Transmission techniques
appear to be an obvious choice in this case, because a signal trans-
mitted through an object contains information about the total attenua-
tion along the path it has travelled. This thesis has investigated a
method which uses a reflection technique for ultrasound attenuation
imaging. Only one dimensional images - A mode - has been considered.
But this can easily be extended to two or three dimensions.

When an ultrasoniic pulse gets reflected from an internal discone-
tinuityg of an object, the resulting echo is determined by the reflec-
tion coefficient at the interface as well as the attenuation of the

subtances in between the transducer and the reflecting point. The

68

69

reflection coefficient is a function of the change in impedance. These
two types of information - attenuation and impedance variation along
the path of propagation - contained in the reflected signal cannot be
separated by the knowledge of reflected signal alone. The additional
information necessary to separate the two types of information is
derived from another reflected signal. This second reflected signal
is obtained by a transmission from the opposite side of the object.
This thesis has developed analytical expressions which would allow
the separation of information about attenuation and reflection
coefficients or the impedance. As a result, the method outlined
can extract both the variation of attenuation coefficient and the
variation of impedance along the path of propagation. The variation
of impedance obtained by this technique should be more accurate than
the conventional ultrasonic impediography methods, because those
methods neglect the attenuation in the medium, whereas the method
outlined in this thesis takes into account the ultrasonic attenuation
along the path of propagation. As a matter of fact, the development
of an imaging technique utilizing the attenuation coefficient vari-
ation is the main concern of this research work.

For the purpose of separating the two types of information
- ultrasonic attenuation variation and the impedance variation
along the path of propagation - the object under investigation was
assumed to have one of the two forms.First, it was assumed that
the body has a layered structure, which consists of homogeneous
layers. Later this was extended to an inhomogeneous medium having a

continuous variation of impedance. The expressions developed for a

70

layered structure! were put to test by using models composed of mechine
oil, milk and acrylic cast.

The experimental results show that this method has not been able
to determine the absolute attenuation coefficients of the internal
material accurately. But this does not hamper its ability to be used
as an imaging techinique, where the relative attenuation coefficient is
needed rather than the absolute attenuation coefficient.

The main factor which governs the accuracy of the result is the
nonlinear beam spreading of the acoustic wave. This could cause the
apparent acoustic attenuation coefficient to be varied along the path
of propagation even in a homogeneous object.

This method of determining the attenuation coefficient has the
advantage of suppressing the multiple reflections of the returned signal.
The reason is that the impulse response function obtained from either
side! of the object is more accurate for reflections closer to the
surface. This is because the power and hence the signal to noise ratio
is less for reflected echoes from the deep lying parts of the object.
But the deep lying parts of the object for one impulse response function
are the near parts of the object for the other impulse response function.
Thus, by comparing the time intervals between echoes of the two impulse
response functions, the echoes due to multiple reflections and noise
can be detected and subsequently suppressed.

The analytical expressions developed for the inhomogeneous case
have not been experimentally ”verified. This has been an extension of

the work done on the layered structures. More experiments are needed

to establish its validity.

71

This method heavily depends on the ability to determine the impulse
response function accurately. The accuracy of the impulse response
function in turn depends on the bandwidth of the transducer. Although
techniques have been developed to determine the impulse response func-
tion using a narrow band source, a wider transducer bandwidth will
certainly improve the accuracy. But too wide a bandwidth can give rise
to another problem. The attenuation coefficient for biological matter
has been found to be dispersive in the frequency domain. Thus a large
bandwidth may result in nonuniform attenuation of various frequency
components, which in turn would invalidate the assumptions made in
this method. Hence, a bandwidth, which makes it possible to determine
the impulse response function accurately, to retain the range resolution
within desired limits and at the same time does not cause too much
variation of attenuation in the frequency band used has to be selected.

By using methods such as_pseude_randem_binary_seguenee§, the
accuracy of the impulse response function may be improved. As this
method allows the use of a higher average power, a considerable improve-
ment in signal to noise ratio can be expected. Such methods, which
improve the accuracy of the impulse response function, may make this

method very attractive in the future developments.

APPENDICES

 

APPENDIX A

ALGORITHM FOR DIRECT IMPULSE RESPONSE FUNCTION CALCULATION

The algorithm given below is used to calculate the direct impulse

response function using the incident and reflected waveforms. The

following notation is used.

x = Xi ; i=1,2,...
y = .1. . i=1,2,

h = h. , i=1,2,

1

X = Xi , i=1,2,

Y = Yi , i=1,2,

H = H1 , i=1,2,

w = “i ; l=1,2,...
A = a1 ; i=1,2,...
T = t ; i=1,2,

. N

Incident signal samples**
Reflected signal samples
Impulse response samples
Discrete Fourier Trans-
form (DFT) of x

Discrete Fourier Trans-
form (DFT) of y

Discrete Fourier Trans-
form (DFT) of h

Hanning filter function
Peak values of the
impulse response function
Epoch times corresponding
to peak values of the

impulse response function.

