The multiple dipole model of the ventricular depolarisation

The multiple dipole model of the ventricular depolarisation

Citation for published version (APA):

Damen, A. A. H., & Piceni, H. A. L. (1971). The multiple dipole model of the ventricular depolarisation. (EUT report. E, Fac. of Electrical Engineering; Vol. 71-E-25). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1971

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THE MULTIPLE DIPOLE MODEL

OF THE VENTRICULAR DEPOLARISA TION

by

Group Measurement and Control

Department of Electrical Engineering Eindhoven University of Technology Eindhoven. The Netherlands

THE MULTIPLE DIPOLE MODEL OF THE VENTRICULAR DEPOLARISATION

by

A.A.H. Damen and H.A.L. Piceni

TH-Report 71-E-25

October 1971

Contents I. In tI'oduction 2. Estimation algorithms 3. Model - dipole functions number of electrodes - number of dipoles

relation of mathematical to physical model

4. Results

5. Conclusion and future developments Appendix

References Figures

THE MULTIPLE-DIPOLE MODEL OF THE VENTRICULAR DEPOLARISATION

Ad A.H. Damen and Hans A.L. Piceni

Abstract

This paper deals with estimation procedures applied to the electrical heart action. The depolarisation wave of both ventricles of the heart is supposed to be represented by a set of current dipoles with time-independent origins and directions. Each dipole timefunction is simulated by a Gaussian curve. The parameters to be estimated are: the peak times and widths of the Gaussian curves as well as the transfer coefficients from dipoles to electrodes. Results of parameter estimation are given on the basis of VCG-curves of 50

healthy persons. The averaged directions of the estimated dipoles show a good correspondence to the anatomic configuration of the heart and the chest. The estimation method is intended as a step towards localizing heart defects.

The authors are with the Department of Electrical Engineering, Technical University, Eindhoven, the Netherlands

1. Introduction

Starting from the measurements of the ventricular depo1arisation carried out by Scher and Young in 1956 [

1J,

Se1vester et a1. in 1964 developed the multiple dipole model, where the dipoles have a fixed origin and direction

[ 2, 3, 4]. In fact, this is still the best physical model of the ventricular depo1arisation and it enables mathematical and numerical extensions to be made. First the electrode potentials were simulated. To this end the transfer from dipole to electrode was evaluated according to the formulae of Wilson and

Bayley [ 5] under the assumption of a homogeneous sphere with eccentric dipoles. For the inverse problem, Bellman [2

J

suggested his quasi1inearisation method and gave results for mode1-to-mode1 adjustments. However,quasi1inearisation is quite a combersome procedure here, because the parameters completely determine the behaviour of the model, while the Gauss-Newton method can be used directly without the roundabout way of the differential equations and it produces the same results (see Appendix).

In 1968, Schloss described a sequential stochastic approximation approach (Kalman filtering) for this problem

[6,

7J ' but as long as we do not have more statistical information about the model parameters and about the noise it will be difficult to implement these theories (See section 2 and 5). His contribution is more in the line of drawing attention to the problem of the observabi1ity of the states according to Kalman's definition.

The difficulty of tackling the inverse problem in, among others, the above ways is the small number of electrodes compared with the number of dipoles. This can be illustrated on the basis of the algebraic formula, which describes the relation between dipole and electrode:

where:

~(t) is a vector of dimension m, containing the electrode potentials a~ a function of the time t.

!(t) is a vector of dimension n, containing the dipole strength-functions of the time.

A is an m x n matrix.

The higher order terms of ~(t) in !(t) are neglected owing to the following

assumptions:

(I)

The wavelengths corresponding to the frequencies in the dipole functions are much longer than the distances between the dipoles and the electrodes. So we may ignore the interaction between the electric and magnetic fields and describe the process quasi-stationarily.

(2) The specific conductivity may be dependent on directions and places, but is hardly influenced by the field strength.

(3) The spatial configuration is time independent.

