I042 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 38, NO. 9. OCTOBER 1991

cibility, quickness of setup, and ease of use, making the system a valuable tool in soft tissue analysis.

REFERENCES

[ I ] D. G. Ellis. “Cross-sectional area measurements for tendon speci- mens: A comparison of several methods.” J. Biomechon.. vol. 2, pp.

121 P. B. Vasseur, R. R. Pool, S. P. Amoczky, and R. E. Lau, “Correl- ative biomechanical and histologic study of the cranial cruciate liga- ment in dogs,” Amer. J. Vet. R e s . , vol. 46, no. 9, pp. 1842-1853, 1985.

131 N. G. Shrive, T. C. Lam. E. Damson. and C. B. Frank. “A new method of measuring the cross-sectional area of connective tissue.” J . Biomech. Eng., vol. 110. pp. 104-108. 1988.

141 T . Q. Lee, B. Walsh. K. J. Ohland. M. I. Danto. and S . L-Y. Woo, “A combined experimental and analytical procedure to determine the cross-sectional area and shape of soft tissues.” Truns. Orthop. Res. Soc.. vol. 34, pp. 190, 1988.

[SI S . L-Y. Woo, C. A. Orlando, M. A. Gomez. C. B. Frank, and W. H. Akeson, “Tensile properties of the medial collateral ligament as a function of age.” J. Orthop. Res.. vol. 4. pp. 133-141. 1986.

175-186. 1968.

Sound Velocity Inversion in Layered Media with Band-Limited and Noise-Corrupted Data

Dong-Lai Liu

Abstract-With reflection data measured from two different incident angles, it is possible to make them “nearly the same” by stretching or contracting the time axis of one of them. In this way, a correspondence is built up between the two travel times, which can be used to calculate the one-dimensional sound velocity profile of the medium. For the spe- cial case that the impulse response of the medium consists of sparse spikes, a spectral fitting procedure is developed which deconvulves the received signal, and gives the exact positions of the spikes. The step- wise sound velocity profile can then be calculated from these positions. In experimental measurements the plane wave assumption made in the analysis is not true, but this can be accommudated with some modifi- cations of the calculation. Results of both computer simulation and of measurements are presented, indicating the validity of these process- ings.

1. INTRODUCTION

Given information about the incident wave and medium prop- erties, it is possible to calculate the scattered (reflected) waves. This is the direct scattering problem. On the other hand. it is also possible to calculate medium properties from simultaneous infor- mation about the incident and the scattered waves. This is the in- verse scattering problem (ISP). The application that we have in mind is in clinical diagnostic ultrasonics. Human bodywall is non- uniform both structurally and acoustically. Properties of this part may not only be of interest by themselves, they can also be used to better focus ultrasonic beams into the organs beneath i t , thus improving the qualities of ultrasonic images.

Manuscript received August 6, 1990: revised December 12, 1990. The author is with the Institute of Medical Electronics, Faculty of Med-

IEEE Log Number 9102485. icine, University of Tokyo, Tokyo 113. Japan.

A great body of literature exists concerning the ISP. Both theo- retically exact and approximate solutions for one-dimensional to three-dimensional problems have been established [ 11-16]. Gen- erally speaking, exact solutions need a lot of mathematics, are dif- ficult to implement and evaluate, while approximate solutions, though they do not have these problems, are usually limited to such cases as weak inhomogeneities and/or small sizes of the object. These conditions being met, approximate solutions will usually be preferred in practice.

Turning to one-dimensional problems, it is found that most au- thors of either exact [4] or approximate [SI, [6] solutions assume impulse responses of the medium as input data to their algorithms. In reality, such data are seldom available (knowledge of impulse response implies data at the whole frequency range, i.e.. from dc to infinity). Instead, one has to work with their filtered and noise- corrupted version. Based upon such considerations, we have tried a new way for solving the ISP for one-dimensional (layered) me- dia. It is noticed, by comparing reflection data measured at two different incident angles, that reflected waves coming from the same part of the medium resemble each other, except that they arrive at different time instants. This suggests that these waveforms can be made nearly the same (according to certain criteria) by stretching or contracting the time axis of one of them. In this way, a corre- spondence can be built up between the two travel times. A simple derivation indicates that it is possible to reconstruct the sound ve- locity profile from such time relationships. For the special case that the impulse response of the medium consists of sparse spikes, a spectral fitting approach has been developed which can almost re- store the filtered and noise-corrupted signal to its original form, differring at most by a factor. In this way the exact positions of the spikes can be obtained, from which the stepwise variations in sound velocity can be calculated. All these are detailed in Sections I1 and 111. Computer simulation and experimental results are described and discussed in Sections IV and V.

