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Bell Telephone Laboratories, Incorporated
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American Telephone and Telegraph Company

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THE BELL SYSTEM TECHNICAL JOURNAL is published ten times a year by the
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Executive Officer; C. L. Brown, President; W. G. Burns, Vice President and

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Copyright (C) 1977 American Telephone and Telegraph Company
THE BELL SYSTEM TECHNICAL .JOURNAL

Vol. 56, No.6, .July-August 1977
Printed in U.S.A.

Reflection, Transmission, and Mode Conversion
in a Corrugated Feed
By C. DRAGONE
(Manuscript received December 14, 1976)

Microwave antennas are often required to carry signals simultaneously over a broad range of frequencies-e.g., the combined TD-2 and
TH common carrier bands encompass a total frequency ratio of about
1.8 to 1 as do the combined 18- and 30-GHz bands. To achieve these
bandwidths, an efficient broadband feed horn is required. The corrugated (hybrid-mode) horn is a leading candidate, but it is not immune
to some cross-polarization coupling, input reflection, and pattern
asymmetry. These problems are introduced mainly by two phenomena:
variation of the dominant mode shape with frequency and mode conversion along the horn taper and at waveguide transitions at the horn
input. Simple formulas for computing the magnitude of these phenomena and their effects on return loss and radiation patterns are
given.
I. INTRODUCTION

Corrugated feeds (also called hybrid-mode feeds) are widely used in
reflector-type antennas because of their excellent radiation characteristics. I - I6 At the frequency Wo at which the surface reactance Xs of the
corrugations becomes infinite, the radiation pattern of a properly designed feed is circularly symmetric, is free of cross-polarized components,
and has low sidelobes. In principle these properties can be obtained over
a frequency range of more than an octave. In fact, one can show that the
835

or Xs = OJ). An evaluation of this effect is given in Section VII. However,
in the experiment, the taper angle a (Fig. 1) was chosen sufficiently small
(a ~ 4°) so that this effect was negligible.
The analysis starts in Section III with a derivation of the asymptotic
properties for large ka of the modes of a corrugated waveguide. The results provide a simple and accurate representation of the modes in a feed
aperture of more than a few wavelengths in diameter. Then, in Sections
V and VI, a first-order derivation of the scattering parameters of a
junction between two waveguides of slightly different characteristics
is given. A simple relation [see, eqs. (83), (84), and (115) to (117)] is found
between the scattering parameters and the coupling coefficients between
the modes on the two sides of the junction. Each coefficient is given,
except for a constant, by an integral of the type

f Is

(E 1 X H 2 *) . iz dxdy,

where S is the junction area, and El and H2 are the electric and magnetic
vectors of the two modes, respectively. In Section IV, this surface integral
is converted to a line integral, thus reducing the calculation of the coupling coefficients to a straightforward exercise. This result is useful also
to calculate the far field of an aperture S illuminated by a mode E 1, since
the far field at a given observation point is, except for a constant, the
coupling coefficient over S between El and the field H2 of a plane wave
having the direction of the observation point. The far-field calculation
is thus reduced to a contour integration.
The calculation of the scattering parameters is carried out in Sections
IV to VI, using the above contour integral. It is found, for instance, that
the input reflection of a corrugated feed connected to a smooth waveguide of the same diameter is simply given by the coefficient
(31 - (3~
PI = - (31 + (3~ ,
(31 and (3~ being the propagation constants in the two waveguides. An

identical formula was derived by Brown 18 from the principle of conservation of momentum, but that derivation is not applicable to the present
problem, which involves hybrid modes.
Finally, Section VII deals with the problem of spurious mode generation in a nonuniform waveguide whose parameters (radius and surface
reactance) vary along the axis, as in Fig. 1. The differential scattering
parameters that give, at any point in a nonuniform waveguide, the local
coupling between the incident mode and the spurious modes are obtained from the analysis of Sections V and VI. By solving the differential
equations specified by the above scattering parameters, we can thus
determine the amplitudes of the spurious modes. An example is provided

CORRUGATED FEED PERFORMANCE

837

of a first-order calculation of mode conversion in a conical waveguide
such as the one in Fig. 1 for z > Zl. The result, eq. (154), is again quite
simple.
II. PRELIMINARY CONSIDERATIONS

For a smooth waveguide, there is a simple relation between the
propagation constant {3 of a mode and the waveguide diameter, but no
such simple relation exists in the case of a corrugated waveguide [see eq.
(20)]. For this reason, the properties of the corrugated waveguide modes
cannot generally be determined as simply as in the case of a smooth
waveguide. Also, the field configuration of each mode varies with
waveguide diameter. There is, however, an important exception. When
the radius a of the corrugated waveguide is sufficiently large, the propagation constant {3 for some of the modes is simply given by
{3 = V(ka)2 - u6m'

(1)

where UO m is the m th zero of the Bessel function J 0 of order zero,
(2)

For all the other modes except one (for this special mode (3 is independent
of a; see Appendix B) one has
{3 =

v (ka)2 - u~m ,

(3)

where U2m is the m th root of the Bessel function of order two,
J 2(U2m) = o.

(4)

Equations (1) and (3) are valid provided a » A, a condition which is
satisfied to a good approximation by most feed apertures. Thus, the
case
ka» 1

(5)

is of considerable practical interest. One finds that as ka -- 00, the
properties of a mode become independent of the surface reactance Xs
of the corrugated walls, except for the mode of Appendix B. Thus, the
field distribution over the aperture of a feed illuminated by a single mode
will be little affected by the surface reactance Xs (which varies fairly
rapidly with frequency) provided ka is sufficiently large. This result, first
pointed out by Thomas,8 is very important for it implies that the aperture field distribution becomes frequency independent for large ka. The
main purpose of this section is to determine the asymptotic behavior of
the hybrid modes for large ka. It is shown that if ka ~ 00 there is over
the aperture of a feed a certain undesirable cross-polarized component,
even if the aperture is illuminated by a single mode, unless of course Xs
838

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

=

CX).

A simple expression for the amplitude of this component is

given.
III. ASYMPTOTIC BEHAVIOR FOR LARGE ka

Consider a disk-loaded waveguide centered around the z-axis, as in
Fig. 1, and assume its parameters a, b, and h are independent of z. Let
r, ¢, z be cylindrical coordinates defined by x = r cos¢ and y = r sin¢.
The separation of the disks, which occupy the region a < r < b, is assumed to be much smaller than a wavelength A,
kh « 1.

(6)

The region between two consecutive disks forms a radial line whose input
reactancejX at r = a is a function of the radial length l = b - a; for ka
» 1, one has approximately

jX = jZo tankl,
where Zo = vi}la/Eo' For a finite number of teeth per wavelength, the
value of l must be corrected. * Because of condition (6) the effect of the
disks can be accounted for adequately by introducing an effective surface
reactance 5 ,12,19

jXs = jX ( 1 - ~),

(7)

where t is the thickness of the disks, and by requiring that the field for
r < a satisfy the boundary conditions

E 1> -0
'"

} for r = a,

(8)

H ;:;:;_ Ez
1>
]'Xs
where E 1>, H 1>, E z are the ¢ and z components of the electric and magnetic
field.
Let (3 be the propagation constant in the z direction,
(3 = k cost'h,

(9)

and assume (h is real, so that (3 < k. The case where (h is imaginary is
considered in Appendix B. Assume the ¢ dependence of E z is given by
cos¢. Then, the field components of a mode that propagates in the z
direction with propagation constant (3 are eiven by
(10)

* See Ref. 17 for the effect of a finite number of teeth per wavelength, which causes a
reduction of the effective depth, l.

CORRUGATED FEED PERFORMANCE

839

(11)
(12)

(13)
(14)
(15)

where
K

= k sinOl.

The boundary conditions (8) give l2
u

J;(u)

'Y=-----

(16)

COSOl 1
1 J;(u)
y=---+---sinOl u 'Y sinOl J 1 (u) ,

(17)

COSOl Jl(u) ,

where

u = ka sinO!,
y = -

Zo
Xs'

(18)

and 'Y is the ratio between the TM and TE components of the hybrid
mode,
A
(19)
'Y =-.
B
By eliminating 'Y from eqs. (16) and (17), one obtains the eigenvalue
equation
(20)

which is eq. (10) of Ref. 12.
The solutions of this equation are now studied for large ka. Both u and
yare assumed to be finite. Then, in the limit as ka --- 00, eq. (20) reduces
to
J'l(U)
( Jl(u)
840

u)2 _1 = o.

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

(21)

We distinguish two cases:
J'l(U)U =

-1

J 1 (u)

(22)

and

J; (u)u = 1.
Jdu)

(23)

According to eq. (16) (with COS01 ~ 1, since 01 - - 0 as ka -- co), these two
cases correspond respectively to
1'=1

(24)

1'=-1.

(25)

and

Using well known recurrence relations between the Bessel functions and
their derivatives, and using conditions (22) and (23), we find
(26)

and
(27)

We conclude that for large ka, eq. (20) possesses the two sets of solutions
u ~ UO m (m = 1, 2, etc.)

(28)

u ~ U2m (m = 1, 2, etc.),

(29)

and

UO m and U2m being respectively the mth root of Jo(u) and J 2 (u). Solutions (28) and (29) are characterized by l' ~ 1 and l' ~ -1; the corresponding modes will be designated, * respectively, HElm and HE;m.
Asymptotic series for u and l' in terms of

1
ka'

(30)

* This mode classification differs from the one by Clarricoats 12 and it was chosen for the
following reason. Here, and in Ref. 17, we are interested in horns whose inner radius a varies
gradually with z, while the wall susceptance y is approximately constant, as in Fig. 1, from
Z2 to Zl. Consider therefore a mode propagating in Fig. 1 from Z2 towards Zl. Clarricoats'
classification assigns in some cases a different name to this mode in different regions of
the horn, even though there will be no discontinuous variation of the mode-field configuration, as it propagates in the horn. On the other hand, our classification based on the
Bessel function roots UO m and U2m, assigns a single name everywhere in the horn. If instead
the frequency is gradually changed the mode of a waveguide of given dimensions will retain
the same name with Clarricoats' classification, whereas this is not always true with our
classification. To understand better these considerations, see also Ref. 25.
CORRUGATED FEED PERFORMANCE

841

are derived in Appendix A under the assumption y ~ co. For the HElm
modes, characterized by 'Y - 1 as ka - co, it is found that
U

=Um = UO m 11 - ~

(k1J'
1
+ ~ [ 1 - ~; (7U 6m + 1)] (k a)

:a -t[1- ~ +
(1

U6m)]

3 ••• )

(31)

3 ••• ).

(32)

and
"V

= 1-

1

U2

Om

{y2 ka1 y28 (4 + u (1)2
ka
- -

-

2 )

-

Om

-

- ~ [ 1 - ~ U5m - ~ (3U6m + 2)]

CJ

For the HE~m-modes, characterized by 'Y - -1, u is given by
(33)
and
'Y = -1 -

u~m {~ :a ... }.

(34)

The x and y components of the electric field are now derived. First
consider the HElm modes. One finds from eqs. (10) to (15), with cosO 1
= 1 and 'Y given by eq. (32), that for large ka the transverse component
of E is given by

E t ;=:; - j k: A [ J 0 (~u ix

)

+ 1:. u2 L J 2 (!... u) (cos2¢ ix + sin2¢ i y )],
4

ka

a

(35)

omitting the factor e - j(3z. Amplitude A is determined by power P carried
by the mode. From eq. (67) with du/dy given by eq. (92) and YJi = A

IAI ;=:;.!VZo~u2_I_
a

7r

a{3 ka Jr(u)

(36)

if P = %.
For the HE~m modes with 'Y ;=:; -1, on the other hand,
E, ;v j k: A [ J, (~u) (cos2q, ix + sin2q, iy) + ... ],
842

(37)

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

where the dots represent terms that vanish as ka - 00. The amplitude
1A 1 for P = %is still given by eq. (36). *
An important property of the field distribution (35) is that Ey - 0 as
ka - 00. Thus, in the limifas ka - 00, the field becomes polarized in one
direction, regardless of the value of the surface reactance Xs (unless,
of course Xs = 0). From eq. (35), the amplitude of Ey is porportional to
the ratio

l

(38)

ka

Therefore, in order that Ey be negligible over the aperture of a feed, it
is sufficient that the aperture diameter be large and the thickness t of
the disks (see Fig. 1) be small compared with their separation.t The far
field of an aperture illuminated by the fundamental mode, the HEn
mode given by eq. (35) for u = UOl = 2.4048, is discussed in Ref. 17. From
a comparison of the radiation patterns of Ex and E y, we find that the
ratio C2 between the maximum value attained by 1Ey 12 and 1Ex 12 (which
occurs on axis) is given by

C2 = 0.14 ( -Y

ka

)2 =

0.14
[ (1 - t/h)ka tan 7r W]2
2 Wo

,

(39)

where Wo denotes the frequency for which y = o. One can easily verify
using this formula that C2 remains less than 0.000316 (-35 dB) over a
frequency range WI < W < 1.93 WI, provided ka > 10 and t/h < 0.1.
Thus, good performance over a wide frequency range is possible,
provided all the power incident at the input of the feed is converted to
the HEn mode. If, however, some of the input power is converted into
some of the HE~m modes, then, according to eq. (37), the field over the
feed aperture will contain a cross-polarized component whose amplitude
is essentially independent of the ratio y/ka. The resulting cross-polarized
component of the far field is discussed in Ref. 17. If WI < W < W2 denotes
the frequency range over which only the fundamental mode (HEn)
propagates, it is pointed out in Ref. 17 that the largest value that W2/WI
can assume is 1.6839; this value is attained for b/a = 1.8309. Cutofffrequency formulas are derived in Appendix D.
In Appendix B, the properties of a surface-wave mode that can exist
in a corrugated waveguide, in addition to the modes of eqs. (35) and (37),
are briefly described.
* In eqs. (35) and (37), only the leading terms for the symmetrical, asymmetrical, and
cross-polarized components are retained.
t Note that from eqs. (7), (8), and (18), y increases with t/h.

CORRUGATED FEED PERFORMANCE

843

IV. COUPLING COEFFICIENT BETWEEN TWO MODES

Suppose the electric field El at the input of a corrugated waveguide
is known, and we want to determine the resulting amplitude of one of
the modes excited in the corrugated waveguide. We have to evaluate a
surface integral of the form

f Is

(E 1 X H;) . iz dxdy,

(40)

where H2 is the magnetic field of the mode whose amplitude is to be
determined. This integral, identical to that involved in determining the
far field radiated in a given direction by an aperture containing the field
Ell is in general difficult to evaluate. However, in many cases, we can
assume that
OEl_ . E
02
-]{31 1,

(41)

f'oJ

where {31 is a constant. This condition is approximately satisfied, * for
instance, in the case of a feed aperture illuminated by a single mode
propagating in the 2 direction with propagation constant {31. We will show
that the above surface integral can be reduced to a line integral which
can be evaluated straightforwardly. We use the symbol (Ell H 2 ) for the
integral (40), and call it the scalar product of the two modes El and
H 2•
If Ell HI and E 2, H2 are two solutions of Maxwell's equations, in free
space,20
(42)

in the absence of sources. Now, let the 2 dependence of the two solutions
be given by
(43)

Then, in eq. (42)
(44)

v = Vt - j({31- (32)iz,
where v t is the transverse part of V. Therefore eq. (42) gives

vdEl X H; + E; X HI) = j({31 - (32)[(E 1 X H;) . iz + (E; X HI)· izl.
(45)

Next, consider a new solution E~, H~ with propagation constant {3~ =
-{31 and with 2 components given by
(46)

* Of course, condition (41) is satisfied exactly by a mode in a cylindrical waveguide.
844

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

(47)

Then the x, y components of E;, H; simply coincide 21 with the x, y
components of Eb -HI,

H;x = -H lx ,

E;y = Ely,

(48)

H;y = -H ly .

(49)

Therefore, replacing in eq. (45) (31, Eb HI with -(31, E;, H; and making
use of eqs. (48) and (49), we obtain
"'V t • (E; X H; + E; X H;) = -j((31 + (32)[(E I X H;)

- (E; . HI)] X i z .

(50)

By adding eq. (45) to eq. (50), we obtain
(51)

(E I X H;) . iz = "'V t • F,
where

We now integrate eq. (51) over a finite area S of the plane z = 0, making
use of the divergence theorem,

J Is

(E I X H;) . iz dxdy =

fc F· ds,
n

(53)

where C is the contour of Sand n is the outward normal. To determine
F . n, let T be a unit vector tangent to C,
T = iz X

n.

(54)

Then, if A and B are two arbitrary vectors,
(A X B) . n = ArBz - AzB n

(55)

where An Br are the components of A and B in the direction of T.
Therefore, from eq. (52), taking into account eqs. (46) to (49),

F· n = -j

A

(31 - (32

[ElrH;z + E;rHlz]

+j

A [E 1z H;r + E;zHlr]. (56)
(31 - (32

CORRUGATED FEED PERFORMANCE

845

Finally, from eqs. (53) and (56), we obtain the desired result,

SIs

(E I , H 2 ) =

(E I X H;) . iz dxdy

= -j

A

rI:' (ElTH;z + E;THlz)ds

{31 - {32 ~ C

A

+j

rI:' (ElzH;T + E;zH IT)ds.

{31 - {32 ~ C

(57)

Thus, the scalar product (coupling coefficient) of two modes EI and
E2 can be determined straightforwardly from the values of EI and E2
on the contour of the aperture S. This result has a number of applications. It can be used, as already pointed out, to determine the far field
radiated by an aperture S with known field distribution El, in which
case H2 is the magnetic field * of a plane wave with propagation vector
k and eq. (57) gives, except for a constant independent of k, the field
component radiated in the direction of k with the polarization of H 2 • In
this article, we are interested in the special case where S is a circular area
of radius a, in which case we can replace in eq. (57) T with ¢, since
T =

if/>.

If Ei, Hi (i = 1, 2) represents a mode of a corrugated waveguide of radius

a, so that for r = a
Eif/> = 0,

ZoHif/> = - jYiEiz,

(58)

then eq. (57) simplifies to
(E I , H 2) = -

~

A

Zo {31 - {32

(Y2 - YI)

rI:' ElzE;z ds.

~C

(59)

Since the modes are characterized for z = 0 and r = a by
Eiz = 7'JiJI(ud cos¢,

(60)

where 7'Ji is the coefficient A of the ith mode, then
a7r
{31
*
(El, H 2) = - Zo {3r _ {3~ (Y2 - YI)7'J17'J2 J I(Ul)JI (U2).

(61)

Note that
(62)

If we assume
Y2 = YI + dy,

(63)
(64)

k.

* There are two cases (two polarizations) that must be considered, for each value of

846

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

from eq. (61) we obtain for the power carried by the mode EI
3

p = 1: (E H) = _ ~ f3la dy TJ2J2(U ).
1

2

b

2Z0 2UI dUI

1

1 1

(65)

1

The derivative dy/duI, which appears in this expression, is calculated
in Appendix C. In the following sections, we choose
(66)
in which case from eq. (65)
.. / 2Z o Ui

ITJd = 'V a 2 f3i a
7r

I dUi I
dy

1

Ji(ud·

(67)

These results are now applied to the problem of a junction between two
different waveguides.
V. JUNCTION BETWEEN TWO WAVEGUIDES OF DIFFERENT SURFACE
REACTANCE

Let two waveguides of different surface reactance, but the same diameter, be jointed at z = o. Assume a single mode incident on the plane
of the junction from the region z < 0 and let E t , H t denote the transverse
field components. To determine the amplitudes of the reflected and
transmitted modes, we expand E t and H t on either side of the junction
in an infinite series of modes, and then require continuity of E t and H t
at the junction. A simple solution for the amplitudes of the scattered
modes is then obtained assuming the difference in surface reactance is
small. This result will be extended in Section VI to the more general case
of two waveguides of slightly different diameter.
Let the transverse fields for z < 0 be represented by a superposition
of the modes of the waveguide occupying the region z < 0,
00

E t = Alel-j/hz + I: Rieiejlhz,

(z < 0)

(68)

(z < 0),

(69)

1
00

H t = Alhle-j/hz -

I: Rihiei/hz,
1

where

are the transverse field components of the incident mode, and

are those of the reflected modes.

CORRUGATED FEED PERFORMANCE

847

Similarly, for z > 0,

f: Tie~e-j{3iz, (z > 0)

Et =

(70)

I

Ht =

f Tih~e-j{3iz, (z > 0),

(71)

where Ti are the amplitudes of the transmitted modes.
We assume that ei, hi are normalized so that

Similarly,
(73)
Since (ei, hi) represents twice the power carried by the ith mode ei, this
power becomes imaginary if the mode is cutoff, in which case eq. (73) for
i = 1 should be replaced with
(ei, hi) = j.

However, in this article the calculation of R i , Ti is restricted to the modes
that are not cut off by the two waveguides.
From eq. (61) with 11i, 11k given by eq. (67),
~--------~------

( r. h') - ~

e,

n

-

{3i
(
') ~ /
UiU~
a2 {3[ _ {3~2 Y - Y V I(3i{3~(dy/dui)(dy' /du~) I·

(74)

and
(75)

where (ei, h~) and (e~, hi) are scalar products defined as in eq. (72)
and
(76)
a being the radius of the two waveguides. In eq. (74) 1/y and 1/y' are the

normalized surface reactances of the two waveguides.
N ow assume y' - y is very small and let
by = y' - y.

(77)

To determine Ri, T i, we require continuity of E t and H t for z = 0,

848

AIel +

f Rnen = Tle~ + f Tne~.

(78)

Alh l -

fI Rnhn = Tlh~ + f2 Tnh~.

(79)

I

2

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

Take the scalar product of the first equation with h~ and of the second
with e~. One obtains, taking into account eq. (73) and assuming that the
mode e; is not cutoff, so that (3; is real,
Al (eI, h;) + Rdei, h;) +

L
n~I,i

AI(e;, hI) - Ri(e~, hd -

L
n~I,i

Rn (en, h) = T i,

(80)

Rn(e;, h n ) = T i .

(81)

Now, assume for the moment that y, y' ~ 00. Furthermore assume none
of the modes under consideration is at cutoff. Then
(en, h~),

(e;, h n )

(i ~ n)

(82)

are small quantities of the same order of oy. Furthermore, as we show
below, this is true also for Rn. It follows that the two sums involving Rn
in eqs. (80) and (81) are of order higher than oy. Therefore, subtracting
these two equations and neglecting terms of order higher than oy,
Ri = -AI (eI, h~) - (e~, hI).
(ei, hi) + (e i,hd

(83)

Adding eqs. (80) and (81), and neglecting terms of order higher than
oy and solving for T i , we obtain
Ti = Al (eI, h~) + (e;, hI) .

(84)

2
Using eqs. (74) and (75), we rewrite eqs. (83) and (84) in the form
Ri = -AI (3i - (3~ yU I (3i
(31 + (3i
ui(31

YI

dy/dui
dy/dul

I'

1
1
(
') ~ lUlu;
1
Ti = Al a2 (31 - (3; y - y 'V (31(3; I(dy/duI) (dy' /du;) I .

(85)
(

)
86

The derivatives dy/dul and dy' /du; are derived in Appendix C.
It is interesting to note from eq. (85) that the reflection coefficient for
the mode i = 1 is simply

R1
(31 - (3~
PI = Al = - (31 + (3~ ,

(87)

which coincides with a formula derived by Brown I8 from a principle of
conservation of momentum. However, that derivation is not applicable
to the present problem, which involves hybrid modes. Measurements
of PI described in Ref. 17 show that this formula, although derived assuming y' ~ y, is quite accurate even for relatively large differences between y and y'.
CORRUGATED FEED PERFORMANCE

849

It is also interesting to note that the following interpretation can be
given to eq. (86). If E t for z = were known, we could determine Ti
simply using the formula

°

Ti = (E t , h~),

(z = 0),

(88)

which follows from eq. (70), in view of the orthogonality relations (73).
Now, if y - y' ;::;:; 0, E t does not differ much from AIel and, therefore, we
might be tempted to write in eq. (88) E t ;::;:; AIel, in which case we would
get
Ti;::;:; AI(et, h}

(89)

Alternatively, since from eq. (71) we also have
Ti = (e~, H t )

for z = 0,

(90)

we might be tempted to assume H t ;::;:; Alh l for z = 0, in which case
Ti ;::;:; Al (e~, hI)'

(91)

N either of the two formulas is correct* even if by ;::;:; 0. However, according
to eq. (84), a correct expression for small by is obtained by taking the
average of the two formulas. We now treat two special cases.
5.1 Limiting case ka

» 1

Assume that both y and y' are finite, but the radius a is very large,
ka » 1,
a condition which is often satisfied near the aperture of a feed. From eqs.
(31) and (33)
. du
1 u
hm - = - - - .
ka-oo dy
2 ka

(92)

Furthermore, for large ka,
1u2

{3a ;::;:; ka -

2. ka '

(93)

since ({3a)2 = (ka)2 - u 2. Therefore,
,
_
1 u? a{3i - a{3i '" - 2.
ka

U[ _1 (Ui)2
'" 2. ka

by,

(94)

since from eq. (92)
(95)

* They are often used, however.22
850

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

Using these results, from eqs. (85) and (86) we obtain
T. = -A
l

UIUi
(y' 2
2
k
Ui - Ul

1

y)

a

(96)

'

R. = A .! UiUl y' - Y
l
14 (ka)2 ka .

(97)

One can show that these formulas are valid even if oy is not small, provided both

L'
ka

and

y

-

ka

are small.
An application of eq. (96) is considered in Section 7.1.
5.2 Case 1/y = 0

At the input of the feed of Fig. 1, the corrugated waveguide is connected to a smooth waveguide (l/y = 0) of the same diameter. We now
wish to calculate the reflection and transmission coefficients of such a
junction. Thus, assume y ~ 00 for z < O. For y ~ 00, there are two types
of modes: TE modes, in which case 'Y ~ 0, and TM modes, in which case
'Y ~ 00. In the former case, from eq. (179) of Appendix C
. dy
hm - = _y2(u 2 - 1)

y_oo du

kau
(ka)2 - u 2 '

('Y ~ 0).

(98)

In the latter case, from eq. (180)
dy
U
~ - y2
du
ka

('Y ~ (0).

-

(99)

Now let the incident mode be a TEll mode. We distinguish two cases
depending on whether the i th mode is a TM mode or a TE mode. In the
former case, from eqs. (85) and (98)

·

11m
y_oo

.!. '
·V (Ul{3i{31
2
- 1) k

R· = -A {3i - {3; ... /
l

1

{31 + {3i
I

(100)

where Ul is the first root of J~ (Ul) = 0,
Ul = 1.8411.

(101)

If, on the other hand, the ith mode is also a TE-mode, from eqs. (85)
and (99),
(102)

CORRUGATED FEED PERFORMANCE

851

10
Pl

-0-0- P2
- - - - t2

m

20

ITI
----.t...
b
I

a

-T-l..

_~~b-'

30

~ = 1.83
a

40

50
0.5

1.4

wlw c

Fig. 2-Reflection, transmission, ~nd coupling coefficients for input junction of Fig. 1.

From eq. (86) we obtain, using eq. (98),

-

hm

T

i

= AI

y-oo

1
.. /
(hu~
, 'V
'2
a{h - a{3i
ka{3i(ul - 1)

1
'
dy~ \

V\

(103)

dUi

where dy' /du~ can be determined using eq. (178), unless y' » 1, in which
case we can use eq. (98) or (99) with y,u replaced by y',u~.
Equations (100) to (103) have been used to calculate the behavior of
a junction with b = 1.8309a. Consideration has been restricted to the
TEn mode and the TMn mode of the smooth waveguide, and the corresponding modes (HEn and HE~l) of the corrugated waveguide. The
results are shown in Fig. 2, where i = 2 refers to the TMll mode (or the
HE~l mode),

_\RI\2

PI2 -

(104)

Al

is the input reflection,

R2\2

P~ = \Al
852

(105)

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

gives the power converted into the TMl1 mode, and
(106)
gives the power converted into the HE~I mode. In Fig. 2, We is the frequency at which y' = 00. The corrugated guide at this frequency behaves
like a smooth guide and, hence,
(107)

PI = P2 = t 2 = O.

The curves of Fig. 2 are useful in determining the practical bandwidth
of the junction of Fig. 1.
VI. JUNCTION BETWEEN TWO WAVEGUIDES OF DIFFERENT DIAMETER

For some applications, to minimize the input reflection of a corrugated
feed, it may be convenient to choose for the smooth waveguide a diameter
different from that of the corrugated waveguide. In this section, the
analysis of Section V is extended to the general case of a junction between
two corrugated waveguides of different diameter. Let a and a' be the two
diameters for z > 0 and z < 0, respectively, and assume again a single
mode is incident on the junction, from the region z < O.
If E t for z = 0 were known, then the transmission coefficients Ti which
appear in eqs. (70) and (71) could be determined at once using the formula*
Ti =

f Ss,

(E t X h;) . i z dS,

for z = 0,

(108)

which follows directly from eq. (70) in view of the orthogonality of the
modes e~, h~ [see eq. (73)]. In eq. (108) S' denotes the circular area

0< r < a'.
Now, for z = 0, E t is given by eq. (68) inside the area

0< r < a,

(109)

and it vanishes for a < r < a'. Therefore, eqs. (108) and (68) give
00

Ti = AI(el, h;) + L Rn(e n, h;),

(110)

n=1

where
(en, h;) =

f Ss

(en X ht) . iz dS,

(111)

S being the circular area (109), which corresponds to the waveguide of
the region z < o.
* Here we are only interested in calculating Ti and R for {3i > 0, {3; > O.

CORRUGATED FEED PERFORMANCE

853

Equation (79), which was obtained by requiring continuity of H t for
z = 0, must be satisfied over the area S. By multiplying this equation
with en and integrating over S, we obtain for n ~ 1
00

-Rn = L Ti(e n, h)

(n ~ 1).

(112)

I

If the coefficients Ti in this relation are expressed in terms of the coefficients Ri using eq. (110), we get for n ~ 1
-Rn = (AI + R I )

f (eb h;)(e n, h;) + Rn f: (en, h;)2

i=1

i=1

00

+ L

Rs L (es , h;)(e n , h}

(113)

i=1

sr"n,1

For n = 1, the second sum of the right-hand side should be omitted and,
furthermore, -Rn should be replaced with Al - R I .
We have thus obtained a system of equations in the unknowns R I , R 2 ,
etc. We solve* them in the limiting case where both a' - a and y' - yare
very small, in which case
(en, h~) NO

for n ~ s

Ri NO

}

(en, h~) - I N O

(114)

'

and therefore the first two terms of the right-hand side of eq. (113) for
n ~ 1 are respectively equal to
Ar[(e n , h~) + (eb h~)]

and Rn. The last term can be neglected. Therefore, eq. (113) gives for
n~1

(n ~ 1).

(115)

RI N! [1 - (er, h~)2]AI N [1 - (er, h~)]AI'
2

(116)

Similarly, for n = 1,

The transmission coefficients can now be determined using eq. (110).
We find for n ~ 1
Tn N -1 [
(er, '
h n ) - (en, hI) AI.
2
I

]

(n ~ 1),

(117)

which is a generalization of eq. (84).
* This derivation is not rigorous, for we neglect to examine the question of convergence
of the summations in eq. (113). However, the validity of the results appears to be confirmed
by the experimental results.
854

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

The coefficients (ei, h~) can be calculated using eq. (57). Ify' - yand
a l - a are very small, we can proceed as follows. The field components
en and h n of the nth mode are considered to be functions of the coordinates r,c/> and of the two waveguide parameters a,Y. Therefore,
ohn
oa,
(118)
Oa
where hn, ohn/oy and ohn/oa are evaluated for a l = a, yl = y, and oy
are oa denote yl - Y and a l - a. a similar relation can be written for e~.
It follows from eq. (118) that (ei, h~) for i =P n is a sum of two terms,
,

_

I

I

h n - h n (r, C/>, a , y)

N

hn +

oh n

Oy

oy +

(119)
since (ei, h n ) = O. The first term is simply the coefficient (ei, h~) calculated for a' = a; it corresponds to a junction between two waveguides
of the same radius, but different surface reactance. The second term can
be interpreted as the coefficient (ei, h~) relative to a junction between
two waveguides having the same surface reactance but different radii
a and a'. Since the term has already been treated in Section V, only the
latter need be considered. If one sets
oh n
Oen
oh n =-oa
oe n =-oa
(120)
Oa
'
Oa
'
and if the c/> variations of both modes are of the type considered in Section
I, then, taking into account that ei4> = 0 for r = a, using eq. (57) we
get
oh~z + Oe ~p hiz ) rjJ=90
(ei, oh n ) -_ 7raoa [.!3n
-] ~ _ 2 ( ei4> !31
!3 n
Oa
Oa
r=a
0

+]. 2 !3i

2

!3i - !3n

( eiz oh ~p + Oe~z hi4>) rjJ=O°'
]
Oa
Oa
r=a

(121 )

As an application, consider y = 00, in which case (ei, h~) can be interpreted as the coefficient (ei, h~) relative to a junction between two
smooth waveguides of radii a and a l = a + oa, respectively. Assume ei
is a TE mode and en is a TM mode, so that for r = a
oh~z

ei4> = eiz = - - = 0,
Oa

(122)

where the last term vanishes because h n is a TM mode. Then eq. (121)
gives
(e u. 0hn) - 7raoa

2

1

2

!3i - !3n

[_.]!3n (~.)
h lZ rjJ=90°

Oa

r=a

(123)

CORRUGATED FEED PERFORMANCE

855

Similarly, interchanging n ~ i in eq. (121) and taking into account that
for r = a
(124)

we obtain
(en, Ohi) =

o.