These assumptions also involve the time independence of matrix~, yet do not

extlude anisotropy, inhomogeneities and boundaries. In short, we only suppose the superposition theorem to be valid.

Especially assumption (3) is a drastic one because of breathing, bloodflow and the motion of the heart itself. This indicates a need for further dis-cuss ion and experimentation.

As soon as the number of electrodes m is larger than the number of dipoles n and we have some values for the elements of ~ from calculations or measurements.

it will be evident that the simplest way of determining f(t) is by means of the pseudo-inverse of A. This has been done several times by, e.g., Rogers [SJ, Barr

[9,

10] and Holt

[II],

using values for the elements of A which had been derived by Barr

[12]

and Pi lkington

ll3

J .

Determination of the elements of A however is still a troublesome task and rather inaccurate because of differences in a~e, race, se,X, dimensions, etc. of the experimental persons. Another complexity ts the enormous number of electrodes that have to be applied and from which the potentials must be recorded simultaneously.

2. Estimation algorithms

Before g01ng into more detail about the heart model, we will motivate the

choice of the estimation algorithms. We were led by the following considerations: (1) The object function is a least square criterion, (See section 4).

(2) The object function is not purely quadratic in the parameters.

(3) The gradient of the object function can be evaluated fairly easily and directly and does not need to be estimated implicitly or explicitly in the estimation algorithm.

(4) At the moment there is hardly any knowledge available about the noise in the system.

(5)

The number of parameters ( >

15)

is rather large.

The lack of noise information (4) hinders stochastic approaches and in combi-nation with point (2) brings us to the us.e of the "hill-climbing" methods. According to the suggestion of Box

[14J

we choose the gradient methods because of point (3). Then there are available the well-known methods of Steepest

Descent, Newton-Raphson, Gauss-Newton and the less familiar algorithms of

Marquardt

[151

,Powell-Fletcher

[16J

and Davidon

[17].

The method of

Newton-Raphson is eliminated because of the evaluation of the second derivative (functional matrix).

The Gauss-Newton method is preferable to the Steepest Descent on account of the quadratic convergence, although, contrary to the Steepest Descent method, the convergence is not guaranteed. So a combination of the two methods in-corporating both properties would be advisable. The Marquardt method satisfies this condition, but a good criterion for the coefficient determining in what measure the Steepest Descent prevails over the Gauss-Newton method, is unknown. Besides, Marquardt needs an inversion of the matrix of the size corresponding to the number of parameters, which takes a great deal of computer time and

What remains is the use of algorithms whi~ start as a pure Steepest Descent and approach the Newton methods after some iterations by updating an estimate

of the inverse functional matrix directly. The revised first method of Davidon, generally called Powell-Fletcher, needs two function and gradient evaluations per iteration in contrast with the second method of Davidon, which uses only one evaluation. We found that the same number of iterations for both methods was necessary to reach the minimum of our object function in a model-to-model adjustment. So in order to save computer time we mainly used the second Davidon method.

When some results become available, itwillbe possible to estimate expectations and covariances of parameters and residuals. We can try to incorporate this a priori infol",ation in subsequent estimations, and here we think more in the way of Foster [18} than in terms of an approach in the time domain like Kalman.

As a matter of fact, we expect difficulties, especially in the time domain,

caused by the rather steep trajectories of the states. Small errors in the initial conditions will result in large deviations of the peak values found by integration (see also Appendix).

3. Model

The potentials as a function of the time measured by S-cher and Young with the help of needles in the myocardium show a rather sharp-edged picture. If, however, we observe a bigger part of the heart from a certain distance, we will perceive the vectorial sum of these angular trapezoids and the resulting

curve-is almost smooth, particularly owing to the large number of trapezoids and their neighbouring peak times. In accordance with statistical considerations we can then choose the Gaussian function as a representation of the total

activity of some smaller parts of the heart.

Hence: f. = h. exp J J (2) where h.

=

height J 2

where is the width

k.

₌

1/0. 0 half

J J

T.