11. CALCULATING SOUND VELOCITY PROFILE FROM T H E

RELATIONSHIP BETWEEN Two TRAVEL TIMES

Referring to the coordinate system shown in Fig. I , travel time t(x) is defined as twice of the time elapsed between the observation of a pulse at the surface and at depth x. Suppose that by some method we have obtained two travel times, t,,(x) and t / , (x) , corre- sponding to two different incident angles, O,,,, and Oh”. Suffixes a and b are used to differentiate the two cases corresponding to dif- ferent incident angles. We do not require that t,, and tl, be known functions of x. x plays the role of a parameter that connects t,, and th. Now let us plot the graph t,, (x) versus th (x) and look at the slope (Fig. 2). With plane waves the equivalent sound velocity for non- perpendicular waves is ~ ‘ ( x ) = C(X)/COS O(x) (see Appendix) where O(x) is the angle between the x axis and the direction of wave prop- agation. Thus,

On the other hand we know that O,,(x) and @/,(x) vary because of sound velocity variations. As a matter of fact, from Snell’s law of refraction for plane waves, we have

0018-9294/91/1000-1042$01.~ 0 1991 IEEE

IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 38, NO. 9. OCTOBER 1991

Fig. I . One-dimensional medium. incident plane wave and the coordinate system.

Fig. 2. A plot of travel times /,,(I) versus t b ( x ) . corresponding to two dif- ferent incident angles B,,,, and ObO. Here .r plays the role of a parameter which connects these two travel times.

Combining this with ( I ) , we obtain

dt,,(x) Jl - sin' O,,(x) JI - c ' (x ) sin' S,,,,/c;

drh(x) 41 - sin' e h ( x ) 41 - c ' ( x ) sin' eM)lC; - - -

from which c(x) can be solved:

This formula indicates that if t,, is given as a function oft/,, together with some auxiliary variables e,,,, em, and c,. then c can be ob- tained as a function of t h . Once this is available, the mapping from th to x can be accomplished via

(4)

and c can be obtained as a function of x . Now the problem is how pairs of travel time can be obtained

from the measured reflection signals, rc,(tct) and rh( t / , ) . One pos- sible way is to employ an approach similar to the DP (dynamic programming) matching technique [7] used in speech recognition. However we will not go into the details here. Instead we will con- sider the case that the medium consists of uniform layers, and that reflection occurs only at the interfaces among thc laycrs, so that the impulse response is a series of spikes. If, by deconvolving the received signal, the accurate positions of these spikes can be re- covered, which may be denoted by T(l l , T,,,, TcJ3, * . . and Thl ,

Th3, . . . , then for the ith section we have (Fig. 2)

111. DECONVOLUTION VIA SPECTRAL FITTING

To attain the above-mentioned goal of recovering the accurate positions of the spikes, we developed a spectral fitting procedure which performs the needed deconvolution.

Let the real impulse response be h ( t ) = E;= I r,6(t - t k ) , whose Fourier transform is H(jw) = E ; = , rk exp ( - ju tL ) . Viewed as a function of w, H ( j w ) is a periodic function, with discrete "fre- quencies" r l , t 2 , . . - , t,,. In practice, we cannot measure h(t) di- rectly. Instead we can only obtain a filtered and noise-corrupted version of it, i.e. r ( t ) = g(t) * h( t ) + m(t) where g(t ) is the probing signal and m(t ) is the noise of measurement. Our problem can be stated as, to estimate h ( t ) from knowledge of g(f ) and r ( t ) . Usually g ( t ) has energy concentrated in a certain frequency band, while m(t) is wide-banded (maybe white). So from r ( f ) and g( t ) we can only get information about H( j w ) in a certain frequency band (U/..

w H ) . To calculate h( t ) in the time domain means to extrapolate H( j w ) from (q. w H ) to the whole frequency domain. Generally this would not be possible, but as discussed above, if h ( t ) consists of sparse spikes, this becomes possible because of the strong pe- riodicity in H(jw) .