(125)

Now, the two modes are characterized by
hiz =

;0 1(~Ui)
71i J

j
(3i a
hi1> = - -Z 71i

r
)
J 1 ( -U'
a I

ui

o

sinet>,

r
-ui
a

coset>,

(126)

(127)

and
enz = 71n J 1 (~Un) COSet>,

.

e n 1> = J71n

(3n a

r
)
J 1 ( -Un
a
r

Un

.
smet>,

(128)

(129)

-Un

a

where the amplitudes 71i and 71n are, because of the requirement (72),
given by
y
71i =

2Zo 1 U[
1
1
y-17ra ~ ~ vur=I IJ 1 (Ui)1
(ka) .

"'" /2Zo 1 Un
71n ='V 7ra ~ V{3;;

1

(130)

~ /-1-

IJ 1 (u n )1 V (ka)'

(131)

From eqs. (123) and (126) to (131), taking into account that J 1 (u n ) = 0,
we obtain the final result
(ei, oh n ) = 2

ka

1
oa
.~
.
va{3ia{3n v U[ - 1 a

(132)

Note that in deriving this relation it has been assumed that oa is sufficiently small so that

856

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

If this condition is not satisfied, we should replace {3n with {3~ in eq.

(132).
Of special interest is the case where ei and en represent the TEll mode
and the TMll mode, respectively. In this case Ui = 1.8411 and, letting
i = 1 and n = 2 for these two modes, we get

ka
oa
va{31a{32 a

(eI, oh 2) = 1.2937 ---.;::===~

(133)

From eqs. (115), (117), (125), and (133) we then obtain for the conversion
coefficients T 2 and R 2

ka
oa
X AI,
(134)
va{3Ia{32 a
where IArl2, IT212, and IR212 represent the incident power, and the
powers transmitted and reflected in the TMll mode. We can verify* that
T 2 is smaller by a factor of 2 than the conversion coefficient given in Ref.
22, which is due to the fact that the assumptions of Ref. 22 imply

T 2 ;;:;:; -R 2 ;;:;:; 0.646

Tn;;:;:; (er, h~),

(135)

rather than eq. (117).
Note that for {32 -- 0, we have a{3~ -- u2voa/a , and therefore

T 2 ;;:;:; 0.646

Va{31U2 (oa)a

3/4.

(136)

Note T2 remains finite even when the TMll mode approaches cutoff
in the first guide.
VII. MODE CONVERSION IN A NONUNIFORM WAVEGUIDE

Typically, a corrugated feed is made of one or more sections of nonuniform waveguide whose surface reactance and radius are functions
of z. Since a nonuniform waveguide does not in general possess a natural
mode of propagation, an incident mode will be scattered in forward and
backward modes. This is true even for a conical waveguide of constant
surface reactance (except when y = 0 or y = co). The analysis of Sections
V and VI gives the differential scattering parameters which allow the
local coupling into forward and backward modes to be determined at any
point in a uniform waveguide. We can thus obtain a set of differential
equations, whose coefficients are given by the above scattering parameters, and which can be solved, at least in principle, for the mode am* In Refs. 22 and 23, the TMll mode was cut off to the left of the junction, and for this
reason there is poor agreement between those measurements and eq. (134), which is not
applicable in this case. However, numerical calculations by Masterman and Clarricoats
agre well with eq. (134) at frequencies well above the cutofffrequency of the TMu mode,
as we may verify from Fig. 11 of Ref. 24.

CORRUGATED FEED PERFORMANCE

857

plitudes. We confine ourselves to a first-order treatment assuming the
total scattered power is much less than the incident power, since this is
the most interesting case if the feed is well designed. It is convenient to
assume for the moment that only y varies with z, in which case the
waveguide can be approximated by a succession of junctions of the type
considered in Section V. Let the HEll mode be incident at the input (z
= ZI). We wish to determine the resulting amplitude T 2 (z) of the HE~1
mode for z = Z2. If the variation ofy is sufficiently slow, we can neglect
reflections and determine T2 assuming the amplitude Al of the HEll
mode is nearly constant. The transverse field of the fundamental mode
is then
(137)

where
d1>1 _ (.J

dz - 1-'1·

(138)

The effect on the HE~l mode of a small variation oy at z = ~ is to produce
at z ;;;:; Z2 a component
(139)
where
d1>2 _ (.J

dz - 1-'2,

(140)

and from Eq. (86),
( ) =
t2~

~ / UIU2
1
'v-.
2
a(31 - a(32
a (31(32 I(dy/du l)(dy/du 2) I

-1

(141)

Note that both (31 and (32 are functions of z. From eq. (139), integrating
from ZI to Z2,

(142)

which assumes a very simple form when ka »1, as discussed in the following section.
7. 1 Conical horn with ka

» 1

Suppose the radius a varies linearly with z, as shown in Fig. 1 for z >
z 1, that the flare angle a is very small, and
858

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

ka(z) » 1

for Z1 < z < Z2. It was shown in Section III that for ka » 1 the properties
of a mode are entirely determined * by y /ka and, therefore, a mode will
propagate without variation of its transverse field distribution if y/ka
= constant. For this reason, it is reasonable to assume that mode conversion will be negligible if
b

(:a) = O.

(143)

Under this assumption, the effect on the amplitude of the HE~l mode
for z = Z2 of a small variation b(y/ka) occurring at z = ~ can be expressed
in the form
dT 2 =

T2(~)b

(:a) A e1

e

j <I>IW -j[<P2(z2)-<P2Wl,

(144)

where, since eqs. (144) and (139) must agree in the particular case a =
constant,
(145)
From eq. (96),
(146)
Therefore,
T 2(Z2) =

-A 1e- j <I>2(Z2)

~1U2 2
U2 -

Ul

r Z2 e j [<I>IW-<I>2Wld (L).

JZI

(147)

ka

Note that, since ka » 1,
a{3i ~ ka -

1 u~

2k~ .

(148)

Therefore, from eqs. (138) and (140)
d
k u~ - uf
dz [4>1(Z) - 4>2(Z)] = 2 (ka)2 .

(149)

Now, a varies linearly with z,
a = (z - zo) tana,

(150)

* Since now we are dealing with a conical waveguide, each mode is a spherical wave
centered at the apex A of the waveguide, and the field distribution over a spherical
wavefront is given to a first approximation (small a) byeqs. (35) and (37). To obtain the
field distribution over a plane z = constant, we must therefore introdce in eqs. (35) and
(37) a factor of exp (-j1/;), 1/; = k(x 2 + y2)/2R, R being the distance of the plane from the
apex.

CORRUGATED FEED PERFORMANCE

859

where Zo is the value of z at the apex of the waveguide. Therefore, from
eqs. (149) and (150)
1 u~ - uf y
-.
2 y tana ka

<I>dz) - 4>2(Z) = - -

(151)

Of particular interest is the case
y = constant.
(152)
Then eq. (147) can be readily integrated. Taking into account eq. (151),
we obtain
2 22t

ana
IT 2 12 = IA I 12 UIU2Y
(2 _
2)4
U2
UI
4

2

l
X 1 - exp [ ]. -1 u~ - uf y ( 1 - -a ) ] 12 ,
2 Y tan a kal
a2

1
where ai = a (Zi). Therefore,

IT 2 12 :s- IA 1 12

2 2 2 t
2
16 UIU2Y
ana

(2

U2 -

2)4
UI

'

(153)

(154)

where it is recalled that U2 = 5.1356 and Ul = 2.4048. For a = 4°, which
is the value chosen in the experiment of Ref. 17, this inequality gives for

y=l

:~::: ;2 0.663610-<, (-41.8 dB),

(155)

which is a very small value for most aplications. For a = 16°, on the other
hand, we obtain -29.8 dB, which may no be negligible.
Note that 1 T212 and IAll2 are, respectively, the powers carried by the
HE~1 mode and the HEll mode.
VIII. SUMMARY

In the feed of Fig. 1, when a TEll mode is incident at the input, some
of the incident power is in general reflected. Furthermore, some power
may be converted to unwanted modes if the corrugated waveguide
supports more than one mode at the input. Additional mode conversion
may take place inside the feed if the variation of the radius and of the
surface reactance is not gradual enough. As a result, a feed will have a
nonzero input reflection and, at some frequencies, unwanted modes may
illuminate the aperture of the feed. The consequences of these unwanted
modes on the radiation characteristics-e.g., enhanced cross polarization-are pointed out in Section III and in Ref. 17, where the theory is
compared with experiment.
These effects can be evaluated to good accuracy using the expressions
derived in this article, as highlighted below. For large ka, eq. (35) ex860

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

presses the field shape for all copolarized hybrid modes of various radial
harmonics with the same ¢ dependence, while eq. (37) corresponds to
the cross-polarized modes. Equation (36) gives the mode amplitudes
required to normalize the power carried by the modes.
A property of corrugated feeds is that the aperture field distribution
does not remain constant with frequency, as in the case of a feed with
smooth walls, but varies because of the frequency dependence of the
surface reactance Xs. Thus, although the desired mode has no crosspolarized component at the resonant frequency of the corrugations, at
other frequencies the desired mode does radiate some cross polarization.
The ratio, C2, between the maximum value of the cross-polarized power
in the radiation pattern and the maximum value of the copolarized power
(which occurs on axis) is given by eq. (39). From eq. (39), it follows that
with large ka and thin disks one can maintain low cross-polarized power
in the radiation pattern from the desired mode over an octave or
more.
At a junction between waveguides of the same diameter but of different surface reactance, eq. (85) gives the general expression for the
mode coupling coefficient to modes reflected from the transition, and
Eq. (86) gives that for modes transmitted forward from the transition.
Equations (97) and (96) are simplifications of eqs. (85) and (86), respectively, which apply for ka » 1. When the input waveguide is smooth,
the mode coupling coefficient is given by eq. (100) for reflected TM
modes, by eq. (102) for reflected TE modes, and by eq. (103) for hybrid
modes transmitted forward from the transition. Since the transition from
smooth to corrugated waveguide is a major source of unwanted modes
in a corrugated horn, eq. (103) is very useful in determining mode purity.
Another important formula is eq. (87), which determines the reflection
coefficient of the dominant mode (return loss) for any transition in Xs
and any ka.
Another source of generation of undesired modes is the mode conversion occurring along the conical taper of a corrugated horn. Equation
(154) gives the mode-coupling coefficient for the transmitted undesired
mode due to a conical taper.
In some cases a step in diameter may be used to match transitions
between different surface impedances; eq. (134) determines the modecoupling coefficients at a step in diameter.
APPENDIX A

Asymptotic Series for u and 'Y in Terms of lIka

We determine the asymptotic series for u and 'Y in terms of
1
ka'

CORRUGATED FEED PERFORMANCE

(156)
861

(167)

F(uo m ) = -1,

we obtain for the derivatives appearing in eq. (166),
= -UO m
( dF)
du U=UOm

(~:~)
d3F)
3

( du

U=UOm

(168)

= -3.

_ - 2 1 + U6m , etc.

U=UOm

UO m

Substituting eqs. (165), (167), and (168) in eq. (166) one obtains F as a
series of powers of l/ka; the coefficients of this series are algebraic expressions in UO m and aI, a2, etc. Similarly, by developing dF/du in a
Taylor series about the point U = UO m , and then using eqs. (165), (167),
and (168), we obtain dF/du as a series of powers of l/ka. Substituting
eq. (165) and the above series expansions of F and dF/du in eq. (164),
we can solve for the coefficients aI, a2, etc. We obtain eq. (31). Substituting eq. (31) in eq. (166) we obtain an expansion of F in powers of l/ka.
Using these results, from

F

l' = - - - = -

cos8,

F,

-=======

Vl- (:J

(169)

2

'

we get eq. (32).
Equations (31) and (32) have been obtained assuming eq. (165), which
corresponds to the limiting case of eq. (28). If, instead of assuming eq.
(165), we assume

u= u~ =

U2m

[1 + (k1a) + (k1a) + ... ],
{3l

{32

2

(170)

we obtain eqs. (33) and (34).
APPENDIX B

Surface Wave Mode

In addition to the modes considered in Section III, there is a mode for
which {3 > k. Thus, since for this mode cos fh > 1, it is convenient to replace 01 with Fh in eqs. (16) and (18). Since
cos jO = cosh fh

sin~Ol ~

= sinh Ol},
Jl(jx) =]Il(x),

(171)

J~Ux) = I~(x),

CORRUGATED FEED PERFORMANCE

863

where II (x) is the modified Bessel function of order 1, we obtain from
eqs. (16) and (17)
I~(u)

')'=---

u

I 1 (u) cosh (h '

(172)

!.1:.]

.:L = _ [.!J'I(U) + cosh 01
ka

ul 1 (u)

(173)

')'u'

U

where u = ka sinh 01 • We can verify from these two equations that u 00 as ka 00. Now, for large u,
eU
I~(u);;::; I 1 (u);;::; _;r.--.
(174)
V 27ru
Therefore eq. (172) gives
')' -

00,

as ka -

00.

From eq. (173), for large u and ka,

L;;::;_!.
ka
u

(175)

Therefore, since u = ka sinh 01 ,

. h 0 ;;::; - -1.
sm
1

(176)

y

We now examine the behavior of the field components for ka - 00.
Taking into account eq. (171), from eqs. (10) to (15), after replacing 01
with ifh, we find for,), = - 0 0 (i.e., for B = 0) that the only nonzero component of the magnetic field is H ¢ and

r)

1
, ( u - cos¢ e- JfJZ •
-ZoH¢ = -.--AI
I
smh 01
a
OR

Therefore, for kr » y

-ZoH¢;;::; A' V~ cos¢ exp[k(r - a) sinh 01 - j,Bz],
r

where A' is a constant. This shows that the field is confined to the near
vicinity of the wall, decaying approximately exponentially from the
wall.
We can show that ka > 1.81, the surface wave mode in combination
with the HElm and HE~m modes comprise the complete set of propagating modes whose Ez azimuthal dependence is cos¢.
APPENDIX C

Derivation of dy/du

From eq. (163),
864

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

APPENDIX D

Cutoff Frequencies of the Modes of Equations (35) and (37)

From eqs. (16) and (17) we get
')'2

+ ')'w - 1 = 0,

where
yu 2

w=

ka cosfh

Therefore, either
')'=

-w+Vw2-=i=4
2

(181)

or
')'=

-w-Vw2-=i=4

(182)

2

respectively, in the two cases of eqs. (35) and (37) (which correspond,
respectively, to ')' - 1 and')' - -1 for ka - 00 ). At cutoff {3 - 0; i.e., cos(h
- 0. For COSOl NO, Y :;6: 0, we have Iwl - 00 and, therefore, from eq.
(181)

')' -

°
{° ify>
if < °'
00

y

(183)
(184)

whereas from eq. (182)

°

if y > .
0, ify <
If ')' = 0, the mode is of the TE type. Now, for a TE mode at cutoff, the
only nonzero component of the magnetic field is Hz and therefore the
surface reactance Xs has no effect on the cutoff frequency. This means
that the cutoff frequency can be determined by replaci~g the corrugated
wall with a smooth wall of radius a, and therefore the cutoff frequency
is determined by the condition:
- ')'- {

00,

°

J~(ka) = 0.

If ')' = 00, on the other hand, the mode is of the TM type and the only
nonzero component of the electric field at cutoff is E z • It follows that if
the disks are very thin (t ""' 0), they can be removed without affecting
the field. Thus, the cutoff frequency can in this case be determined by
replacing the corrugated waveguide with a smooth waveguide of radius
b. It is thus determined by the condition
866

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

Copyright (c;) 1977 American Telephone and Telegraph Company
THE BELL SYSTEM TECHNICAL JOURNAL

Vol. 56, No.6, July-August 1977
Printed in U.S.A.

Characteristics of a Broadband Microwave
Corrugated Feed: A Comparison Between
Theory and Experiment
By C. DRAGONE
(Manuscript received December 14, 1976)

A corrugated feed with nearly ideal radiation characteristics from
17 GHz to 29 GHz has been built using a novel fabrication technique.
The bandwidth of single-mode operation was maximized by properly
choosing the input parameters of the feed. As a result, only the fundamental mode can propagate at the input from 19 GHz to 28.8 GHz.
In this frequency range, the far field is essentially polarized in one
direction. At frequencies higher than 28.8 GHz, there is a cross-polarized component caused by an unwanted mode. An approximate calculation of the power in this mode is given. A simple formula for the
input reflection coefficient is provided. Results are included that show
how to compute mode conversion in a conical taper, cross polarization
from a corrugated horn, including contributions from spurious modes,
and the reflection coefficient from the smooth-guide to corrugatedguide transition. Comparison of theory and experiment shows good
agreement.
I. INTRODUCTION

It has been shown by Thomas! that under certain conditions the field
over the aperture of a corrugated feed is virtually constant over a very
wide frequency range. For this behaviour to occur, the radius a of the
aperture must be much larger than a wavelength, i.e.,

ka » 1(k = 2;)

(1)

and furthermore, the aperture must be illuminated by a single mode. If
both conditions are satisfied, we can show that the field distribution is
essentially independent of the surface reactance of the corrugated horn
wall, X s , and, as a consequence, it is little affected by the variation of
Xs with frequency. This result is very important, for it implies that it
869

x

a = 4°

(z)

--- --z=o
ala) = 0.629 em
b(a) = 1.161 em
a(z2) = 3.144 em

b(Z2) - a(z2);;;:;:\/4 AT 22.9 GHz

z, = 45.7 em
z2 = 78.7 em

Fig. I-Dimensions of the corrugated horn.

is possible to design a feed so that its radiation pattern is circularly
symmetric and polarized in one direction over a very wide frequency
range I- 3 (an octave or more). The most difficult condition to satisfy to
obtain such large bandwidth is the requirement that a single mode be
excited in the feed. This requirement is discussed in a separate article. 4
Here, after summarizing certain results of that article, we describe the
results of an experiment. A long feed, with a flare angle of only 4 0 , was
fabricated using a special technique described in Appendix D. At the
input of the feed, which is shown in Fig. 1, the waveguide dimensions (the
corrugated depth I and the radius a; see Fig. 1) were optimized so as to
maximize the bandwidth of single mode propagation. As a consequence,
unwanted modes were cut off (at the input) over the frequency range
WI < W < 1.6839wI,

(2)

where WI = 19 GHz and 1.6839wl = 28.8 GHz. The input reflection and
the far field were then measured from 17 GHz to 35 GHz. The input reflection agrees very well with a simple formula given in eq. (42) and derived in Ref. 4. Over the frequency range (2), the far field is essentially
polarized in one direction. At frequencies higher than 28.8 GHz, however,
a strong cross-polarized component is caused by an unwanted HE~I
mode, which is excited primarily at the input of the feed. A simple expression for the power converted into this mode is given in Ref. 4.
870

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

II. PRELIMINARY CONSIDERATIONS

In the experiment described in Section V, the feed is excited at the
input by a linearly polarized TEn mode. Therefore, consideration will
be restricted to the modes arising for this particular excitation. 2- 8
Consider a disk-loaded waveguide centered around the z axis as in Fig.
1, and let r, ¢, z be cylindrical coordinates defined by x = rcos¢ and y
= r sin¢. Assume for the moment that the waveguide parameters h, a,
b, and t are independent of z. The separation h of the disks, which occupy the region a < r < b, is assumed to be much smaller than a wavelength A
kh « 1.

(3)

The region between two consecutive disks forms a radial line whose input
reactance jX at r = a is a function of the radial length l = b - a; for ka
» 1,
jX ~ jZo tankl,

(4)

where Zo = vi PO/EO. Because of condition (3), the effect of the disks can
be adequately accounted for by introducing an effective surface reactance 2

.

. ( t)

}Xs =}X 1 - h '

(5)

where t is the thickness of the disks, and by requiring that the field for
r < a satisfy the boundary conditions

E cp

~o

} forr=a,

(6)

Ez

Hcp~- .X
}

s

where Ecp, H</H Ez are the ¢ and z components of the electric and magnetic
field.
2. 1 The HE'm and HE' 1m modes

The properties of the hybrid modes 2- 8 of a corrugated waveguide are
determined by the radius a and the surface reactance Xs of the waveguide. In general, there is no simple relation 2 between the propagation
constant (3 of a mode and the two parameters a and Xs. If, however,
condition (1) is satisfied and furthermore

L« 1

ka

'

CORRUGATED FEED EXPERIMENT

(7)
871

where

Zo

y= - X '

(8)

s

then we can show2 ,4 that for all the modes except one, described in Appendix B of Ref. 4, the propagation constant is independent of Xs and
is simply given by
(3a;:;;:; v(ka)2 - u 2 ,

(9)

where u is a constant that is either the m th root of J o(u) = 0, or of J 2(U)
= o. Thus, either
(10)
or
(11)
The modes to which eq. (10) applies will be called HElm modes and those
corresponding to eq. (11), HE;m modes. The HElm modes approach
asymptotically, for large ka, the field distribution 2 ,4
E= J

o(~u) e-

j

{3z ix,

(12)

whereas the HE;m modes approach the distribution
E = J 2 (~u) (cos24>ix + sin24>iy )e-j {3z.

(13)

The absence of a z component in eqs. (12) and (13) is due to the fact that
this component vanishes like l/ka, as ka -- 00.
The HElm modes, which are given by eq. (12), have the important
property that the electric field is polarized in one direction. Of special
interest is the fundamental mode (m = 1) characterized by
u = Ul = 2.4048.

(14)

Note that both eqs. (12) and (13) are frequency independent.
Thus, over the frequency range in which both conditions (1) and (7)
are satisfied, the field distribution of an aperture illuminated by the HEll
mode is essentially frequency independent. 1 ,2 If, however, only condition
(1) and not condition (7) is satisfied, then we must add 4 to the right-hand
side of eq. (12) a component of the type (13), so that
E;:;;:; J o (~U) ix - I' - 1 J
a
1'+1

(cos24>ix + sin24>iy),
2(~U)
a

(15)

where the factor e - j{3z has been omitted. Both I' and U are functions of
ka and X s , and, therefore, they vary with frequency. If one develops 1',
872

THE BELL SYSTEM TECHNICAL JOURNAL, JULY -AUGUST 1977

U

in power series of y and l/ka one finds 4
U = Urn { 1 -

'Y =

~ y (k~) ... }

1- u;( ~ k~ - ~ u~ (4+ u~) (k1J .. -J.

(16)
(17)

2.2 Mode conversion in a conical horn 4

Consider a conical horn of constant surface reactance and flare angle
a, as shown in Fig. 1, from z = Zl to Z = Z2. If a is sufficiently small, and
if the input of the horn is excited in the HEll mode, then the resulting
field inside the horn is very nearly a spherical wave 1 ,3 originating from
the apex of the horn and given by

Ee- ihR ,

(18)

where E is given by eq. (15) and R is the distance from the apex of the
horn. Since a is small,
r2

R-;:;::;z+-

(19)

2z'

r being the radial distance from the axis and z the axial distance measured from the horn apex. From eq. 17, the parameter 'Y which appears
in eq. (15) is a function of both y and ka; assume Iyl ~ 00, O. Then, since
ka increases with z, we have from eq. (17) that 'Y varies with z, and
therefore the field distribution (15) does not remain constant with z. This
variation is accompanied by generation of unwanted modes, an effect
that will be negligible only if a is sufficiently small.
To determine how small a should be, consider the special case ka (z 1)
» 1 treated in Ref. 4. Let Pc be the total power converted from the desired mode into the HE~l mode and let Po be the power incident at the
input. Then, we find that for z = Z2
Pc = Po X 3.393 (10-3)y2 tan 2all - e N 12,

(20)

t/; = 10.295 - y - (1 _ a(zl))

(21)

where
ytanaka(zl)

a(z2) .

Therefore,
(22)
where the equality sign is attained for t/; = (2n + 1)7r. In the experiment,
a = 4 0 , in which case for y = 1 we find Pc ~ 6.636 X 10- 5 (-41.8 dB),
which is negligible for most practical purposes.
CORRUGATED FEED EXPERIMENT

873

III. RADIATION CHARACTERISTICS2,6,7

Let the aperture of the horn be of sufficiently large diameter that the
far field is simply proportional to the Fourier transform F of the aperture
field. Also let a be so small that we can neglect the phase variation caused
over the aperture plane by the variation of R in (18). Then, by taking the
Fourier transform of eq. (15), we obtain for the field distribution at a
great distance D from the aperture, for example, using the contour integral method of Ref. (4),
F = No(u,v)ix +

I'1'+1
- I N2 (u,v) (cos2</>ix + sin2</>iy),

(23)

v = ka (Z2) sinO,

(24)

where

and </>, 0 are spherical coordinates defined by x = D sinO cos</>, y = D sin

osin</>. Equation (23) gives, except for a factor independent of </>,0, the
far-field pattern. For small y/ka we have
U

= Ul = 2.4048.

In this case, from eqs. (23) to (25) the cross-polarization ratio C between
the maximum value of lEy I, which occurs for v = 3.67, and the maximum
value of lEx I, which occurs for v = 0, is given by
C2 == lEy I:ax = (0.26)2

lEx Imax

(I' - 1)2.
I' + 1

(26)

From eq. 17, the asymptotic value of C2 for large ka is
C2 -- 0.14

(:a)

2.

(27)

If ka » 1, but y/ka is not small, C2 has the behaviour of Fig. 2.
Equation (23) assumes that the aperture of the horn is illuminated
by the HEu mode. If, in addition to this mode, there is also some HE~l
mode of power Pc, then we have in addition to the component in (23),
a component4

"./,

-eJ'I"

'" /PcJ¥(u)
)(.
.. )
X 'V
2 _
N 2(U,V COS</>lx + sln</>ly ,
P OJ 1 (u)

(28)

where Pc, Po are the powers carried by the two modes and t/; is their difference in phase for z = Z2. For ka » 1,
U r;:; U ~ = 5.1356.
874

(29)

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

-10.-----------------------------------------------~~
Ul
-l
W

en

U

w

o

~

«
~

z

o
I«
N

a:

«
-30
-l

f2
I
Ul
Ul

o

a:

u

-40
0.30
y/ka

Fig. 2-Cross-polarization peak versus normalized surface susceptance.

Therefore, from eq. (25) with J 2 (u) = 0,

Ni (u,v) = (v2a~uu2J2(V)Jr(u))2

(30)

whose maximum value occurs for v;:;;;:; 4.356 and is
(31)
Because of the component (28), we have for 'Y = 1 that the ratio C2 between the maximum value of lEy 1 2 , which now occurs for v = 4.4 and
the maximum value of lEx I is
C2 = 0.194 Pc .

(32)

Po

Note from eq. (23) that the normalized radiation pattern of the HEll
mode for ka » 1 is simply given by
P(O) =

[u 2 uJo(v)v ]2
2 -

2

(33)

with u = u 1 = 2.4048 and v = ka sin O. If 01 and O2 denote the values of
8 for which P(O) = 0.5 and P(O) = 0.1, then from eq. (33)
ka sin 01 = 2.078 (3-dB point)

(34)

and
CORRUGATED FEED EXPERIMENT

875

(35)

ka sin ()2 = 3.597 (10-dB point),

res pecti vel y.
IV. FEED DESIGN 2 ,4,9

The feed of Fig. 1 is now described. Its measured characteristics are
discussed in Section V. Let \W2, wl1 be the frequency range over which
only the dominant mode, the HEll mode, propagates at the input (z ~
0). Let Wo be the frequency at which the surface reactance is infinite at
the aperture; then
(36)
At the input, the ratio between the radii b and a is optimized for maximimum W2/Wl' This is shown in Appendix A to require
b(O) = 7.0155 = 1.8309.
a(O) 3.8317

(37)

Then*
W2 = (W2)
= 1.6839,
WI
wI MAX

(38)

and the surface reactance can be shown to vanish for W = W2,
Y = 00 for z = 0, at W = W2.

(39)

In the experiment, the frequency WI was chosen equal to 19 GHz, which
gives W2 = 28.8 GHz, a(O) = 0.556 cm, and b(O) = 1.161 cm, as we obtain
from eqs. (37) and (38) and the condition
kb = 7.0155 at W = W2,

(40)

shown by Fig. 9 of Appendix A.
From Fig. 1 the feed consists of two parts joined at z = ZI. From z =
o to z = ZI, the outer radius is kept constant, so the cutoff frequency of
the HE~l mode remains constant, as shown in Appendix A, where the
relations between a, b and the cutoff frequencies of the various modes
of eqs. (12) and (13) are derived. Note that from z = 0 to z = ZI the radius
a increases, which implies that at any given frequency the surface reactance decreases with z. The importance of this requirement was first
realized by Bryants. From z = z 1 to the aperture, the surface reactance
remains constant with z. The frequency Wo at which it is infinite was
chosen in the experiment
* To put this ratio in perspective, the highest frequency of the 6-GHz (TH) common
carrier band is 1.732 times the lowest frequency of the 4-GHz (TD-2) band; similarly, for
the 18- and 30-GHz band3, the ratio is 1.695.
876

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

(41)

Wo = 1.21 WI.

Over the frequency range IwI, w21 only the HEll mode is excited at the
input, other modes being cut off. Therefore, since the variations of a (z)
and b (z) from the input to the aperture in Fig. 1 are sufficiently gradual
to ensure negligible generation of unwanted modes, we have that the
aperture is essentially illuminated by a single mode, the HEll mode, over
the above frequency range. The radiation characteristics for WI < W <
W2 are, therefore, given accurately by eq. (23) with a = a (Z2) and u given
by eq. (16) for m = 1.
From eq. (39), the input surface reactance vanishes at W2. Therefore,
since the corrugated waveguide is connected at the input to a smooth
waveguide of the same diameter, the input reflection is essentially zero
at W = W2. For W ~ W2, however, there is a reflection which, as we shall
see in Section V, is given accurately by
2
2 = ({31 - (3'l)
{31 + {3; ,

(42)

1P 1

where {31 is the propagation constant of the TEll mode and (3; is that of
the HEll mode. Equation (42) is derived in Ref. 4.
At frequencies higher than W2, some of the incident power is converted
into the HE;l mode. If Pc denotes the converted power and Po denotes
the incident power, then

Pc _ [
1
Po '" a ({31 - {3~)y

]2(3~(u{3I2 -

1) ,

(43)

where u = 1.841 and {3~ is the propagation constant of the HE;1 mode.
Some of the incident power is also converted into the TMll mode of the
smooth waveguide. If P~ denotes this power, which is reflected by the
junction, we have 4
P~ _ ({32 - {3~)2 {32{3I 1
(44)
Po '" ({31 + {3~)2u2 - 1 k 2 '
where (32 is the propagation constant of the TMll mode. The above
coupling equations are derived in Ref. 4 assuming Iyl » 1.
In the experiment described in the following section, the corrugated
waveguide is connected at the input to a smooth waveguide whose radius
gradually increases from a relatively small value a' to the final value a(O).
The initial value a' is sufficiently small so that the TMll mode is cut off
and, as a consequence, the power P~ is reflected back towards the junction where it is converted into the HE;1 mode of the corrugated waveguide. The total power converted into the HE;lmode is thus in general
different from Pc. It varies approximately between the two values

CORRUGATED FEED EXPERIMENT

877

o~------------------------------------------------~

(f)

...J
W

OJ

U

~ -20
~

(f)
(f)

o
z -30

...J

a:

:J
Iw

a:

-40

FREQUENCY IN GIGAHERTZ

Fig. 3-Return loss of corrugated horn of Fig. 1 vs frequency.

depending on the phase angle of the TMu mode incident on the junction.

v. EXPERIMENT
A corrugated feed of the dimensions shown in Fig. 2 was built using
a technique described in Appendix C. Its measured reflection coefficient
agrees* very well with eq. (42), as shown in Fig. 3. Its radiation characteristics are shown in Figs. 4 through 6. From 17.5 GHz to 32 to GHz, the
radiation patterns are in good agreement with eq. (33), as shown in Fig.
4, which compares eq. (33) with two measured patterns of lEx 12 in the
plane ¢ = 45°. Furthermore, from 17.5 GHz to 32 GHz, there is little
difference between the patterns of lEx 12 in the two principal planes (¢
= 0 and ¢ = 90°) and the pattern for ¢ = 45°. The difference is altogether
negligible at Wo = 23 GHz, which is the frequency for whichy = 0 at the
aperture. Figures 5 and 6 show a few examples of patterns measured in
the two principal planes. The variations with frequency of the beamwidths 2(h and 28 2 (respectively, the 3-dB and 10-dB beamwidths) agree
very well with eqs. (34) and (35), as shown in Fig. 7, where the measured
beamwidths in the plane ¢ = 45° are compared with the calculated
values.
Finally, the cross-polarized component Ey is very small over the frequency range of eq. (2), as shown in Fig. 8. For w > 28.8 GHz, however,
* Note that the measured reflection coefficient includes a small reflection due to a
transition from rectangular to circular waveguide, which was connected at the input of
the feed during the measurement.
878

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

o~--------------------------------------------------~

10

. - -CALCULATED [EO. (37)]
15

/'

20

25

30

---17.5GHz
35

40

45

5~~----------~--------~--~~~L---~~~----------~20
ka sin e

Fig. 4-Calculated and measured radiation patterns vs normalized angle at ¢ = 45°.

the normalized peak C2 of Ey increases rapidly with frequency as expected because of the HE~l mode, which is excited at the input for w >
28.8 G Hz. The solid curves and the dashed curves in Fig. 8 correspond
to eqs. (27) and (43), respectively. The agreement with the measurements
is satisfactory, taking into account that eq. (43) is not expected to give
exactly the total power converted into the HE~l mode, for several reasons.
In the first place, eq. (43) assumes a very large number of disks per
wavelength,

h« A,
CORRUGATED FEED EXPERIMENT

879

whereas in the experiment, h ~ 0.137A at 30 GHz; the effect of a finite
number of teeth is briefly discussed in Appendix B. In the second place,
eq. (43) assumes that only the TEu mode is incident at the input,
whereas in practice also the TMu mode is incident, for the reason
pointed out in Section IV. The total power carried by the TMl1 mode
is approximately given by eq. (44).
o~--------------------=-~--------------------~
- - ¢=oo

- - - 9=90°

-10

-20

-30

BEAMWIDTH (0) IN DEGREES

Fig. 5-Measured radiation patterns at 1> = 0° and 1> = 90° (principal planes).