=

peak time J

In the past, one has often met the difficulty that a small deviation in the starting values of the Gaussian function resulted in large errors after integration of the differential equations describing these states. This is now avoided by using the state function directly instead of implicitly in a differential equation.

Fourier analysis of the sum of Gaussian curves gives a frequency contents of the electrode signals, represented by the following formula:

h exp j exp

[

_ ..!.

w" ] 2 k. J \' 2 2 2n

f

w 2 ] La . • h. -k exp - -k +

L La ..

a. lh . hI cos I _[ I 2 I I i W(T.-T )2Tf

V

i exp - -W ( - + - ) ] because . 1J J . . J J J 2 say (2Tfo f) j" I 1J 1 J . J 1 k /l 2 k_j kl 2 2

La ..

h . . 1J J J +

L

_{La .. a·lh.h l cos w("-'l)} j

"1

1J 1 J J Oms < ,. < 100 ms J

There is quite an acceptable resemblance to the frequency contents of a recorded ECG-curve as is seen in figure 1.

According to Barr et al. [19J the minimum number of electrodes is about 24 in order to be able co reproduce all surface potentials with an

in-accuracy smaller than the noise level. However, while using so many signals it was tried to find the dipole functions by means of the pseudo-inverse of

!,

and it was found that the rank of the inverted matrix did not correspond

to the number of dipoles but was near to three. This means that the matrix to be inverted was nearly singular, so that small errors would involve completely different results. This is due to the fact that the origins

of the dipoles are close together compared with the distances between dipoles and electrodes. If all the dipoles had the same origin and the medium was

.* homogeneous, isotropic and infinite, then the rank of! would be exactly three. All deviations from these assUmptions enlarge the rank of ! , but experience with

the familiar one-dipole model shows that three signals contain the main part of the information about the depolarisation wave.

These considerations suggest the use of only three electrode signals. In that case a large amount of information is available and another extra advantage is that results can be applied in the clinics directly because of the usual equipment for three VCG-signals.

The above decisions about dipole functions and number of signals bring about an upper limit of the number of parameters and thus also of the number of dipoles. We have seen that above 50 Hz the electrode signals contain hardly important

information for our model. Without noise, a sample time of 10 ms. would be

sufficient for our model to represent the signal according to Shannon's theorem. Three signals lasting 100 ms. (the average length of the QRS-complex) would then give us about 30 data points, which are to be approximated by 30 non-linear functions of the parameters. Taking into account some measurement error and noise, it is evident that no more than about 20 parameters can be estimated. As we will see, each dipole description needs about 4 parameters so that five dipoles is the limit.

As said before, it is especially matrix~ which is hardly known and causes the

inaccuracy of the model. So the first thought would be to estimate those elements

too.

When doing this, the estimation algorithm has every freedom and simulates the electrode signals by an arbitrary sum of arbitrary Gaussian curves. We found that the least square deviation can be made smaller than 0.3%. However, we cannot expect that there will be a close relationship between this mathematical

model and the physical reality. So, omitting information about the transfer matrix A forces us to put in some other a priori knowledge.

For healthy hearts the time sequence of the depolarisation wave is welL known, and this suggests the exchange of some time information for spatial knowledge.

The peak times and sometimes the width of the dipole curveH are g~""u ~LA"U values in our algorithms, which estimate!:.. too.

Anoth.er complication of estimating

A

is losing the possibility of estimating explicitly h., the height of the dipole function, because these parameters are

J

now absorbed in A. The elements of A are defined by both the transfer conditions and the dipole strength.

What remains to be done is to mark out the heart segments that we want to examine. The most significant parts are septum, apex, left and right ventrical walls, and because the left ventricle is the largest and most active muscle, we divide it into two segments.

One can divide the heart by a flat plane into two more or less symmetrical parts and such a section is shown in figure

2.