The details of our procedure are as the following. Having prac- tical calculations in mind, we consider sampling in both time and frequency domain. First we determine ( T ~ . p I ) by minimizing

N

&TI, p I ) = C J ~ ( j y ) - ple-/w"l)', I = I

This minimization problem can be effectively solved using FFT. Specifically, for a fixed T ~ , to minimize J ( T ~ , pI ), it can be shown that p , should take the following value:

and the corresponding minimum is J ( T ~ , p , ) = E;"= I HH* - N p i where H * ( j w I ) is the complex conjugate of H( j w , ) . Now to min- imize J(T~, p I ) with respect to T ~ , we can calculate p I using (6) for every T ~ . and choose the one that results in the maximum p f . This is nothing more than calculating the IFFT of H( j w , ) [zero-pad the data for frequencies outside of (aL. w H ) ] and choose the one that has the largest absolute real part. The position of that point gives T ~ . while the amplitude gives p I . Next, the term ple - 'w ' r ' is sub- tracted from H( j q ) , and the same process is repeated, giving suc- cessively (7?, p z ) , ( T ~ , p 3 ) , * . , etc. This can be terminated by observing the decrease in J , or equivalently, the magnitude of p , . Utilizing the fact that zero-padding in the frequency-domain cor- responds to over-sampling in the time-domain, it is also possible to determine the 7's down to a fraction of the sampling interval.

To obtain H( j w ) from r ( t ) and g ( t ) , we have simply put H ( j w ) = FFT{r(t)} /FFT{ g ( r ) } , although some more complicated pro- cessing such as Wiener filter could be used.

IV. COMPUTER SIMULATION

To confirm the validity of all the above-mentioned processings, and to investigate the influences of multiple reflections and mea- surements noises, computer simulation studies have been con-

.-

I044 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 38. NO. 9. OCTOBER 1991

Theta = 0 deg 0.2 I I

-0.2 I , -0.2

0 IB 28 36 40 58 66 micro-sec.

I .5 I .0

';ii 0.5 4 0.0 6 ' 2 . 5

0

GI 7i-1.0 I

-1.5

Result of Deconvolution

-2.E ' I' , I 0 I@ 20 30 40 SE 66

Solid=OdeQ D a s k l 5 d e g (Used

Spectrum of Rec. I

I 6 5 I0 15 20

Frequency(MW

Sound Velocity vs. Depth 1.7 I

0 5 I6 15 26 25 30 35 40 Solid=Original Dash=Recon. (mm)

Fig. 3. Results of computer simulation. On the top left panel is shown an example of simulated reflection signals (the incident angle is 0"). whose FFT spectrum is on the right panel. The left bottom panel shows the result of deconvolution. where the reference signal g( t ) has been cut out from the tirst pulse in the received signal. The solid line on the bottom right panel shows the assumed profile of sound velocity. while the dash line is the result of reconstruction from travel times.

ducted. With reference to practical tissue parameters. we assumed a five-layer medium whose parameters are as the following:

c, = 1.52 mm/ps (water) p,, = I .OO g/mL cI = 1.35 mm/ps (fat) pI = 0.92 g /mL

c2 = 1.69 mm/ps (muscle) pz = 1.07 g /mL

c3 = 1.52 mm/ps (blood) p3 = 1.06 g /mL

c4 = 1.56 mm/ps (liver) p4 = 1.06 g /mL.

d,, = 5 mm

d, = 10 mm

d? = 10 mm

dl = 10 mm

(7)

Impulse responses of the medium under plane wave insonification are calculated by Goupillaud's method [8] for incident angles e,,, = 0" and BhO = 15". For nonperpendicular incident angles, the sound velocity profile is first transformed into a perpendicular- equivalent profile, and the calculation is carried out likewise.

The impulse response is (numerically) sampled at a rate of 320 MHz. It is then convolved with an appropriate waveform g ( t ) , and the result is resampled at a rate of 40 MHz (simply pick one out of 8). White Gaussian noise of appropriate amplitude is added to this result to give the simulated r ( t ) . The reason that two sampling rates are used is that, although the Nyquist frequency of g( t ) is quite low so that a low sampling rate (40 MHz) can be used for r ( t ) . the time resolution that is needed in doing inverse scattering is much higher. It should be noted that, using a sampled signal, it is possible to resolve much finer time variations than the sampling interval itself.