Note, finally, that even if the total power converted into the HE~l mode
is calculated accurately, to determine the resulting cross-polarization
peak C2, the difference in phase between the HEu and HE~l modes must
be determined.
880

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

O~----------------------~-----------------------.

-10

-20

-30

-40

BEAMWIDTH (0) IN DEGREES

Fig. 6-Measured radiation pattern at ¢ = 0° and ¢ = 90° for 32 GHz (effect of HE~l mode
on pattern symmetry is evident at upper frequency limit of operation).

VI. CONCLUSIONS

Using a novel fabrication technique, described in Appendix C, which
can be applied at very high frequencies, a corrugated feed of small flare
angle was fabricated. Its input reflection, found to be given accurately
by the simple formula

11P 2=({3I-{3~)2
{3I + {3~ ,
remained less than -30 dB from about 24 to 32 GHz. Over the frequency
range WI < W < W2, the far field was essentially polarized in one direction;
CORRUGATED FEED EXPERIMENT

881

20~----------------------------------__----------~

rilw 15
a:

19
w

o

~
(f)

I

I-

o

~

~
<{

~ 10

5~

16

______

~

__
____________
__________
20
25
~

~

~

__

~

____

~

30

FREQUENCY IN GIGAHERTZ

Fig. 7-Calculated and measured 3-dB and lO-dB beamwidths.

its pattern is simply given by

At frequencies higher than 28.8 GHz, the far field contained a crosspolarized component, as predicted by the theory of Ref. 4.
It was shown that a maximum bandwidth, expressed as the ratio W2/Wl,
of about 1.68 can be achieved for the type of corrugations considered
here. By using the corrugations of Ref. 10, which, however, are difficult
to realize at high frequency, greater values of W2/Wl may be achieved.
Both the input reflection and the cross-polarization ratio can be improved by increasing the thickness t of the disks at the input. Curves of
1p 12 and C2 as a function of frequency for different values of t /h are given
in Fig. 2 9f Ref. 4. In the experiment, t /h was kept constant for reasons
of simplicity, since our main concern was to verify the results of Sections
II through IV and of Refs. 1 and 4, and to demonstrate the feasibility of
the fabrication technique described in Appendix D.
882

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

Ul
-.J
W

a:J

U -10
w
o
~
N

U

::,,: -20

:5
"-

CALCULATED
(HE'll )

z
o

f-

~ -30
a:
<!

-.J

o

"-

o

J, -40

Ul

o

a:

u

o

o

o 0

FREQUENCY IN GIGAHERTZ

W2

(28.8 GHz)

Fig. 8-Calculated and measured cross polarization; these are maximum values of cross
polarization (see Section III, eq. 26).

ACKNOWLEDGMENT

The author wishes to acknowledge the assistance of W. E. Legg who
carried out the measurements.
APPENDIX A

Cutoff Frequencies of the HE'm and HE~m Modes

To determine eqs. (37) to (40) it is necessary to establish the relation
between a, b and the cutoff frequencies of the various modes of eqs. (12)
and (13). Assume the thickness of the disks is very small. Consider first
the cutoff frequencies Wcm of the HElm modes, which are characterized
for large ka by the field distribution of eq. (12). In the vicinity of the
cutoff frequency wcm , we have, for anyone of the above modes,
cos Om = 0,

(46)

where k cos Om is the propagation constant in the z direction,
cos 8m =

V1 - (:~2 .

(47)

For W ~ wcm , we can show 2,4 that the mode degenerates into either a TE
mode or a TM mode according to the following rule:
TE mode, if y > O'}.
TM mode, if y < 0,
CORRUGATED FEED EXPERIMENT

(48)

883

BOTH HE" AND
HE'" PROPAGATE_ .........
.........

'" '"
7.0155
(TM'2

REGION WHERE ONLY
THE HE,,- MODE
CAN PROPAGATE

....... BOTH THE HEll - MODE

3.8317
(TM,,)

AND THE "SURFACE WAVE"

- - - - - --!'=-F'-'=:J;,,=::::;;"

MODE PROPAGATE

I

f
b

:t=1111111I111111I1

T-r-----I

1

I

1111111111111111

1

(TE" )
ka

Fig. 9-Region where only the HEll mode propagates. Note: As w varies for a given a, b,
a point ka,kb moves along a straight line through the origin.

In the former case, since the only nonzero component of the magnetic
field of a TE mode near cutoff is Hz, we have H 1> = 0, and therefore the
second of the two boundary conditions (6) can be ignored, which implies
that Wcm is independent of y. The corrugated waveguide can be replaced,
therefore, with a smooth waveguide of the same diameter whose cutoff
frequencies for the TE modes are given by the roots of J~(ka),
J~(ka) =
884

°for =
W

Wcm

(y > 0).

(49)

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

(b)

(a)

I

f
~

I

1
l
I

1
l

~

~

(e)

(d)

(e)

I

\

Fig.lO-Aluminum and brass disks assembled on mandrel. (a) Before machining. (b) After
machining exterior surface. (c) With electroformed wall of copper. (d) After machining
interior surface. (e) After etching away aluminum disks.

In the latter case, y < 0, the only component of the electric field for
is Ez and, therefore, the very thin disks can be removed without
affecting the field. The cutoff frequencies therefore coincide with those
of the TM modes in a smooth waveguide of radius b, and are given by
the roots of J1(kb),
(50)
J1(kb) = for W = w cm , (y < 0).
4
Analogous considerations are valid for the HEimmodes, except that
now, instead of the rules (48),

W N Wcm

°

TM mode, if y > O,}
TE mode, ify < 0, '

(51)

and therefore the cutoff frequencies w~m are given by the conditions
J~(ka) = 0 for W = w~m' ify <
J1(kb) =

°

°for = w~m' ify > 0.

(52)

W

CORRUGATED FEED EXPERIMENT

885

(e')

(b')

Fig. II-Modification of assembly to obtain strong attachment to teeth of electroformed
wall.

These results are illustrated by the diagram of Fig. 9 showing in the ka,kb
plane the region* where only the HEll-mode can propagate. We can see
that W2/Wl is maximum when b/a = 7.0155/3.8317 = 1.8309, in which case
W2/Wl = 1.6839, as pointed out in Section I before eq. 2.

APPENDIX B

Effect of a Finite Number of Teeth Per Wavelength

In Section IV, we assumed an infinite number of teeth per wavelength-i.e.,

h« >..,
a condition which seldom holds in practice. For instance, in the experiment

h > 0.13716>..
for W > 30 GHz. The effect of a finite number of teeth is not difficult to
evaluate approximately if
ka » 1,

In this case, in fact, the field in the vicinity of the corrugations can be
considered to be locally the field of a plane wave reflected by a corrugated
plane tangent to the actual corrugated surface. The angle of incidence
* In Fig. 9 we have not indicated a region for kb < 3.8317 where mode propagation can
occur, since that region is of no interest to us here. (The HEn mode is cut off for kb <
3.8317).
886

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

is 90° - (h, where
{31

cos 81 = k.
Thus, from Ref. 9, we find that a corrugated waveguide of radius a »
'A with a finite number of teeth is equivalent to one with h « 'A, but with
slightly larger inner radius a',
In2

a';:;:;a+-h,
7r

and the same outer radius b. Although this result is strictly valid only
if a » 'A, measurements l l have shown that it is valid approximately even
for ka;:;:; 4.
APPENDIX C
Fabrication Technique

Even if the aperture of a feed is illuminated by a single mode, the far
field contains a cross-polarized component with amplitude proportional
to [eqs. (4), (5), and (27)]

ILl
= I
1
I
ka
(1 - t/h) ka tankl .

(53)

To minimize this cross-polarized component, it is important that the
thickness t of the disks be much smaller than their separation h,

t« h.

(54)

Since h is always appreciably smaller than 'A/4, and since typically the
depth l of the grooves is not much different from 'A/4, condition (54)
implies

t « 'A/4

(55)

t « l.

(56)

Corrugated feeds are difficult to fabricate. When a corrugated feed
is electroformed, a mandrel of aluminum or other material is first prepared, and then the corrugated feed is electroformed around the mandril,
which is then removed with a solvent. However, at high frequencies, say
at frequencies higher than about 10 GHz, condition (55) demands that
t be very small (much less than 0.318 em). Then, taking into account (56),
the above technique cannot be used.
Figures 10 and 11 illustrate a technique that can be used at very high
frequencies, as high as 100 GHz, and which allows very small thicknesses
t to be realized. First a set of disks of aluminum and brass is assembled,
as shown in Fig. lOa, to form a single block whose outside surface is then
CORRUGATED FEED EXPERIMENT

887

Copyright IC) 1977 American Telephone and Telegraph Company
THE BELL SYSTEM TECHNICAL ,JOURNAL

Vol. 56, No.6, July-August 1977
Printed in U.S.A.

Acoustic Properties of Longitudinal
Displacement in Vocal Cord Vibration
By K. ISHIZAKA and J. L. FLANAGAN
(Manuscript received January 12, 1977)

We examine the acoustic significance of longitudinal displacement
in the self-oscillatory behavior of the vocal cords, and inquire into the
need for representing this detail in speech synthesis. We use computer
techniques and a previously derived model of the vocal cords to study
the contribution of longitudinal displacement to the total acoustic
volume velocity generated at the vocal cords. This volume velocity is
the effective sound source for production of voiced speech. From computational results, and from speech sounds synthesized by the programmed model, we find that the contribution of longitudinal displacement is not significant perceptually, and is not essential for
modeling the dominant acoustic properties of voiced speech.
I. VOCAL-CORD MODEL

In earlier work I ,2 we derived an analytical model for the self-oscillatory
motion of the human vocal cords. We consider the displacing tissue of
each cord to be approximated by two stiffness-coupled masses (see Fig.
1). For normal (nonpathological) conditions of phonation, the oscillator
is bilaterally symmetric, and the mechanical constants of the opposing
cords are identical. The left-hand mass pair (denoted mI, m;) constitutes
the bulk of the firm cord tissue, while the smaller right-hand mass pair
(m2, m~) represents the more flaccid mucous membrane covering of the
firmer tissue. Each mass has associated with it a restoring stiffness and
a resistive loss. All the stiffnesses and resistances are substantially
nonlinear,l and in the original work, these elements act to oppose lateral
motion (x-direction) only. The restriction to lateral motion still permits,
of course, phase differences in the motion of the coupled masses. Lateral
displacement of each mass pair determines the cross-sectional area of
opening at each position. If the length of the cords, or glottal opening,
is taken as f g, then the cross-sectional glottal areas are taken as rectangular shapes whose areas are Agi = 2fgXi, i = 1, 2, where the factor
889

I.---

VOCAL __
CORDS

-J

Fig. 1-Two-mass model of the vocal cords. Translational displacement is permitted in
lateral (x) and longitudinal (y) directions_

2 arises from the bilaterally symmetric cord configuration. These
cross-sectional areas determine the acoustic properties of the glottal
volume current UgS ' which enters the cord orifice (from the subglottal
system), and that which leaves it Uge (to pass into the larynx tube). The
latter volume velocity is the effective sound source for all voiced speech
sounds. The air pressure just to the left of (beneath) the vocal cords is
the subglottal pressure PSg, and the pressure just to the right of (above)
the cords, at the entrance to the vocal tract, is Pt. The differential
pressure (Psg - P t ) is the potential that creates the glottal volume currents.
The resulting volume currents depend upon serial acoustic impedances
dictated by Agl and Ag2 and, hence, upon the cord motion, which, in turn,
is conditioned by the intraglottal pressure distribution in the orifice and
by the transglottal pressure (Psg - P t ). These serial acoustic impedances
also are nonlinear (and flow dependent), and represent the mass (inertance) of air contained within the glottal orifice and the associated
resistive flow losses. 1
Additionally, there is another potential influence upon the glottal flow,
namely, the volume of air displaced by the vibrating mass pairs. In
general, this volume displacement can have components associated with
lateral and longitudinal motion. In the original work, components of
890

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

r

oz

TRACHEA

GLOTTIS

VOCAL

TRACT

G)

=i

o
Z
»
r
o

Rc

---'U g

- U g2

- Ugl

c

RV1/2

Lgl/2

Lgl/2

Rvl/2

R12

Rv2/2

Lg2/2

Lg2/2

Rv2/2

Re
-Ugi

- U gs

en
"'U
r

»

()

m
$:

m
Z

-i

Z

Psg

UY2 (

+

o<
»
r

()

()

a
:::0
o
<
05

1

Agl

i

Ag2

:::0

»
-i

(5
Z

co
CD

""""

Fig. 2-Equivalent acoustic circuit including lateral and longitudinal displacement-volume
velocities.

t

Pt

Table I -

Values of impedance components of Fig. 2
R

c

= 137 ~,I U~l1l
•

P (

R 12 ='2

2~'

Serial
Impedances

Longitudinal
Components

Lateral
Components

1
1 )
AT-AT
IUg2
g2
gl

1

g2 ) I I
AR e =P- -2- - ( 1 Ug
2 Ag2Al
Al
Ru1 = 12~£lddA2h
Lg1 = pdd gl,

Ru2 = 12~£ld2/A22
Lg2 = pd 2/ g2

dy
Uy1 = 2t'g(d 1 + d 2) dt'

dy
Uy2 = 2t'g(d 1 + d 2) dt

d
Ux1 = d 1 dt (Agt>,

d
Ux2 = d 2 dt (A g2)

= 2t'gd 1 dx/dt,

= 2t'gd 2 d x2 (dt
Cg2 = Ag2d 2/pC 2

Cg1 = Ag1ddpc 2
G - S 1] -1
1 1 pc2

V

AWo

2c p p'

Sl = 2(£g + 2X1)d1

G -S
2 -

2

1]-1~
2c p

---;;G2

p

S2 = 2(fg + 2X2)d 2

p = 1.14 X 1O-3gm/ cm3, air density

Constants
(for vocal
system,
moist air
at body
temperature) *

~

= 1.86 X 1O- 4dyne-sec(cm 2 , kinematic-coefficient of viscosity.

c = 3.5 X 104 cm/sec, sound velocity
1] = 1.4, adiabatic constant
A = 0.055 X 1O-3cal/cm-sec-deg, coefficient of heat conduction
WO = 27r(1000), mid audio range radian frequency

cp = 0.24 cal/gm-degree, specific heat at constant pressure.

* From J. L. Flanagan, Speech Analysis, Synthesis and Perception, second edition,
New York: Springer Verlag, 1972.

glottal current corresponding to rate of volume displacement (both
lateral and longitudinal) were neglected.
II. ACOUSTIC CIRCUIT

Recognizing that the cord dimensions are very small compared to
sound wavelengths at the frequencies of interest, and that all mechanical
velocities are small compared to the sound velocity, we derive a onedimensional equivalent circuit for the acoustic quantities involved. Its
complete form is shown in Fig. 2. The values of all impedance elements
are given in Table 1.
The serial elements (top branch in Fig. 2) are identical to those of our
original work!, and relate to time-variation of the acoustic impedance
of the glottal opening.
892

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

-Ugl

- U gs

Fig.3-Simplified equivalent acoustic circuit, including longitudinal displacement
currents.

All shunt elements relate to rate of displacement of air volume by the
moving cord masses. The time variation of all shunt quantities is also
determined by the motion of the cord masses.
Lateral motion of the cord masses (normal to the direction of glottal
flow) displaces air volume at the rate of
Uxi = 2fg d i -dXi cm 3/'
s, l = 1, 2,
dt
where Xi is the lateral displacement and d i is the depth (thickness) of
the cord element (mass). Again, the factor 2 arises from the two bilaterally opposing cords. The acoustic compliances, C 1 and C2, represent
the compressibility of the small air volumes contained between the opposing cords and the conductances G 1 and G 2 represent the heat-conduction loss at the tissue surfaces of the cords.
Longitudinal motion of the cord masses is assumed to occur cophasically and to be translational only. In this regard, consider the y- motion
of the locked masses to be opposed by a nonlinear spring and loss similar
to that of hI and rl. The effective surface area exposed to the transglottal
pressure difference is taken to be the product of cord length and total
cord thickness, fg(d 1 + d 2 ). No cavity compliances or losses are associated with the longitudinal motion, and the longitudinal contribution to
the total volume velocity is
Uyi = 2fg(d 1 + d 2 )

~~ , i = 1,2.

In other words, Uy1 and Uy2 are equal and oppositely poled.
Notice that in the earlier formulation,I,2 the absence of the shunt elements imposes the constraint U ge = U gs = U g. The presence of the
shunt elements (all time-varying with displacements that are determined
by the equations of motion for the mechanical system which, in turn, is
forced by the intraglottal and transglottal pressures to close the feedback
loop of the oscillator) makes the input flow Ugs and the output Uge
typically different.
LONGITUDINAL DISPLACEMENT IN VOCAL CORD VIBRATION

893

15
/a/

X,

10

0
-5

~

15

N
I

X2

a

~

10

IZ

w

:2:
w

u

0

c..

-5

«-.J
~

0

15
Y
10

/ ..... Ps = 8 em H2O

,. ,._---1-----,/

,/
,/

0
-5
0.3
Ag

1«

0.2
0.1

w

a:

«

0
-0.1
0

10

20

30

40

50

60

70

80

TIME IN MILLISECONDS

Fig. 4-Computed mechanical behavior of the vocal-cord/vocal-tract model. The vowel
configuration is /a/.

A recent related study3 examined the influence of the Uxi upon Ugf .
The present study considers separately the effects of the Uyi . For this
purpose, the circuit of Fig. 3 is a simplification of Fig. 2.
III. PURPOSE OF PRESENT STUDY

In the original work,1,2 we made estimates of the volume displacement
currents, based upon long-wave assumptions and one-dimensional sound
propagation, together with what we believed to be reasonable physiological estimates of cord velocities (compared with volume velocities
responsive to transglottal pressure). We concluded that displacement
currents are of second order, and in the original work we chose to neglect
them in favor of elucidating dominant principles. The original formulation, therefore, treated only lateral displacement as it affects the serial
glottal impedances. As a matter of completeness, we more recently have
894

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

CORD
MIDLINE

OSCILLATION
BUILDUP

I

/34

35

/a/

33

CORD
MIDLINE

I
I
31

30
8

,

m2

,

/

I

,~Jd2
I
dl

c;:J

I,

I

,

ml

\

~
I

6

y=Omm--~--~~~----L-----L-----~----4-----~____~

o

Fig. 5-X- Y trajectories for initiation of oscillation. Trajectories are for pellet positions
shown in insert.

returned to a quantitative examination of these assumptions. A first
study,:3 now completed, considered the importance of the shunt branches
that represent the lateral components of volume velocity generated by
the displacing masses-that is, from the volume current sources Ux1 and
Ux2 . The results of the study support the original assumptions, and show
the lateral components to be second order by comparison to the currents
actuated by the pressure difference acting across the glottal opening.
The present study examines the contributions of the longitudinal
displacement to the total glottal volume velocity (specifically, the contributions of U y1 and Uy2 ) and the importance of longitudinal displacement to the self-oscillatory dynamics of the cord model and to
sound perception.
We take the longitudinal restoring stiffness ky typically to be the same
as the lateral restoring stiffness kI, namely 80 kdynes/cm. The longituLONGITUDINAL DISPLACEMENT IN VOCAL CORD VIBRATION

895

CORD
MIDLINE

OSCILLATION
STEADY STATE

I

I

I a/

(Fa = 125 Hzl

I

o

/ TIME IN ms
/

/

y = 3.8 mm----

3

5~

6

CORD
MIDLINE

I

m2

I I
I
It::::±::::r...1.d2

IQ
I \ 1
I m,

d,

y=Omm

Fig. 6-X- Y trajectories for steady-state oscillation of the cord model.

dinalloss (or damping ratio) is also taken to be similar to the lateral one,
namely (y = (1 = 0.2 These values are based upon clinical observations. 9
We examine these choices subsequently. Further, since the longitudinal
and lateral motions of the cord masses are considered to be translational
only, no rotational behavior is included. Still further, while the lateral
translations of the coupled masses ml and m2 can have large (and
physiologically natural) phase differences, their longitudinal translations
are considered to be cophasic, and the internal coupling stiffness is assumed to act only for lateral motion. The lateral and longitudinal motions are, therefore, coupled only through the acoustic variables that
determine the oscillator forcing functions. In the course of our discussion,
we will indicate comparisons to actual physiological data to assess the
realism of these assumptions.
896

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

10

/ a/

PI

0
-5
-10
20
0",
I

15

§
w

a:

10

:::J
(J)

w

a:
~

0
20
Psg

15

./ Ps (8 em H2 O)
/

/ __ t __

10

"
0 ""
>-

'=

u

""

1500

0

1000

>'"

500

Ug

....Jw on

w E
22
:::J
....J

0

0

>

-500
0

10

20

30

40

50

60

70

80

TIME IN MILLISECONDS

Fig. 7-Computed acoustic qualities for the vocal-cord/vocal-tract model. The vowel
is /a/.

IV. RESULTS OF COMPUTER SIMULATIONS

The vocal-cord model, as represented by Fig. 3, was combined with
a transmission-line formulation of the vocal tract that we have used
previously in speech synthesis studies. 4 The programmed vocal tract
contains 20 sections which, in addition to the classical acoustic elements,
represents the yielding soft walls of the tract and sound radiation from
the yielding walls. This formulation is based upon measurements of
tissue impedances that we reported earlier.5 Also included for the present
study is a transmission-line representation of the subglottal system. Six
sections of line represent the trachea, bronchi and lungs, as previously
described. 6 We implemented the entire system in terms of difference
equations programmed on a laboratory computer by techniques we have
described in detail previously.l,2
LONGITUDINAL DISPLACEMENT IN VOCAL CORD VIBRATION

897

/::.:::'~-'PARTICLE 2

APPROXIMATE
MIDLINE

f~

"'PARTICLE 1

\

"'- - - PARTICLE 3

4

zo

6 ~ - - - - TIME IN 1.25 ms INCREMENTS

(JJ

:::o

FO=100HZ
)
( Ps = 8 CM H 2 0

E
E

::.
>

o

~
6

1

2

4

3

5

(AFTER BAER)

x (1 mm/DIVISION)

Fig. 8-X- Y trajectories observed from excised larynx of dog (after Baer).

Most of the data reported here are for the vocal tract configured in
the shape for the neutral (schwa) vowel I a/. Some data are also included
for the vowels Iii and la/.
A first step is to ascertain if the cord oscillator, so arranged for longitudinal motion, performs realistically when compared with observations
on the human larynx. A second step, then, is to determine the acoustic
significance of the volume displacement current arising from longitudinal
motion.
Throughout these calculations, the laryngeal parameters are set to
the "standard" values used earlier for phonation by a man's voice in the
chest registerl (i.e., neutral glottal area Ago = 0.05 cm 2 , cord tension
parameter Q = 0.78, d 1 = 0.25/Q cm, d 2 = 0.05/Q cm). Recall that the
Q parameter scales the values of mass and stiffness and, hence, also the
values of the d i . Phonation is initiated by raising the lung pressure P s
smoothly from zero to the standard value of 8 cm H 2 0. The pressure is
elevated in a lO-ms interval.
4.1 Mechanical behavior

As the lung pressure is elevated, the model commences a buildup of
oscillation. After four or five transient swings, the oscillation settles into
898

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

TIME IN MILLISECONDS

Fig. 9-Subglottal pressure variation measured on a human subject (after Sawashima).

a steady state behavior with a fundamental frequency (pitch) determined
by the model parameters (the tension parameter Q has the dominant
effect on pitch 1 ). The initial 80 ms of this synthetic phonation is illustrated in Fig. 4 for the mechanical variables.
The top two curves show the displacements x 1 and x 2 of mass pair m 1
and mass pair m2, respectively. The first collision of each mass pair is
indicated by the first flat, negative-going portion of the displacement
waveforms. For the Af,'() = 0.05 cm 2 value, this occurs for Xi = -0.0178
cm. Note, too, that x 1 leads X2 in phase by the order of 60°, which is
25
ky = 40 kdyne/ em

20

/a/

[x~y TRAJECTORIES]

I

15

10

0

1 20

N

I

a

ky= 80kdyne/em

~

15

I-

z

w

10

~

w
u

«

.J
c..

~

0

0
20
ky = 120 kdynel em

15
10

0
0

10

20

30

40

50

60

70

80

90

TIME IN MILLISECONDS

Fig. lO-Effect of longitudinal restoring stiffness ky upon the longitudinal displacement,
y. Data show oscillation buildup for a lung pressure" Ps that is raised smoothly from zero

to 8cm H 2 0.

LONGITUDINAL DISPLACEMENT IN VOCAL CORD VIBRATION

899

10
ky = 40kdyne/cm

Pt

/a/

0

-5
-10
10

0

ky = 80kdyne/cm

N

I

]
w

0

cr:
::J
(f)
(f)

w

-5

cr:

c..

-10
10
ky = 120 kdyne/c m

0
-5
-10
0

10

20

30

40

50

60

70

80

90

TIME IN MILLISECONDS

Fig. ll-Effect of ky upon Pt.

consistent with observations from high-speed motion pictures of the
human vocal cords. The third trace shows the longitudinal displacement,
y, which bulges upward as P s is raised. The y motion is roughly sinusoidal. The lower trace shows the net area of glottal opening Ag (namely
the rp.inimum of AgI and A g2 ). The y-displacement is seen to lead in
phase the Ag wave, again consistent with the upward, rolling motion seen
in high-speed photography of the real cords.
An x-y plot of the buildup transient portrays the behavior perhaps
more graphically. Figure 5 shows the Xl vs y and the X2 vs y values with
time as the parameter. Imagine pellets fixed to the lower and upper inner
edges of one simulated cord, shown by the inserted anterior-posterior
view of Fig. 5. The trajectories of the two pellets are plotted for the oscillation buildup. The y-axis is broken and re-originated at y = (d l +
d 2 ) = [(2.5 + O.5mm)/Q] = 3.8 mm. The flat portion of the tracks, along
the vertical midline, reflect collision with the opposing vocal-cord
mass.
After several initial swings, the oscillator settles into a steady-state
behavior. One cycle of this trajectory is shown in Fig. 6. The steady-state
pitch frequency in this case is Fa = 125 Hz, or a period of T = 8 ms.
4.2 Acoustical behavior

The corresponding acoustical parameters, calculated for the same
buildup period, are shown in Fig. 7. The acoustic pressure at the input
900

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

o

4

02::1

Omm ____L--Lt~~-----J----~-----L----~I____~I__--.. x
-~ O.2mm ~-

Fig. 12(a)-Steady-state X- Y trajectory for ky = 40 kdyne/cm.

to the vocal tract P t is shown in the top trace. It reflects strongly the
eigenfrequency structure of the tract, in this case configured for /a/ and
having formant frequencies of approximately 500 Hz, 1500 Hz, 2500 Hz
... The transglottal pressure (Psg - Pd, which is the forcing function
for the y- motion and the pressure potential for the volume flow through
the glottal opening, exhibits a pronounced pitch-synchronous variation.
Its peak values, in fact, approach twice the lung pressure value of P s =
8 cm H 2 0. Recall that Ps is the lung pressure input to the simulated
LONGITUDINAL DISPLACEMENT IN VOCAL CORD VIBRATION

901

6
5
4

o

~
I

0.2 mm

i
Omm----L-~~~

__

~

____

~

____

~

____

~----~----+

Fig. 12(b}-Steady-state X- Y trajectory for ky = 80 kdyne/cm.

subglottal system, representing trachea and bronchi. But notice that
the mechanical y-displacement (Fig. 4) does not respond with this detail.
(Neither do the x 1 and X2 displacements respond to high-frequency detail
in their forcing functions-that is, the mechanical system, being masscontrolled, filters out this detail.)
The sub glottal pressure Psg (the pressure just beneath the vocal cords)
also exhibits a pitch-synchronous fluctuation, but of somewhat less
amplitude, namely about ±20 to ±30 percent of the mean subglottal
pressure. Its positive peaks correspond to the closing epochs of the glottal
port. The calculated volume velocity passing the glottal opening Ug
(bottom trace) appears as a traditionally shaped, pulsive waveform. This
wave is similar to that calculated in previous work (without longitudinal
motion) but differs in that its values are modified by the effects that U y1
and Uy2 couple into the pressure variables. That is, Uy1 and Uy2 can
influence Psg and P t and, hence, Ugl . The latter three variables, in turn,
close the oscillator feedback loop by constituting the forcing functions
902

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

y

5
4

o

I

0.2 mm

i
Omm----~--~+_----~----~----~----+_----+_----x

Fig. 12(c}-Steady-state X- Y trajectory for ky = 120 kdyne/cm.

for the lateral displacement. As was the case in the mechanical variables,
the VI! flow does not reflect a temporal fine structure comparable, say,
to the (PsI! - P t ) waveform. The resistive and inertive components of
the glottal impedance (i.e., the serial components in Fig. 3) act effectively
as a low-pass filter. It is not unusual, however, to see pronounced temporal structure that corresponds to the lowest eigenfrequency of the vocal
tract, especially for low, back vowels (such as /a/), or for tightly articulated sounds.
A next question, then, is how do these mechanical and acoustical
quantities, resulting from the model with longitudinal displacement,
compare with physiological data.
4.3 Comparisons to physiological observations

One qualitative comparison can be made for the mechanical displacement behavior. Baer 7 performed studies on the excised larynx of
LONGITUDINAL DISPLACEMENT IN VOCAL CORD VIBRATION

903

CORD
MIDLINE

y

OSCILLATION
BUILDUP

I
I

22

/a/
(NO SUBGLOTTAL SYSTEM)

2~

32~-~-.l-

31
~-+-_~ 36
9

29

37

8

23
22

30

y=Omm----~~~--~----~----~----~----~----~x

Fig. 13-0scillation buildup without subglottal system.

a dog in which he fixed pellets to the displacing tissue and made optical
observations under stroboscopic illumination. While his pellet positions
do not correspond exactly to our mass-pair corners, we can roughly
compare his observations with our data. Figure 8 shows x-y trajectories
for one set of conditions for the dog larynx that approximates values used
in human phonation (namely P s = 8 cm H 2 0, Ug = 275 cm 2 /s, and Fo
= 100 Hz). Particles (pellets) 2 and 3 are of interest. While the vibratory
excursions of the excised dog larynx are larger than those we calculate
with the model, the qualitative motions are gratifyingly similar. One
question that arises is how much does the longitudinal (vertical) displacement depend upon the choice of longitudinal stiffness constant.
We shall examine this question in more detail subsequently.
Another comparison can be made in the acoustic domain-namely,
to the subglottal pressure variation Psg shown previously in Fig. 7. Sawashima 8 has measured the subglottal pressure during phonation in a
904

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

CORD
MIDLINE

OSCILLATION
STEADY STATE

I

/a/
(Fo= 123 Hz)

1
5

(NO SUBGLOTTAL SYSTEM)

4

~

__

~O

/TIME IN ms
./

3.8mm-----

I

6;?

4

3

I~f---+----~
12

I
I

rr

0.2,mm
I

y=Omm--~--~~----~----~----~----~I----~I-----..x
-,../

0.2 mm

h-

Fig. 14-Steady-state oscillation without subglottal system.

human subject. One of his results is shown in Fig. 9. The qualitative
correspondence to the model calculation appears relatively good, and
the acoustic interaction among the simulated cords, vocal tract, and
subglottal system is realistic.
4.4 Effect of longitudinal stiffness constant

In view of uncertainties in the measurement of the stiffness constants
in physiological preparations, it is important to examine how critical the
value of ky (the longitudinal restoring stiffness) is to the oscillatory behavior of the model.
For the bulk of our studies, we have taken ky equal to our typical
"standard" value of kI, namely 80 kdynes/cm.I We have also used the
standard value for the damping ratio, y =
= 0.2. This choice is based
upon the physiological measurements on cord tissue conducted by Ka-

r rl

LONGITUDINAL DISPLACEMENT IN VOCAL CORD VIBRATION

905

20r---~-----------------------------------------------.

/a/

Y - DISPLACEMENTS

15

x - Y TRAJECTORIES
I

10

o~=-~~

__

~

____

~

____

~

____

~

____

~

____

~

__

~~

__

~~

g 20r----------------------------------------------------.