The sectioning plane corresponds to the plane perpendicular to the polar vector [21J and can be found by factor analysis of the electrode signals by means of the intrinsic components

[22J

All electric activity. is almost restric;:ted to this pl\lIle because devi~ting

depolarisation fronts are compensated by symmetrical ones (vectorial sum). The

.-~~

depolarisation wave is started in two points in the septum (crosses in figure 2) as though the walls of the right and left ventricles were not yet grown together in the septum. Here, too, the left side prevails and produces about 80% of the total electrical activity, which is compensated partly by the right half, so far as can be seen from the outside.

The expansion of the depolarisation wave in the septum occurs during the total QRS-time, and we were only able to determine the activity of the septum between the moment of starting and the depolarisation of the rest of the heart.

The wavefronts at certain times are sketched in figure 2, and one can see that especially in the right and left ventricle segments the gradient and hence the direction of the supposed dipole vector is not perpendicular to the wall surface contrary to what the Selvester-Bellman model indicates.

Finally, in table 1 a survey of the segments and corresponding peak times is

The peak times correspond to a 100 ms QRS-duration and are adapted linearly to shorter or longer QRS-complexes, because we assumed that the velocity of the depolarisation wave is the same for different hearts and that only differences in dimensions cause the different peak times.

Table 1

Septum 20 ms

2 Apex 40 ms

3 Left Ventricle 1st part 50 ms

4 Left Ventricle 2nd part 70 ms

4. Results

The VGG-signals have been obtained according to the Frank lead system, and were provided to us by the Institute of Medical Physics at Utrecht (Netherlands). This Institute also informed us about the QRS-duration, the beginning of the complex

[21],

and the diagnosis of the cardiologist concerning each patient. With the model above explained we tried to minimise:

e =

f

~

-

2!) T

~

- 2!) dt

ri:t.

dt

x 100%

where the integral is over one QRS-complex

:t.(t)

=

VGG-signals

2!(t) = model signals = 2!(t,

i)

= parameter vector containing the elements a .. and in some cases

1.J k., T ••

J J

(4)

We started to estimate only the a .. for 50 healthy persons. In this case the

1.J

error function e is quadratic in the parameters a .. , which means that

Gauss-1.J

Newton leads to a simple one-iteration process (pseudo-inverse) and therefore the algorithm does not need any initial. values for the parameters. So we did not force the algorithm to look in the neighbourhood of some values, but gave it every freedom. The fixed values k. related to a 100 ms QRS-complex

J

and adapted in the program to deviating durations, together with the results are given in table 2.

Next we selected 13 patients with a QRS-duration of approximately 100 ms. and used the Davidon algorithm to estimate a .. and k .• The initial values

1J J

were the averaged values for the above 50 patients (See table 2). Further we gave the Davidon method the possibility of estimationg also T. for the

J above 13 patients, starting from the last-found values for a .. and k.

1J J

(See table 2). The resultant relative error is ah<ays found to be about 1%. The values for A are illustrated better by means of vectors (figure3) than by a series of numbers. If the medium were homogeneous, isotropic and infinite and the heart were a mere point, these arrows would represent the directions of the dipoles directly. Because of the anatomical resemblance between the patients we expect that the estimated directions will not show too many differences among the patients.

We also looked for correspondence to the original Bellman-Selvester model values and simply evaluated the vectorial sum of those dipoles (See table 2 and figure 3):

20-dipole model 5-dipole model

I

2 + the half of 8, 9, 10, 17, 20 II 11, 12, 13 + the half of 8, 9, 10 III

14, 15, 16 IV

18, 19 + the half ~f 17, 20 V

Figure 4 shows for an arbitrary patient how the electrode signals are built up by the estimated curves with prefixed T values.

5. Conclusions and future developments

A careful look at table 2 shows us that, when also estimating the parameters k., the variances of the a .. become smaller. Obviously the fixed values

J 1J

of k given in table 2 for the first estimation were not the right values for our model. This was to be expected because those fixed values of k. were

J obtained from the 20-dipole model in a very rough way. Along the same lines we can state that the fixed values of the parameters T. were the right values

J

for our model.