Shown in Fig. 3 are the simulated signal r<,(f, ,) and its spectrum. On the bottom left panel the results of deconvolution are shown of both r<,(r<,) and rh(f , , ) . g ( r ) is taken from the first pulse in r ( f ) , so it also contains noise. The frequency range used for spectral fitting

is (2.2 MHz, 4 .8 MHz). The sound velocity profile calculated from the spike positions is shown on the bottom right panel of Fig. 3 (dash line), which coincides with the original one (solid line) quite well.

We also notice from the results that multiple reflections are to- tally invisible compared with the primary reflections. It has been confirmed that only larger parameter variations will bring the mul- tiple reflections out. However, as the parameter variations assumed here are comparable to those found in human tissue, we expect that the influence of multiple reflections will be small when doing in- verse scattering on human body.

V . EXPERIMENTAL MEASUREMENTS

We have measured (Fig. 4) the reflection from an acrylic plate (a kind of plexiglass), using a linear-array transducer. This trans- ducer has 128 elements, each spaced 0.64 mm apart. It is put at about 50 mm above the plate, and both are immersed in water (23.3"C). The elements are numbered from left to right by 0, I . . . . , 127. Firing the 20th element alone, the reflected signals are received by each element and recorded (sampled at 40 MHz with 10-bit precision, 4096 points long each). The signals and the esti- mated pulse positions are shown in Fig. 5.

In processing these data we have found some discrepancy be- tween our analyses and the actuality, which originates in the plane wave assumption. In experimental measurements we use single ele- ments as transducers and receivers, which are identical and are very small in size. So the waves are better approximated by spherical waves, and the incident and reflected rays should be symmetric, as shown in Fig. 6 . In this case, the time interval T between the two observed pulses depends on cos 0 , (Fig. 6) in a different way. Fig.

IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 38, NO. 9. OCTOBER 1991

128 elements

acrylic

1045

Fig. 4. Experimental setup for inverse scattering measurement. The linear array transducer has 128 elements, each spaced 0.64 mm apart. It is put at about 50 mm above an acrylic plate. and both are immersed in water (23.3"C). The 20th element is fired, and the reflection signals received by each element have been sampled at 40 M H z with IO-bit precision.

10 """

I5 V "

25 V;

35

40

45

50

0.8 2 8.6 b + 0.4 z 0.2

8.8 g-a.2 'C-0.4 -

1st pulse

~

1.2

6 1.0

+ O.a

$ 0.6

0.4

0

IC

-

.$ 0.2 -

2nd pulse

:11-.;1 -8.6 ' ' ' ' ' ' ' E . @ ' ' ' ' ' '

2E 25 38 35 40 45 58 20 25 38 35 48 45 50 Line Number Line Number

Fig. 5 . Received signals and the estimated pulse positions. The first 2500 points have been skipped. and the next 1000 points (25 ps) are shown here.

7 shows a plot of In T versus In cos 8 , , calculated for a typical setting. It is observed that a straight line fits to this curve fairly well (but not exactly). This linear relationship is found to hold good for various configurations and for multi-layers. So generally we can write T - coso( 8 , where a is a parameter which depends on the configuration in a complicated way. This modification to the de- pendency of travel time on cos 8 can be incorporated in our pre- vious analyses, by using

(8)

in place of ( 1 ) . However, because a depends on the unknowns, the calculation is performed iteratively. Here is an outline of the pro- cedure of the calculations carried out in the present experiment, i.e., to calculate c,,. d,,, c , , d , (Fig. 6) from pulse positions T,,,, T,? and Thl, T,,, measured under two distances I,, and /, between the transmitting and the receiving elements:

i) Initialization. Solve c, and do from

I046 IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 38, NO. 9. OCTOBER 1991

Fig. 6. In actual measurement point-like transducers are used for trans- mitting and receiving sonic waves. The interval T between the two pulses depends on the angle 0, differently from the plane wave situation.

