Direct synthesis of the Gaussian filter for nuclear pulse amplifiers

NUCLEAR

INSTRUMENTS

AND

METHODS

138 (I976) 85-92; ©

NORTH-HOLLAND

PUBLISHING

CO.

D I R E C T S Y N T H E S I S OF T H E GAUSSIAN F I L T E R F O R N U C L E A R P U L S E AMPLIFIERS S. O H K A W A , M. Y O S H I Z A W A and K. H U S I M I

Institute for Nuclear Study, University of Tokyo, Tokyo, Japan Received 27 December 1975 and in revised form 5 July 1976 The Gaussian pulse forming networks which can be suitably realized by active filters are derived based on the frequency domain analysis. The frequency characteristics and the output waveforms of the results are calculated, and show the favourable features of the filter for nuclear pulse amplifiers. Calculations on the effects of the rise time variations of the detector signal upon the pulse peak height are also carried out.

1. Introduction Nuclear pulse shaping has previously been studied theoretically based on the matched filter theory. This study put forward the idea that a network having a " c u s p " shape time response to a step-voltage yields the best possible signal-to-noise ratiol'2). However, the spectral resolution of a detector amplifier system is not only dependent on the noise characteristics of the signal source but on many factors such as pile-up effect, baseline shift and also rise time variation of the detector signal 2,3). Another theoretical study based on the indeterminacy principle 4'5) for time width and frequency width of a signal, presented a Gaussian shaping when there is no frequency and time width limitation and also a sine-square shaping when there is a time width limitationS). These studies seem to have been stimulated by the theoretical interest on the pulse waveform for reducing pile-up errors in high speed data transmission. In nuclear spectroscopy, the Gaussian shaping is also expected to have favourable features, which will be discussed briefly with respect to the characteristics of filters composed of a CR differentiator and integrators. The formula used to evaluate the noise filtering performance of various shaping networks is V.

1

27r

0

N (o)) f2 (o)) do),

ENV = V,/hp, where V. is the root mean square noise voltage at the amplifier output. N(o)) is the output noise power spectrum of the detector preamplifier combination and f(o)) is the angular frequency characteristics of the amplifier6'7). The symbol ENV represents the equiv-

alent rms noise voltage of the amplifier, where hp is the peak height of the output response to a unit input voltage. In the presence of a noise spectrum of the form A N(o)) = ~ - ~ + B , the optimum ENV of the Gaussian filter is 1.119 (AB) + and is considerably better than that of the C R RC filter3). In addition the fall time of the pulse measured from the top to a point where the tail of the pulse decays to a noise level of the amplifier is much smaller than that of the C R - R C filter. Thus the Gaussian shaping has a potential superiority when used in nuclear amplifiers for realizing a high resolution and high counting rate spectroscopy system, although at present, a relatively slow nuclear A D C seems to limit the maximum rate of data acquisition. The true Gaussian waveform cannot be achieved by a physically realizable network. It is well known that a cascade connection of a CR differentiator and an infinitely large number of RC integrators with the same time constant approximates the true Gaussian filter2). But because of a slow convergence of the approximation, this method needs a lot of RC filters when a good Gaussian waveform is required. In practice a CR differentiator followed by four RC integrators is usually considered adequate in this approach. However it does have a slowly decaying tail similar to that obtained with C R - R C shaping and so would be expected to give similar pile-up errorsS). A further straightforward repetition of C R integration will not only be uneconomical because the improvement is expected to be small, but also inadequate because buffer amplifiers necessary for this cascade integration are also noise sources. Recently it has become common practice to use an

86

S.

OHKAWA

active filter as a pulse forming network of the nuclear pulse amplifier. Some of these works seem to improve this problem by introducing a complex pole pairg'l°). A systematic study for obtaining the optimum pulse shaping network has been carried out by Blankenship and Nowlin 11,12). Their method of network synthesis is based on the line-pole constellation, adding a real pole for suppressing the excess peaks and an imaginary zero pair for improving the symmetry of the waveform. However, the network obtained contains inductors which require special care for reducing the temperature dependent variation 13). All of the above mentioned studies are methods which belong to the time domain analysis by looking at the waveform, and the criterion of approximation seems to be somewhat empirical. However, the method described in the present work is the frequency domain synthesis, using the property that the frequency characterstics of the ideal Gaussian waveform is also Gaussian, as shown by the Fourier transform14). The frequency domain synthesis is superior to the other approach in that the problem can be dealt with in a strictly mathematical sense. The analysis is also intended to show that the pole locations obtained can be realized using active filters considering the convenience of practical use. The nuclear amplifiers now available are well developed in many respects. However, the detector is also improved remarkably and it requires a large shaping time constant for the amplifier. Therefore, a solution to the problem of obtaining a high quality filter, which is suitable not only for high resolution but also high counting rate measurements, is of major importance. The aim of this work is to obtain a very close approximation to the true Gaussian waveform by using a network not as complicated as the conventional systems employed for this purpose.