Furthermore we would like to stress that among healthy persons with a 100 ms QRS-complex, the variances in the parameters k. are very small.

J

Regarding the directions we can say that there is a good agreement between our estimates and the geometrical configuration of heart and chest. The differences between the perpendicular directions (Selvester-Bellman model) and our estimates can be explained as follows:

In the plane perpendicular to the polar vector, especially dipole 2 and 5 are deviating. This may be due to the fact that indeed the dipole directions are related to the wavefronts in the segments concerned, and that the wavefronts behave differently because of the thick wall of the left ventricle in relation

to the right ventricle as shown in figure 2.

We have the impression that the differences in dipole directions between the patients are caused by different orientations of the heart in the chest. When comparing several patients we need a reference and we hope to find this in the estimated left ventricle dipole and the plane of symmetry. That plane will be determined by factor analysis of the VCG-signals.

We also intend to test this model on the ground of diseases.

As has been said before in section 2, we want to derive some information about the density functions of the parameters and use this in further estimates.

Acknowledgement

The authors wish to thank Dr. J.H. van Bemmel of the Institute of Medical Physics for providing the VCG-sig~als, Ir.·"J.J.H. van Nuenen and A.F.A.M. Gruythuysen for the help in implementation and

pro-gramming, and Professor Dr. Ir. P. Eykhoff for reading the manuscript and offering suggestions.

Appendix

Quasi-linearisation in a nutshell:

Suppose X(t) is the output of a process and~(t) is an estimate of it. Adjust parameters in such a way that ~ approaches X as close as possible in the least square sense, so minimise:

For the sake of simplicity, measurement matrices and weighting matrices are omitted, which does not disturb generality.

The relation between x and the parameters ~ is given by

!

= .!.~, ! , t)

A Taylor expansion provides:

• N+ 1 (N N )

x =.!.~,!,t+

N

Where Nand N+I denote iteration numbers.

I

~N+I

- !N) + •••

N

The solutions of this set of varilinear differential equations is:

N+I

1.1 (t) 1.2(t) ~(O)N+I 2. (t) x

= +

!N+I _{Ol. I} !N+I ₀

The minimum of e is found by setting

oe

o.!(O) = 0

~=

_oS

_-

0 (3)

Substitution of (2) in (3) gives a set of linear algebraic equations. which can be solved by matrix inversion. Then a new iteration starts.

If we can prove that equation (I) is the Taylor expansion of the function .! directly. then there is no difference between quasi-linearisation and

Gauss-Newton in case the solution of the differential equation is known.

This is true in our situation.

x = x <.!(O).

!.

t)

x

= f <.!.

!.

t) = ~ (x(O).

!.

t) N+I N x = x + ox ox

(4)

N Differentiate (4): .N+ 1 • N x = x + (f;lN+I

-l)

+ ₍₅₎ N

Mostly.! is found by simultaneous integration of

.N+ 1 (N N ) x =f,!,~.t+ + N N = f (x • ~ , t) +

I

I

:~T

1 N - N Of

+-1

_ol

_N ~N+I

-l)

+ _ _ Of -::

I

o,!(O)T N Of

because f = f

C,!.

~. t) one knows that - - - : =

or

o,!(O)T

Finally, substitution of (4) gives:

which equals (I), Q.E.D.

The objection may be that solving the differential equation takes as much

time as evaluating exponentials like exp .[-

~i

(t-Ti)Z.]. However, when we utilise Gauss-Newton we do not need to evaluate the Gaussian functions so often.

The iteration formula for our problem becomes: .§.N+ 1 =.!iN +

f

_{ox ox}

T

-=- -=-

dt a.§.

ol

-1

f

T

ox ox

_-

-- -- d t

a! O!T

and

can be evaluated purely analytically without determining Gaussian functions first, and

References

[ I

j

A. Scher and A. Young, "The pathway of ventricular depolarization in the dog", Circul. Res., vol. 4, pp. 46 1-469, 1956.

[2] R. Bellman, C. Collier, H. Kagiwada, R. Kalaba and R. Selvester, "Estimation of Heart Parameters Using Skin Potential Measurements", Communications of the A.C.M., vol. 7, pp. 666-668, November 1964.