<"

I

o

15

~

10

!a/

ill

:2"
ill

u

5

<l:

-.J

"-

~

o

20.----------------------------------------------------.
15

/ i/

10

O~=_~

o

____

10

~

____

20

~

____

30

~

____

40

~

____

50

~

____

60

~

__

70

_L~

_ _~~

80

90

TIME IN MILLISECONDS

Fig. 15-Effect of vowel configuration upon longitudinal displacement.

neko,9 who found the stiffness constants for lateral and for longitudinal
displacement to be similar. Both stiffnesses are also taken to have the
same nonlinearity, namely, a cubic nonlinearity in the restoring force
(see Ref. 1).
To assess the model's sensitivity to large variations in ky, we calculated
the buildup of synthetic phonation for ky = 40,80, and 120 kdynes/cm.
The resulting y-displacements for these values is shown in Fig. 10, and
the sound pressure at the entrance to the vocal tract is shown in Fig. II.
Also, the x-y trajectories are shown in Fig. 12a, b, and c.
As would be expected, the greatest influence in this variation is reflected in the y- displacements. The "softer" ky gives larger dc displacement and smaller peak-to-peak vibratory excursions. The P t data
indicate that the variations in acoustic behavior and glottal excitation
are very small. The fundamental pitch is sensibly the same for all cases,
namely 125 Hz. This factor is almost completely dominated by the lateral
motion. In auditory assessment of the output synthetic sound, the differences are virtually imperceptible, suggesting that the longitudinal
displacement current is insignificant for speech synthesis.
4.5 Effect of subglottal system

A side issue, of some interest in passing, is the effect of acoustic interaction between the subglottal system and the cord oscillator. If the
906

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

/a/

4

o

/ TIME IN ms
/
/
/

7J?

:rI
0.2 mm

I
Omm--~--~~~----~----~----~----~----~---.

Fig. 16(a)-X - Y trajectory for the vowel fa/.

trachea-bronchi-lung system is removed and the lung pressure applied
directly to the cord input (as a zero-impedance source; i.e., Psg becomes
a pressure "battery" equal to P,.J, then the temporal structure previously
reflected in Psg is eliminated, the transglottal pressure excursions simply
equal P t , and the longitudinal component of cord displacement is lessened. This is illustrated by the x-y trajectories for oscillation buildup
and steady state shown in Figs. 13 and 14. For this case, ky is reset to the
typical value 80 kdynes/cm. Note the slight lowering of the fundamental
frequency to 123 Hz.
4.6 Effect of vowel configuration

It also is instructive to consider the influence of vowel configuration
upon the cord model, as presently formulated. Such studies were made
in detail in the original work. 1 We therefore compare synthesized results
for the vowel la, a and i/o (In this case the longitudinal stiffness constant
LONGITUDINAL DISPLACEMENT IN VOCAL CORD VIBRATION . 907

/ a/

4

3

6

;r-I

0.2 mm

I
Omm ____L--L~~--~----~----~----~I-----LI--__..x

-~ 0.2mm ~-

Fig. 16(b)-X- Y trajectory for the vowel /a/.

was set to ky = 88 kdynesfcm through an inadvertent keypunch error.
Because the differences are so small, it did not seem worthwhile to recompute the data for ky = 80 kdynesfcm.)
The y-displacements are shown in Fig. 15, and the x-y trajectories for
one period of steady-state oscillation are shown in Fig. 16a, b~ and c. The
longitudinal displacement is not greatly affected by vowel configuration,
but the constricted articulations fa, if clearly lead to slightly greater
longitudinal peak-to-peak excursions than does the open-pipe (neutral)
vowel faf. This typically is owing to the greater acoustic interaction at
the eigenfrequencies for the configurations with higher acoustic impedance levels, which in turn leads to greater transglottal pressure differences. This is well reflected in the acoustic variables resulting from
this calculation. The corresponding acoustic quantities are shown in Figs.
17 thru 20. In these data, note especially how the tract eigenfrequencies
are manifest, including in the synthetic output sound pressure from the
908

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

iiI

6

3

8

Omm----L---~~----~--~----~----_7----_T----.x

Fig. 16(c)-X- Y trajectory for the vowel Iii.

mouth, Pout. Note, too, that the resulting steady-state pitch frequencies
are about 125 Hz for lal and lal and about 120 Hz for Iii.

v. SIGNIFICANCE OF LONGITUDINAL DISPLACEMENT
Having established that the cord oscillator (with lateral and longitudinal degrees of freedom) appears to behave realistically, we consider
the next questions:
(i) Is the longitudinal motion significant or necessary for proper
self-oscillatory operation of the model?
(ii) Is the acoustic volume velocity contributed by the longitudinal
motion physically or perceptually significant?
So far as the purposes of speech synthesis are concerned, we answer both
of these questions in the negative.
What we wish to do, therefore, is compare the mechanical and
acoustical behavior with and without longitudinal displacement. Because
the longitudinal effects are coupled only through Uy1 and Uy2 , the latLONGITUDINAL DISPLACEMENT IN VOCAL CORD VIBRATION

909

15
/a/
10

§
N

I

~
~
f-

z

w
~

0
-5
15

w

u

<l:

10

..J
Q..

~

0

0
-5
0.3

1 0.2
~
<l:
w
c:
<l:

0.1
0
-0.1
80

0
TIME IN MILLISECONDS

Fig. 21-Computed mechanical behavior of the cord/tract model without y-displacement.

eral-motion-only condition (that is, the original formulation of the
model) is conveniently realized by setting Uy1 = U y2 = O. This condition,
with no longitudinal displacement flow, yields the mechanical results
shown in Fig. 21. These results are essentially identical (at least to three
significant figures) to the corresponding quantities in Fig. 4. In particular,
note that the crucial A JJ waveforms are virtually identical.
We can now examine pertinent acoustic quantities. Calculation of the
conditions with y- displacement yields the result shown in Fig. 22. The
figure shows one cycle of the steady-state oscillation. Recall that Uge is
the total volume velocity at the larynx tube entry to the vocal tract. U g
is the flow component through the actual opening of the glottis. Note
that Uge is non-zero during the time the cords are actually closed, corresponding to an upward (vertical, longitudinal) displacement of air
volume that adds positively to UJJ • Similarly, later in the cycle, downward
longitudinal displacement subtracts from Ug . The difference between
these volume velocities is

by virtue of the assumption of cophasic longitudinal motion. This difference is plotted on an XIO enlarged scale in Fig. 22. The peak value
912

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

800
Ugl

600

/ a/
Fa = 125 Hz

400
200

a

g 800
~

>- 600
~

u
0

-l
W

400

> 200
w
::::;

:)

-l

a

0

>

40
20
0
-20
-40
0

4

8

10

TIME IN MILLISECONDS

Fig. 22-Glottal volume velocities calculated with y-displacement (refer to Fig. 3).

of the difference for this condition is on the order of 10 cm3 /s, or about
l/fiO of the peak value of the total Uge . This result is not just peculiar to
this range of volume velocity, but rather it scales comparably at louder
and softer phonation. For example, if the lung pressure Ps is doubled
say to 16 em H 2 0, the longitudinal displacement current increases because the transglottal pressure and the longitudinal displacement increase. But the Ug flow also increases and remains far and away the
dominant quantity.
The amplitude spectra of these quantities provide convenient correlation with auditory percepts. The spectra for Uge and Ug are shown in
Figs. 23a and b. A close comparison shows the differences to he less than
2 dB, an amount that is not significant perceptually. The more relevant
comparison is obtained when the effect of y- motion is eliminated (by
removing U y1 and U y2 ). The corresponding glottal waveform for no ydisplacement is illustrated in Fig. 24. It is denoted Ug ;. Also reproduced
is the Uge with y~displacement. Further, the difference between the
longitudinal displacement and lateral-only conditions (Uge - Ug ;) is
shown on an XI0 enlarged scale. During the glottis-closed time, this
difference is identical to the (Uge - Ug ) difference of Fig. 22, because
U g = O. During the glottis-open time, the (Uge - Ug ;) difference differs
from the (Uge - Ug ) difference. In other words, Ug differs from Ug ;
LONGITUDINAL DISPLACEMENT IN VOCAL CORD VIBRATION

913

o~--------------------------------------------~

Ugl

/a/

-10

-20

(f)

uj -30
CD

u

w

o

z

~

o

-40

~

~
...J

~ -50
-60

-70

o

8
TIME IN MILLISECONDS

-80L-____L -_ _ _ _L -_ _ _ _L -_ _ _ _L -_ _ _ _L -_ _ _ _

o

~

____

~

__

~

3

4

FREQUENCY IN KILOHERTZ

Fig. 23(a}-Amplitude spectrum of U/: e. which includes the effects of y-displacement.

essentially by the influence that Uy1 and Uy2 have upon the transglottal
pressure difference (Psg - P t ).
Again, the more perceptually relevant comparison is to the amplitude
spectrum. The spectrum of Ug ; is given in Fig. 25. A close comparison
to the Uge spectrum of Fig. 23 shows the differences to be less than about
2 dB. Auditory assessment of the output synthetic vowels shows them
to be indistinguishable even in close comparison.
VI. CONCLUSION

In view of these results, we conclude that realistic acoustic behavior
(which is needed in speech synthesis) can be obtained in the cord model
without the additional complexity of longitudinal displacement. Longitudinal displacement is not necessary for realistic self-oscillation of
the model. The important vertical phase differences in the two-mass
motion are adequately duplicated by lateral displacement only, as is the
significant acoustic interaction between vocal tract and vocal cords.
Further, the rate of volume displacement owing to longitudinal motion
914

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

-80~--------------------------------------------~

lal
-70

-60

Vl

uj -50
CD

~
o
z
o

ill

-40

::J
f-

::i

~ -30

~ 1:::~

-20

~

~ 200

OL-______

-10

!l
~

________

~

o

8
TIME IN MILLESECONDS

OL-____L -_ _ _ _

~

____

~

____

~

____

~

____

o

~

____

~

__

~

3

4

FREQUENCY IN KILOHERTZ

Fig. 23(b)-Amplitude spectrum of Ug calculated with y-displacement.

is clearly perceptually not significant and need not be represented with
added detail.
These conclusions about the mechanical and acoustic behavior have
a corollary in a comp~nion study on the rate of displacement of air volume owing to lateral motion only. 3 This contribution was examined by
making use of the shunt branches in Fig. 2 that include U x1 , U x2 . Calculations and computer simulations showed that the contribution to
glottal volume velocity of the air extruded from the glottal port by lateral
tissue displacement is barely discriminable in a differential auditory
comparison. In fact, the perceptual effect for the lateral volume displacement is just slightly larger than for the longitudinal displacement.
Both are quite second-order in importance.
We have found in the present study that proper acoustic and oscillatory behavior of the model does not depend significantly upon longitudinal displacement. The longitudinal motion is relatively insensitive to
acoustic loading and to changes in longitudinal stiffness. The longitudinal motion influences fundamental frequency only slightly. What,
LONGITUDINAL DISPLACEMENT IN VOCAL CORD VIBRATION

915

700

/a/
600
400
200

"'5

0
40

~

>-

f0-

20

G

0

...J
W

0

w

-20

>

::?:

'WITHOUT V-DISPLACEMENT

::J

...J

0

>

-40
600
Ug £

400

WITH V -DISPLACEMENT

200
0
-200
0

2

4

6

8

TIME IN MI LLiSECONDS

Fig. 24-Comparison of waveforms for Uf{f' which includes y-displacement current,
and UI:;' which is calculated without y-displacement.

then, are the critical and sensitive parameters of the cord model? In other
words, what parameters are most influential upon the perceptual attributes of Uge , since the end product-the output sound-depends
directly upon Uge ? The results of our earlier work can be combined with
the insights obtained here to consider this question.
The original study showed that the intra-glottal pressure distribution,
and the fluid flow laws used to deduce it, are quite important to proper
oscillatory behavior, to proper generation of the Uge flow, and to realistic
acoustic interaction between the vocal tract and vocal cords. To a large
extent this pressure distribution determines how the pitch frequency
varies with subglottal pressure and with articulatory configuration. The
mass-stiffness product (i.e., the natural frequency of the mechanical
system) is quite dominant in determining pitch range. Subglottal pressure, assuming it to be above an initiation threshold of several cm H 2 0,
is primarily correlated with sound intensity, a relatively noncritical factor
for voiced sounds. Mechanical parameters such as cord thickness,
damping ratio, and nonlinearity are relatively noncritical except as they
influence duty factor and "flow chopping" at collision (which yields a
broad-spectrum Uge function). None of the mechanical variables, lateral
or longitudinal, reflects the temporal fine structure of the acoustic
variables, but both must and do reflect the open-close cycles of the vi916

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

O~----------------------------------------------,

fa /

-10

-20

'WITHOUT DISPLACEMENT CURRENTS

en

Ld -30

CD

U
w

o
z
-

w

-40

o

:)

f-

::::i

~ -50

~ ~:::~

-60

~ ~20t

-70

nl

~.

°0~------~--------~8
TIME IN MILLISECONDS

-80~

____~____~____~____~____~____~____~__~

o

3

4

FREQUENCY IN KILOHERTZ

Fig. 25-Amplitude spectrum of Ug ;, the glottal volume velocity without y-displacement.

brating cords. The realistic phase differences in motion of the upper and
lower edges of the cords (m2 and ml in the model) allow phonation
smoothly over a wide range of input impedances to the vocal tract (both
inductive and capacitive), and this behavior can be obtained satisfactorily by permitting lateral displacement only of the stiffness-coupled
masses. The computational complexity of anything more detailed does
not seem necessary from the standpoint of duplicating realistic acoustic
behavior, which is the objective in speech synthesis.
On the other hand, if the objective were a detailed study of tissue deformation (as might be the case in simulations for clinical diagnosis or
for representing pathological conditions) then the computational complexity of longitudinal displacement might be considered. In such a case,
the vocal-cord model should be treated as a more distributed system.
For the representation and synthesis of normal speech, however, these
details do not appear perceptually significant and are not needed to
represent the dominant properties of vocal-cord vibration.
LONGITUDINAL DISPLACEMENT IN VOCAL CORD VIBRATION

917

Copyright © 1977 American Telephone and Telegraph Company
THE BELL SYSTEM TECHNICAL JOURNAL

Vol. 56, No.6, July-August 1977
Printed in U.S.A.

Faulty-Trunk Detection Algorithms Using
EADAS/ICUR Traffic Data
By J. s. KAUFMAN
(Manuscript received November 1, 1976)

A class of algorithms for detecting abnormally short-holding-time
trunks has been developed that utilizes individual trunk data available
in EADAS/ICUR (Engineering and Administrative Data Acquisition
System/Individual Circuit Usage Recorder). This data consists of a
two-dimensional statistic that compresses the raw trunk measurements-the state of the trunk (busy or idle) sampled every 100 or 200
seconds-into a manageable form. Because this data is essentially a
sufficient statistic for the stochastic process used to model the (unobservable) trunk state measurements, one of the algorithms developed
is Wald's sequential probability ratio test. Two of the algorithms developed have been implemented in ICAN (Individual Circuit Analysis
Program), and are currently being used to test trunks associated with
the No.1 crossbar, No.5 crossbar, crossbar tandem (lXB, 5XB, XBT),
and step-by-step switching machines. The focus in this paper, however,
is on the modeling and analysis aspects of the problem, and only slight
attention is paid to the various trade-offs and real-world constraints
encountered in implementing the algorithms.
I. INTRODUCTION

A message trunk, the basic connecting link in the switched telephone
network, provides the communication path between switching machines
as well as certain call setup capabilities, such as supervision, signaling,
and ringing. For an important class of trunk faults that cause call failure,
the trunk is released by the switching system upon customer abandonment and is again available to fail another call. As a result, a single undetected faulty trunk of this type can fail a significant fraction of the
offered attempts to a group and will characteristically have an abnormally short holding time.
Because of their potential service impact, significant efforts have been
made to understand and quantify the impact that such abnormally
919

short-holding-time trunks have on central office and network service. 1- 4
It is now widely understood as a result of these studies that this generic
trunk fault results in a fraction of service attempts "killed," which is out
of all proportion to their number in the trunk population. Consequently,
traffic data available from new and existing traffic data-acquisition
systems has been viewed in the light of increasing interest in trunk-fault
detection. In particular, with the advent of the Bell System EADAS/ICUR
(Engineering and Administrative Data Acquistion System/Individual
Circuit Usage Recorder) system,5 it was natural to ask whether the new
individual trunk data available could be used to detect such "killer"
trunks.*
This paper discusses the theoretical aspects of a class of killer-trunk
detection algorithms that utilize the individual trunk traffic data
available in EADAS/ICUR. These algorithms were designed for, and
practical versions of them are presently implemented in, the ICANt
portion of the EADAS/ICUR system. We focus here, however, on the
problem formulation, modeling, and analysis aspects of the algorithms
without bringing in many of the diverse factors and trade-offs encountered in the actual implementation.
Because the holding time of a trunk affects the statistical properties
of the trunk data in EADAS/ICUR, it is natural to formulate the killertrunk detection problem as a problem in the testing of statistical hypotheses. In this context our modeling effort is basically an attempt to
precisely define the state of a trunk (normal or killer) and expose the
relevant underlying distributions. Well-known aspects of the theory of
hypothesis testing are then applicable and immediately suggest a number
of different tests. Sequential tests are naturally considered since the
EADAS/ICUR data evolve sequentially in time. Questions about the robustness of the models, and the structure and performance of statistical
tests, are addressed using standard analytic tools.
The material in this paper has been organized into six major sections,
whose content we briefly describe. After considering the data available
in EADAS/ICUR (Section II), we proceed to model a trunk (Section 3.1),
motivate an appropriate set of statistical hypotheses suitable to our
problem (Section 3.2), and briefly review several classical tests for deciding between statistical hypotheses (Section 3.3). With these preliminaries out of the way, we develop individual trunk algorithms based
solely on individual trunk data. Proceeding in a heuristic manner, we
use the individual trunk data to "derive" an ad hoc killer-trunk-detection
* The term killer trunk has been widely adopted in referring to a faulty switchingmachine-accessible trunk in a group whose average holding time is substantially smaller
than the average group holding time.
t Individual circuit analysis program-a software program that analyzes much of the
EADAS/ICUR traffic data and maintains the system data base.
920

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

algorithm (Section 4.1). Although the insight gained in proceeding in
a heuristic fashion is significant, we shift our emphasis in Section 4.2 and
rigorously derive an optimal test statistic. It is interesting to find that
the ad hoc statistic is essentially one of two symmetric statistics which
comprises the optimal test statistic. The relationship between these
individual trunk statistics is further explored in Section 4.3.
In Section V we factor grouping information (which essentially
identifies all trunks common to a trunk group) into the picture, and
develop detection algorithms tailored to trunks associated with the
No. 5 crossbar switching machine. This development necessitates
modeling the 5XB trunk-group selection procedure, and several results
due to Forys and Messerli 2 are utilized. In Section VI we shift our discussion to the performance of the 5XB group algorithms, deriving approximate expressions for the mean statistic update and mean detection
time in Sections 6.1 and 6.2, respectively. The paper concludes in Section
6.3 with an approximate analysis of the false-alarm probability of the
5XB group algorithms.
II. EADASIICUR DATA

The structure of a killer-trunk detection algorithm is largely dependent on the type of individual trunk measurements available. * In
EADAS/ICUR, the raw (unobservable) data consists of the state of each
trunk (busy or idle) every 100 or 200 seconds. Fortunately, the data accumulations available essentially summarize all the relevent information
in the raw data.
2. 1 Switch-count and transition data

The data available from the EADAS/ICUR system, which can be used
to distinguish between normal and killer trunks, is obtained by sampling
individual trunks at 100 or 200 second intervals. This data consists of
periodic accumulations (typically hourly, two-hourly, or three-hourly)
of both the Busy states, and the State transitions. The busy state accumulation is usually referred to as the switch count. For the 200-second
sampling option with a one-hour accumulation period, the switch-count
is an integer between 0 and 18. A state transition occurs whenever the
state of a trunk (busy or idle) is different at two successive scans. For the
200-second sampling option with a one-hour accumulation period, the
number of state transitions is an integer between 0 and 17.
If we denote the ith scan during an accumulation period in which m
scans occur by Xi, and let 0 and 1 correspond to trunk idle and trunk
* Until very recently, almost all trunk-traffic measurements were obtained on a group
rather than on an individual trunk basis.
TRUNK-DETECTION ALGORITHMS

921

TRUNK- 1
STATE

0

0

o }S SWITCH COUNTS
3 TRANSITIONS

In
0

0

In I

400

800

CALLS ON NORMAL
TRUNK

1200

1600

SECONDS

0

0

0

0

0

0

o } 25WITCH COUNTS
3 TRANSITIONS

SAMPLING-e
EPOCHS

n ~~ n 0 0
I

0

400

CALLS ON KILLER
TRUNK

I

800
SECONDS

1200

1600

Fig. I-leUR data.

busy, respectively, the available data may be written
m

(i) n (m) = L Xi (switch count)
i=l
m

(ii) t(m) = L IXi - xi-d (state transitions).
i=2

Thus, the raw (unavailable) data in the form of a binary sequence,

is compressed into the two statistics n(m) and t(m).
Because the holding time of a killer trunk is generally on the order
of a few tens of seconds, it should have substantially more state transitions than a normal trunk, for a given switch count. Figure 1 illustrates
the sampling process on both a normal and killer trunk. * For the purposes of this figure individual calls are represented by rectangles, call
durations correspond to the width of the rectangles, and a half-hour
accumulation period with the 200-second sampling option is used.
We note in passing that for the 200-second sampling option, very little
information is lost by "compressing" the raw data Xm = (Xl, . ·,x m ) into
the two statistics n(m) and t(m). Thus, normal conversation lengths tend
to be in the vicinity of 3 to 4 minutes and, hence, with the 200-second
sampling option, we expect that only adjacent samples are significantly
* The realizations shown in Fig. 1 are more or less typical for a 5XB trunk group with a
mean group holding time of approximately 4 minutes operating at about 40-percent occupancy, and having a killer trunk with a mean holding time of approximately 1 minute.
922

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

correlated. If Ixd~l is Markovian, then [n(m),t(m)] is almost a sufficient
statistic ([n(m),t(m),xl,x m ] is sufficient) for Xm (see Section 4.2). Note
also that for a trunk in the killer state, succesive samples should be essentially independent (for both sampling options).
2.2 Grouping information

In addition to the switch-count and transition data available from the
EADAS/ICUR system, we are also able to utilize a system map to identify

(i) all trunks common to a trunk group, and (ii) the trunk-selection
procedure* associated with the trunk group. It turns out that using this
grouping information,t in addition to the switch-count and transition
data, enhances the detection potential considerably.
Thus, we divide the class of algorithms into two types according to
whether or not grouping information is utilized. The first type, which
uses only switch-count and transition data is referred to as an individual
trunk algorithm. These individual trunk algorithms are applicable to
all trunks-including two-way trunks-independent of the type of
switching machine they are associated with. They do however assume
knowledge of the trunks nominal holding time. The second type of algorithm uses the grouping information in addition to the switch-count
and transition data and is referred to as a group algorithm. Group algorithms are "tailored" to a specific kind of trunk-selection procedure
and, hence, apply to trunk groups associated with specific switching
machines. For the purposes of this paper, the trunk-selection procedure
considered is random selection of idle trunks. This procedure models
the selection procedure of trunk groups associated with the 5XB
switching machine. Group algorithms generally apply only to one-way
trunk groups.
III. PRELIMINARY CONSIDERATIONS

In attempting to quantify the intuitive notion that a killer trunk will
exhibit more transitions than a normal trunk (see Fig. 1), for a given
switch count, it is natural to consider the transition probabilities:
P1,o = PIXt+T = O/Xt = 11
and

PO,l = PIXt+T = l/xt = 01,
* The map in EADAS/ICUR indicates the type of switching machine that the trunk group
is associated with, and this allows us to model the trunk-selection procedures (see
Ref. 2).
t We will consistently use "grouping information" to refer to both the identification
of all trunks common to the group and the trunk-selection procedure associated with the
group.
TRUNK-DETECTION ALGORITHMS

923

where Xt denotes the state of a trunk at epoch t and r denotes the sampling interval. Of course, to evaluate these transition probabilities, we
must be concrete about how we model a trunk.
Before tying ourselves down to any specific model, however, it is useful
to view these conditional probabilities in a canonical form. Thus, suppose
we begin by assuming only that the binary valued process Xt is stationary.
We have then the following simple result:
Lemma 1: Let Xt be a binary valued, stationary random process and let
p and R(·) denote its mean and covariance function, respectively.
Then,
(l) P1,o(p,r) = (1 - p) 1 - R(O)

.

{

R(r)}

(1)

(ii) pP1,o(p,r) = (1 - P)Po,l(p,r).

(2)

Proof: Part (i) is a consequence of the definition of R(·). That is,
R(r) = E(XtXt+T) - p2
= P(Xt = 1,xt+T = 1) - p2,

where
p = E(xt) = P(Xt = 1).

Part (ii) follows from the two identities:
p = pP1,1(p,r) + (1 - p)Po,dp,r)

and
1 = P1,dp,r) + P1,o(p,r).

A consequence of this result is that uncorrelatedness and independence
are equivalent:
Corollary 1: For the process in lemma 1, Xt,Xt+T are independent if and
only if R(r) = o.

Note that the dependence of R (.) on p has been suppressed for convenIence.
3. 1 Modeling an individual trunk

A particularly simple way to model a trunk is as the server in a single
server loss* system with a Poisson arrival process and an exponential
service time distribution. This model is commonly denoted by
M/M/1-10ss. 6 Let x t denote the state of the server:
* In a loss system, customers who are blocked depart without waiting.
924

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

_ {I
Xt 0

if server is busy at epoch t
if server is idle at epoch t

and R{o) the covariance function of Xt. It is easily shown 7 that for the
M/M/1-10ss system,
R{r) = R{O) exp \-(A + ~hj,

where Aand ~ are the mean arrival and service rates, respectively. Thus,
P1,o(p,r) may be written as:
P1,o{p,r) = (I - p) {I - exp (1-~:) },

(3)

where the trunk occupancy p is equal to A/{A + ~). Throughout this paper
we will be concerned with r = 100 or 200 seconds and a nominal holding
time 1/~ in the vicinity of 3 minutes. The mean holding time of a killer
trunk 1/~* will always be expressed as l/r~ with r typically in the range
5 to 15. Thus, if we denote ~r by S, we may write the transition probability P\xt+r = O/Xt = 1j for a trunk with mean occupancy pas
P1,o(p,r) = (1 - p) [ 1- exp

C-~SJ

l

(4)

where r = 1 corresponds to a normal trunk. (Since P1,o = 1 - p implies
that XnX2n° ° ° are independent, we will assume independence for r
sufficiently large in subsequent sections.)
Figure 2 is a plot of P1,o vs p corresponding to S = 10/9 (200-second
sampling and a 3-minute mean holding time) for several values of r. P1,o
is essentially equal to 1 - p for r ~ 5. Figure 3 is a similar plot of P1,o vs
p corresponding to 100-second sampling and a 3-minute mean holding
time (8 = 5/9)~ In this figure P1,o is essentially equal to 1 - p for r ~

7.5.
Before putting too much emphasis on the transition probabilities in
Figs. 2 and 3, it is prudent to consider the effect of factoring more realistic
assumptions into the single server loss model. Thus, while the Poisson
arrival process assumption is probably a reasonable assumption for a
trunk in a 5XB trunk group (random selection of idle servers), it poorly
models the overflow nature of the traffic offered to trunks in a 1XB/XBT
trunk group. t In the latter case, it is more appropriate to model the input
stream to a trunk as a peaked process. 6 Figure 4 is a plot of P1,o vs p
parameterized by the peakedness (z) of the input stream. This figure
is based on an expression for P1,o derived for a GI/M/1-10ss modelt with
t The trunk-selection procedure for lXB and XBT trunk groups is essentially a twosided ordered hunt. 2
t GI/M/l-loss denotes a single server loss system with a renewal process input stream
(GO and an exponential (M) service time distribution.

TRUNK-DETECTION ALGORITHMS

925

1.0~---------------------,

GI/M/l - LOSS MODEL

- - KILLER TRUNKS (r = 5)

fl- 1 = 180 SECONDS

Z = 1,2,5

T= 200 SECONDS

0.8

Z= PEAKEDNESS

0.6
a
c...

0.4
./

NORMAL TRUNKS~~-/

0.2

OL-_ _

o

~

___

20

~

___

40

~

___

60

~

___

80

~

100

MEAN TRUNK OCCUPANCY IN PERCENT

Fig. 4- I -- 0 transition probability for the GI/M/I-loss model with a switched Poisson
arrival process-~OO-second sampling option.

a switched Poisson input stream (commonly used to model overflow
traffic).8 Appendix A contains several details on the model and derivation. It is clear that the effect of peaked traffic on the transition probability is very small (z = 1 corresponds to a Poisson stream).
Recent data 9 indicates that the service time distribution of a normal
trunk has a coefficient of variation significantly greater than 1 (the exponential case). Thus, in Appendix A we derive the covariance function
of the server process Xt for an M/G/I-Ioss! model with a mixed exponential
type of service distribution. Figure 5 is a plot of P1,o vs p parameterized
by the coefficient of variation of the mixed exponential service distribution. We see that increasing the coefficient of variation has a noticeable
effect on the transition probabilities, but the effect is to increase the
discrimination between the normal and killer-trunk transition probabilities.
Thus, it would appear that the transition probabilities based on the
M/M/l-Ioss model are reasonably robust to perturbations in the trunk
model. In addition, one suspects that using these transition probabilities
in a detection scheme, which exploits the basic differences between killer
and normal trunk transitions, might lead to a conservative design.
t M/G/I-loss denotes a single server loss system with a Poisson (M) input stream and
a general (G) service time distribution.

TRUNK-DETECTION ALGORITHMS

927

1.0.r---------------------,
M/G/l - LOSS MODEL

fL -1 = 180 SECONDS

7KILLER TRUNKS
(r = 10)

T = 200 SECONDS

c = COEFFICIENT OF
VARIATION

0.8

0.6

0.4

0.2

O~

___

o

~

___

~

___

40

20

~

___

~

80

60

___

~

100

MEAN TRUNK OCCUPANCY IN PERCENT

Fig. 5- 1-- 0 transition probability for the M/G/1-lossmodel with a mixed exponential
service time distribution-200-second sampling option.

3.2 Testing statistical hypotheses-a basic idea

Suppose a trunk has constant mean occupancy p (EXt = p) and we
observe it for h seconds during which n switch counts and t 10 1 - 0 state
transitions accumulate. We may write
·m

n =

:L Xhr

k=l

and
m

t10 =

:L [Xkr - X(k-l)r]-,
k=2

where
m = ~ and z- = {O
T
1 z<O
z~O

Hence, we have
E(n) = p

(~:)

(5a)

and
E(t1O) = (m - l)E([xkr - X(k-l)r]-) =
928

(~ - 1) pP1,o0

(5b)

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

Thus, for example, a trunk at 20-percent occupancy (sampled at the
100-second rate) would accumulate 108 switch counts, on the average,
in 15 hours. The mean number of 1 -- 0 state transitions in this time
interval corresponding to a normal trunk is 43, but the corresponding
mean number of 1 -- 0 state transitions for a killer trunk (during the
same time interval) is 86. (Referring to Fig. 3, we see that P1,0(p = 0.2)
is 0.40 and 0.80 for a normal and killer (r ~ 7.5) trunk, respectively.)
With this example in mind, it is natural to consider a detection scheme
of the following type:
(i) Wait until we accumulate no switch counts on a trunk.
(ii) When n = no, compare the accumulated 1 -- 0 state transitions

t 10 to some threshold Tt(iii) If tlO > T j , decide that the trunk is a killer, otherwise decide that
the trunk is normal. [tlO is not directly available (see Section 2.1), but
it may be estimated by t/2.]
If the trunk occupancy were known and fixed, this scheme would appear to be very reasonable. The analogy to the usual scheme suggested
for deciding between a fair and a biased coin is clear: no is the number
of (hopefully) independent experiments (analogous to the number of
coin tosses), with each experiment having just two possible outcomes:
the scan which follows the switch count is either 0 or 1. Thus, each switch
count is associated with a 1-- 0 state transition ("heads") or a 1--1 state
transition ("tails").
From the point of view of statistical hypothesis testing, we are thinking
of two underlying states:

Null hypothesis H o: P 1,0(p) = P trunk normal
Alternate hypothesis HI: P 1,0(p) = p* trunk killer
Thus, our intuition suggests that a threshold test of the type sketched
above is natural for distinguishing between H and HI. We will see
(Section 4.1) that a (nonoptimal) test of this form arises naturally from
pursuing the coin tossing analogy further.

°

3.3 Problem formulation-sharpening the focus

To simplify matters, assume to begin with that
(i) The nominal mean holding time 1/J.L is known.
(ii) The trunk occupancy P is known and constant.
(iii) The switch-count and transition data accumulations n(m) and
t(m) are continuously available (scan by scan).