2. Synthesis of the Gaussian network function The detector signal is integrated by a charge sensitive preamplifier before it is fed to a main amplifier. Therefore, the pulse forming network of the amplifier is assumed to be composed of an ideal differentiator followed by a network whose impulse response is the Gaussian waveform. The frequency characteristics of the true Gaussian waveform 1)

f (t) = ao e -~'2/°2

is given by _10-2 ~j2

F(oo) = ao x/(2=) tre ~

,

(2)

et al.

where ao is a constant and tr is also a constant corresponding to the rms deviation of the normal distribution. Let us assume that the transfer function H(s) of the Gaussian filter can be expressed as follows: (3)

H ( s ) = Ho/Q(s),

where Ho is a constant and Q(s) is a Hurwitz polynomial. Since the roots of the equation Q (s)= 0 are the poles of the Gaussian filter, the network to be synthesized will be composed of poles only without zero. The amplitude and the phase of the frequency characteristics are closely related. However, the problem is to obtain the network function H(s) from the amplitude characteristics only, without any information of the phase characteristics. Referring to the mirror image pole-zero constellation of an all-pass network, it is seen that the product H ( j o o ) . H ( - j c o ) is given by the amplitude characteristics only and is expressed as follows (appendix A) /4(j~o)./4(-j~o)

= [F(~o)] 2 .

(4)

By using eqs. (2) and (3), the above relation is transformed into --' ( H - -o ) ' tr2 e~2 Q(joo).Q(-jtn) = 2= \ a o a / e ,

(5)

or

Q(s)'Q(-s)

, (.0j.

= -e - ¢2s2, 2= \aocr/

(6)

where s=j~o. By a proper normalization, eq. (6) can be rewritten finally as Q(p).Q(-p)

(7)

= e -p2,

and p = crs,

(8)

Without loss of generality. The exponential function in eq. (7) is approximated by finite terms of the Taylor series as

p4 p6

Q(p)'Q(-p)

p2n

= l - p 2 + _ _ _ _ + ... + ( - 1 ) " - - . 2! 3! n!

(9)

It is necessary to factorize the right-hand side of the equation into the same form as the left-hand side and to obtain a network function Q(p), which is a Hurwitz polynomial. The Hurwitz polynomials for n - - I and n - - 2 are easily obtained as follows:

DIRECT

SYNTHESIS

OF T H E G A U S S I A N

87

FILTER

1) n = l : From the relation

Q(p).Q(-p)

= l-p2

IMAGINARy AXIS

2]

,n=7

= (l+p) (l-p).

p- plane

//? 5 r/ i ",',, 4

Q(p) is obtained as

,'F ,' I' ~ 3

Q(p) =

f/fj :, J~xJ

l+p.

This network function has the real pole at - 1 on the p plane.

I

-2

2) n = 2: From the relation

i

0

, ' , ( ; T -I

REAL AXIS

, L , ~ x

4

Q(p)'Q(-p) = l-p2 + P._ 2! = ½j-x/2 + x / ( 2 + 2 x / 2 ) x (x/2 - x / 2 + 2 x / 2 )

Q(p)

' k

p+p2] x p+p2).

-2) Fig. 1. Pole constellations of the G a u s s i a n filters.

is obtained as

Q(p) =

~

l

The C R - R C filter has the waveform (x/2 + x/(2+Zx/2) p + p 2 ) .

f(t)

=(~R)

e -t/cR,

(10)

This network function has the conjugate pole pair at

#(2#2+2)

[-1 + (#2-1)].