[ 3

J

R. Selvester, C. Collier and R. Pearson, "Analog Computer Model of the Vector-cardiogram", Circulation, vol. XXXI, pp. 45-53, January 1965.

[4J R. Selvester et al., "Digital computer model with distance and boundary effects", American Heart Journal, vol. 74, pp. 790.-80.8, December 1967.

[ 5

J

F. Wilson and R. Bayley, "The electric Field of an Eccentric Dipole in a

Homogeneous Spherical Conducting Medium", Circulation, vol. I, pp. 84-92, 1950.. [ 6

J

H. Schloss, "Sequential Stochastic Identification of Myocardial Parameters",

Mathematical Biosciences, vol. 2, pp. 139-144, 1968.

[ 7] H. Schloss, "Computation of Solutions to the Inverse Problem of Electrocar-diography", Comput. BioI. Med., vol. I, pp. 193-198, April 1971.

[8J C. Rogers and T. Pilkington, "Free-moment Current Dipoles in Inverse

Electro-cardiography", IEEE Trans. on Biomedical Eng., vol. BME-I 5,. pp. 312-323, o.ctober 1968. 9] R. Barr and T. Pilkington, "Computing Inverse Solution for an On-o.ff Model",

IEEE Trans. on Biomed. Eng., vol. BME-16, pp. 20.5-214, 1969.

o.J

R. Barr, T. Pilkington, J. Boineau and C. Rogers, "An inverse Electrocardio-graphic Solution with an On-o.ff Model", IEEE Trans. on Biomed. Eng., vol. BME-17, pp. 49-57, January 1970..

I] J. Holt, A. Barnard, M. Lynn and P. Svendsen, "A study of the Human Heart as a Multiple Dipole Electrical Source", Circulation, vol. XL, pp. 687-718, November 1969.

12] R. Barr, T. Pilkington, J. Boineau and M. Spach, "Determining Surface Potentials from Current Dipoles, with Application to Electrocardiography", IEEE Trans. on Biomed. Eng., vol. BME-13, pp. 88-92, April 1966.

13]

T. Pilkington, R. Barr and C. Rogers, "Effect of Conductivity Interfaces", Bulletin of Mathematical Biophysics, vol. 29, pp. 705-709, 1967.

14J M. Box, D. Davies and W. Swann, "Non-Linear Optimization Techniques", Edinburgh, Oliver & Boyd, 1969.

15

J

D. Marquardt, "An algorithm for least-square estimation of nonlinear para-meters"', J. Soc. Industr.-Appl. Math., vol. II, pp. 431-441, June 1963.

J 6 J R. Fletcher and M. Powell, "A rapidly convergent descent method for mini-mization", The Computer Journal, vol. 6, pp. 163-168, 1963.

17 ] W. Davidon, "Variance algorithm for minimization", The Computer Journal, vol. 10, pp. 406-410, 1968.

18J M. Foster, "An Application of the Wiener-Kolmogorov Smoothing Theory To

Matrix Inversion", J. Soc. Industr. Appl. Math., vol. 9, pp. 387-392, September 1961. 191 R. Barr, M. Spach and H.G. Giddens, "Selection of the Number arid Positions

of Measuring Locations for Electrocardiography", IEEE Trans. on Biomed. Eng., vol. BME-18, pp. 125-138, March 1971.

20] D. Gabor and C. V. Nelson, "Determination of the resultant dipole of the heart from measurements of the body surface", J. Appl. Phys., vol. 25, pp. 413-416. April 1954.