With these assumptions, it is an easy matter to conceptually describe
TRUNK-DETECTION ALGORITHMS

929

the "optimum" scheme for deciding between the two simple hypotheses,

H 0: Trunk normal (mean holding time 1iJ.t)
HI: Trunk killer (mean holding time l/rJ.!),

with it understood that "trunk" refers to one of the specific models described in Section 3.1 (for concreteness assume the M/M/1-loss
model).
Let Xm = (Xl,' ',x m ) be the sequence of trunk states up to and including the mth scan, and let the available data be (as before)
m

m

i=1

i=2

n(m) = LXi, t(m) = L IXi - xi-II·

Let
Pim(n,t) = P(n(m) = n,t(m) = t/Hd i = Oar 1

and let
fm(n,t) = Plm(n,t).
POm(n,t)

The joint probability distributions P im (n,t), i = 0,1, are well defined
for any specific trunk model, but they may be nontrivial to derive. f m (. ,.)
viewed as a function of the vector random variable [n(m),t(m)] is referred to as the likelihood ratio statistic and plays a central role in the
theory of statistical hypothesis testing. More specifically, the optimum
test (in a variety of senses) for deciding between two simple hypotheses
involves suitably comparing f m to a threshold (or thresholds) in order
to make a decision.
We briefly review two optimum tests, the Neyman-Pearson (fixed
sample) test lO and Wald's sequential probability ratio test (SPRT), using
notation appropriate to our (discrete) problem.
3.3. 1 The Neyman-Pearson test

Suppose a and {3 denote the type 1 and 2 errors* of the test,
Choose HI if f m ~ T
Choose Ho if fm < T,
and suppose a' and {3' denote the type 1 and 2 errors of any other test
(requiring m samples) for deciding between Ho and HI. Neyman and
* The type 1 and 2 errors, (\' and {3, are often referred to as the probability of false alarm
and the probability of miss, respectively ((\' = probability of choosing HI given H is the
true state, (3 = probability of choosing H 0, given HI is the true state.)

°

930

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

Pearson's classical result is: if a' ~ a, then {3' ~ {3. Thus, of all tests requiring m samples and having a false-alarm probability not exceeding
a, the likelihood ratio test achieves the minimum probability of miss
(maximum probability of detection). Since a = P( f m > T /H 0), choosing.
a sample size m and threshold T to achieve a ~ ao requires knowledge
of the (conditional) distribution of f m' Similarly, having chosen m and
T, calculating (3 = P(f m < T/H 1 ) requires the distribution of fm (conditioned on HI). Note also that with such a fixed sample test, we decide
in advance to accumulate exactly m samples before making a decision.
In many contexts, data accumulates sequentially in time, and rigidly
requiring m samples-independent of the particular realization that is
unfolding-is not an optimal strategy.
3.3.2 Wald's sequential probability ratio test

Using Wald's SPRT,10,11 we continue to update fk' k = 1,2,··· and
defer a decision as long as fkdT o,T1). We make a decision the first time
fk falls outside the interval (T o,T1 ). Thus,
if To < fk < T1

k = 1,2,···,m-1

and fm ft (To,T1 ),
then choose HI if f m ~ T 1
and choose H o if fm ~ To.
Clearly the stopping time m of the SPRT is a random variable, and the
mean of m (given either hypothesis) is a measure of the time it takes to
reach a decision. (Under a wide variety of circumstances, the SPRT terminates with probability 1.) Let Edm) (i = 0 or 1) denote the mean
stopping time, given that hypothesis i is in effect. Given a SPRT with type
1 and 2 errors a and (3, and with mean stopping times Eo(m) and E 1(m),
consider any other test (sequential or not) with type 1 and 2 errors a'
and (3', and with mean stopping times E~(m) and E~(m). The SPRT has
the following optimal character10
if a' ~ a and {3' ~ (3,
then E~(m) ~ Eo(m) and E~(m) ~ E1(m).
Thus a SPRT is superior to a fixed sample test, if both tests have the same
type 1 and 2 errors, in the sense that on the average it reaches a decision
more quickly (under either hypotheses).
In sharp distinction to the fixed sample test, the thresholds To and
T 1 required to approximately achieve specified type 1 and 2 errors are
trivially determined.!l On the other hand, even determining the mean
and variance of the stopping time is often a difficult chore.
TRUNK-DETECTION ALGORITHMS

931

In Section 4.2, we explicitly calculate the SPRT* for the simple hypothesis testing problem described at the beginning of this section.
Before looking at this optimum test, however, we describe an ad hoc
algorithm which is very robust and consequently attractive from a
practical point of view.
IV. INDIVIDUAL TRUNK ALGORITHMS

A basic underlying assumption in this section is that the normal mean
holding time of a trunk is known. Thus, if the algorithms in this section
are designed relative to a normal mean holding time of 3 minutes, they
will not discriminate between normal trunks having a mean holding time
in the vicinity of 40 seconds,t and an actual killer trunk with the same
mean holding time-both of these trunks will be detected as killer
trunks.
The rationale for studying this type of detection problem is two-fold:
from the practical point of view the simplicity of implementation and
general applicabilityt of these algorithms is attractive, and EADAS/ICUR
can flag trunk groups which should not be studied by the killer trunkdetection algorithms (thus preventing false alarms on normal shortholding-time trunks). From the theoretical point of view, it was natural
to consider this problem before factoring group information into the
picture.
Another modeling assumption used in this section (as well as in subsequent ones) is that the arrival process is stationary within data accumulation intervals, but the mean arrival rate may change arbitrarily from
one accumulation period to another. Since we use equilibrium analysis
(e.g., in calculating PI,o) we assume, in effect, that equilibrium is achieved
instantaneously.
4. 1 An ad hoc algorithm

The essential idea of the test suggested in Section 3.2, is to decide on
the state of a trunk by comparing the number of 1 -- 0 state transitions
(t1O) to some threshold Tr, conditional on having accumulated a fixed
number of switch -counts. We heuristically* proceed to derive such a test,
using a standard likelihood ratio formulation, and explicitly take into
account the time-variability of traffic.
Let Xm = (xv· ·,x m ) correspond to the (unobservable) binary se* Based on the M/M/l-Ioss model for a trunk.
t Trunks in special-purpose trunk groups (credit checking, weather, etc.) will typically

have mean holding times in the vicinity of 40 seconds.
t The individual trunk algorithms can be used to test any trunk-regardless of the type
of switching machine the trunk is associated with.
* The distributional assumptions made in this section are intuitively motivated, but
cannot be rigorously justified. We examine these assumptions carefully in Section 4.3.

932

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

quence of trunk states during an accumulation period in which m scans
occur. Let t lO (m) and n(m) be the number of 1- 0 state transitions and
switch counts associated with X m . Denote the conditional probability
P(tlO(m) = tln(m) = n)

for a normal and killer trunk by P(tln) and P*(tln), respectively. These
conditional distributions depend, of course, on the trunk's occupancy
and on the particular trunk model we have in mind. [p* (tin) also depends on the killer parameter r.] However, for the purposes of the heuristic development of this section, we do not precisely define which trunk
model we have in mind.
Since each switch count is associated with either a 1 - 0 or a 1 - 1
state transition with probabilities PI,o and PI,1 = 1 - PI,o, respectively
for a normal trunk, and since we expect successive transition events on
a trunk to be essentially independent, * it seems reasonable to assume
that P[tlO(m) = tln(m) = n] for a normal trunk is binomially distributed
with parameters nand PI,o. This same argument applies to a killer trunk.
Denote the binomial distribution with parameters nand p by b (k ;n,p)
k = 0" . ',n, where
b(k;n,p) = (~) pk(l - p)n-k.

Thus, we may think of a trunk with occupancy p during an accumulation
period as having a conditional distribution
P[tlO(m) = tln(m) = n] = b[t;n,PI,o(p,r)],

(6)

with r = 1 and r = ro corresponding to the normal and killer states of the
trunk. (Recall that PI,o(p,r) is essentially independent of r for r ~ ro with
ro = 7.5 and 5.0 for 100- and 200-second sampling, respectively.) With
these assumptions, we may think of testing the two simple hypotheses:
Ho: P(tln) = b(t;n,PI,o)

PI,o = P I ,o(p,l)

HI: P(tln) = b(t;n,P~,o)

P~,o = PI,o(p,ro).

If the 1 - 0 transition and switch -count accumulations for two successive

and contiguous accumulation periods are (tl,nl) and (t2,n2) respectively,
we assume that
P(tI,t2I n l,n2) = P(tr!n l)P(t2In 2)'

The idea here is that the only dependence between the two successive
* The idea is that if significant correlation extends only one or two scans back, then
successive transition events (events "triggered" by switch counts) should be essentially
independent.
TRUNK-DETECTION ALGORITHMS

933

bit streams x~ = (Xl, • ·,x m ) and x~ = (Xm+l,· ·,X2m) is essentially due
to the dependence between Xm and Xm+l.
Thus, if we denote the transition and switch-court accumulations for
the ith accumulation period (in which mi scans occur) by [tlO(mJ,n(mi)]
during which the trunk has occupancy Pi, we have
P[tlO(ml) = tv· ·,tlO(mk) = tk \n(ml) = nv· ·,n(mk) = niJ
k

=

rr b[ti;ni,PI,O(Pi,r)],

(7)

i=l

where Pi i = 1,.· ·,k are the occupancies for the k accumulation periods.
If tk = (tv· ·,tk) and nk = (nl, . ·,nk) consider the likelihood ratio:
n(

£

~

/

nk

) _ rrk b[ti;ni,PI,O(Pi,rO)]
.
i= I b [ti;ni,P1,O(Pi, 1)]

(8)

Denote the log likelihood ratio t log f(tk/ nk) by i(tk/ nk) and note
that

where

"'( ./ .) - 1 b[ti;ni,P1,O(Pi,rO)]
og [
].

f tc nc -

b ti ;ni,P1,O(Pi, 1)

The expression i(tJni) can be written as
i(tJnd = a(Pi)ti - a(pdni

with
a ()
P = 1og

1 - PI o(p,l)

,
1 - PI,o(p,ro)

(9a)

and
a(p)=a(p)+log

PIO(p,ro)
, (
).
PI,o p,l

(9b)

Thus, we have
(9c)
Unfortunately, the occupancy in the ith accumulation period (Pi) is
unknown and hence equation (9c) cannot be used as a test statistic. One
obvious "fix" is to estimate Pi by Pi = nJmi, where ni and mi are the
t I > Tiff g(e) > g(T) if g is monotone increasing, so the tests! > T and g(f) > g(T)are
equivalent.

934

THE BELL SYSTEM TECHNICAL JOURNAL, JULY -AUGUST 1977

switch count and the number of scans, respectively, during the ith accumulation period. In stationary traffic, the estimate

Pi = .Li nj / .Li mj
)=1

)=1

would be used [a-{pd == I/Vl a-{pd if mj = m for allj].
Corresponding to the sequence of accumulations, {t1,ni,mJ i = 1,2,.
we define ri and Ri, i = 1,2,. by
0

0

0

0

(lOa)

and

Ri = R i- 1 + ri with Ro = o.

(lOb)

Thus, we arrive at the sequential test:
(i)

Compute R i , i = 1,2,.

0

0

and defer making a decision as long as To

< Ri < T 1.

(ii) If i = k corresponds to the first accumulation period for which
Ri ft (T o,T 1), then
Rk ~ To ==> Trunk normal
Rk ~ T 1 ==> Trunk killer.
If we ignore the fact that we are estimating Pi by Pi, and by assuming that
the various assumptions made are valid (see Section 4.3), we identify the
above test as Wald's SPRT and as such To and T1 can be calculated as
refollows:l1 to approximately achieve type 1 and 2 errors, a and
spectively, a + < 1, choose

/3,

/3

To = log

(_/3_)
I-a

(10c)

T1 = log

(1 : /3).

(10d)

and

Throughout this section, we have assumed that the 1 ~ 0 transitions
(t 10) are available when, in fact, only the total transitions (t) are available.
It should be clear that tlO can differ from t/2 by at most ±%. To be precise, let tlO{m), tOl (m) be the number of 1 ~ 0 and 0 ~ 1 state transitions
corresponding to a bit stream Xm = (Xl,
xm). If n{m) is the switch
0

0

0,

count corresponding to X m , then we have

= tlO{m) + tl1{m) + Xm

(lla)

n{m) = tor(m) + tl1{m) + XI,

(lIb)

n{m)

and

TRUNK-DETECTION ALGORITHMS

935

where t 11 (m) is the number of 1 -- 1 state transitions. Therefore
tlO(m} + Xm = tOI(m} + Xl,

which together with t(m} = t01(m} + tlO(m} yields
tlO(m} = ~ t(m} + (Xl ~ Xm)

(I2a)

tOI(m} = ~ t(m} _ (Xl ~ Xm).

(12b)

and

Thus, we can write the statistic update (eq. lOa) as
a

i-an +

(X I ~ Xm) a.

[It is easy to show that E[(XI - xm}a(p)] = 0.]
We conclude this section with an interpretation of the statistic update.
Rewriting the statistic update as
r = (a - a)tlO - a(n - tlO)

and using eq. (I1a), we obtain
r = (a - a}tlO - at11 - ax m.

(13)

Now, from eqs. (9a) and (9b), it is clear that a > a > o. Thus, each 1 --

otransition is weighted positively (evidence of a killer) while each 1-1 transition is weighted negatively (evidence of a normal trunk). This

is an intuitive explanation of the fact that the random walk (eq.
(lOb»
Rk = Rk-l

+ rk

has a positive drift if the trunk is a killer and a negative drift if the trunk
is normal.
The fact that the update assigns a negative weight (-a) whenever the
last bit (xm) is 1 uncovers a modeling deficiency. Recall that in eq. (6)
we assumed
P(tlO(m} = t/n(m) = n} = b(t;n,P lO },

even though Xm = 1 can not contribute to an observable 1 -- 0 transition.
In this way we effectively modeled in a bias towards making "trunk
normal" decisions. We can easily correct eq. (6) by conditioning on
whether Xm = 0 or 1, obtaining:
P(tlO(m) = t/n(m) = n) = (1 - p)b(t;n,PlO) + pb(t;n - I,P IO }.

Now, proceeding as before in formulating the log likelihood ratio yields
936

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

a statistic Rk , where

and

where rk is defined by eq. (lOa),

qk = 1og

1+ (~)(1-~)(
1 )
1 - Pk
nk
1 - P ,O(Pk,rO)
1

(14)

C~kpJ (1 - ~:) C_ 0(Pk,l))
1

1+

Pl

°

and Pk, tk, and nk are the trunks occupancy estimate, 1 -- state transitions, and switch count, respectively, during the kth accumulation
period.
Thus, we obtain our original test statistic with the correction term qk
added on. Note that qk ~ 0, qk -- as Pk -- 0, and qk -- a as Pk -- 1,
which is just the type of behavior expected, to offset the bias term in

°

rk·

Having heuristically developed an ad hoc sequential algorithm that
is intuitively appealing and easily implementable, it is natural to ask:
how does it compare to the optimum sequential algorithm? In the following section, we rigorously develop an optimum sequential test.
4.2 An optimal algorithm

Consider the two simple hypotheses:
Ho: Trunk normal (mean occupancy p, mean holding time I/Jl)
HI: Trunk killer (mean occupancy p, mean holding time l/roJl).
The optimum test for deciding between the two hypotheses-in the sense
of minimizing the mean decision time-for given type 1 and 2 errors, is
Wald's SPRT (see Section 3.3), and it is based on the likelihood ratio
statistic fm(t,n) given by
/J
{,m

(

t,n

)

= P*(t(m) = t,n(m) = n) .
P(t(m) = t,n(m) = n)

(15)

Thus, it is clear that the ad hoc test described in Section 4.1 is not optimal, based as it is on an assumed conditional distribution,
P(tlO(m) = t/n(m) = n).

Before proceeding to study eq. (15), we must define the trunk model
precisely. In the developments that follow, we model a trunk as the server
in an M/M/1-loss system (see Section 3.1). The model implies that the
sequence of trunk states Xt, t = k T, k = 1,2,· .. is Markovian. Note that
TRUNK-DETECTION ALGORITHMS

937

although this appears to be a reasonable model for a normal trunk with
200-second sampling, it ignores the conditional dependence "2 samples
back," which is more important for 100-second sampling-e.g., Xt given
Xt-r is independent of Xt-2r for the M/M/1-loss model. Taking this dependence into account in a trunk model would not be useful however,
since the data needed to implement dependence "two scans back," is not
available.
Since we are modeling the sequence of trunk states as a binary valued
Markov process Xkn k = 1,2,.· ·,in equilibrium,iit is clear that this process
is characterized by 8 = (PI,O,PO,I), where PI,o and PO,1 are the transition
pro babilities
P(Xt+r = O/Xt = 1) and P(xt+r = l/xt = 0),

respectively. (In general, a binary valued Markov process Xkn k = 1,2,.··
in equilibrium, can be characterized by any two of the three quantities
p, PI,O,PO,I. For our special Markov process (based on the M/M/1-loss
model), both PI,o and PO,1 and hence the process itself is determined by
p alone.) Now having observed any m-tuple of the samples, which we
denote by Xm = (XI,"· ·,x m), it is trivial to show that the statistic

is a sufficient statistic for O. Thus, except for the initial and terminal
states (Xl and x m ), the transition and switch-count accumulations
summarize all the "relevant information" in X m .
Our hypothesis-testing problem can now be formulated as follows:
Xt,X2,·· is a binary-valued Markov chain in equilibrium with parameter
o= (PO,I,PI,O) or 0* = (P~,I'P~,O). That is, our two states are
H o: Ixil Markovian, characterized by 0 = (PO,I,PI,O)
HI: Ixi! Markovian, characterized by 0* = (P~,bP~,O).

Now, because (t(m),n(m),xI,x m) is a sufficient statistic for 0, we know
that the likelihood-test statistic based on the raw (unobservable) data
Xm = (XI,"· ·,x m) will be expressible in terms of t(m), n(m), Xl and Xm
only. Thus, instead of studying eq. (15), we proceed (for simplicity) to
study the likelihood-ratio statistic:
P*(Xm )
f(x m) = log P(Xm) .
A

(16)

In Appendix B we study i m(t,n) and find that it differs from 1(xm) only
in an end-effect term . .In l(xm ) this term depends on Xl and X m , whereas
in 1m (t,n) the corresponding term is a function of t and n.
Since
938

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

m

P(Xm ) = P(Xl) II P(XdXi-l),
i=2

we may write
SO

(1 -

P*(Xl)
P~o
P~O)
f(xm) = log-(-) + tlolo g -p + tulog
P
P Xl
10
1 - 10
A

+ tOl log -P~1 + too log
POI

(1 - P~1) . (17)
1 - POI

Note that a trunk with mean occupancy p is busy and idle with probability p and 1 - p respectively, independently of the state it is in (normal
or killer). Thus,

P*(Xl)
.
log - - - = 0 and eq. (17) can be WrItten
P(Xl)
i(x m ) = [(a - a)tlO - atul + [(iJ - b)tOl - btoo],

(18)

where the parameters band iJ are defined by

I (11 -- POI)
p*

(19a)

iJ = b + log -P~1) ,

(19b)

b = og

01

(POI

and the parameters a and a are defined as in Section 4.1 (eqs. (9a) and
(9b)). P~,1 and P~,o correspond to PO,l(p,r) and Pl,o(p,r) with r = roo
Before discussing the symmetric structure of the optimum statistic
[eq. (18)], we examine the PO,l characteristics for the M/M/1-loss model.
Using eqs. (2) and (3), we can obtain Po'! vs mean-trunk occupancy p for
a normal (r = 1) and killer (r = ro) trunk. Figures 6 and 7 are plots for
the 200- and 100-second sampling option, respectively, with a mean
holding time of 180 seconds. It is clear from Fig. 6 that a 0 -- 1 transition
is just marginally more likely to occur on a killer trunk than on a normal
trunk with a 200-second sampling rate. Although, the difference in the
o -- 1 transition probabilities between a normal and killer trunk increases substantially with the 100-second sampling rate, it is clear that
these differences are still quite small-compared to the spread between
the PI,o and P~,o plots (see Figs. 2 and 3). Note that eqs. (2) and (4) show
that
-ros
1- exp-P~,1 __
1- P
(19c)
- - - - - > 1 for ro > 1
PO,I
-s
1- exp--

1-p

TRUNK-DETECTION ALGORITHMS

939

and, hence, we have {3 > b > O. Using eqs. (4) and (19c), we see that
P~,dPO,1 = P~,O/PI,O and, therefore,
(19d)

{3-b=o:-a

Equation (18) shows that the optimum statistic is the sum of two
symmetric statistics:
(i) The statistic [(0: - a)tlO - atllL which is essentially the ad hoc
statistic (see eq. (13) and related discussion).
(ii) An additional statistic [({3 - b)tOl - btoo], which weights 0 ~ 1
transitions positively (evidence of a killer) and 0 ~ 0 transitions negatively (evidence of a normal trunk).

Note that by interchanging the role of 0 and 1 in either of these two
statistics, we obtain the other-b is obtained from a and (3 is obtained
from by replacing PI,o with PO,I.
By using eq. (1Ia) and the analogous equation

0:

m - n(m)

= too(m) + tOI(m) + x~, (x~ = 1 - xm)

(20)

in eq. (18), the optimum statistic can be written
l(xm ) = [o:tlO(m) - an(m)] + [(3tOl(m) - b(m - n)] + el(x m ),

(21)

where the end-effect term edxm) is given by
el(x m ) = aXm + bx~.

To implement l(xm ) with only t(m) and n(m) available, necessitates
estimating both tlO(m) and tOl(m) by t(m)/2. That is, using eqs. (12a)
and (12b) in eq. (21) yields.
i(xm ) = [ IX t(;) - an(m)] +

Ip t(;) - b[m - n(m)]) + e(x"xm),
(22a)

where

b) (Xl + xm) + b

a = ( -2-

t

(22b)

or

(0: (3)

+ - t(m) - (a - b)n(m) - bm + e(xl,x m ).
f(x m ) = - 2
A

t Recall that a -

(23)

{3 = a - b [eq. (I9d)].

TRUNK-DETECTION ALGORITHMS

941

In the development of the ad hoc algorithm, we assumed that the
statistics corresponding to successive accumulation periods are independent. We conclude this section by examining the independence assumption and with some remarks on implementation.
Thus, turning to multiple accumulation periods, suppose x~, i =
1,2,. . ·,k are the (unobservable) bit streams for k successive (and contiguous) accumulation periods, where x~ = (X(i-l)m+l,"· ·,Xim). Assuming stationary traffic, and noting that Ixd?:I is Markovian, we can
write

and, therefore,

i(x~," . .,x~) =

t i(x~) + kf.1log {P*(Xim+l/Xim)/P*(Xim+l)}, (25)

i=l

i=l

P(xim+dxim)/P(Xim+l)

where P(.) and P* (.) denote the distribution under H 0 (trunk normal)
and HI (trunk killer), respectively. But, as we have seen,

-roS}

P*(xim+dxim) = 1 - exp {- P*(Xim+l)
1-p

--~----'----....:..;..;.'--

(26)

and hence Xim+ 1 and Xim are essentially independent for ro sufficiently
large. Therefore,
P*(Xim+l/Xim) = P*(Xim+l)

and, hence, eq. (25) may be written as
....

k

k-l

k.....

f(x m," ••, x m ) = L f(x~) - L I(Xim;Xim+l),
i=l
i=l

where
(27)

is recognized as the mutual information random variable, which plays
a central role in information theory.1 2 -1t is well known 12 that (under Ho)
EII(xim;xim+l)1 is non-negative, and hence to ignore the end-effect
term
k-l

L I(Xim;Xim+l)

i=l

by implementing the statistic
k

.

L i(x~)

i=l
942

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

would tend to make a normal trunk look more like a killer trunk on the
average. In Appendix C, however, we show that the mean end effect
EII(xim;xim+l)} is negligibfe compared to the mean statistic update
Ell(x~}.
The "optimal" sequential algorithm is implemented in the same
manner as the ad hoc algorithm (see eqs. (lOa) and (lOb)) except that
now, corresponding to the sequence of accumulations (ti,ni,mdi =
1,2,· .. , we define ri by
ri =

(ex ; (3) ti - (a - b )ni - bmi.

The term ti denotes the total number of transitions in the ith accumulation interval. As is the case for the ad hoc algorithm (which corresponds
to b = f3 = 0), the weights are functions of the trunk occupancy estimate
Pi = nJmi. In a practical nonstationary environment, no claims of optimality are made or implied. The term "optimal" is applicable only in
the context of the equilibrium (e.g., stationary) model with known trunk
occupancy.
4.3 The ad hoc algorithm reviewed

The assumption that tlO(m) conditioned on the switch count n(m)
is binomially distributed, is the basic assumption in the development
of the ad hoc statistic. Although this assumption is incorrect (as we will
soon see), the ad hoc statistic is essentially (except for an end-effect term)
one of two symmetric statistics whose sum is the optimum statistic. Our
purpose in this section is to examine the binomial assumption and to
explain the relationship found between the ad hoc and optimal statistics.
Since the optimal statistic was developed for a trunk modeled as a
server in an M/M/1-loss system, it is natural to examine the binomial
assumption (eq. (6)):
P[tlO(m) = t/n(m) = n] = b[t;n,P1,o(p,r)],

(28)

where P1,o(p,r) is given by eq. (4) in this context. Consider a killer trunk
with r sufficiently large and suppose Xm = (XI," • •,x m ) is the bit stream
for a killer trunk during some accumulation period. Then, for all practical
purposes [see eq. (26)], the trunk states Xi i = 1,2,·· ·,m are independent
and identically distributed Bernoulli random variables:
P(Xi = x)

= {P

1- p

~f x = 1

lfx = O.

Thus, it is clear that the switch-count distribution on a killer trunk is
the binomial:
TRUNK-DETECTION ALGORITHMS

943

P(n(m) = n) = b(n;m,p).

(29)

Now suppose Am{t,n) denotes the number of binary m-tuples having
1 -- 0 transitions and n ones. If Xm = (Xl,· ·,x m ) is a seexactly t
quence of trunk states for a killer trunk with n(m) = n, it is clear that
each such sequence has probability
P(xm ) = pn(l - p)m-n

and therefore
P(tlO(m) = t,n(m) = n) = Am (t,n)pn(1 - p)m-n

(30)

for a killer trunk (with r sufficiently large). Equations (29) and (30) show
that for a killer trunk,

(31)

where we have used the fact that
(32)

It is interesting to note that while our assumed distribution for a killer
trunk (28) differs from the correct distribution (31)-note that (31) is
independent of p-there are some interesting similarities. For example,
the assumed distribution peaks in the vicinity of (n + 1)(1 - p) and has
mean equal to n(l - p)t whereas the true distribution peaks in the vicinity of (n + 1)[1- (n/m)] and has mean equal to n[l- n/m]. Note that
for "typical" realizations (Xl, • ·,x m ), we have

-mn =p
and, hence, the two distributions have the same general location and
scale. [In fact, expression (31) is a hypergeometric distribution, which
converges to (28) as m -- 00 if n = pm (Ref. 13).] Thus, although incorrect, the binomial distribution approximates the true distribution of the
killer trunk.
The following result helps to put the relationship between the ad hoc
and the optimal statistics in perspective.
Lemma 2: If lxi/ is a binary state stationary Markov chain with transition probabilities PO,1 and PI,o, and if Xm = (Xl,· ·,x m ), then we
t P1,o(p,r) -- 1 - p as r -- roo

944

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

have
P(Xm ) = b(tlO;n,P 1,0) x b(tO,l;m-n,PO,l) x q,

(33a)

where
(33b)

Proof:
m

P(Xm ) = P(X1) II P(XdXi-1)
i=2

= P(X1) PL~o P~V P~~l P~~oo
= PLloo (1 - P 1,0)t ll P~~l (1 - P O,l)t oo X P(X1).
Using eqs. (IIa) and (18) to express t11 and too in terms of tlO and tOb
respectively, yields the result.
Thus, given a binary state stationary Markov chain lxii, it is clear from
the above lemma that the log likelihood ratio
fi(
{, Xm

) - I
-

P*(xm )

og P(x

m

)

formulated for the two hypotheses
H o: Ixd Markovian, characterized by (P O,1,P 1,0)

H 1: IXi I Markovian, characterized by (P~,bP~,O)

is the sum of three terms:
fi(

{, Xm

) = I og b(tlO'n
"p~,0) + I og b(t01;m - n,P~ ,1) + I og (q*)
- .
b(t1O;n,P 1,0)
b(t01;m - n,PO,l)
q

The first term is the ad hoc statistic (O'tlO - an), the second term is the
additional statistic [(3t01 - b(m - n)], and the third term is an end-effect
term (ax m + bx~).
log (q*) = ax
q

m

+ b XCm + log P*(X1)
= ax + bx c
P(xd
m
m,

since, P* (x d = P(x 1) = p.
The ad hoc algorithm, although based on the approximate binomial
distribution, is very attractive for a number of practical reasons:
(i) For the 200-second sampling option, it is essentially optimum in
a practical sense, since the P O,l characteristics for a normal and killer
trunk are not far enough apart to exploit (see Fig. 6).
(ii) When we exploit the grouping information for the 5XB trunk
TRUNK-DETECTION ALGORITHMS

945

group in Section V, it will become obvious that the additional part of the
optimum statistic is quite sensitive to the trunk occupancy and, hence,
to the trunk-selection procedure modeled. (The sensitivity of the "additional" statistic to trunk occupancy stems from the fact that PO,1 is
"almost" proportional to p.) On the other hand, we will see that the ad
hoc 5XB algorithm is relatively robust to minor perturbations in the
trunk occupancy (and therefore to the trunk-selection procedure) and
hence might be expected to perform well in a rea15XB environment.
V. THE 5XB TRUNK-GROUP ALGORITHMS

In addition to utilizing individual trunk switch-count and transition
accumulations the 5XB group algorithms exploit the following:
(i) The identity of all trunks common to a group.
(ii) The trunk-selection procedure.

The resulting 5XB group algorithms typically are faster* than their
individual trunk counterparts and are also less sensitive to the groups
nominal holding time.
5.1 The 5XB trunk-group model

For the purposes of this paper, we model a 5XB trunk group (with all
trunks normal) as an M/M/N-loss model with random selection of idle
trunks. 2 The same assumptions apply if the group contains one or more
killer trunks, but in this case we assume that killer trunks have a mean
holding time equal to 1/r that of the normal mean holding time. In addition to being convenient theoretically, this idealized model has also
been very useful in developing the 5XB group algorithms presently
implemented in leAN.
If all N trunks are normal, the random selection rule implies that all
trunks have the same mean occupancy. In Ref. 2, the birth and death
equations for the above model with a single killer trunk were solved in
closed form, and in Ref. 14 this was generalized to an arbitrary number
of killer trunks. These analytic results turn out to be quite useful, and
in what follows we will need the following results derived in Ref. 2.
Theorem 1: For the above 5XB trunk group model having a single killer
trunk with parameter r and an offered load of a erlangs, the blocking
probability B(N,a, r) and the mean occupancy p; (N,a,r) of the killer
trunk are given by:
* For given type 1 and 2 errors, the group algorithms typically have a considerably
smaller mean decision time than their individual trunk algorithm counterparts.
946

THE BELL SYSTEM TECHNICAL JOURNAL, JULY -AUGUST 1977

(i) B(N,a,r) =

NB(N,a)
Nr - (r -l)a[l - B(N,a)]

(34a)

(ii) p;(N,a,r) =

1
rN
1 + - - r[l - B(N - 1,a)]
a

(34b)

where B(N,a) is the usual erlang B blocking associated with an MI
MIN-loss system with all trunks normal and an offered load of a
erlangs.

It is easy to see that the occupancy Pr of each of the N - 1 normal
trunks must satisfy the conservation equation:
rp; + (N - l)Pr = a[l - B],

(34c)

and the trunk-group occupancy ¢; is defined by:
*

¢r =

P; + (N - l)Pr
N
.

(34d)

(For a 5XB trunk group having a killer trunk with parameter r: P; and
Pr denote the mean occupancy for a killer and normal trunk and ¢; denotes the mean group occupancy.)
Although eqs. (34a) through (34d) define an implicit relationship
between p;(N,a,r) and ¢;(N,a,r), it will be very useful to have a simple
explicit relationship. If the blocking term in eq. (34b) is ignored and if
we "associate" ¢; with alN, an approximation suggested is:
* .

Pr =

¢;

*.

r - (r - l)¢r

(35a)

This approximation, although quite good for large N, is rendered obsolete by the following exact result:
Theorem 2: Consider a 5XB trunk-group model with all trunks normal
and mean-group occupancy ¢. (We will let ¢ denote the (mean) group
occupancy for a 5XB trunk group with all trunks normal.) If one of the
trunks is replaced by a killer with parameter r, then
P; = p(¢,r),
where,
p(¢,r) = r - (r - 1)¢

(35b)

Of course
a[l - B(N,a)]
¢ = --=--------=N

TRUNK-DETECTION ALGORITHMS

947

where B(· , .) is the usual erlang B blocking expression.
This surprising result, which follows easily from eq. (34b), is proved
in Appendix D. As a consequence of this theorem, "5XB group occupancy" will be used to denote the occupancy of a 5XB group with all
trunks normal.
The mean occupancy of the normal trunks in a 5XB trunk group
model having a single killer trunk no longer is given exactly by 1>. But
the following result, derived in Appendix D, shows that 1> is a good approximation.
Theorem 3: Consider a 5XB trunk group model with N trunks having
a single killer trunk with parameter r ~ 1. The mean occupancy (Pr)
of the N - 1 normal trunks satisfy
r -

1> ~ Pr ~

{

(~) (r - 1)1>}
r _ (r _ 1)1>

X 1>,

(36)

where 1> is the mean-group occupancy with all trunks normal.