2 3) n = 3 ~ 7 : It is highly complicated and quite difficult to perform a higher order Hurwitz factorization over n - - 3 by an algebraic method, hence the problems are dealt with by numerical analyses. The pole constellations obtained are plotted on the complex p plane as shown in fig. 1, and are listed in table 1. When the order of the right-hand side of eq. (9) is 2n, the number of poles is also n. 3. Time constant definition and the characteristics

The values of the equivalent noise voltage, the rise time immunity and the fall time of the output response of a filter are dependent on the shaping time constant. When these characteristics are compared by using the optimum time constant, the C R - R C filter, which has the worst N/S ratio, also has the best rise time immunity in the CR-(RC)" filter series3). Fortunately the ENV is a fairly slowly changing function of the time constant in the region of the optimum time constantS). Therefore for comparing with different waveforms, it seems convenient to make use of an equal-area time constant to the C R - R C filter instead of the optimum time constant (appendix B).

and the peak height (l/e) is at t=CR, so that the area of the pulse with peak height 1 is given by

ScR=efo°(-~R) e-'/CRdt=e. CR.

(11)

The Gaussian waveform expressed by eq. (1) has the peak height a o at t = 0, and the area of the pulse with peak height 1 is given by S~ =

e -~-¢2/~2 dt -- ,J(2rc) ~.

(12)

TABLE 1 Pole locations of the Gaussian filters. n=3

Ao Al w~ A2 w2 As w3

n=4

1.2633573

n=5

n=6

1.4766878

1.6610245

1.1490948 1.3553576 1.4166647 1.5601279 0.7864188 0.3277948 0.5978596 0.2686793 1.1810803 1.0603749

n=7

1.6229725 0.5007975

1.2036832 1.4613750 1.2994843 0.8329565

1.4949993 1.0454546

1.2207388 1.5145343

1.2344141 1.7113028

88

S. OHKAWA et al.

Both areas SCR and So are equal, when ~r is given by e

a = - - "

,,/(2 ~)

C R = 1.0844%,

(13)

where To is the C R time constant. The frequency characteristics a n d the impulse responses o f the G a u s s i a n filter are calculated numerically with respect to the above time constant definition. The transfer function H ( s ) can be expressed as k

+ w?}

Ao gl H (s) =

i= ~

(14)

k

IF]

+ w/}

i=1

when the n u m b e r o f poles n is odd, a n d k

]7I

08[

(15)

k i=,

H(s)=

However, r e m a r k a b l e i m p r o v e m e n t s are observed with an increase o f n. The impulse responses o f the filters are shown in fig. 3 as a function o f the time n o r m a l i z e d with % . The time derivatives o f the impulse responses are shown in fig. 4. It is clearly seen from these figures that the a p p r o x i m a t i o n becomes better with increasing n. Fig. 5 shows the c o n t r i b u t i o n s o f each pole on the waveform o f n = 7 separately. The m a i n pulse is formed as a resultant o f an exponential decay and three d a m p i n g oscillations, each o f which has a different a m p l i t u d e , a different frequency and a different phase. It is interesting to note that these c o m p o n e n t s cancel out one a n o t h e r at the tail o f the main pulse. This is m o r e clearly seen in fig. 6, which shows the tails o f m a i n pulses for n from 3 to 7 in a magnified scale. The impulse response of the n e t w o r k of n = 3 has a large

0.7-

2 + W/}

0.6

i=1

0.5

when n is even. In these equations, A 0 denotes the real pole, A i a n d W, denote the real a n d the i m a g i n a r y parts o f the ith complex pole pair on the p plane respectively and k denotes the n u m b e r o f complex pole pairs. The n u m e r a t o r s o f eqs. (14) and (15) are the constants for n o r m a l i z i n g the value o f H ( s ) to 1 when s = 0 . The frequency characteristics o f the G a u s s i a n filter are shown in fig. 2 as a function o f the frequency normalized with ~o0(= l / t o ) , in which n is the parameter. The true G a u s s i a n frequency characteristics are also shown in the figure for c o m p a r i s o n . The simple filter o f n = 3 has a p o o r a t t e n u a t i o n in high frequency regions.

n=3 4567

0.4 0.3 0.2 0.1 0 -0.

[

-0.~

p

i

i

I

I

I

I

I

f

0

I

I

5

~

I

I

I

I

I

I

I

I

I

I

I

I

I

l

J

,

J

,

,

I0

TIME(1/T 0 )

Fig. 3. Impulse responses of the Gaussian filters.