21] H.C. Burger, "Heart and Vector", Eindhoven, Philips Technical Library, 1968. 22

J

T.Y. Young and W.H. Huggins, "The IntrinsiC component. Theory of

[23J J. v. Semmel et al., "Statistical Processing Methods for Recognition and Classification of vectorcardiograms", Proe. Xlth Intemational Vector-cardiography Symposium, North Holland Publishing Company; 1970.

[24]

D.M. Detchmendi and R. Sridhar, "On the experimental determination of the dynamical characteristics of physical systems, "Proc. National Electronics

"

Figure 2

Section of the heart ventricles in the plane perpendicular to the polar vector. The symbols ~ denote the directions of the expected dipoles lathered from the depolarisation wave fronts, which are roughly indicated by dotted lines. The symbols ~are the directions perpendicular to the outside wall.

Figure 3

Estimated and perpendicular directions.

t--= estimate a, 50 patients only a .. 1.J ~=

_"

_b, 13 patients only a .. 1.J ~=

_"

_{c, 13 patients only a ..} 1.J

<>--=

perpendicular directions. Figure 4 and k. J k. and T. J J

The estimated Gaussian ·curves for an arbitrary patient. Summation of the dotted curves gives the VCG (full lines). Notice that I and V have always the same sian, also III and IV.

Figure I

The frequency contents of a QRS-complex of a healthy person (full line). Used is standard lead I.

~::-,~~

J

j

(f)

...

c

(!)

....

'

c

o

u

>-u

c

(!) ::J 0-(!) L

4--i

o

\ \ \

,

I

,

,

_,

,

_,

,

\ \

,

,

_,

,

_,

\ \ \ \

,

\ \ \ \

\

\ \

\.'-50

100

-~

frequency (Hz)

-25-:1

j! , ;

'"I

i

) , ,

, "1

" _.

,

" 'I

";

~ "

right

ventricle

apex

-26-, ,

"

,

OJ - - ' CJ 'C -+oJ C OJ

>

I

I

I

I

I

I

I

,1 1

!

I

j

I

,

I

: .... .':' ,

ttl

\

. . ,,'." .... " ;,

,

N '-J

,

a

b

c

• « i"

.

fixed 1:' 'faluea k. 50 vee; 'III Oa1 1 A _~A= el!Stimated. 13 VCG '0 A and k l'A= estimated. 'Il k= 13 VeG 'e:

A,k and t" 'I1A=

estim&hd. ~k= ~1:. Sel.,.eter-Bellman A. Bodel 20 40 50 1.54 0.73 0.85 1-0.8._ f-1. 7. .9~ 1-0.6 _ 1.1 11,,1_ 1-1.0 -6.0 5.4 1_0 . 8 2.4 9.6 --- ... --0.5 ... 2.1

_..

_~J!.... _1.4 -6.6 6.2 1.17 0.98 1.01

L-.1....f.

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TH-Reports:

EINDHOVEN UN:I:.YER,S:I:.n: Olf TECHNOLOGY THE NETHER,LA,NDS

DEPARTMENT OF ELECTRICAL ENGINEERING

1. Dijk, J., M. Jeuken & E.J. Maanders

Al~ ANTENNA FOR A SATELLITE COMMUNICATION GROUND STATION

(PROVISIONAL ELECTRICAL DESIGN). TH-report 68-E-01. March 1968. 2. Veefkind, A., J.H. Blom & L.H.Th. Rietjens

THEORETICAL AND EXPERIMENTAL INVESTIGATION OF A NON-EQUILIBRIUM PLASMA IN A MHD CHANNEL. TH-report 68-E-02. March 1968. Submitted to the Symposium on Magnetohydrodynamic Electrical Power Generation, Warsaw, Poland, 24-30 July, 1968.

3. Boom, A.J.W. van den & J.H.A.M. Melis

A COMPARISON OF SOME PROCESS PARAMETER ESTIMATING SCHEMES. TH-report 68-E-03. September 1968.