* N is selected such that there are adequate number of zeroes on

either side of the signal samples to prevent aliasing.

** The duration of the incident signal is very small compared to
m

that of the reflected signal. Hence there are only a small number

01°an

72

73

x and y are the data collected in the experiment. This algorithm

calculates the sample values hi of the impulse response function and

detepts jts peaks by using the amplitude detection scheme given in

Appendix C.

was used to determine the Discrete Fourier Transform and the Inverse

Discrete Fourier Transform. This as well as the other algorithms

were implemented using the Cyber machine at the Michigan State

University. The algorithm is as follows.

1.
2.
3.

Calculate the DFT X of the incident signal samples x.
Calculate the OFT Y of the reflected signal samples y.
Calculate the filter function w given by

2 . .
Wi COS 134—12310) for io -Ai < i < io +Ai

and N+2-io-Ai < i <N+2-io-Ai
wi = 0 otherwise

where-is the centre frequency of the incident signal spectrum

and J: is Wdth.

Calculate the transfer function H given by

... _1
Hi ”i x

Find the inverse DFT of H, h. h ds the direct impulse response

function.
\——-——-

Call amplitude detection algorithm (Appendix C) for determination

of peak values and epoch times.

FFTCC routine resident in the Hustler Auxiliary Library,

APPENDIX B

ALGORITHM FOR FREQUENCY RESTORED IMPULSE RESPONSE FUNCTION CALCULATION

The algorithm given below is used to calculate the frequency

restored impulse response function using the incident and reflected

signal samples. The notation given in Appendix A applies here too.The

algorithm is as follows.

1.

Follow the algorithm for direct impulse response function calcula-

tion .

. If the original bandwidth of the spectrums of x,y and h, given by

(ZNA-t).1 where(Zilis_the—sampling_time, is sufficient for restora-
tiQn_Ihen_go_tn_stap.7.

. Find N' such that the required bandwidth is given by (2N2At)'1.
. Let

Hi 0 for 1 2 , . . . N

. Let

Conjugate( Hi) for i= ”2+1, . . . N'

I
ll

 

. Let

N = N'

. From step 1, the positions of the impulses and their magnitude

estimates are known. These values are now used to estimate the
unreliable frequency components.
a. Form Pi’ i= 1,2,...N , where

k
P. = Z. aj exp(-2T1j.i.tj/ N.At)
.=1

(.4

74

 

 

10.

75

b. Form the new estimate of the transfer function

Hi = Hi + (1 - Ni ) Pi 1=1,2,...N

. Find the inverse DFT of the transfer function H. This is the

improved impulse response function.

. Call the amplitude detection algorithm (Appendix C) to find the

improved estimates of echo magnitudes and epoch times.

Steps 2 through 9 may be repeated if more accuracy is needed.

 

 

APPENDIX C

ALGORITHM FOR SOFTWARE IMPLEMENTED AMPLITUDE DETECTION SCHEME

This algorithm is used to.deteet the éEEE‘Vsiuag and the corres-
ponding epoch times of the impulse response function. This is not
similar to any hardware implemented amplitude detection scheme. The
_input to the algorithm is in the form of a seguence(fi;} i=1,2,...N,
where hi are the sample values of the impulse response function
calculated using the algorithms given in Appendices A and B. The
output is in the form of two sequences aj and tj, j=1,2,...k. aj is

the j th peak value detected and tj is its position in time. The

algorithm is given by the flow diagram shown ieJEigure A.1.

76

 

77

 

 

 

 

Do i =
91 = h

 

 

 

 

 

 

 

 

II
|-|
U
\I

00 j

 

3"
IV

If

 

i=i+1

II
0

Let 9i+j

If hi

 

 

 

IV
23'

Let gi-j = O

 

 

 

 

 

 

 

 

 

 

 

 

l<N-7 N :
Y
k = 0
1
Do 1 = 10, N-1O
If 91% 0
Then
k = k+1
ak=91
tk = 1
END

FIGURE A.1 Flow diagram of the amplitude detection scheme.

10.

11.

BIBLIOGRAPHY

Beretsky, I. , "Detection and characterization of Atherosclerosis
in a human arterial wall by Raylographic technique, an in-vitro
study", Ultrasound in Medicine, Volume 38, Plenum Press, 1977.

Beretsky, I. , Farrell, G. , Lichtenstein, B. , "Raylography,

A pulse echo technique with future biomedical applications",
Proceedings of the 20 th Annual Meeting of the American Institute
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