1.0 ~

0.6 0.5 0.4

i 0 -f

0.5 g

0.2

i °o -0.2 G

-0.5

i ~4 ~

-

0

5 I0 FREQUENCY (oJ/cdo)

Fig. 2. Frequency characteristics of the Gaussian filters.

\

-0.4

,

o

,

,

, , ,

,

,

,

,

5

,

,

,

,,

,

,

,

TIME (t/T O)

Fig. 4. Time derivatives of the impulse responses.

,

,,

~o

DIRECT

SYNTHESIS

OF

THE

undershoot because there occurs no cancellation of excess peaks. The cancellation occurs when n is larger than 4 and the larger the number of poles n, the better the cancellation. The undershoot is very small when n = 7 , because excess peaks of three damping oscillations of different frequencies are staggered out. The effects of rise time variation of the input pulse upon the output waveform of the Gaussian filters are shown in fig. 7 for n = 3 and 7 as a function of the time normalized by to, in which the parameter r, is the time constant of the exponentially increasing waveform of the input pulse normalized by %. Fig. 8 show plots of the loss in peak height for n = 3,4,5 and 7 as a function of ~ . In this figure, the loss in peak height of the Gaussian filter when L is 0.2 is 1.89% for n = 3 and 1.55% for n = 7 and is considerably better than that of the C R - R C filter, which is calculated as 2.5% by Hatch3).

GAUSSIAN

The equivalent noise voltage ENVeq of the Gaussian filters for the equal-area time constant are shown in table 2. Both the rise time immunity and the noise performance become better with an increase of the number of poles, when compared to using the equalarea time constant. The optimum time constants %0 and the optimum equivalent noise voltages E N V o p are also shown in the table. 4. Gaussian active filter design For realizing the ideal differentiator of the main amplifier, it is common practice to use a differentiator with a proper time constant, derived from the zero at the origin and a real pole. The differentiator with a 0.8 0,7 0.6 I

0.6

Ao

p- plan

x (AzW=]

°ill

0.5

2]

(A3W

j

~(AjWn)

123

:2\XX~.A

n: 7

0.3

F-

14

J

a. 0.2

× IA~WO

TA L

0.2

~(A2W2)

i

'~ 0.1

-2I

o

--o.,it,:::,y/' ,

-o. I - o .2

i

i

i

i

i

i

I

I

I

5

o

A2W2)

-0.2 -o .3

n= 3 Tr = 0

0.4

0.3 w

89

FILTER

I

I

I

I

I

i

i

T I M E ( t / ~ - O)

i

i

;

i

i

i

i

i

i

5

i

i

i

i

i

i

i

i

i

i

i

i

i

i

I0

TIME(I/T o )

i

F i g . 7. Dependence of rise-time variations of step input pulse upon the output waveforms of the filters for n = 3 a n d 7. IO0

F i g . 5. Contributions of each pole for the output waveform.

I(54I 95

tJJ

~9o

2 . 5 x I (3 I-

n=7

EL
J

a_

0 85

\4

-2.5× I ~5~

80 -5.0xl

i

(h

oLTT L / 7.5x

i

I0

i

0

]

i

i

i

L

I

f

I

I

I

I

I

I

I

I

I

5

I

I

I

I

I

IO

TIME(I/T

0 )

F i g . 6. Tails of the output waveforms for n from 3 to 7.

]

i

i

i

F

0

I

I

I

I

0.2 OA 0.6 0.8 TIM E('Cr)

I

1,0

F i g . 8. Losses in peak height of the filters for n = 3, 5 a n d 7.

i

90

s. OHKAWA et al.

TABLE 2

Equivalent noise voltages o1 the Gaussian filters for the equalarea and the optimum time constants. n

ENVeq/(AB) ~

ENVop/(AB)+

top/to

3 4 5 6 7

1.2212 1.2044 1.1973 1.1928 1.1891

1.1997 1.1660 1.1517 1.1439 1.1389

0.8291 0.7538 0.7274 0.7164 0.7115

A i .~.

=

1),

(19)

Qi = W/A~.