Theorems 2 and 3 are proved in Appendix D, where an exact expression
for Pr is also derived. These results are special cases of general results
obtained for the random selection model. 14
5.2 Exploiting the 5XB Grouping Information

To simplify matters, we assume that a trunk in a 5XB group* with
mean-group occupancy 1> has mean occupancy p(1),r) given by
p(1),r) = r - (r - 1)1> '

(37a)

where r = 1 corresponds to a normal trunk. Thus, if a group has no killer
trunks, all normal trunks satisfy P = 1> and eq. (37a) with r = 1 yields the
correct occupancy. If, however, the group has a killer trunk, then all
normal trunks satisfy inequality (36) so eq. (37a) with r = 1 is an approximation that increases in accuracy with the size of the group. Of
course eq. (37a) is exact for a (single) killer trunk in a 5XB trunk
group.
It is clear from eq. (37a) that
p(1),l)
p(1),r) = r - (r - l)p(1),l)

(37b)

* We use "5XB group" and our idealized model of a 5XB trunk group interchangeably.
948

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

and hence p(4),r) is typically much smaller than p(4),l). (For 4> = 0.50 and
r = 10, p(4),r) = 2/11 p(4),l). Thus, it would appear that considering the
1 ~ 0 transition probability as a function of the 5XB group occupancy
would effectively "spread" the Pl,o characteristics in Figs. 2 and 3 further
apart. That is, for a given 4>, we propose comparing
P l ,o[p(4),l)] and P~,o[p(4),r)]

[rather than Pl,o(p) and P~,o(p), as in Section IV].
Denoting the composition P l ,o[p(4),r)] by P l ,o(4),r), we have

P1,o(q"r) =

(1 - p(q"r» (1 - exp (1 --;~,r»))'

(38)

which is plotted in Figs. 8 and 9 for the 200- and 100-second sampling
options, respectively. The normal holding time used in these figures is
180 seconds, and the killer-trunk characteristics are drawn for r = 5,10,
and 15.
The increased "spread" between normal and killer Pl,o characteristics
obtained in this way is simply a consequence of exploiting the distinctly
different occupancies of a normal and killer trunk in a 5XB trunk group.
Figure 10 is a three-dimensional sketch of the composition of Pl,o and
p. Because all normal trunks in a 5XB group have the same mean occupancy, we see that a single Pl,o vs 4> characteristic suffices to describe
1.or:::::::~;;;;;;;;;;===---------l

KILLER TRUNKS--0.8

0.6
a
Cl.

0.4

5XB TRUNK GROUP MODEL
(RANDOM SELECTION OF IDLE TRUNKS)

0.2

W1 = 180 SECONDS
T=200SECONDS

O~

o

____

~

______

20

~

40

______

~

____

~

60

______

80

~

100

MEAN GROUP OCCUPANCY IN PERCENT

Fig. 8- 1 - 0 transition probability for the 200-second sampling option.

TRUNK-DETECTION ALGORITHMS

949

l.°r~~::.:::::::::::==-------------'

O.B

KILLER TRUNKS---

0.6
0

a:0.4

5XB TRUNK GROUP MODEL
(RANDOM SELECTION OF IDLE TRUNKS)

0.2

",,-1 = lBOSECONDS
T = 100 SECONDS

0
0

20

40

60

100

BO

MEAN GROUP OCCUPANCY IN PERCENT

Fig. 9- 1-+ 0 transition probability for the IOO-second sampling option.

P1.0 'P PLANE:--MiMil - LOSS
MODEL

)rooo::~------T--¢

/

/

/

//

~7L __ - p.¢PLANE: 5XB GROUP
//

MODEL

p(¢.r)

Fig. lO-Sketch of the composition of P1,o(p) with p(¢,r).

all normal trunks. This fact allows us to translate the individual trunk
algorithm's development to this 5XB context with essentially only notational changes.
950

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

1.0 ~:::::::~;;;;;;;::===---------I

0.8

0.6
a

--------

NORMAL TRUNK
(r = 1)

..........

NORMAL TRUNK .............
FOR A GROUP
........ ,
WITHP.-l = 180 SECONDS)
................

a.

.......

0.4

0.2

"

-.;:

5XB TRUNK GROUP MODEL
(RANDOM SELECTION OF IDLE TRUNKS)
p.-l = 45 SECONDS
T = 200 SECONDS

MEAN GROUP OCCUPANCY IN PERCENT

Fig. 11- 1 -- 0 transition probability for the200-second sampling option (mean group
holding time = 45 seconds).

Figures 11 and 12 are plots of eq. (38) drawn for a normal group mean
holding time of 45 seconds. * The normal trunk characteristic corresponding to 180 seconds is shown in dashed lines. We see that with 5XB
grouping information factored into the picture, considerable discrimination exists between both normal trunk characteristics as well as between the normal trunk having a holding time of 45 seconds and the killer
trunks. The discrimination that exists between the normal trunks permits us to make the 5XB group algorithm adaptive to the group mean
holding time. [Although we will not pursue this topic, the basic idea is
that L.jtlO(j)/L.jn(j) (sums are over all trunks in the group) is an estimate
of Pl,O and can be used to decide which (of several) normalpl,O characteristics constitutes H 0.]
5.3 The ad hoc and the optimal 5XB group algorithms

We assume that the mean group holding time is known and consider
formulating a hypothesis-testing problem similar to that in Section 4.l.
Thus, we denote P(tlO(m) = t/n(m) = n) by P(t/n) and consider the two
hypotheses:
* For normal holding times in the vicinity of 45 seconds, a killer parameter r in the range
3 to 5 probably is typical. An r of 10 or 15 in this context is unrealistic.
TRUNK-DETECTION ALGORITHMS

951

1.0 r~~;;;;;;;===---------I

0.8

0.6
a

NORMAL TRUNK
FOR A GROUP WITH
p..-l = 180 SECONDS

cC
0.4

-----------..._..

---- --

-- ......

' ......

5XB TRUNK GROUP MODEL
(RANDOM SELECTION OF IDLE TRUNKS)

0.2

"~

JL -1 = 45 SECONDS
T = 100 SECONDS

o~

o

____

~

______

20

~

40

______

~

______

60

~

80

____

~

100

MEAN GROUP OCCUPANCY IN PERCENT

Fig. 12- 1 --. 0 transition probability for the 100-second sampling option (mean grouping
holding time = 45 seconds).

H o: P(t/n) = b[t;n,P I ,o(¢,l)]
HI: P(t/n) = b[t;n,PI,o(¢,r)],

r ~ roo

(39)

There are two differences between this formulation and the one in
Section 4.1:
(i) The trunk occupancy p in Section 4.1 is replaced by the group
occupancy ¢.
(ii) The alternate hypothesis HI is composite since PI,o(¢,r) for r ~
ro are distinct.

The approach taken in dealing with (ii) is a natural one often
adopted;l1 since PI,o( ¢,r) is monotone increasing in r Ithis follows from
eq. (38) upon noting that r/[1 - p(¢,r)] = r + ¢/(1 - ¢)I, then testing
between H and the simple alternate hypothesis

°

HI: P(t/n) = b[t;n,PI,o(¢,ro)],

say with type 1 and 2 errors a and {j, respectively, implies that if the true
state of nature is H I with r = r~ > ro the resulting type 2 error will not
exceed {j. With this approach, we can simply translate the ad hoc algorithm results developed in Section 4.1 to this 5XB group context by
making the appropriate changes in notation.
952

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

Thus, the ad hoc 5XB group algorithm can be described as follows:
corresponding to the sequence of data accumulations (t{,n{) i = 1,2,· ..
for the jth trunk in a group of N trunks, where t{ and n{ are the 1 - 0
transition and switch-count accumulations, respectively, during the ith
accumulation period in which mi scans are made, define sf and S{ i =
1,2,· ..; j = 1,· .. ; N by
(40a)

and

S{ = S{-I + s{ with Sb = 0,

(40b)

where ¢i is the group's occupancy during the ith accumulation period.
The sequential test for the jth trunk in the group, j = 1,· . ·,N is defined
by
(i) Compute Sf, i = 1,2,· .. and defer making a decision as long as To

< sf < T 1.

(ii) If i = k corresponds to the first accumulation period for which

S{ rt (To,T I ),
then

S{ ~ To ~ trunkj is normal
S{ ~ TI ~ trunk j is a killer.
The weights a(¢) and a(¢) are defined by
a ()
¢ = I og

1 - PI o(¢,I)
,

1 - P1,o(¢,ro)

(41a)

and
(41b)

where PI,o(¢,r) is defined by eq. (38).
Just as in the individual trunk algorithm, the actual occupancy required to choose the weights a and a is unknown and must be estimated.
Thus, the group occupancy ¢i during the ith accumulation period is
estimated by
1 N
.
¢i = N L ni/mi
A

(42)

j=l

Note that ¢i is a "better" estimator than Pi = nJmi (the estimator used
in the individual trunk algorithm) in the following sense: given a 5XB
group with all trunks normal and mean-group occupancy ¢ (in equilibrium), we have
TRUNK-DETECTION ALGORITHMS

953

(i) E(P) = p = cp
(ii) E(¢) = cp
(iii) var(¢) < var(p).

In addition to the better occupancy estimate available on a group basis,
the fact that the group P 1,0 characteristics are "flatter and broader" then
the individual trunk P 1,0 characteristic implies that the group algorithm
more faithfully tracks the required weights, than does the individual
trunk algorithm.
The ad hoc 5XB group algorithm has the same pleasant intuitive interpretation that the ad hoc individual trunk algorithm had (see eq. (13)
and related discussion). It is also easy to show how the optimal individual
trunk algorithm development of Section 4.2 carries over to the 5XB
group context.
Thus, consider the two states of a trunk to be described by:

H o: \xi/ Markovian, characterized by 0 = (P O,1,P1,0)
HI: \xi/ Markovian, characterized by 0* = (P~,l'P~,O)'

where P1,0 = P 1,0(</>,I), P~,o = P 1,0(cp,ro), and [see eq. (2)]
P

(A.)
p(cp,r) P (A. )
0,1 'P,r = 1 _ p(cp,r) 1,0 'P,r

(

43

)

with p(cp,r) defined by eq. (37a). The assumptions that lead to a consideration of these two statistical hypotheses as a model of the normal
and killer states of a trunk can be found in Section 4.2.
Proceeding as in Section 4.2 leads us to the optimum statistic l(xm )
for distinguishing between the two simple hypotheses under consideration:
P*(X1)
f(x m ) = [(a - a)tlO - atl1] + [btoo - ({3 - b)tod + l o g - - ,
P(X1)
A

-

-

-

(44)

where the parameters a and a are defined by eqs. (41a) and (41b) and
the parameters b and ~ are defined by
- I 1 - Po ,l(cp,rO)
b = og
1 - P O,l(cp,l)

(45a)

~ = b + log P O,l(cp,l) .

(45b)

and
P O,l(cp,rO)

As in eq. (39), the alternate hypothesis is really composite since
[P O,l(cp,r), P 1,0(cp,r)] for r ~ ro are distinct. Since P O,l(cp,r) is monotone
decreasing in r, the approach discussed earlier of treating HI as a simple
954

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

hypothesis with r = ro is followed. The parameters 6 and ~ are defined
a bit differently than were the parameters band {J for the optimal individual trunk algorithm (see eq. (20a) and (20b)) in order to obtain nonnegative weights. Thus, in the individual trunk algorithm context, we
had P~,l > PO,l (eq. (19c)), but in the present 5XB group context we have
PO,l > P~,l. The reason for this "flip-flop" is easy to see: for a given group
occupancy ¢, we are now contrasting the 0 -- 1 transition probability
for two trunks which differ, not only in their hang-up rates but also in
their occupancies as well. Thus, the difference in the two occupancies
dominates the effect that the hang-up rates alone have. Roughly
speaking, the 0 -- 1 transition probability of a trunk is approximately
equal to its occupancy (conditioning on the last scan has little effect)
and hence since p(¢,ro)« p(¢,I) it is clear that we shmlld have PO,l(¢,r)
< P o,r(¢,I). Figures 13 and 14 are plots of PO,l(¢,r) for the 200- and
100-second sampling options, respectively. In both figures, the killer
trunk characteristics have been plotted for r = 5, 10, and 15 and a normal
mean holding time of 180 seconds is assumed. These figures are very
insensitive to the assumed normal mean holding time, since they essentially reflect eq. (37a), which is independent of the mean group
holding time.
The "additional" statistic which appears in eq. (44),

btoo - (~ - b )tOl
1.0r-----------------------""
5XB TRUNK MODEL
(RANDOM SELECTION OF IDLE TRUNKS)

p.-1 = 180 SECONDS
T = 200 SECONDS
0.8

0.6

0.4

0.2

20

40

60

80

100

MEAN GROUP OCCUPANCY IN PERCENT

Fig. 13- 0 -- 1 transition probability for the 200-second sampling option.
TRUNK-DETECTION ALGORITHMS

955

1.0.-----------------------,.,
5XB TRUNK GROUP MODEL
(RANDOM SELECTION OF IDLE TRUNKS)

f.L- 1 = 180 SECONDS
T= 100SECONDS

0.8

0.6
o·

a..

0.4

0.2

0~~§5====~

o

20

____

-L____-L____~

40

60

80

100

MEAN GROUP OCCUPANCY IN PERCENT

Fig. 14-0 -- 1 transition probability for the 100-second sampling option.

shows that 0 -- 0 transitions are weighted positively (evidence of a killer
trunk*) and 0 -- 1 transitions are weighted negatively (evidence of a
normal trunk). This additional statistic is strongly influenced by the
occupancy of a trunk, and only slightly by its hang-up rate.
Note also that the term log P*(XI)/P(XI) is nonzero in the 5XB context
SInce
1

r

-(r:

1)",

{
r - (r - I)</>

if Xl = 1
if Xl = O.

VI. PERFORMANCE OF THE 5XB GROUP ALGORITHMS

In common with all sequential detection algorithms, the time required
by the killer-trunk detection algorithms to reach a decision (trunk normal or killer) is a random variable. In this section, we obtain an approximate formula for the mean time required by the 5XB group algorithms to reach a decision. This result is used to contrast the performance
* Killer trunks in the 5XB group model have very low occupancy, and hence 0 -- 0
transitions are likely.
956

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

of the ad hoc and optimal algorithms as well as to point out the considerable effect that the sampling rate has on each. In addition, we also
obtain an approximate expression for the false-alarm probability of the
5XB group algorithms. The analysis for the individual trunk algorithms,
although differing in several respects from the group algorithms, involves
the same sort of considerations and is omitted.
The analysis in this section assumes a server system in equilibrium
and, therefore, the mean trunk-group occupancy ¢ is assumed constant.
In addition, to simplify the analysis, we assume that ¢ is known; an approximate analysis which does not require this assumption is sketched
in Section 6.1. A consequence of this assumption is that the algorithm
weights are treated as constants rather than random variables. This
assumption is not unreasonable because for multiple-hour accumulation
periods var(1)) is quite small (1) is the switch-count estimate of ¢). (Var(1»
has been derived for an M/M/N-Ioss system.1 5 )
6. 1 Mean statistic update

Corresponding to a sequence of trunk states Xl,X2, • ',Xm in an accumulation period with m scans, define a sequence of transition updates
Z2,' ',Zm by

6

Zn =

(~

if

{- 6) if
(a - a)
if

-a

if

(Xn-bXn) =

(0,0)
(Xn-bXn) = (0,1)

(46)

(Xn-l,X n ) =

(1,0)
(Xn-l,X n ) = (1,1)

The optimum 5XB statistic (eq. (44)) may therefore be written:
"

m

P*(x m )

t=2

Xm

f(x m ) = .L Zi + log P(

)

(47)

In practice the end-effect term cannot be implemented and all the
transitions in eq. (44) must be estimated in terms of t(m), n(m), and m.
Thus, if we denote the implementable version of eq. (44) by Sm(¢,ro), t
use eqs. (11), (12), and (20) in eq. (44), and drop all end-effect terms we
obtain

(48)

t ro is the value of the killer parameter used in defining the alternate hypothesis and
hence the algorithm weights (see eqs. (41) and (45».

TRUNK-DETECTION ALGORITHMS

957

Now using eqs. (11), (12), and (20) once again, we see that 8 m (cp,ro) may
be written
8 m (cp,ro) = ~ Zi + (bx~ - aXm) + (ex + ~) (Xl - Xm).
i=2
2

(49)

Since the 5XB group is in equilibrium, Z2, • ·,Zm are identically distributed and it is easily verified that their common mean is given by
E(z) = (ex PI,o - a)p + (b - Po,I~)(1 - p).

(50)

We recall that p (eq. (37a)) as well as the transition probabilities (eqs.
(38) and (43)) are functions of the group occupancy ¢ and the state r of
the trunk. The weights a, ex, b and ~ are only functions of cp for a specified
choice of the parameter (ro), which characterizes the alternate hypothesis. Thus, the mean statistic update for the optimum 5XB statistic and
its "implementable version" is given by:

" m )} = (m -1)E(z) +E {l oP*(xd}
E{e(x
g-P(XI)

(51a)

E{8 m (cp,ro)} = (m - I)E(z) + b(1 - p) - ap.

(51b)

and

Note that it is easily shown that

¢\

E {IOgP*((XI)} = -log [ro - (ro -1)cp] + r(~ log ro
P Xl)
r - r - 1 cp

which is negative for r = 1 and positive for r ~ roo
Although the increments (transition updates) defined in eq. (46) are
identically distributed, they are not independent. In fact, since the state
sequence {xd has been modeled as a Markov chain (see Section 4.2), it
is easy to see that the sequence {zd defined by (46) is also a Markov chain.
Relabeling the four natural states,

b, -(~ - b), (ex - a), -a,
of the chain {Zi} by
0,1,2,3,
respectively, it is easily seen that the one-step transition matrix 7r for
this chain is given by
7r =

958

(1 -:0" P~,j P~,o 1_°Pj,O)
1 - Po, I

Po, I

o

0

0
PI,o

0
1 - PI,Q

'

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

l-P O,l

(a)
OPTIMAL STATISTIC

(b)
AD HOC STATISTIC

Fig. 15-State diagram for the ad hoc and optimal 5XB group statistics. (a) Optimal
statistic. (b) Ad hoc statistic.

where 7ri,j = P(zn = j/Zn-l = i). Of course, the stationary distribution
P satisfying P7r = Pis
P = [(1 - p)(1 - Po,r), (1 - P)PO,b pPI,o, p(1 - PI,o)].

The state diagram corresponding to the Markov chain \zd is shown in
Fig. 15a. If 6 = {3 = 0, we obtain the ad hoc algorithm, for which the sequence \zd is a three-state Markov chain, with natural states 0, a - a
and -a. The state diagram for this chain is shown in Fig. 15b.
Figures 16 and 17 are plots of the mean statistic update vs group occupancy for the implementable version of the 5XB group algorithms.
(Log base 10 is used in this paper.) These figures are drawn for a killer
trunk with parameter r = 10. Also shown is the corresponding plot for
a normal trunk (r = 1).
We close this section by indicating an approximate analysis of E(z)
which doesn't assume that ¢ is known. Thus, in practice, ¢ is unknown
TRUNK-DETECTION ALGORITHMS

959

and is estimated by ¢. Hence eq. (46) should read:
Zn

= (,6(<1>0)

if

(xn-J,x n )

= (0,0)

and

¢ = ¢o.

Conditioning on ¢, we have
E(z) = E1E(z/¢)1,

where
E(z/¢) = b(¢)P(Xn-I = O,xn = O/¢) +""".

Now by assuming that (Xn-I,X n ) and ¢ are independent,* we get eq.
(50) with the constant weights a(¢),. " " replaced by the mean values
Ela(¢)l,""". The mean values can be approximated in either one of two
ways:
(i) Ela(¢)l == a[E(¢)] = a(¢), which obviously amounts to assuming
¢ is known.

var(¢) d 2a t

(u) Ela(¢)l == a(¢) + - , - d A2
2.
¢
••

A

l
A_

rj>-rj>,

which factors the available variance of ¢ into the picture.
6.2 Mean time to detection

The basic structure of all the detection algorithms in this paper are
the same: a statistic Si is evaluated at the end of the ith accumulation
period, i = 1,2,. "" and a decision is made the first time that the sum SI
+ S2 +""" falls outside an interval (To,TI). Presumably, the random walk
type statistic Si has a negative drift under Ho (trunk normal) and a
positive drift under HI (trunk killer). Wald's SPRT always has the appropriate drift: if H 0 and HI correspond to the probability distributions
Po(w) and PI (w), respectively, and if f(w) is defined by
PI(w)
f(W) = log-Po(w) ,
A

then Eli(w)l < 0 under Ho and Eli(w)l > 0 under HI. The proof is immediate by using the inequaiityI2
- LPi log Pi < - L Pi log qi,
i

i

* For reasonable-size trunk groups, we expect very little dependence b.etween the
sampled state process of an individual trunk (Xl,· •• ) and the group process ¢.
t This is a Taylor expansion to second order.
TRUNK-DETECTION ALGORITHMS

961

where Ipd and Iqi\ are distinct probability distributions. In Appendix
E, we show that both the ad hoc (6 = ~ = 0) and the additional (a = a
= 0) parts of E(z) (eq. 50) have the appropriate drift. This in turn shows
that both the ad hoc and additional parts of the 5XB group statistic (eq.
(51a)) have the appropriate drift.
Suppose Y bY2, .. are i.i.d. random variables with common mean J.£
and consider a random walk
n

Sn =

2: Yi

n = 1,2,· ..

i=I

with absorbing barriers at To and T I . If J.£ is small compared to To and
T I , then the mean stopping time (mean number of steps to absorption)
E(n) is approximately given by
PoTo + PIT I
J.£

where Po and PI are the probabilities of absorption at To and Tb respectively. This follows from Wald's identityll
E(Sn) = J.£E(n)

if we approximate the mean value of the random walk at absorption by
PoTo + PIT I .

In our detection theory context, Po and PI correspond to {3 and 1 {3, respectively, if the trunk is a killer ({3 = probability of miss) and 1 a and a, respectively, if the trunk is normal (a = probability of false
alarm). If we denote the mean number of accumulation periods needed
to reach a decision under H 0 and HI by E ( TN) and E (Tk ), respectively,
and assume (i) successive statistic updates are independent and (ii) the
mean statistic update is small compared to To and T I , we obtain
E(TN) == (1 - a)To + aT I
EISm (c/>,ro))

(52a)

E(T ) == {3To + (1 - (3)T I
k
EISm (c/>,ro) I '

(52b)

and

where the mean statistic update EISm (c/>,ro) I is evaluated for r = 1 in (52a)
and for r ~ ro in (52b).
The assumption that successive statistic updates are independent is
not strictly true if the successive statistic updates are contiguous.
However, one expects that the slight (end-effect) dependence will not
give rise to very much error.
Let Tko be the time required to decide (incorrectly) that a killer trunk
is normal. Similarly, let Tkl be the time required to decide (correctly)
962

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

that a killer trunk is a killer. Then we have
E(T,J = (3E(Tho ) + (1 - (3)E(Th I)'

which suggests that
(53)
The moments of the conditional stopping times Tho and Thl can be obtained by using a well known technique of Wald's,* and one finds that
approximation (53) is reasonable if Tl and (-To) are sufficiently
large.
In our faulty-trunk detection context, a type 1 error ("false alarm")
may result in the misuse of craft resources (e.g., testing a perfectly good
trunk). A type 2 error ("miss") on the other hand, will result in an increased time to detection. Assuming type 1 and 2 errors of 10-6 and 10- 2,
respectively (realistic implementation values), implies approximate
thresholds To = -2 and Tl = 6 (formulae 4.5c and d, log base 10). With
these parameter values, the mean detection time E(Th I) can be approximated by
E(T ) --'-hI

-

Tl

E1S m (¢,ro)l

(54)

For a normal trunk, the mean statistic update is comparable to To and
hence expression (52a) isn't applicable, nor is it needed since the 5XB
group algorithm reaches a decision on a normal trunk after one or two
updates.
Figures 18 and 19 are plots of the mean-detection time vs group occupancy for the implementable versions of the 5XB group algorithms.
[The dashed line portion of Figs. 18 and 19 indicates where approximation (54) involves considerable error (e.g., the region in which
E1S m (¢,ro)l is a significant fraction of Tl)'] It is apparent from these
figures that
(i) The mean detection time for both the ad hoc and optimal algorithms is enhanced by using the 100-second sampling option. This enhancement is far more pronounced for the ad hoc algorithm.
(ii) The optimal algorithm is "faster" than the ad hoc algorithm. This
contrast is greater for the 200-second sampling option.
6.3 False alarm probability of the 5XB group algorithms

If B(a,(3) and A (a,(3) are the test thresholds that result in type 1 and
2 errors a and (3, respectively, for a SPRT, Wald showed l l that using
* See eq. 158 and 159 in Appendix A.5.2 of Ref. 11. The method ignores (as usual) the
"excess over the boundaries" and hence yields approximate results.
TRUNK-DETECTION ALGORITHMS

963

50r------------.~----------------------_.

KILLER TRUNK PARAMETER, r = 10
SINGLE-HOUR ACCUMULATION,
m = 18
UPPE R TEST TH R ESHO LD, T 1 = 6.0

40

FALSE-ALARM PROBABILITY ~ 10- 6
DESIGN, ro = 5
T = 200 SECONDS

fL -1 = 180 SECONDS
<f)

~

30

a
I

~
~

t 20

~
PTIMAL

10

o~

______

~

_

"EXCESS OVER
BOUNDARY"
IS NON~NEGLIGIBLE
\

"""-. . . . . . """---1- ______ ______
____ ______ ---------~

o

~

40

60

~

80

~

100

MEAN GROUP OCCUPANCY IN PERCENT

Fig. I8-Mean time to detection for the 5XB group algorithms-200-second sampling
option.

thresholds To and T I defined by*

To = log

(1 ~ a) and Tl = log (1 : /3) (assuming a + /3 < 1)

yield type 1 and 2 errors a' and f3' which satisfy
R':s _/3_
fJ

-I-a

and

a':S _a_
-1-/3

The proof of this result is trivial and depends only on the assumption
that the SPRT terminates with probability 1. This assumption is satisfied
by a wide class of SPRTs, 16 including the case of interest to us, where the
underlying distribution is "Markovian". Thus, the probability of false
alarm (a') for a SPRT (an example of \Yhich is the "optimal" 5XB group
algorithm) satisfies a' ~ 10- T1 .
Because the ad hoc algorithm is not a SPRT (the true underlying distribution is not binomial, see Section 4.3), we may wish to study the
consequences of using the above thresholds appropriate for a SPRT. To
do this, we can proceed in (at least) two distinct ways:
(i) consider the statistic S m(I¢,ro) to be the basic update in the algorithm.

* This assumes that the log likelihood ratio is used in defining the SPRT.
964

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

20.-------.------------------------------.
KILLER TRUNK PARAMETER, r = 10
SINGLE-HOUR ACCUMULATION, m = 36
UPPER TEST THRESHOLD, Tl = 6.0
FALSE·ALARM PROBABILITY = lOG

15

DESIGN, ro = 5

Ul

T= 100SECONDS

::l

[L-l = 180SECONDS

a:

0
I

-~

10
---"EXCESS OVER BOUNDARY"
IS NON·NEGLIGIBLE

;Z

t:
w

,

......

', .............
.............

O~

o

______

~

20

____

~

' ...... ...........

- _----- ---

.......
.......
______

~

40

______

..........

~

60

______

80

~

100

MEAN GROUP OCCUPANCY IN PERCENT

Fig. 19-Mean time to detection for the 5XB group algorithms-lOO-second sampling
option.

(ii) Consider the transition update ZJXi-bXi) to be the basic update
in the algorithm.

In either case, we study a process of the form Sj = ~{=l Xi, j = 1,2,.· .
with absorbing barriers To < 0 < T 1• The advantage of proceeding as in
(i) above is that the increments Xi may be assumed to be i.i.d. We briefly
sketch this approach.
Consider a random walk Sj, j = 1,2,.·· with i.i.d. increments {xJ If
n is a stopping time associated with Sj, j = 1,2,. .., Wald's fundamental
identity ll,17 is given by
E{e Snt X (t )-n} = 1 (all t satisfying IX (t) I ~ 1),

(55)

where

is the moment generating function corresponding to the common distribution of the increments. If PTa and PTI denote the probabilities of
absorption at To and T 1, respectively, then rewriting (55) in a standard
way yields

Now if the Xi take on both positive as well as negative values with nonzero
probability and have a nonzero mean, then the equation
x(t) = 1
TRUNK-DETECTION ALGORITHMS

965

has only one nonzero root ho with the property that ho and E(x) have
opposite sign.* That is, E(x) > 0 ~ ho < 0 and E(x) < 0 ~ ho > o. Assuming that the excess of Sn over the boundaries is small, eq. (56) yields
the standard approximation 11, 17

PT

e hoTl - 1
-'------

o - ehoTl _ e hoTo

(57a)

and
1-

e hoTo

PT = - - - - 1

ehoTl _ e hoTo

(57b)

Note that the probability of false alarm (type 1 error) corresponds to P T1
if the random walk increment is that of a normal trunk [8 m (cp,ro), r =
1]. Similarly, the probability of miss (type 2 error) corresponds to PTo
if the random walk increment is that of a killer trunk [8 m (cp,ro), r ~
ro].

To use these approximations, we must compute the moment generating function
(58)

by using the joint distribution p(t,n) derived in Appendix B. [Note that
our discussion applies equally well to the additional_and optimal statistics. In general, we need E!exp [8 m (cp,ro),.L]J, where 8 m (cp,ro) is given
by eq. (48).] Choosing the test thresholds To and Tl for the ad hoc algorithm according to the Wald SPRT formulae [eqs. (10c) and (10d)]
typically results in PTo < {3 and P T1 < [Y.
VII. SUMMARY

A class of killer-trunk detection algorithms has been developed that
use the individual trunk usage and transition accumulations available
in EADAS/ICUR. Because this data is essentially a sufficient statistic for
the Markov chain used to model the (unobservable) sampled data, one
of the algorithms developed is Wald's celebrated SPRT.
The detection algorithms developed can be partitioned in two natural
ways:
(i) By sampling rate (100 or 200 seconds).
(ii) According to whether grouping information is used.

The algorithms which do not use grouping information are applicable
to all trunks (one way or two way) independent of the type of switching
machine used. A version of one of these individual trunk algorithms is
currently in use in leAN, testing trunks on 1XB, XBT and step-by-step
* See Appendix A.2.1 of Wald's original treatise (Ref. 11).
966

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

switching machines. The algorithms that exploit grouping information
detect killers more quickly but are tailored to a specific switching machine. A "group" algorithm of this type is currently being used to test
trunks associated with 5XB switching machines.
In the course of this study several problems of independent interest
were studied. These include:
(i) The server covariance in a M/G/l-10ss and GI/M/l-10ss system.
(ii) The structure of the likelihood statistic that arises in testing

simple hypotheses characterized by a binary valued Markov chain.
(iii) The occupancy of nonidentical trunks in a random-selection
(Markovian) loss system.
The major conclusion in this study is, of course, that accumulated
switch-count and state-transition data on individual trunks (based on
sampling intervals on the order of a normal holding time) can be used
to reliably detect abnormally short holding time trunks. Moreover the
(near) optimal sequential detection algorithms using this accumulated
data are easily exhibited, simple in structure, and intuitively appealing.
VIII. ACKNOWLEDGMENTS

Various stages of this study benefited from discussions with A. E.
Eckberg, R. L. Franks, S. Horing, E. J. Messerli, and T. E. Rutt.
APPENDIX A
Sensitivity of the Transition Probabilities to Modeling Assumptions

To get an idea about the sensitivity of the algorithms to some of the
modeling assumptions, the transition probability P1,o = P(Xt+T = O/Xt
= 1) was studied for the following two cases:
(i) M/G/l-10ss, where the service distribution function F(·) is the
mixed exponential given by

F(t) = 1 - d1e- x1t - d 2e- x2t

t ~0

(ii) GI/M/l-10ss, where the arrival process is the switched-Poisson
process 8 commonly used to model overflow traffic.

Because the methods used to obtain P1,o for these two models differ,
we discuss these models separately.
A.1 The M/G/1-loss model

An observer viewing the server in an M/G/l-10ss system sees an alternating sequence of busy and idle intervals. The busy intervals are
distributed according to some distribution F(.) and are independent.
TRUNK-DETECTION ALGORITHMS

967

The idle intervals are exponentially distributed with mean A-1 (A = mean
arrival rate) and are independent. Thus, the sequence of alternating busy
and idle intervals constitutes an alternating renewal process. In this
context, the conditional probability
P 1,o = P(Xt+T = O/Xt = 1),

(59)

where
I
x - {
t 0

if server is busy at epoch t
if server is idle at epoch t

has already been studied,18 and we have the following result:

Theorem 4: Consider an M/G /1-10ss system in equilibrium with a service time distribution F(t) having Laplace transform f* (s). If P~,o(s)
denotes the Laplace transform of P 1,o( T), then we have
P* (s) 1,0

-

Jl[I - f*(s)]
sis + A[I - f*(s)]l

(60)

where A and Jl are the mean arrival and service rates, respectively.
Proof: See Section 7.4 of Ref. 18 [t;(s) = A/S + A andf~(s) = f*(s)].
For the mixed exponential service time distribution mentioned above
let
and
X2 =

2Jl(I - d),

0 ~ d ~ 1.

If T is distributed according to this two-parameter (Jl,d) family of distributions, then
(i) E(T) = Jl- 1•

(ii) var(T) = Jl- 2

(2 ~ 0), where 0= 4 d (1 - d).

(iii) c(T) = (J(T) =
E(T)

(2 -0 0) 1/2.