Slope=0.725 - 10 ? -

-70 -60 -50 -40 -30 -20 -10 0 ~ c g [ c o s i t h e t a i ~ ~ (x10^-31

Fig. 7. A plot of In T versus In cos 8 , . calculated for d,, = 50 mm. d , = 10 mm. c,, = I .5 (mm/ws). c , = 2.7 (mm/ps), I = 0 - 20 mm. according to the geometry shown in Fig. 6. A straight fits well to this curve. sug- gesting the relationship T - cos'" 0 , .

and put

ii) Calculate c , from

(;)2'1x = I - cf sin' O , , ~ ~ / C ~

I - c i sin' e,/c;

iii) d, is obtained from the simultaneous equations

do tan e:, + d, tan

co sin

= 4 / 2

= c , sin e;,"

T,, dl +--- d0 d0 - - 2 c , COS e(:l COS e:, CO COS old

In T,', - In T:, In cos O h , - In cos e,,, f f =

where the three unknowns are d, , O,h, and e,;,. In the special case of t9c,o = 0, we have d , = c , T,,/2.

iv) Numerically evaluated a by

where T:,, TL, e,,,, and O h , are calculated using the obtained cI and d , (Fig. 6).

v) Repeat steps ii) through iv) until convergence is achieved. Only part of the data have been processed. Each pair of two

reflections can be used to calculate the profile of sound velocity as a function of depth, and we have selected the pairs (20, 30), (20, 32), . . . , (20, 50), where the numbers represent the receiving ele- ment number. We have only used the signals received by even- number elements because of some technical reasons. The results (average & standard deviation) are the following:

CO = 1.471 + 0.004, do = 51.84 + 0.13

cI = 2.72 + 0.02, d , = 10.4 k 0.07.

The real value of the thickness d , of the acrylic plate is measured to be 10.3 mm, while the sound velocity, calculated from the time interval between the two pulses in the signal received by the 20th element, is found to be 2.69 mm/ps . The sound velocity of water co is somewhat different from the expected 1.492 mm/ps at 23.3"C, which might have been caused by the existence of a thin acoustical lens on the surface of the array transducer. However, in general the results are very reasonable.

VI. CONCLUSION A N D DISCUSSION

This communication proposes a new way for solving the ISP for layered structures. It is observed that by matching the waveforms of two reflection signals measured at two different angles, a rela- tionship between the two travel times can be established, and using such a relationship, the profile of sound velocity can be recon- structed as a function of depth. When the impulse response of the medium consists of sparse spikes, a spectral fitting procedure can be applied, which deconvolves the received signal, and gives the exact positions of the spikes. The stepwise profile of sound velocity can then be reconstructed from these positions. Experimental mea- surements show some discrepancies from the plane wave analysis, but with slight modifications of the theory we applied our approach to measured data, and obtained reasonable values for the sound velocity and thickness of an acrylic plate. Both computer simula- tion and experiments have shown satisfactory results, confirming the validity of these analyses and processings.

In our processing no special attention has been paid to multiple reflections. Results of simulation study show that multiple reflec- tions may be negligible for the soft tissues found in human body. Another problem is the generation of shear waves, which might happen if there is a bony interface. We did not consider it here because we are mainly interested i n soft tissues. We have not per- formed error analysis in this communication, which certainly need be done in the near future. Inverse scattering problems are known to be ill-posed [9], [IO], in the sense that small errors in the mea- sured data are amplified in the process of inversion. In our study we have observed that small errors in the estimated spike positions give rise to large errors in the reconstructed sound velocity profile, but the results are not that bad as to be unacceptable. Presently we are planning to perform experiments on more complicated objects, including media with nonparallel plane interfaces.

APPENDIX

In this appendix we consider the incidence of a plane wave on a flat plate, from which two reflection pulses are observed, one from the front surface and another from the back surface. Investigated

IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 38, NO. 9, OCTOBER 1991 I047

Fig. 8. Geometry for calculating the time interval between the two ob- served pulses, when the incident wave is a plane wave.

is the dependence on propagating angle 61 of the time interval T between the two pulses. Referring to Fig. 8, take a wavefront dr for example. It impinges on the plate at point A , and a reflection d2 and a refraction $ 2 are generated. d3 propagates on, and after reflection at B and refraction at C , emerges as d.,. Now dI is re- flected not only at A , but also at C (because it is infinitely large). The reflection at C will be denoted as ds, which is on the same plane as 42. The time interval T between t # ~ ~ and d5 is calculated from the time difference tABC - rDC. With the help of Fig. 8, we can write down

2d 2d tan 61 sin 6, T = - - C I COS 61 CO

Using sin 6 0 / ~ 0 = sin 6, / c , the above expression can be reduced to

2d COS 61 T = - CI

So we see that the time interval is proportional to cos 6,, and the equivalent sound velocity is 2 d / T = cI /cos 6,.