(16)

where a o is the equal-area time constant (e/x/2n) defined by eq. (13). There are several circuit configurations of the active filter for realizing a complex pole pair. The circuit shown in fig. 9 is one of them. This circuit has the advantage that the gain can be varied by adjusting the coupling resistance R3, and this has no effect upon the location of the pole of the filter. The voltage transfer function of the circuit is given as

~(s)

GO TO 2 C2 R2'

( aoZo ~/(4C2R2

complex pole pair is also realizable, but seems to have no special usefulness for nuclear amplifiers. Therefore the pole constellation with a real pole such as n = 5 or n = 7 is suitable for the shaping network of this type. The actual Gaussian filter is composed of the differentiator followed by two (n= 5) or three (n=7) active filter sections with complex poles. The time constant CoRo of the differentiator is given by the real pole as

Co Ro = ao'co/Ao,

Let the real and the imaginary part of the complex pole be At and W,. Then A t and W~ are expressed as follows

From these relations, C1 and C2 are obtained as a function of R 1 and R2 as

Ct = (4C2R2~ \

1

R, / O~+l'

c~ = (2-~) °°'r° A,

(20)

It is preferable for circuit tolerance to choose nearly the same value for C l and C 2. When C~ is equal to C2, R2 is expressed in terms of R~ as R2 = R , (1 +

Q2)/4.

(21)

The circuit has the freedom to select the value of R~ for optimizing the amplifier system considering the gain and the coupling resistance R 3 . Fig. 10 shows the circuit details of the Gaussian filter used in a pulse amplifier. The peak amplitude of the output response to the input of a unit step pulse is given by

v2/v~ peak gain

\R3/ 1 + sC1 R1 + sz C 1 R

1C2

=(-~7o)(Rl~3G7,kR3/I

R2 I.Sk

The complex pole pair is obtained from the denominator of eq. (17) as

s = (2C-~z)[--l+---j / ( 4C2Rz

~

500

~AoJ CI

!"°l ',

I.Sk I

0

: operational

U

:

unit

gain

amplifier

buffer

Fig. 9. Active filter amplifier unit.

(22)

(A2.w2)

i

'

Fig. 10. Gaussian filter amplifier.

I

J (i"°1 [z3.w3] I,Sk ,'vvv~,

[A, . w ~ ]

: J I I

I

,

91

D I R E C T S Y N T H E S I S OF T H E G A U S S I A N F I L T E R

where l/A7o is the amplitude loss due to the differentiator and G 7 is the peak amplitude of the impulse response of the filter for n = 7 (fig. 3) and is 0.39. Therefore the peak gain of this circuit is 13.19 for the input of a unit step pulse.

5. Discussion The single stage active filter with a complex pole pair is now widely used as a pulse forming network of nuclear pulse amplifiers, because the output waveform is better than that of the active filter with real poles. But the undershoot of the impulse response becomes remarkable when the pulse tail returns to baseline more quickly. It is also necessary to pay attention to the frequency characteristics of the filter. The single stage active filter is inadequate for reducing a high frequency component of noise. The active filter followed by an RC integrator considerably improves the waveform 9,,o). This is because excess peaks are smoothened out by the integration averaging. The pulse forming network derived here has a multistage construction of active filters. The contributions of each stage not only improve the Gaussian approximation of the main output pulse but also cancel out the unnecessary excess peaks which appear at the tail of the pulse, as stated before. The cancellation by staggering excess peaks is more effective and positive than by a simple averaging. The amplifiers constructed by applying this design have been proved to show excellent performance as expected, and are now standard electronic units of this institute. The Gaussian filters described here need more circuit components than the networks reported until now. However, this method of synthesis is certainly a way of using active elements efficiently and is one of the best solutions for obtaining a high quality filter which is required by the recent advance of nuclear spectrometry. The authors wish to express their thanks to Dr. G. Madueme* for his kind advices on the manuscript.

Appendix A Let us consider the pole-zero constellation of an all-pass network, in which zeros in the right half-plane of the complex frequency s are the mirror image of the poles in the left half-plane. The transfer function G(s)

is given by

G(s) = P ( - s)/P(s),

where P(s) is a Hurwitz polynomial' s). The vector drawn from a pole to a point co on the imaginary axis is always identical in magnitude with the vector drawn from its mirror image zero to the same point o). Therefore the amplitude of G(j~o) is constant and equal to 1 for all frequencies. On the other hand, the phase characteristic of G(je)) is given by the sum of the phase angles of the vectors from poles and zeros. From the above discussion, it is seen that the product P(jco).P(-jco) is given by the square of the amplitude characteristic of the frequency response of P(jco), because phase angles of P(jm) and P ( - j c o ) are cancelled out.

Appendix B The exponential function e -p2 of eq. (7) can be expressed as e-P2=

lim

1-

.