Thus, the mean is fixed at Jl- 1 and the coefficient of variation satisfies
c ~ 1, with equality occurring when 0 = 1, which corresponds to the
M/M/I-Ioss system.
Equation (60) is easily inverted for this family of mixed exponential
distributions, obtaining the following result:
P 1,o(p, T) = (1 - p) +

rl + 0
r2 + 0
e rrllT +
e r2llT ,
r1 (rl - r2)
r2(r2 - r1)

(61)

where
968

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

rl = -

a+2) + [ (I - 0)(1 + a) + (a)2]112
"2
(-2-

r2 = -

(a ; 2) - [ (1 - 0)(1 + a) + (~rr
o= 4d (1 - d),

a = -pI-p

and p = >../(>.. + 11) is the mean occupancy of the server.
In Fig. 5, P 1,O(p,T) [eq. (61)] is plotted vs p for several different values
of c (c = 1, 1.5, and 2) assuming
(i) 11- 1 = 180 seconds
(ii) T = 200 seconds.

Also shown is a plot of P 1,o vs p for a killer trunk with r = 10 [11 in (61)
is replaced by rll]. The normal trunk PI 0 characteristic with c = 1 corresponds to the M/M/l-loss system.

A.2 The GIIMI1-loss model

The covariance function R(·) for the GI/M/l-loss model with a
switched Poisson arrival process has the form:
(62)
where the coefficients Ci and the exponents Wi are messy expressions
involving the three switch parameters w, 'Y, and>" (Ref. 8) and the mean
service rate 11. The derivation of this covariance function is straightforward but tedious and is therefore omitted. (For the switched Poisson
arrival prccess, the Markovian state equations can be solved for
P(Xt,Xt+T)' where Xt = state of server at epoch t.) Our purpose here is
to explain how eq. (62) was used in generating Fig. 4.
If p is the mean occupancy of the server (p = E(xd) and a is the offered
load in a GI/M/l-loss system, then it is easy to show that the peakedness6
z satisfies

P)

1z=a ( p-.

(63)

For GI/M/l-loss system the call congestion is ¢(Il) and is related to the
time congestion (p) by a(1 - ¢(Il)) = p. So using z(J1) = 1/[1 - ¢(J1)] a yields the result. 2 (¢(.) is the L.S. transform of the interarrival time
distribution.) Therefore, specifying p and z uniquely determines a.
Hence, with a and z known, we obtain the equivalent random parameters
and use the three-moment match to obtain the switch parameters (see
TRUNK-DETECTION ALGORITHMS

969

Ref. 8). Using this procedure, we obtain R(r) vs p paramete'rized by z.
Equation (1) then yields Pl,o.
APPENDIX B
The Likelihood Statistic Based on the Observable Data

For ease of derivation, the likelihood statistic derived in Section 4.2
was based on the raw data Xm = (XI,"· ·,x m) rather than on the observable
data [t(m),n(m)]; t(m) and n(m) are defined in Section 2.1. We will now
study i m (t,n) and verify that the two statistics differ only in their endeffect terms. We will also examine the end-effect term based on t(m) and
n(m) and show how it "tracks" the end-effect term based on Xl and
Xm·

We begin by expressing the probability of Xm = (Xl,· ·,x m ) in terms
of t(m), n(m), Xl, and Xm:
Lemma 3: If IXi} is a binary state stationary Markov chain with transition probabilities Po, 1 and Pl,o and if Xm = (Xl,· ·,x m ), then
P(xm ) = P(Xl) X Q(Xl,X m ) X Pt;J(l - P l ,0)n-(t/2)
X Pt/2(1
- P 0,1 )m-n-(t/2) ,
0,1

(64)

where
1-x m )/2 p-(x 1+x m )/2 p-(x 1-x m )/2 p(xJ+x m )/2+1
Q( X b Xm ) = p(x
1,0
1,1
0,1
0,0

(65)

and
P (X 1) =

{p

1- p

~f Xl = 1 .
If Xl = 0

Proof: This result is obtained from lemma 2 using eqs. (12a) and

(12b).
For convenience, we introduce the following notation

(ii) g (~;m - n,Po,l) =

pUr(1 - P O,1)m-n-t/2.

(iii) Px,y(t,n) = P(t(m) = t,n(m) = n,Xl = X,Xm = y).
970

THE BELL SYSTEM TECHNICAL JOURNAL, JULY -AUGUST 1977

(iu) SX,y(t,n) = number of binary m-tuples satisfying t(m) = t, n(m)
= n, x I = x, and Xm = y.

Thus, we can write
I

I

P(t(m) = t,n(m) = n) = L 2: PX,y(t,n)
x=o y=o

(66)

and
Px,y(t,n) = f (~;n'PI'o) g (~;m - n,Po,I) P(x)Sx,y(t,n)Q(x,y).

(67)

Equations (66) and (67) imply that
P(t, n(m) = t, n(m) = n}

{t
t P(x)SX,y(t,n)Q(x,y)} (68)
x=Oy=O

= f (!;n,PI,o) g (!;m - n,Po,I)

2

2

and therefore it is easily seen that:
... (

)_ I

P*(t(m) = t,n(m) = n)

em t,n - og P(t(m) = t,n (m) = n )'
=

[a~-an] + [(J~-b(n-m)] +E(t,n),

(69)

where

E(t,n) = log

{

I

I

x~Oy~o p*(x)sx,y(t,n)Q*(X,y)}
.
I

(70)

I

2: 2: P(x)Sx,y(t,n)Q(x, Y)

x=o y=o

Comparing eqs. (70) and (22), we see that im(t,n) and i(xm) differ
only in their respective end-effect terms. The following result is perhaps
a bit surprising:
Lemma 4: e(t,n) = f(x m) if t is odd.
Proof: todd ==> Xl :;6: Xm ==> e(O,l) = e(l,O) = a + b/2 (see eq. (22b)). todd
==> So,o(t,n) = SI,I(t,n) = so E(t,n) for t odd may be written:

°

(1 - p)So IQ*(O,l) + pSI oQ*(l,O)
,
E( t n ) = I o g '
,
(1 - p)So,IQ(O,l) + pSI,oQ(l,O) .

Now using (4) E(t,n) for t odd can be manipulated into the following
form
E(t,n) = log (1 - p)So,IQ*(O,l) + pSI,oQ*(l,O) .
(1 - p)So,IQ(O,l) + pSI,oQ(l,O)
TRUNK-DETECTION ALGORITHMS

971

but PO,t/P 1,0 = P~,IIP~,o = p/1 - p which completes the proof.
If t is even then Xl = X m ,
e(O,O) = b

and
e(l,l) = a.

For even t, eq. (70) can be written as

E( t,n ) -- I og {

8 0 ,0 + 8 1,1

)8 +( 1 - P )8
(11- POI
P

p

0,0

10

}.
1,1

Thus, for even t, E(t,n) is a complicated function* of t and n. Note
however that E(t,n) ~ 0 and
a
E(t,n) = { b

if 8 0 ,o(t,n) = 0
l'f 8 1,1 (t,n ) = 0

It can be shown that

t
n--

81,l(t,n) = _ _ _2_
SO,o(t,n)

t

m -n--

2

if (t,n) is such that 81,1 and 8 0 ,0 are nonzero.

APPENDIX C
The End Effect EI/(xim;xim+1)1

The end effect
EII(xim;xim+dl = H(Xim+l) - H(Xim+t/Xim),

where

and
H(Xim+1), H(Xim+t/O) and H(Xim+t/1)

are the binary entropy function ,71 (x) evaluated at p, P O,1 and P 1,o, respectively (7I(x) = -x log x - (1 - x) log (1 - x), 0 ~ x ~ 1).
Table I exhibits EII(xim;xim+1)1 and Eli(xm)l as a function of the
trunks occupancy p-for a normal trunk, with a single-hour accumula972

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

Table I -

End effect as a function of occupancy

(T = 100 seconds)

(T = 200 seconds)

p

EIl(xim+ l;Xim)\

Elf(xm}l

EII(xim+l;xim}1

Elf(xm}l

0.10
0.20
0.30
0.40
0.50
0.60

0.04
0.05
0.04
0.03
0.02
0.01

-1.36
-1.60
-1.50
-1.21
-0.84
-0.47

0.01
0.01
0.01
0.01
0.00
0.00

-0.22
-0.20
-0.15
-0.09
-0.04
-0.01

tion period (m = 36 and 18 for the 100- and 200-second sampling options,
respectively). The M/M/l-loss model is used with I/J.l = 180 seconds. It
is clear that the mean end effect is negligible compared to the mean
statistic update.
-

APPENDIX D
Occupancy Formulae for a Random-Selection Loss System

Theorem: Consider an N server Markovian loss system with random
selection of idle servers and
(i) N - 1 servers with mean service rate J.l.
(ii) 1 server with mean service rate rJ.l (r > 0).
(iii) Mean arrival rate A.
Let Pr and P; denote the mean occupancy of the servers with mean
service rates J.l and rJ.l, respectively, and let B denote the blocking (call
congestion). Also let </J and </J - denote the carried load per server in an
N server and N - 1 server Markovian loss system, respectively, (assuming all servers have rate J.l)' given an offered lead a = AI J.l. That is,
</J = a[1 - B(N,a)]!N and </J- = a[1 - B(N - l,a)]!(N - 1),whereB(· , .)
is the Erlang blocking formula. Then
( l.) P* =
r

</J

(71)

r - (r - 1)</J

(ii) B(N,a,r) =

B(N,a)
[B(., .) is the Erlang
r-(r-l)</J

(72)

blocking formula].

Proof:

... )
(r - (r - I)[NI(N - 1)]</J)
(Ul
</J ~ Pr ~
</J.
r - (r - 1)</J

(73)

(W.) Pr = (r-(r-l)</J-) </J.
r - (r - 1)</J

(74)

TRUNK-DETECTION ALGORITHMS

973

(i) This part of the theorem follows immediately from eq. (34b)

1

*

Pr =

rN

1+a

- r[l - B(N - 1,a)]

by using the well known Erlang B recursion 6
B(N,a) = ~ B(N - 1 a)
1-B(N,a) N
'

and the expression for the mean group occupancy
a[l - B(N,a)]

~--'----'-----'--"-=¢

N

.

(ii) This part of the theorem follows immediately from eq. (34a) by
using the above expression for mean-group occupancy.
(iii) The lower bound part of eq. (73) follows from eq. (34c) by noting
that 13(N,a,r) ~ B(N,a)t and using eq. (71) of this theorem. Thus, we
obtain
r¢ ) + (N - l)Pr ~ a[l - B(N,a)] = N¢
r - (r - 1 ¢

which can be arranged to yield the lower bound in eq. (73). The upper
bound in (73) is an immediate consequence of (74) since ¢- ~ ¢. Thus,
it remains to prove (74).
(iv) We prove this part as follows: Using eqs. (71) and (72) in the
conservation eq. (34c),
rp; + (N - l)Pr = a(l -

B)

(75)

yields
a(r - 1)(1 - ¢) + (N - r)¢

P =
r

(76)

(N - l}[r - (r - 1)¢]

Therefore, eq. (74) holds if and only if
)
[r - (r-1¢

_]

X¢=

a(r-1)(1-¢)+(N-r)cp
(N - 1)

.

(77)

The right-hand side of eq. (77) can be rewritten as:
rhs =

a
N(N - 1)

{r(N - 1 - a) + a - B[N - (a + l)r + all

t This follows eq. (34a) by noting that a[l -

974

(78)

R(N,a)j ~ N.

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

using
a[l - B(N,a)]

¢=

N

.

Similarly, using ¢- = a[l - B(N - 1,a)]/(N - 1), the left-hand side of
eq. (77) can be written as:
lhs = Ir - (r - l)(a/N - 1)[1 - B(N - 1,a)]j(a/N)[1 - B(N,a)]. - (79)
Now using the Erlang B recursion formula in eq. (79) and rearranging
terms yields eq. (78). Thus, eq. (77) and consequently eq. (74), of the
theorem is proved.
APPENDIX E

The Sign of the Mean Statistic Update

The mean statistic update of the optimal 5XB group algorithm is given
by eq. (51a):
A

-

P*(X1)

-

Elf(xm)l = (m - 1)I(aP1,o - a)p + (b - (3Po,d(l - p)l + log P(X1) .

The mean update of the ad hoc and the additional statistics correspond
to 6 = ~ = 0 and a = a = 0, respectively. The following lemma is needed
to study the sign of Eli(xm)l:
Lemma 5: If 0 < q < p < 1 define

q)

1-

p

a = log ( - - and a = a + log - ,
1-p
q
then

a
q <- <po
a

Proof: Consider the log likelihood ratio statistic i(m) for this simple
hypothesis testing context: the observed process is n(m) = L~l Xi, where
Xi are i.i.d. Bernoulli random variables with P(Xi = 1) given by q and p
under H and H 1, respectively. Therefore,

°

f(m) = log
A

b(n;m,p)
b(n;m,q)

= an - am

and hence the mean of i(m) is m(aq - a) under H o and m(ap - a) under
HI. But as we noted in Section 6.2, Elfl is negative under H o and positive
under HI in a general discrete setting. * Thus, we must have
aq - a < 0
TRUNK-DETECTION ALGORITHMS

975

Copyright '0 1977 American Telephone and Telegraph Company
THE BELL SYSTEM TECHNICAL ,JOUHNAL

Vol. ,'i6, No.6, ,July-August 1977

Printed in U.S.A.

Cancellation of Polarization Rotation in an Offset
Paraboloid by a Polarization Grid
By T. S. CHU
(Manuscript received July 20, 1976)

The polarization rotation properties of the field radiated from a
polarization grid have been found to be similar to those in the aperture
of an offset parabolic antenna. This observation suggests broadband
cancellation of the polarization rotation in a large offset reflector by
the opposite prerotation of the incident feed radiation via a polarizationgrid. Reduction of cross polarization from -24 dB without the grid
to -39 dB with the grid wires is predicted in a numerical example. A
previously unexplained polarization rotation measured using an offset
parabolic grid is shown to be in good agreement with calculation.

I. INTRODUCTION

Current interest in frequency reuse through orthogonal polarizations
creates a strong incentive for improving polarization properties of antennas. Cross-polarized radiation from an offset reflector! is often regarded as a blemish on an otherwise excellent antenna, which offers both
low sidelobe level and good impedance matching. 2 Although the cross
polarization can be minimized using a large effective f /D ratio, the corresponding requirements of small offset angle and large feed aperture
are not always convenient in applications. Recently a trimode feed horn 3
has been discussed as a means of reducing the cross-polarized radiation
of offset reflectors; however, a multimode arrangement is inherently
narrow in bandwidth. Our purpose here is to propose a broadband
method for reducing the cross-polarized radiation of the offset reflector
antenna. This scheme is based upon cancellation of polarization rotation
due to reflector curvature! by opposite rotation generated by a polarization grid. 4
In Section II we examine the similarity between cross polarization in
the aperture of an offset reflector and that of a polarization grid. The
condition of cancelling the first-order cross-polarization terms is then
977

Xp

~-

I

I
Ii
I
I

-7POLARIZATION VECTOR
/'
ROTATES IN THE
SAME DI RECTION

2a

I'

i
I

!

I ,I'
~

/

L ___--=__~__=-_________' -/-'_ _

f

REFLECTOR
APERTURE

xp,
x'

«.«.:_<v<::>_

~-

Be

~

\

, ;!:5-~'e\--------'-----.yp

Y'

IY,

_ _ __

z'

'

-~

~

\
"

REF LECTOR AXIS

I

~---------~----------J
Fig. I-Geometry of offset reflector.

deduced. In Section III numerical calculations of practical examples are
given.
In the appendix the previously unexplained measured polarization
rotation* of a parabolic grid 5 is calculated. This example is given here
as experimental evidence of agreement with the predicted rotation of
radiation from a polarization grid.
II. CANCELLATION OF POLARIZATION ROTATION

Let us first briefly review the salient properties of the cross-polarized
field in the aperture of an offset paraboloid as shown in Fig. 1. For a
balanced feed radiation,
COS ¢'

Ef = F(8', ¢')

()' 1=
[

sin ¢'

sin ~'] exp(-jp)
¢'

p

,

(1)

cos ¢'

where (p, 8', ¢') are spherical coordinates with respect to the z' axis and
(1)', ¢') are the corresponding unit vectors. Since the reflected field from
the paraboloid is

* The design was initially proposed by Comsat for the Comstar satellite.
978

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

where n is a unit vector normal to the reflector surface. The principal
and cross-polarized field components in the reflector aperture can be
written respectively:1

xp

M = E r ." =

YP

F(()', ¢') [. 0' . 0
,. 2
sm sm 0 cos 1> - sm ¢'
tp

. (cos ()o + cos ()') - cos 2 ¢'(1 + cos ()o cos ()')]

(2)

YP
F(()',¢') [. 0' . () . ,
N = E r ." = =t=
sm sm 0 SIn ¢
Xp
tp

- sin ¢' cos ¢'(1 - cos 0')(1 - cos ()o)],

(3)

where t = 1 + cos 0' cos 00 - sin 0' sin ()o cos ¢', M2 + N2 = F2j p2, and N
vanishes when 00 = o. The offset angle ()o is between the feed axis and the
reflector axis. The sign combination in eqs. (2) and (3) indicates that the
rotation of the polarization vector due to offset in a paraboloidal aperture
has the same magnitude and is in the same direction as illustrated in Fig.
1 for any orientation of the incident linear polarization. The projection
of the intersection of a circular cone (with vertex at the focus) and the
offset paraboloid onto the xpYp plane is a circular aperture with center
Xc = ---'----.::....--

2{ sin ()o
cos ()o + cos ()c

(4)

2{ sin Oc
cos ()o + cos ()c '

(5)

and radius
a=----=-----=------

where Oc is the half angle of the cone. Equations (4) and (5) will be used
later to obtain the relations in eqs. (12) and (13).
Radiation from transmitting and reflecting wire grids can be obtained
by magnetic and electric current sheet models, respectively. The principal and cross-polarized components are 4
p = -C [1 - cos 2 ¢' (1 - cos 0') - sin ()' cos ¢' tan 0]

(6)

X = ±C[sin ¢' cos ¢'(1 - cos 0') + sin ()' sin ¢' tan 0],

(7)

where C is a proportionality constant, 0' and ¢' are the spherical coordinates of the feed (z') axis, and 0 is the angle between the conducting
wires and the x'y' plane as shown in Fig. 2; the expressions inside the
brackets are identical to those of eqs. (9) and (10) in Ref. 4, provided one
makes the following substitutions: ¢' = ¢ - 90° and 0 = 90° - 'Y. The
changes of notation are made for the purpose of comparison with eqs.
(2) and (3). The upper and lower signs in eq. (7) correspond to the
transmitting and reflecting cases. The orientations for the transmitting
PARABOLOID ANTENNA

979

XI'

\

X'

\
\
\

\

TRANSMITTING---- y,-7'&--r---I,,----------l~Zp _ _ _ _ ~
POLARIZATION
"
VERTEX OF
",- REFLECTING

PARABOLOID

POLARIZATION

POLARIZATION GRID WITH
CONDUCTING WIRES
PARALLEL TO THE x' z' PLANE

Fig. 2-Configuration for cancellation between polarization rotations of an offset paraboloid and a polarization grid.

and reflecting polarizations together with a given grid geometry are
shown in Fig. 2 where the conducting wires are parallel to the plane of
the figure.
One notes that the leading terms, which contain first power of ()', in
eqs. (3) and (7) have the same sinusoidal dependence on ()' and ¢'. Furthermore, the sign combination in eqs. (6) and (7) indicates that the
transmitting and reflecting orthogonal polarizations rotate in the same
direction, opposite to the rotation in the aperture of an offset paraboloid.
Let us take the first-order approximation-i.e., cos ()' ~ i-in eqs. (2),
(3), (6) and (7), but sin ()' is retained. Then the cross polarization in eq.
(3) normalized with respect to eq. (2) cancels that in eq. (7) normalized
with respect to eq. (6):
)(]V
P M

.

()o

- + - = 0 If 0 = - .
2

(8)

Polarization rotation of the radiation from a wire grid, as predicted
by eqs. (6) and (7), also explains data measured using a cylindrical re980

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

flector made of a curved wire grid. The report on this experiment5 misinterpreted the polarization rotation as a consequence of diffraction and
the offset geometry. A comparison between the calculated and measured
polarization rotations of this case is given in the Appendi~.
III. NUMERICAL EXAMPLES

Since the cancellation of polarization rotation discussed in the preceding section only eliminates the leading terms, it is of interest to determine the residual cross polarization. Assuming that the offset reflector
is located in the far zone of the radiation from a wire grid, as'shown in
Fig. 2, the principal and cross-polarized components in the reflector
aperture can be written
P = F(t9')[l - cos 2 ¢p (l - cos 8p ) + sin 8p cos ¢p tan (0]

(9)

X = =rF(8') [sin ¢p cos ¢p (1 - cos 8p ) - sin 8p sin ¢p tan (0], (10)

where F(8') is the feed-radiation pattern and
8' = COS-I [cos 8p cos 80 + sin 8p sin 80 cos ¢p].

(11)

The above equations are simply a decomposition of the grid radiation
into the two orthogonal components of a balanced feed whose axis
coincides with the paraboloidal axis. The expressions inside the brackets
of eqs. (9) and (10) are of the same form as those of eqs. (6) and (7); but
8p and ¢p are the spherical coordinates with respect to the paraboloidal
(zp) axis instead of the feed axis, and (0 = (80 - 0) is the angle between
the conducting wires and the xPyp plane, as shown in Fig. 2.
To relate eqs. (9) and (10) to the normalized aperture coordinates r
= (Pala) and ¢a, the following expressions can be obtained with the aid
of eqs. (4) and (5):
8 = 2 tan-I

[v(r sin 8e sin ¢a)2 + (sin 8 + rsin 8e cos ¢a)2 ]
0

cos 8e + cos 80

p

(12)
¢p = tan

-I [

r sin 8e sin ¢a
]
.
sin 80 + r sin 8e cos ¢a

(13)

Numerical examples of several combinations of parameters (80 , 8e and
(0) have been calculated for the principal and residual cross-polarization
components from eqs. (9) and (10). The feed pattern has a gaussian shape
with 10-dB taper at the edge of the reflector. The principal polarization
is close to unity (0 dB) around the center of the reflector aperture. The
maximum, calculated, residual cross polarization is given in Table I for
a number of examples. Fig. 3 shows a plot of both principal and cross
polarizations for the case 80 = 50°, 8e = 20°, and (0 = 25°. Only half of the
PARABOLOID ANTENNA

981

Table I 00

Cross Polarization in the aperture of an offset reflector

(Deg)

Max. Residual Cross
Pol. With Grid
(dB)

Max. Cross Po1.6
Balanced Feed
Without Grid (dB)

25
23
27
30
30
45
45

-38.6
-36.4
-36.1
-38.0
-30.9
-34.3
-40.5

-24.0
-24.0
-24.0
-22.5
-18.0
-17.5
-20.0

0('

f

(Deg)

(Deg)

50
50
50

20
20
20
20
30
20
14

60
60
90
90

aperture needs to be shown for each polarization because of the symmetry. The maximum residual cross polarization in this case is -38.6
dB, reduced from -24 dB for the same reflector aperture illuminated
by a balanced feed without a polarizer. Keeping the same set of parameters, 00 = 50° and Oc = 20°, the calculated residual cross polarization
PRINCIPAL POLARIZATION ( d B ) - \ -

CROSS POLARIZATION (dB)

Fig. 3-Relative amplitude levels in the aperture of an offset paraboloid. 00 = 50°, Oc

= 20°, f = 25°.
982

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

becomes -36.4 dB for E = 23° and -36.1 dB for E = 27°. These results
indicate that the residual cross polarization is not overly sensitive to a
slight departure from the optimum orientation of E = 00 /2.
The examples for 00 = 60° and E = 30° show residual cross polarizations of -38.0 dB and -30.9 dB for De = 20° and 30°, respectively. The
second-order terms are not quite negligible at Oe = 30°; however, the
cancellation of cross polarization appears to be significant.
IV. DISCUSSION

In view of the residual second-order (1 - cos 0') terms, the half-cone
angle of the reflector subtended at the focus should not exceed about
20° in order to take full advantage of the cancellation. If the reflectors,
such as in an offset cassegrain configuration, do not cause significant
cross polarization, the conducting direction of a polarization diplexing
grid should be oriented to avoid introducing any polarization rotation,
as discussed in Ref. 4.
When 00 - Oe is less than about 30° and E = 0 = 00/2, the feed horn
associated with the polarization reflected from the grid will introduce
some blockage, as shown in Fig. 2. The blocking problem can be eased
by using a smaller value of E. This practical difficulty may prevent optimum orientation of the grid wires for reflectors of small offset angle,
and hence reduce the effectiveness of the cancellation. The practical
application of this scheme should indeed lie in reflectors with large offset
angle.
The expression E = 00/2 implies that the grid wires are approximately
. parallel to the tangent plane at the center of the offset reflector. One
notes the similarity between this case and a symmetrical small-coneangle paraboloid illuminated by a grid-covered feed.
ACKNOWLEDGMENT

The author is indebted to M. J. Gans and D. C. Hogg for valuable
discussions, and to Mrs. Diane Vitello for assistance with the computation.
Appendix
Polarization Rotation of an Offset Parabolic Grid

An offset cylindrical-reflector system fed by two line sources of pillbox
type (as shown in Fig. 4) was proposed 5 as a dual-polarized antenna with
an elliptically shaped coverage pattern. The reflecting system consists
of a vertically polarized grid attached to the surface of a parabolic cylinder, the front surface of the grid having the same curvature as the cylindrical reflector. The measured data 5 showed excellent orthogonality
(within 1°) between vertical and horizontal polarizations over the whole
PARABOLOID ANTENNA

983

BEAM DIRECTION

... -

_ - OFFSET
PARABOLIC
CYLINDER

POLARIZED GRID

GRID-- __
FOCAL LINE

t,

x

'/

VERTICALLY - - POLARIZED
LINE SOURCE

---

z
/

~~//j\

I,",
\
\

--PARABOLIC
CYLINDER
FOCAL LINE

.__--GRID
VERTEX

+""'-- PARABOLIC
CYLINDER
VERTEX

I
HORIZONTALL Y
POLARIZED
LINE SOURCE

Fig. 4-Configuration of a dual-polarized cylindrical reflector antenna (Ref. 5).

6.8° X 3.4° (3 dB) elliptical beam. However, significant polarization
rotations, in the same direction for both polarizations, were observed.
Maximum rotations of about 2° occur on the major axis (xz plane) of the
half-power ellipse of the beam. No adequate explanation was given for
this measured polarization rotation. Here we explain the rotation using
electric and magnetic current sheet models for the grid.
The first-order approximation of polarization rotation by a wire grid
is simply sin () cos ¢ cot 'Y (eq. 7 in Section II or eq. (10) in Ref. 4), where
ois the angle off the beam axis, cos ¢ is unity in the xz plane of maximum
rotation, and 'Y the angle between the conducting wires and the beam
axis. Since the wires of the curved grid have a variable direction, the
rotation ~ in the plane of maximum rotation can be obtained by averaging cot 'Y over the parabolic curve, y2 = 4{z.
.
sin () f E dz V (dy)2 + (dz)2
sm()
dy
~ = - - f E cot 'Y ds = ---;:-=-~=:::::::::======:;--fEds
fEv(dy)2+(dz)2.

(14)

where E is the aperture field distribution. Let u = y/2{, eq. (14) be
transformed into
984

THE BELL SYSTEM TECHNICAL JOURNAL, JULY -AUGUST 1977

Fig. 5-Measured angles of nominal vertical and horizontal polarizations around -3-dB
pattern contours of 6.8 0 X 3.4 0 for a dual-polarized cylindrical antenna with a parabolic
grid; + indicates locations of measurement.

r

sin ()

1

E Vf+U2 u du

JO.1763

(15)

~=-----------1

r

E Vf+U2du

JO.1763

where the upper and lower limits of integration are obtained from y =
2f tan 1/;/2 with 1/; = 90° and 20°, respectively. ~ is not sensitive to the
aperture distribution. Numerical calculation gives ~ = 0.61 sin () for both
cases when E is assumed to be uniform and when E has a 10-dB edge
taper-i.e., when
E = -0.396 + 4.746

(;f) - 4.034 (;f)

2.

(16)

The quadratic form has been chosen to perform the integration in closed
form. When () = 3.4°, ~ = 0.0362 rad = 2.07°; agreement between this
calculated value and Wilkinson's measured rotation shown in Fig. 5 is
indeed very good.
Furthermore, eqs. (6) and (7) in Section II indicate that the transmitted and reflected orthogonally polarized fields from the grid rotate
in the same direction, just as in the measured rotations. The planes of
measured maximum and null rotation also agree with the predictions.
Since the two polarizations have the same sense of rotation, orthogonality
is preserved. However, if this rotation occurs in the feed radiation,
nonorthogonal elli ptically polarized radiations from the reflector will
result.
REFERENCES
1. T. S. Chu and R. H. Turrin, "Depolarization Properties of Offset Reflector Antennas,"
IEEE Trans. Ant. Propag., AP-21 (May 1973), pp. 339-345.

PARABOLOID ANTENNA

985

Copyright © 1977 American Telephone and Telegraph Company
THE BELL SYSTEM TECHNICAL JOURNAL

Vol. 56, No.6, July-August 1977

Printed in U.S.A.

Ten Years of Power Aging of the Same Group
of Submarine Cable Semiconductor Devices
By A. J. WAHL
(Manuscript received January 18, 1977)

The active devices in the first Bell System transistorized submarine
.cable system (SF) are unpassivated, diffused-base, germanium transistors and oxide-passivated, silicon diodes. At the date of this writing
these devices have accumulated over 550 million device hours of powered operation in service and on the aging rack without failure or impaired device performance for a demonstrated failure rate of less than
0.00045 percent per thousand hours (4.5 FITS) with 90-percent confidence. This paper reports details of the behavior of 500 of these devices
that have reached ten years of controlled power aging. The overall results indicate that initial reliability objectives are being achieved and
that the semiconductor devices should not be expected to limit the
desired life span of SF submarine cables.
I. INTRODUCTION

In the initial stages of the project, the decision to use germanium
transistors in the first Bell System transistorized submarine cable was
a rather bold gamble to take advantage of their then superior frequency
capability while also attempting to mitigate any reliability inferiority
to silicon transistors. When the decision was made, in the time period
of 1960-61, silicon transistors were clearly the wave of the future and,
although still inferior to germanium transistors in frequency capability,
had already indicated their reliability superiority. Nevertheless, the
results reported here, together with the results of regular aging and actual
service experience thus far, offer rather convincing evidence that the
desired reliability objectives are being achieved. The first submarine
cable using these transistors, a relatively short cable with 136 repeaters,
has been in service over eight years. A transatlantic cable with 363 repeaters has been in service nearly seven years. No semiconductor device
failure nor degradation toward failure has yet been observed in any of
the systems that have no redundancy and in which the failure of even
987

42r-----------------------------------------~----~

~r

40

l?

Z

l?

<t:

a:

~ 36
<t:

/

34

/

~-

/

/

....

Jt'

-

,
38

,1'

,-

"

•

- LINE OF NO CHANGE

32

BEFORE AGING

Fig. I-Colledor-base breakdown volLage in volLs aL 100 /lA for L2287 transistorsbefore and after 10 years of power aging at 75 mW.

one device could disable the system. At the time of this writing the device-hours of service in these cables totals more than 250 million for both
transistors and diodes. On the aging rack these devices have accumulated
over 300 million device-hours of power aging-also without failure. On
the basis of these results the demonstrated failure rate, with 90 percent
confidence, is less than 0.00045 percent per thousand hours (4.5
FITS).

The method of providing semiconductor devices for the SF submarine
cable system has been described previously.1 As a further part of this
program, a hundred devices of each of the five basic device codes of
transistors and pn diodes have been retained on extended power aging.
A milestone was observed in this program when the aging time on these
devices reached ten years. The primary purpose of this paper is to show
in a condensed way the behavior of these devices across ten years of
power aging.
The transistors are of diffused-base germanium with no passivation
or junction protection on the active chip. The diodes are of diffused
silicon with grown oxide passivation over the junctions. All devices are
encapsulated in hermetic packages.
988

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

1.4~-----------------------------------------------------'

-

./
/.

1.2 -

-

;/

1.0 -

./
,/

(.:J

z

0.8-

(.:J

-

.;/

~ 0.6-

. .L---

«

cc

UJ

I-

/,1'

0.4 I-

0.2f-

LINE OF NO CHANGE

/

1-/
0/1 I
o

/

I

0.2

I

0.4

I

I

0.6

I

I

0.8

I

I

1.0

r

I

1.2

I

1.4

BEFORE AGING

Fig. 2-Collector-base reverse leakage current in J.1A at 15 V for L2287 transistorsbefore and after 10 years of power aging at 75 m W.

II. DATA COLLECTION

All transistor parameters are measured before the devices are placed
on long-term power aging, which is a period of about six months for
candidate devices. From previous work with devices of this kind, we
found that device current gain was the only parameter expected to show
any consistent aging trend, and it is also the most critical parameter for
the repeater amplifier circuit. For practical testing reasons during
long-term aging, the dc current gain hFB was measured. From the
standpoint of the repeater circuit, however, the critical parameter is the
low-frequency ac current gain hfb rather than the dc current gain h FB .
The two are not the same, but they bear a relationship to each other, as
described in any standard text on transistor theory. If one shifts, the
other shifts in the same direction.
POWER AGING

989

2.1

r--------------------------,

1.9

1.7

l:J

~ 1.5

l:J

«
cr:
w

I-

~ 1.3

1.1

0.9

0.9

1.1

1.3

1.5

1.7

1.9

2.1

BEFORE AGING

Fig. 3-Emitter-base breakdown voltage in volts at 100 /-lA for L2287 transistors-before
and after 10 years of power aging at 75 mW.