4. Eykhoff, P., P.J.M. Ophey, J. Severs & J.O.M. Oome

AN ELECTROLYTIC TANK FOR INSTRUCTIONAL PURPOSES REPRESENTING THE COMLEX-FREQUENCY PLANE. TH-report 68-E-04. September 1968.

5. Vermij, L. & J.E. Daalder

ENERGY BALANCE OF FUSING SILVER WIRES SURROUNDED BY AIR, TH-report 68-E-05. November 1968.

6. Houben, J.W.M.A. & P. Massee

MHD POWER CONVERSION EMPLOYING LIQUID METALS. TH-report 69-E-06. February 1969.

7. Heuvel W.M.C. van den & W.F.J. Kersten

VOLTAGE MEASUREMENT IN CURRENT ZERO INVESTIGATIONS. TH-report 69-E-07. September 1969.

8. Vermij, L.

SELECTED BIBLIOGRAPHY OF FUSES. TH-report 69-E-08-.. September 1969.

-30-9. Westenberg, J.Z.

SOME IDENTIFICATION SCHEMES FOR NON-LINEAR NOISY PROCESSES. TH-report 69-E-09. December 1969.

10. Koop, H.E.M., J. Dijk& E;J. Maanders

ON CONICAL HORN ANTENNAS. TH-report 70-E-IO. February 1970. II. Veefkind, A.

NON-EQUILIBRIUM PHENOMENA IN A DISC-SHAPED MAGNETOHYDRODYNAMIC GENERATOR TH-report 70-E-II. March 1970.

12. Jansen, J.K.M., M.E.J. Jeuken & C.W. Lambrechtse THE SCALAR FEED. TH-report 70-E-12. December 1969. 13. Teuling, D.J.A.

ELECTRONIC IMAGE MOTION COMPENSATION IN A PORTABLE TELEVISION CAMERA. TH-report 70-E-13. 1970.

14. Lorencin, M.

AUTOMATIC METEOR REFLECTIONS RECORDING EQUIPMENT. TH-report 70-E-14. November 1970.

IS. Smets, A.J.

THE INSTRUMENTAL VARIABLE METHOD AND RELATED IDENTIFICATION SCHEMES. TH-report 70-E-15. November 1970.

16. White Jr., R.C.

A SURVEY OF RANDOM METHODS FOR PARAMETER OPTIMIZATION. TH-report 70-E-16. February 1971.

17. Talman, J.L.

APPROXIMATED GAUSS-MARKOV ESTIMATORS AND RELATED SCHEMES. TH-report 71-E-17. February 1971.

18. Kalasek, V.K.

MEASUREMENTS OF TIME CONSTANTS ON CASCADE D.C. ARE IN NITROGEN. TH-report 71-E-18. February 1971.

19. Hosselet, L.M.L.F.

OZONBILDUNG MITTELS ELEKTRISCHE ENTLADUNGEN. TH-report 71-E-19. April 1971.

-31-20. Arts, M.G.J.

ON THE INSTANTANEOUS MEASUREMENT OF BLOODFLOW BY ULTRASONIC MEANS. TH-report 71-E-20. May 1971.

21. Roer, Th.G. van de

NON-ISO THERMAL ANALYSIS OF CARRIER WAVES IN A SEMICONDUCTOR. TH-report 71-E-21. August 1971.

22. Jeuken, P • .]. C. Huber &.C.E. •. Mulders

SENSING INERTLAL ROTATION WITH TUNING FORKS. TH-report 71-E-22. September 1971

23. Dijk, J.B., & E.J. Maanders

BLOCKING IN CASSE GRAIN ANTENNA SYSTEMS TH-report 71-E-23, September 1971. A review.

24. J. Kregting, & R.C. White, Jr. ADAPTIVE RANDOM SEARCH

TH-report 71-E-24, September 1971 25. A.A.H. Darnen & H.A.L. Piceni.

THE MULTIPLE DIPOLE MODEL OF THE VENTRICULAR DEPOLARISATION. TH-report 71-E-25, October 1971.