(24)

m ~

Hence eq. (7) is approximated as follows

Q ( p ) ' Q ( - p ) = (1 - --~-)".

(25)

The function Q(p) is given from this equation as Q(p)=

l+

.

(26)

This relation shows that the Gaussian filter can be approximated by a cascade connection of many RC integrators and the C R - R C filter is the special case of this when m = 2. The function Q(p) is expressed as Q(p)=

1+

.

(27)

The denominator of the transfer function of the C R - R C filter is 9_.(s) = (1 + s C R ) 2 .

(28)

Comparing eq. (27) with eq. (28), the relation

p = x/2 Zo s * On leave from the Institute of Physics, University of Uppsala, Sweden.

(23)

(29)

is obtained, where ro is the CR time constant. The

92

s. OHKAWA et al.

TABLE 3 Values of E N V and percentage loss in peak height o f the o u t p u t pulse against the rise time o f the input pulse, for the equal-area time constant, with reference to the values for the o p t i m u m time constant. Filter Loss in ENVeq/ type peak height a (AB) ~

req/ro

ENVop/ (AB) +

rop/Zo

1.0 0.7357

1.359 1.215

1.0 0.5773

N/S ratio when compared to using the optimum time constant. However when compared to using the equalarea time constant, both the rise time immunity and the N/S ratio become better with an increase of the CR integrator. This is shown in table 3 in which the C R - R C filter is compared with the C R - ( R C ) 2 filter as an example. References

CR-RC CR-(RC) 2

2.402 2.055

1.359 1.236

The loss in peak height is calculated for the pulse whose time constant o f the rise time is 0.2to. The values for the o p t i m u m time c o n s t a n t are quoted from ref. 3.

parameter in eq. (8) is also obtained as a = ,,
(30)

This time constant definition however is too large compared with the value given by eq. (13) and seems to be inadequate, because the function Q(p) of eq. (26) approximates the Gaussian frequency characteristics only when m is much larger than 2. Full widths at half-maximum of triangular, rectangular and trapezoidal waveforms are just the same, when areas of these waveforms with the same peak height are equal. The full width at half-maximum of the CR RC shaping is 2.446 to, whereas that of the true Gaussian is 2.554 ro when the time constant is defined by eq. (13). In the CR-(RC)" filter series, single integrator shaping has the best rise time immunity and the poorest

1) R. Wilson, Phil. Mag. 41 (1950) 66. 2) E. Fairtein and J. H a h n , Nucleonics 23, no. 7 (1965) 56; 23, no. 9 (1965) 81; 23, no. 11 (1965) 50; 24, no. 1 (1966) 54. 3) K. Hatch, IEEE Trans. N u c h Sci. NS-15, no. 1 (1968) 303. 4) H. L. L a n d a u a n d H. O. Pollak, Bell Sys. Tech. J. 41 (1961) 65. 5) T. H o s o n o a n d S. O h w a k u , Inst. Elect. Com. Engrs. J a p a n 48 (1965) 1394 (in Japanese). 6) E. Baldinger a n d W. Franzen, Advances in electronics and electron physics (Academic Press, New York, 1956) vol. 3, p. 256. v) M. T s u k a d a , Nucl. Instr. a n d Meth. 14 (1961) 241. 8) p. W. Nicholson, Nuclear electronics (J. Wiley, New York, 1974) p. 109. 9) F. S. Goulding, D. A. Landis a n d R. H. Pehl, Semiconductor nuclear-particle detectors a n d circuits, U.S. Nat. Acad. o f Science, publication 1593 (1969) p. 455. 1o) T. K u m a h a r a , J A E R l - m e m o 4795, J a p a n A t o m i c Energy Res. Inst. (1972) p. 135 (in Japanese). i J) j. L. Blankenship a n d C. H. Nowlin, IEEE Trans. Nucl. Sci. NS-13, no. 3 (1966) 495. i2) C. H. Nowlin, IEEE Trans. Nucl. Sci. NS-I7, no. I (1970) 226. ~3) D. R. D u n n , R. C. Kaifer and McQuaid, IEEE Trans. Nucl. Sci. NS-I9, no. 1 (1972) 461. 14) j. A. Stratton, Electromagnetic theory (McGraw-Hill, New York, 1941) p. 290. is) F. F. Kuo, Network analysis and synthesis (J. Wiley, New York, 1962) p. 221.