Previous work with transistors of this kind had also indicated that any
interruption of the bias on the device would cause the current gain to
suffer a disturbance that would tend to obscure the real aging trend. For
this reason, during long-term power aging, only the dc current gain was
measured, and the measurement was made very precisely without disturbing the bias on the device. From the standpoint of device parameters
directly relevant to circuit performance, therefore, the complete picture
of the aged transistor cannot be known until the device is removed from
long-term aging and all device parameters have again been measured.
For such transistors, the aging time interval is ten years rather than six
months.
Since the aging trends of diode parameters were not disturbed by
change of electrical bias for testing purposes, most of the sensitive diode
parameters were tested during aging. The diodes, therefore, were not
removed from the aging rack at the end of the ten-year aging time. Interim aging information on these diodes has already been published. 2
990

THE BELL SYSTEM TECHNICAL JOURNAL, JULY -AUGUST 1977

Fig. 4-Ernitter-base forward voltage in volts at 15 rnA for L2287 transistors-before
and after 10 years of power aging at 75 rn W.

III. PRESENTATION OF THE AGING RESULTS

A rather simple and clear picture of device aging behavior can be
shown in the scatter plot in which the initial value of a device parameter
is plotted vs the final value across the aging period for each device. For
the transistors in this case, the information obtained at times between
the initial and final values is not available for reasons discussed previously. For the diodes the information is available, but not significant
since general aging trends, where they exist at all, are monotonic. For
simplicity and consistency, therefore, all the ten-year aging results are
shown here in the form of scatter plots.
IV. COMMENTS ON THE RESULTS

4.1 General

In most cases where any detectable parameter shifts have occurred
across ten years of aging, the magnitudes of the shifts have been small
POWER AGING

991

o.994,...-----------------------r--"-]

0.990
END OF LIFE LIMIT FOR - __ _
AVERAGE OF ALL DEVICES

0.986
<.::I
Z
<.::I

<l:

a:
~

0.982

u..

<l:

0.978

----LINE OF NO CHANGE

0.974

0~1~0-.9~7-4-~-0.~97-8-~-0-.9~8-2-~-0~.9~86-~-0-.9~9-0-~-0~.994
BEFORE AGING

Fig. 5-Low-frequency common-base current gain for L2287 transistors-before and
after 10 years of power aging at 75 mW.

and very uniform among all the devices, almost as if the devices had
shifted in unison. Such behavior immediately raises the suspicion that
the apparent shift may actually be an offset in testing rather than a real

z

~

<.::I

tz
w

0.9820

a:
a:

::J

u

w

~

0.9818

en

Z

o

~
~

o
u

0.9816

ol,.~--~~--~~--~~--~~--~~
o
0.5
1.5
2.5
3.5
5.5
TIME IN YEARS

Fig. 6-Typical behavior of dc common-base current gain for diffused-base germanium
transistors on long-term aging.

992

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

6~------------------------------------------~

I-

51-

f---

.. ..

4f---

t:J

· . : ..
..
. .. ·..·· ··· ...

I-

Z

I : : : ::

t:J

~ 3 f---

LlJ

t;::
«

I-

21-

f---

If---

I-

o

I

I

o

I

I

I

I

I

3

I

I

I

I

6

4

BEFORE AGING

Fig. 7-Noise figure in dB at 5.5 MHz for L2287 transistors-before and after power
aging at 75 mW.

shift. Unless the testing offset can be definitely confirmed, however, the
shift must be accepted at face value. In the latter case, the uniformity
of the shift makes the extrapolation very simple for more devices and
longer time. The extrapolated result is then quite obvious and comfortable. Possibly the only deviation from this situation is the behavior
of the leakage current of the L2320 diodes. Further discussion of this case
follows under the heading of that particular diode.
4.2 L2287 transistors

L2287 transistors are split into 38A, 38B, and 38D codes for use in the
first two stages of the amplifier and in the oscillator of the SF repeater.
The two amplifier stages operate at power levels of 25 and 75 mW, respectively. The oscillator operates at a lower power level. During aging,
all of these devices are held at a power level of 75 m W, com posed of 5 V
and 15 rnA. The extended aging program has included one hundred of
these transistors.
As shown in Fig. 1, the collector-base breakdown voltage of these
transistors showed almost no discernible change after ten years of power
POWER AGING

993

1.8,-------------------------,

/

1.6

./

1.4

~c;J
~t!

l'J
Z
l'J

<!

cr:
w
f-

/"

,.

1.2

l.L

<!

/

1.0

/

/

~----LINE

OF NO CHANGE

0.8

o~~~__~~~__~~__~~~~
o

0.8

1.0

1.2

1.4

1.8

BEFORE AGING

Fig. lO-Ernitter-base breakdown voltage in volts at 1 rnA for L2288 transistors-before
and after 10 years of power aging at 0.99 W.

aging, with the exception of one device which showed a decrease of about
1.5 V. This device showed an increase in collector-base leakage current
of about 0.15 J.£A, as seen in the lower part of Fig. 2, and a decrease in the
emitter base forward voltage of about 5 m V, as seen in the lower part of
Fig. 4. These changes, however, would have no effect whatever on the
operational reliability of the device. No attempt has been made to determine the cause of the changes since all parameters of the device are
still well within initial specification limits.
Figure 2 shows that the lower values of collector-base reverse-leakage
current show no discernible change while those above about 0.5 J.£A show
a very slight upward shift. Such a small shift, however, is of no practical
significance.
The emitter-base breakdown voltages of most of the devices show
remarkable stability, as depicted in Fig. 3. Only those above about
1.6 V show a small amount of scatter in the behavior. With the one exception, previously noted, the emitter-base forward voltages showed
essentially no discernible change after ten years of power aging, as seen
in Fig. 4.
Perhaps the most gratifying result of this extended aging program is
the behavior of the low-frequency common-base current gain h fb , as
POWER AGING

995

0.425 r - - - - - - - - - - - - - - - - - - - - - - - - - - ,

/

/.

0.415

LINE OF NO CHANGE----7·

/..
/,.

0.405

/r:

t9
Z

t9

«

/"

a:
w

t;: 0.395
«

/.'~

/-0

0.385

/Ji
0.375 /

•

O~.~~--~~~~~~--~~~~
o 0.375
0.385
0.395
0.405
0.415
0.425
BEFORE AGING

Fig. ll-Ernitter-base forward voltage in volts at 90 rnA for L-2288 transistors-before
and after ten years of power aging at 0.99 W.

shown in Fig. 5. From the beginning, this parameter was expected to
show a significant aging trend. Even under the conservative assumption
that the aging of this parameter is linear with time, Fig. 5 indicates that
in ten years of aging the average of the parameter has shifted only about
a quarter of the way to end-of-life for the group. Actually, the aging of
this parameter is known to occur at a decreasing rate with increasing
time, as shown in Fig. 6 for a typical case of the aging of the dc common-base current gain. With similar behavior of the ac common-base
current gain, the end-of-life for the group of devices would appear to be
at some indeterminate time far beyond forty years.
Only one-third of the L2287 transistors (those for the first stage of the
amplifier) are required to meet the initial noise figure specification limit
of 4 dB at 5.5 MHz. Figure 7 shows that in fact nearly all of these devices
still meet this limit after ten years of power aging. In the evolution and
refinement of techniques for measuring the noise figure over the ten-year
period of extended device aging, it was more important to provide the
best and most realistic noise figure data than to maintain reference to
the past. Other experience in aging indicates that if the other device
parameters remain well-behaved, the noise figure does not change.
996

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

0.990.-----------------------------,

...

0.988

END OF LIFE LIMIT FOR-- - _
AVERAGE OF ALL DEVICES

0.986

t9

<t:
LL

0.982

<t:

r

0.980

~.

~----LlNE
0.978

0.976

./

./
.7
./

t9

I-

. ...

.;?

0.984

~

cr:
w

1/

OF NO CHANGE

.-;7

./

BEFORE AGING

Fig. 12-Low-frequency common-base current gain for L2288 transistors-before and
after power aging at 0.99 W.

4.3 L2288 transistors

The L2288 transistors, coded as 38e and operating at a power level
of 0.99 W, are used in the output stage of the amplifier in the SF repeater.
During aging, all of these devices are held at this same level of power,
composed of 11 V and 90 rnA. The extended aging program has included
one hundred of these devices.
Figure 8 shows that the collector-base breakdown voltage of these
transistors has very uniformly shifted downward by about a quarter volt
after ten years of power aging. This is the kind of result which arouses
suspicion that the apparent shift may actually be due to an offset between tests conducted over ten years apart. However, since the fact of
testing offset could not be definitely established, the shift must be accepted at face value. Even if it is real, such a shift is of no practical significance. Under a simple linear extrapolation the voltage would be decreasing at the rate of 1 V in forty years, and a decrease of at least 30 V
would be required before even the first device would begin to impair
system performance.
POWER AGING

997

48~-------------------------------.

LINE OF NO CHANGE --'-~-7/

~

., /

".:/
..~. .y

461-

.-:

t:l

O~

Z

t3 44-

«

O~}

a:

UJ

l-

-

ll.

NO;

«
42 -

, :7
40~

~/~_______IN_I_TI_A_L_SP_E_CI_F_IC_AT_I_ON
__
Ll_M_IT__~

/

o~,
o

I

40

I

I

42

I

I
44

I.

I
46

I

48

BEFORE AGING

Fig. 13-Second-harmonic distortion in dB below fundamental for L2288 transistorsbefore and after power aging at 0.99 W.

In Fig. 9 the lower values of collector-base leakage current (below
about 4 IlA) show no discernible change after ten years of power aging ..
Those above about 4 IlA show, if anything, a very slight decrease.
The emitter-base breakdown voltages shown in Fig. 10 indicate very
consistent behavior, with the possible exception of one device whose
voltage decreased by about 0.1 V and another whose voltage increased
by about 0.05 V, again with no significant effect on circuit or system
reliability since the emitters are always forward biased.
The emitter-base forward voltages shown in Fig. 11 indicate a small
but consistent decrease of about 1 m V after ten years of power aging.
Such a decrease is not significant enough to argue about whether it is
real or an offset in testing.
The behavior of the low-frequency common-base current gain shown
in Fig. 12 is again very gratifying. An increase in this parameter was
expected, but the magnitude of increase has turned out to be smaller than
expected. If the first ten years of aging serve as any indication, the end
of life liinit as an average for all devices will probably never be reached
for all practical purposes.
Harmonic distortion for these devices was determined by measuring
the level of the second and third harmonic at the output with a sine wave
input at a frequency of 1.0 MHz at a power level of 5.0 dBm. Measurement precision is not in the same class with that of the dc and low-frequency parameters, especially for the initial data. During the ten-year
998

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

73.-----------------------------------------~

-

.

71 -

. ~/

•• .1'" •

. .(, ...

69 r-

,£! ('.: : .:.

LINE OF NO CHANGE--- __

"'7:... :••
•/ ..• til. a.

-y' •• •

/ . • ••"1

/O~: O.'! °

/ ..
/.

65 f-

63 f-

61-/

/

/

/
INITIAL SPECIFICATION LIMIT

O~
I
I
I
I
I
I
I
I
I
I
o "~6~1--~--~63--~--~6-5--~--~6~7--~--~69--~--~71
BEFORE AGING

Fig. 14-Third-harmonic distortion in dB below fundamental for L2288 transistorsbefore and after power aging at 0.99 W.

period of extended aging, the precision of measurement was improved
considerably. Nevertheless, the aging results shown in Figs. 13 and 14
should be considered merely indicative that the distortion parameters
are quite well behaved and that after ten years of power aging, they still
meet all of the device initial requirements.
4.4 L2317 diodes

The L2317 diodes, coded as 467 A, are large area pn junction protective
devices used in the power path of the SF repeater. During normal system
operation, each device merely draws reverse-leakage current. In the event
of a cable fault, however, it would conduct a large transient of reverse
current to protect the repeater circuit from excess voltage. These devices
are aged with 13-V reverse bias, which is a few volts below breakdown.
Although the ten-year aging results are again presented in the form of
initial and final values for each parameter, a large number of intermePOWER AGING

999

11.2 .------------------------.~/
LINE OF NO CHANGE---".,/

-

//

11.1 c-

/

11.0 r

//

c.:J
Z
~

10.9-

<!

a:

w

I-

~

10.8 -

//
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BEFORE AGING

Fig. 19-Breakdown voltage in volts at 2 rnA for L2320 diodes-before and after 10 years
of aging at 2 rnA in breakdown region.

diate data points taken during the ten years show no significant excursions away from the end points shown.
Figure 15 shows the behavior of the breakdown voltage of one hundred
L2317 diodes across ten years of power aging. A very consistent upward
shift of about 0.01 V is readily apparent but is too small to be of any
practical significance. The behavior of the reverse leakage current shown
in Fig. 16 also exhibits a small but very consistent upward shift of about
1 nA, while Fig. 17 shows a consistent downward shift of about 2 m V in
the forward voltage.
4.5 L2318 diodes

The L2318 diodes, coded as 468A and 468B, are used for surge protection in the oscillator circuit of the SF repeater. Each device consists
of two silicon pn junction diode chips mounted on a common metallic
heat sink and oppositely poled so that only a forward characteristic is
observed with an applied voltage of either polarity. During long-term
power aging, a 60-Hz voltage is applied to make the diode conduct 10 rnA
rms in each direction. The forward voltage drop is measured at two
1002

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

36~------------------------------------------~

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BEFORE AGING

Fig. 20-Reverse leakage current in nA at 8 V for L2320 diodes-before and after 10
years of aging at 2 rnA in breakdown region.

current levels in each direction during long-term power aging.
Figure 18 shows the behavior of the forward voltage of one hundred
L2318 diodes in one direction and at one current level across ten years
of power aging. No discernible aging can be seen. The behavior of the
other forward voltages is not shown because they look no different.
4.6 L2320 diodes

The L2320 diodes, coded as 467B, are silicon pn junction limiter devices used in the third stage of the amplifier of the SF repeater. Operation
is at 2.0 rnA in the breakdown region, and the devices are aged under
these power conditions.
Figure 19 shows the behavior of the breakdown voltage of one hundred
L2320 diodes across ten years of power aging. No discernible shift in this
parameter is evident.
In Fig. 20, a definite upward shift in the reverse-leakage currents can
be seen. This is the one case where a significant amount of scatter is
encountered in the magnitudes of the parameter shifts among the
hundred devices. The data suggests that percent change rather than
POWER AGING

1003

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BEFORE AGING

Fig. 21-Forward voltage in volts at 200 rnA for L2320 diodes-before and after 10 years
of aging at 2 rnA in breakdown region.

incremental (delta) current change is probably the appropriate way to
regard the behavior of individual devices. A leakage-current magnitude
of at least 100 p,A would be required in anyone device to produce even
the start of any impairment in circuit operation.
For this particular situation, even the aging of 100 devices for ten years
still leaves some degree of uncertainty in how best to interpret the behavior in the tails of the distribution where the device would be found
with potentially the greatest change in leakage current. In consideration
of this situation and in consideration of the fact that a long cable might
typically have 400 of these devices compared to the 100 shown here, it
has been conservatively estimated3 that with high (90 percent) confidence, a 100-p,A leakage current would not be reached in anyone device
in less than 60 years. 3
According to Fig. 21, the forward voltages of these diodes have shown
a small but very consistent downward shift of about 1 m V in ten years
of power aging. Such a shift is of no significance to system reliability.
1004

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

V. GENERAL COMMENTS

The regular procedure for producing submarine cable semiconductor
devices can be divided into three general stages: manufacturing,
screening, and aging. A selection process is employed in each of these
stages. The devices in this extended aging program progressed through
the first two stages in the usual way, but did not have the benefit of
having completed the standard aging stage of about six months with
ensuing selection before beginning the extended aging period. Nevertheless, any of these devices, even after ten years of power aging, could
still be considered candidates for cable use.
For the next generation of submarine cable (SG), the first of which was
laid across the Atlantic Ocean in the early part of 1976, silicon transistors
have been used throughout. Probably no new systems will ever be designed with germanium transistors. Nevertheless, the reliability record
thus far of the germanium transistors used in the SF system is rather
impressive in its own right.
VI. ACKNOWLEDGMENT

Over many years, a number of people contributed to this project. To
list all of them would not be possible and to list only part of them would
not be fair. Overall, however, the project stands as a shining example of
team effort between Bell Laboratories and Western Electric.
REFERENCES

1. A. J. Wahl, W. McMahon, N. G. Lesh, and W. J. Thompson, "SF System: Transistors,
Diodes, and Components," B.S.T.J., 49, No.5 (May-June 1970), pp. 683-698.
2. L. E. Miller, "Reliability of Semiconductor Devices for Submarine Cable Systems,"
Proc. IEEE, 62, No.2 (February 1974), pp. 230-244.
3. J. A. Tischendorf, private communication.

POWER AGING

1005

Joseph S. Kaufman, B.E.E., 1965, Pratt Institute; M.S.E.E., 1966
and Ph.D., 1970 (computer, information, and control engineering),
University of Michigan; Assistant Professor of Electrical Engineering
and Computer Science, Columbia University 1971-1973; Bell Laboratories, 1973-. Since Joining Bell Laboratories, Mr. Kaufman has been
concerned with the modeling and analysis of telephone traffic systems.
Member FAS, IEEE.

A. J. Wahl, B.S., 1942, University of Kansas; Ph.D., 1950, Princeton
University; United States Air Force, 1951-1953; Bell Laboratories,
1950-1951, 1953-. Since 1953, Mr. Wahl has been engaged in various
phases of transistor development. His early work was in the area of
surface effects relating to device behavior, particularly in regard to device
reliability. Later he was concerned with providing the semiconductor
devices for the Telstar satellites and has since supervised the provision
of transistors and diodes for submarine cable systems. He is currently
supervisor of a group concerned with the development of microwave
transistors and submarine cable devices. Member, IEEE, Sigma Xi.

1008

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

Abstracts of Papers by Bell System Authors
Published in Other Journals
COMPUTING
A "DOGLEG" Channel Router. D. N. Deutsch, Des. Automation Conf. Proceed., No.
13 (June 1976), pp. 425-433.
A new algorithm for interconnecting two sets of terminals
across an intervening channel is presented. The routing is done on two distinct layers, one
for horizontal paths, the other vertical. Typical channels (300 terminals, 100 nets) are
routed within one track of the mathematical bound (density) in a few seconds.
Impact of On-Line Systems on a Literature Searching Service. D. T. Hawkins, Spec.
Some experiences with on-line literature
Libr., 67 (December 1976), pp. 559-567.
searching systems in the Bell Laboratories Libraries and Information Systems Center
during 1975 are described. A total of 604 sessions, averaging 15 minutes each, occurred.
Average cost per session was approximately $22, including connect time, communications,
and off-line printing. The actual cost of many sessions, however, was $10 or less. On the
average, on-line searching costs $1/minute and 10¢/off-line print. A search on a specific
subject took an average of two sessions. Details on types of searches performed, the method
of charging costs to the user, and some effects on library services are given, as well as some
of the advantages of using an on-line system for information retrieval.
LTX-A System for the Directed Automatic Design of LSI Circuits. G. Persky, D.
N. Deutsch, and D. G. Schweikert, Des. Automation Conf. Proceed., No. 13 (June 1976),
pp. 399-407.
L TX is a minicomputer-based design system for large-scale integrated
circuit chip layout which offers a flexible set of interactive and automatic procedures for
translating a circuit connectivity description into a finished mask design. The system
encompasses algorithms for two-dimensional placement, string placement, and channel
routing.
Pro-An Automatic String Placement Program for Polycell Layout. G. Persky, Des.
Automation Conf. Proceed., No. 13 (June 1976), pp. 417-424.
PRO is a minicomputer-based string placement program that reorders cells in polycell rows such that channel
routing in the adjacent channels needs fewer routing tracks, and is not blocked by cyclic
constraints. PRO utilizes cell reflections and pairwise neighbor exchanges, simultaneously
monitoring both channels adjacent to the row being reordered.

ELECTRICAL AND ELECTRONIC ENGINEERING
Birefringent Coupler for Integrated Optics: Comment 2. A. Albanese and J. A. Arnaud,
A proof is given of the impossibility to realize
Appl. Opt., 15 (September 1976), p. 2029.
lossless and one-directional couplers.
Control of Limit Cycles in Recursive Digital Filters by Randomized Quantization.
R. B. Kieburtz, V. B. Lawrence, and K. V. Mina, Proc. 1976 IEEE Int. Symp. Circuits
Systems, April 1976, pp. 624-627.
The different types of limit cycles that can occur
in single second-order sections respond differently to efforts to reduce them. Length 1
(dc) limit cycles play an important complicating role, especially in cascaded filters. We
describe a simple method, involving random requantization of multiplier outputs, which
can reduce or eliminate limit cycles in digital filters. We compare limit cycle and roundoff
noise in three typical low-pass filters with and without the method.
Impurity and Substrate Diffusion in a Thin Contact Layer. R. P. Goel and F. E. Bader,
J. Electrochem. Soc., 123, No.8 (August 1976), pp. 1242-1245.
Based on simplifying
assumptions, diffusion phenomena of substrate and impurity through a thin contact layer
have been compared. Results are presented in terms of nondifTI.ensionalized parameters
k (a measure of diffusion rate, time, and contact thickness) and 1/1 (film thickness/contact
thickness). A single master curve is shown to represent each diffusion process.

1009

Low-Noise GaAs MESFETS. B. S. Hewitt, et aI., Electron. Lett., 12, No. 12 (June 10,
1976), pp. 309-310.
GaAs MESFETs with optimum noise figures of 1.6 dB at 6 GHz
have been fabricated by projection photolithography. An equation has been developed
for the calculation of optimum noise figure which gives good agreement between calculated
and measured values.
Measurement of Land to Plated-Through-Hole Interface Resistance in Multilayer
Boards. J. N. Hines, 14th Annual Proceed., 1976 Reliability Phys. Symp., April 20, 1976,
pp. 132-134.
Thermally induced cracks in the bond between an innerlayer land and
barrel will occur during the lifetime of an MLB if processing is faulty. A technique to detect
incipient failures that features four-terminal resistance measurements with a minimum
of bulk resistance to overcome the insensitivity of series circuit measurements is presented.
Mechanical-Chemical Technique for Removal of Epitaxial Spikes. L. E. Katz and
W. C. Erdman, J. Electrochem. Soc., 123, No.8 (August 1976), pp. 1249-1251.
A new
mechanical-chemical spike removal technique has been devised which effectively removes
epitaxial spikes on silicon substrates. A mask damage technique was used to evaluate masks
subjected to spike removal. Results show effective spike removal as well at no subsequent
mask damage.
Millimeter Wave Integrated Circuit Technologies. R. H. Knerr and L. F. Moose,
Microwave J., 19, No. 11 (November 1976), pp. 23-24.
Introduction to a two part series
of short articles authored by the participants in a panel discussion on "Millimeter Wave
Integrated Circuit Technologies" at the 1976 IEEE-MTT-S International Microwave
Symposium.
A New Digital Echo Canceler for Two-Wire Full-Duplex Data Transmission. K.
H. Mueller, IEEE Trans. Commun., COM-24, No.9 (September 1976), pp. 956-962.
A
new approach to echo canceling for two-wire full-duplex data transmission is proposed.
The canceling signal is directly synthesized from the binary data, using a transversal filter
approach, and the usual multiplications are replaced by additions and subtractions, thus
allowing efficient operation of a large number of taps as required for the canceling of distant
echoes. As a specific application, a system processing one sample per baud is discussed
where timing signals at both communicating stations are assumed to be synchronized.
Optimization of Digital Postdetection Filters for PSK Differential Detectors. R.
D. Gitlin and K. H. Mueller, IEEE Trans. Commun., COM-24, No.9 (September 1976),
pp.963-970.
A digital transversal postdetection filter for use in differential PSK detectors is proposed, and a method for optimizing the weighting coefficients is presented.
It is shown that even though the detector output signal is quadratic, the optimization
problem can be formulated in a way similar to the well-known MMSE equalizer for linear
signals.

MATERIALS SCIENCE
Application of Weibull-type Analysis to the Strength of Optical Fibers. B. K. Tariyal
and David Kalish, Mater. Sci. Eng., 27 (January 1977), pp. 69-71.
The strength distribution for 400 measurements on 0.61-m-Iength fused silica optical fiber specimens exhibits a bimodal behavior. The cumulative failure probability is presented for Weibull-type
distributions and is analyzed by assuming that the strength distribution is comprised of
two distinct populations having finite lower and upper strength limits. This procedure
provides the best prediction of the strength in long lengths.
Many-Body Effects in Transition Metals: Role of the Density of States. G. K. Wertheim and L. R. Walker, J. Phys. F Metal Phys., 6, No. 12 (December 1976), pp. 22972306.
The effects of structure in the density of electron-hole pair excitations is examined by numerical integration of the many-body screening formalism. The results are
applied to Ir and Pt.
Oxidation State of Tungsten in the Na x W0 3 Bronzes. G. K. Wertheim, M. Campagna,
.J.-N. Chazalviel,* H. R. Shanks t , Chem. Phys. Lett., 44, No.1 (November 15, 1976) pp.
50-52.
X-ray photoemission spectra of W 4f electrons in vacuum-cleaved cubic
Na x WO:l covering the entire composition range give no evidence for the existence of
multiple W valence states in the bulk. Strong oxidation effects are observed in air exposed

1010 THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

surfaces.
* Laboratoire PMC, Ecole Polytechnique, France. t Ames Laboratory, Iowa
State Univ.
Thin Film Interaction in Aluminum and Platinum. S. P. Murarka, I. A. Blech, and
H. J. Levinstein, J. Appl. Phys., 47 (December 1976), pp. 5175-5181.
Aluminum and
Platinum interacted very rapidly with the formation of several intermetallics (namely
Pt3A12, Pt5Ah, PtAI 2, PtAh and PtAI 4 ) which resulted in large volume changes and loss
of adhesion. The AI-Pt interaction rate was dependent on the annealing ambient being
higher in forming gas, argon or helium than in vacuum or air.

MECHANICAL AND CIVIL ENGINEERING
Free Vibrations of a Beam-Mass System With Elastically Restrained Ends. R. P.
Goel, .J. Sound Vib., 47, No.1 (.July 1976), pp. 9-14.
The vibration problem of a beam
with an arbitrarily placed concentrated mass and elastically restrained against rotation
at either end is solved. The effects on eigenfrequencies of the system produced by varying
the mass, stiffness, and position ratios are presented.
Tranverse Vibrations of Tapered Beams. R. P. Goel, J. Sound Vib., 47, No.1 (July
1976), pp. 1-7.
Transverse vibrations of linearly tapered beams, elastically restrained
against rotation at either end, have been investigated. Results for the first three eigenfrequencies, with different values of stiffness ratios and taper ratios are presented. Cases
of a tapered cantilever beam with a concentrated mass at the free end and spring hinged
at the other end have also been presented.
PHYSICS
Absence of the Low-Temperature Specific-Heat Anomaly in bcc 3He. D. S. Greywall,
37, No.2 (July 12, 1976), pp. 105-107.
The specific heat of bcc :lHe has been measured
along three isochores for temperatures T between 50 mK and the melting curve. The data
for T < 0.:3 K can be described well by the function Cu = aT-2 + "IT:l and are inconsistent
with the existence of the low-temperature specific-heat anomaly reported previously by
others.
Digital Normalization of Iodine Filter Structure in Quasielastic Light Scattering.
K. B. Lyons and P. A. Fleury, Phys. Rev. Lett., 47 (November 1976), pp. 4898-4900.
A
molecular iodine filter may be used to reduce the elastic component in a light scattering
experiment but also introduces extraneous structure into the spectrum. A technique is
reported here to compensate quantitatively for this structure by a digital normalization
scheme.
Enhanced Optical Fluorescence by Resonant Mossbauer Excitation. C. P.
Lichtenwalner, H. J. Guggenheim, and L. Pfeiffer, Phys. Lett., 56A, No.2 (March 8, 1976),
Enhanced optical Eu 2+ fluorescence in Eu-doped CaF 2 crystals is obpp. 117-118.
served following resonant absorption of Mossbauer "I-rays by Eu 3+ nuclei. The effect is
due to scintillation fluorescence from Eu 2 + ions excited by conversion electrons from
decaying 21.6 keY 151Eu.
Particle Brownian Motion Near a Hydrodynamic Instability. J. B. Lastovka and J.-P.
We consider the alteration
Boon,* Phys. Rev., 14, No.4 (October 1976), pp. 1583-1586.
of the normal Brownian motion of seed particle brought about by the enhanced velocity
fluctuations that exist near the threshold of a hydrodynamic instability. We utilize our
results to quantitatively predict the fluctuation induced effects for the threshold region
of the Benard convective instability.
* Universite Libre de Bruxelles.
Pressure-Induced Valence Change in Cerium Phosphide. A. Jayaraman, W. Lowe,
The pressure-volume relationship for cerium moL. D. Longinotti, and E. Bucher.
nophosphide (CeP) has been determined to 200 kbars pressure. From the data, we conclude
that there is an electronic transition involving a change in the valence state of Ce from 3+
towards the 4+ state near 100 kbar pressure. Qualitative observations of the reflectivity
of CeP under pressure support the valence transition.
ABSTRACTS 1011

The Use of Our Specially Designed Nonachromatic Microscope Objective to Examine Tracks in Nuclear Emulsions. J. S. Courtney-Pratt, Opt. Eng., 15, No.4 (JulyAugust 1976), pp. 379-383.
We have shown that our specially designed nona chromatic
objective allows us to produce sharp images of particle tracks in nuclear emulsions, for
visual study and/or photographic recording, with a lateral resolution of about one micron,
and a depth of field many times greater than can be achieved with conventional objectives.

1012

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

Copyrightc<;:) 1977 American Telephone and Telegraph Company
THE BELL SYSTEM TECHNICAL JOURNAL

Vol. 56, No.6
Printed in U.S.A.

B.S.T.J. Briefs

Adhesive Sandwich Optical Fiber Ribbons
By M. J. SAUNDERS and W. L. PARHAM
(Manuscript received April 15, 1977)

In 1974 a proposal was made to put optical fibers together into easily
handled units for optical communication purposes. 1 This proposal
suggested "the use of fiber ribbons consisting of linear arrays of fibers
embedded in a thin, flexible supporting medium as components of a cable
for fiber transmission systems." This note is a brief description of fiber
ribbons made by sandwiching fibers between two layers of polyesterbacked adhesive (adhesive sandwich ribbons).
The machine for making these ribbons has evolved around two

Fibers

Razor Blade Cutter
Controlled Spacing

To Tape Reel

i

Powered Excess Tape Take-Up Reel
Pulls Ribbon Through Cutter
Ribbon Take-Up Reel

Fig. I-Schematic diagram of adhesive sandwich ribbon machine.

1013

motor-driven pressure rollers. Figure 1 is a schematic diagram of the
machine. Twelve fibers from payout reels are directed through the fiber
guide where they are made contiguous. The fibers then pass between the
two pressure rollers and are sandwiched between two layers of polyester
backed adhesive tape supplied from large reels. After emerging from the
pressure rollers, the sandwich is cut by a blade cutter to a width somewhat larger than the width of the 12 fibers. The excess tape that has been
cut away from the fiber-carrying ribbon is attached to a reel that supplies
the power to pull the tape through the system. The ribbon with the fibers
is either wound on a reel or is permitted to fall, under zero tension, into
a container so that loss measurements can be made. Currently, we are
making kilometer lengths of ribbon at speeds of about 0.2 m/s. However,
shorter lengths of ribbon have been made at a speed of 0.8 m/s. We have
made about 100 adhesive sandwich ribbons, including ribbons for the
Atlanta Fiberguide System Experiment,2 varying in length from 0.2 to
2km.
The added loss of fibers in these ribbons, caused by microbending,3
is quite small. The average added loss of the fibers in the ribbons used
to make the two cables, each 1 km in length, for the Atlanta Fiberguide
System Experiment was 0.95 ± 0.07 dB/km for 137 fibers and 0.55 ± 0.05
dB/km for 132 fibers. 4 Some fiber breakage occurred, caused both by
defects in the machine and by the fibers being wound improperly on the
reels.
We thank M. J. Buckler for making the loss measurements, E. D. Knab
for much of the design work, and R. Sabia for originally suggesting the
concept of the adhesive sandwich ribbon.
REFERENCES
1. R. D. Standley, "Fiber Ribbon Optical Transmission Lines," B.S.T.J., 53, No.6

(July-August 1974), pp. 1183-1185.
2. J. S. Cook, J. H. Mullins, and M. I. Schwartz, "An Experimental Fiber Optics Communications System," Conf. on Laser and Electro-Optical Systems, San Diego,
California, May 25-27,1976. Sponsored by IEEE and OSA.
3. D. Gloge, "Optical-Fiber Packaging and Its Influence on Fiber Straightness and Loss,"
B.S.T.J.,54, No.2 (February 1975), pp. 245-262.
4. M. J. Buckler, unpublished work.

1014

THE BELL SYSTEM TECHNICAL JOURNAL, JULY-AUGUST 1977

magazine :: Bell System Technical Journal :: BSTJ V56N06 197707