ACKNOWLEDGMENT

Iida of Fujitsu Co. is gratefully acknowledged for measuring the reflection data. Timely instructions from Prof. Saito and many free discussions with Prof. Tanaka have been very helpful. Comments from anonymous reviewers have made the manuscript more concise and readable.

REFERENCES

[6] P. C. Pedersen, 0. J . Tretiak, and P. He, “Numerical techniques for the inverse acoustical scattering problem in layered media,” in Acoust. Imaging, Vol. 12, E. A. Ash and C. R. Hill, Eds. New York: Plenum, 1982, p. 443.

[7] H. Sakoe and S. Chiba, “Dynamic programming algorithm optimi- zation for spoken word recognition,” IEEE Trans. Acoust. , Speech, Signal P r o c . , vol. ASSP-26, pp. 43-49, 1978.

[8] J . F. Claerbout. Fundamentals of Geophysicul Datu Processing, McGraw-Hill, 1976, pp. 145-161.

191 K. Sobeczyk. Stochastic Wave Propagation. New York: Elsevier,

VNU Science 1985, pp. 157-158.

[ IO] A. G . Tijhuis, Electromagnetic Inverse Profiling. Press, 1987.

Inspiration Produced by Bilateral Electromagnetic, Cervical Phrenic Nerve Stimulation in Man

L. A. Geddes, G. Mouchawar, J. D. Bourland, and J . Nyenhuis

Abstract-Eddy-current stimulation of both phrenic nerves at the base of the neck in human subjects was carried out to provide inspiration resulting from tetanic diaphragm contraction. The inspired volume ob- tained was in excess of spontaneous tidal volume.

Rhythmic tetanic stimulation of one phrenic nerve at the base of the neck in man with skin-surface electrodes was demonstrated by Sarnoff er al. [ l ] in 1950 as a means of providing artificial respi- ration. Subsequently, i t was found that transchest electrodes could be used for the same purpose [2]. Although effective, the method has not gained wide popularity for two reasons 1) the optimum site for electrode placement is difficult to locate, and 2) the skin sen- sation is objectionable to most subjects, particularly if the elec- trodes are not in the optimum location and the current has to be higher to achieve adequate inspiration. In previous papers we de- scribed the use of a pulsed magnetic field to twitch the hemidia- phragm with a coil placed on the right lateral lower chest of the dog [3] and in another study with the coil placed at the base of the neck in man [4]. In both cases the hemidiaphragm was twitched and a brief inspiration was produced. The history of magnetic stim- ulation [SI reveals that there has been no report in which the dia- phragm has been contracted tetanically to produce a sustained in- spiration by a repetitively pulsed magnetic field. This short communication describes the first application of this technique.

To create a localized induced eddy current in the vicinity of the cervical phrenic nerves, we used two double coils placed at the base of the neck, as shown by the inset in Fig. 1. This technique is slightly different than that described by Ueno et al. [6], Mouchawar et al. [7], and Nyenhuis et al. [8] in that our coils were not in the same plane. Each double coil consisted of eight turns of

I. M. Gel’fand and B. M. Levitan, “On the determination of a dif- ferential equation from its spectral function;^ A ~ ~ , . , M&, sot, Trans., Ser. 2 vol. I , p. 253, 1955. R. G. Newton. “Inverse scattering. 11. Three dimensions,” J. Math. Phys . , vol. 21, p. 1698, 1980. R. K. Mueller, M. Kaveh, and G. Wade, “Reconstructive tomogra- phy and applications to ultrasonics,,, Pror, IEEE. vol. 67, p, 567, 1979. G. N. Balanis. “Inverse scattering: Determination of inhomogene- ities in sound speed,” J. Math. P h y s . , vol. 23, p. 2562, 1982. S. M. Candel. F. Defillipi. and A. Launay, “Determination of the inhomogeneous structure of a medium from its plane wave reflection response. Part 11: A numerical approximation,” J. Sound Vibration. vol. 68, no. 4, p. 583. 1980.

1 /4 in copper ribbon with internal and external diameters of 1 in and 2.5 in. The circumferential end of one coil was joined to the circumferential end of the other coil and the coils were wound in opposite directions to cause the current to flow in the same direc-

Manuscript received November 7, 1990; revised February 21, 1991. The authors are with The William A. Hillenbrand Biomedical Engineer-

IEEE Log Number 9102486. ing Center, Purdue University, West Lafayette, IN 47907.

0018-9294/91/1000-107$01.~ Q 1991 IEEE