Losses and Impulse Response in Parabolic Index
Fibers With Square Cross Section

Transverse Coupling in Fiber Optics—Part |:
Coupling Between Trapped Modes

Performance Models of an Experimental
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Multiplexer With Concentration
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On the Behavior of Minimax Relative Error
FIR Digital Differentiators

Interframe Picturephone® Coding Using
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Simultaneous Measurements of Depolarization
by Rain Using Linear and Circular
Polarizations at 18 GHz

F, T. ANDREWS, JR. J. M. NEMECEK
S. J. BUCHSBAUM B. E. STRASSER
|. DORROS D. G. THOMAS
D. GILLETTE W. ULRICH

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Explicit expressions are derived for the phase constant, the speczfic-
group-delay constant, and the rms width of the impulse response for two-
dimensional or square media having a transverse variation of index of
refraction according to n = ni(1 — faye" — 40,2"), in which x ts the
transverse dimension, au and a, are constants with |a,| <<a, and
(n — m1) K 1. Use is made of an approximation which the author has
previously shown yields significant results.

The results are applied to fibers with graded-index variation, clad by an
additional medium of index n = ny(1 — A). The ideal index gradient, a
near-parabolic profile, gives delay distortion orders of magnitude less
than for the conventional fiber with a step-change in index at the core-
cladding boundary. However, it 1s shown that several forms of 5-percent
error in the ideal gradient yield improvement of the order of 50 compared
with the conventional clad fiber. The delay distortion is shown to be very
sensitive to the exact index distribution in the vicinity of the deal distribu-
tion but increasingly insensitive to perturbations in the index distribution
as that distribution departs more and more from the ideal.

l. INTRODUCTION
Optical fibers have assumed considerable importance for potential
use as transmission media wherever wire pairs or coaxials are now used.
177

We give here some theory related to increasing the information
capacity of such fibers.

Conventional fibers have a core of uniform index of refraction n,
surrounded by a cladding of slightly lower index of refraction. The
cladding serves to isolate the outer fiber surface from the optical field,
which is confined to the core, thus permitting the fiber to be handled
and bundled into cables without affecting the information trans-
mission. More recently, fibers having continuously graded index of
refraction, with maximum on the fiber’s axis and lower valucs at
increasing radial distances, were proposed and realized in practice.!-3
Graded-index fibers have image-focusing properties? and _ provide
modulated-carrier transmission with less delay distortion than con-
ventional fibers having a step-index change at the core-cladding
boundary.*~® Recent experimental work verified the potential of low
delay distortion.’ Current studies of graded-index fibers from Corning
Glass Works show that total transmission losses under 10d B/kilometer
and low delay distortion (approximately one nanosecond per kilometer)
can be realized simultaneously in graded-index fibers.*:® This brings
into focus the need for detailed knowledge about delay distortion
related to the shape of the index gradient in the transverse plane. The
following sections relate to that need. This study was done in parallel
with that of D. Gloge and E. A. J. Marcatili.!° The present paper
presents an approximate analysis that may be extended to a wide
variety of index distributions with closed-form solutions for the
important wave-propagation constants.

Section JI summarizes the approximate method of calculation,
yielding in turn the phase constant, the specific-delay constant, and
the impulse response.

This paper is concerned with the delay difference between signals
launched simultaneously in the various propagating modes; the fiber-

(z) all propagating modes are excited equally at the fiber input,
(ic) the fiber structure is uniform along its length, yielding no
mode conversion,
(117) there are negligible losses or, equivalently, the same loss for
all modes, and
(tv) material dispersion due to variation of the bulk index of
refraction versus wavelength is ignored.

A very few modes very near cutoff are not accounted for; this is be-
lieved to yield negligible error, and the same assumption was made by
Gloge and Marcatili.”

It is found that the delay distortion of graded-index media is a very
critical function of the index distribution in the vicinity of the near-
parabolic distribution which gives the lowest delay distortion. For
index distributions increasingly far from the ideal, the performance is
less and less sensitive to changes.

For fixed percentage error in fabrication of the index distribution,
the delay distortion is linearly dependent on A, the fractional index
difference between core center and the edge of the guiding region
where the index becomes

This is true even near the ideal index distribution, where previous
work assuming no fabrication error indicated the delay distortion
varied as A?.1!

The approach taken here is based on Ref. 5, which gives an approxi-
mate method for deriving the phase constant and other relevant
quantities for wave propagation in the generalized media.

with | f(z)| «<1. For guiding media, f(x) is predominantly negative;
x = 0 at the central axis. Here we assume f(x) is independent of the
longitudinal coordinate. In accordance with Ref. 5 we derive a param-
eter a., which measures the transverse field width, using

The normal mode field may be considered composed of plane-wave
components traveling at an angle a to the longitudinal axis; from
Ref. 5,

in which m is the mode number. The mth-order mode has (m + 1)
extrema in the transverse cross section. The maximum angle that
such plane-wave components can have is set by the fractional index
difference A [defined in (2)]; any components with a larger than

exceed the critical angle for total internal reflection and are unguided.
Thus eqs. (7) and (6) taken together establish a maximum modal
index mMmax, With (m+ 1)max transverse extreme, controlled by a,
and A,
The specific group delay is
_ dp

Po dw’ (8)
where w = 27f is the angular frequency. The range of values 7 can
assume run from the value with m = 0 to tmax which is

It is convenient to compare the specific group delay for the guided
mode 7 to that for an infinite medium of index n, using

The work of Ref. 5 related specifically to two-dimensional wave-
guides. We extend this here to three-dimensional guides in Cartesian
coordinates. We note the relation between the propagation constants
has the form

where e, is the dielectric constant, Bo is the free-space phase constant,
and 8., B21, and B22 are the longitudinal and transverse wave numbers
respectively. In our case, 821 and Bz. are small compared to 6,. The
value of 8, depends only on the sum of the squares of the two transverse

for the square guide. This replaces (m+ 1) in the two-dimensional
guide. The maximum (m + 1)max can be reached with a variety of
field distributions given by various values of m, and m, in (14).

To compute the impulse response we first note in solutions for
specific group delay 7 from (8) that all combinations of modes having
the same m or p in (14) have the same specific group delay. Hence
the fiber output at a given delay associated with p will be P(p), for
equal power into the fiber in each mode, where

The relation between ¢ and p is obtained from (10) and (8). The range
of ¢ over which (16) is valid is found by inserting the minimum and
maximum values that p may assume,

For many purposes the value of tmin corresponding to pmin may be
taken as zero since pmax >> pmin; thus

The effect on transmission system performance is approximately the
same for various shapes of the impulse response if the rms width is
the same ;! this applies in the region where the fiber impulse response
degrades the system signal-power requirements by only 1 or 2dB. We

We now give the results for the index distribution according to
eq. (1). The guide has width 2a. We specify that at x = a, f(x) =—A,
leading to eq. (2) and

Equation (29) can be used to eliminate the last factor in (36). For
the first term of (34), representing the major response due to a,x” in
(1), the maximum value of ¢ is

For the second term of (34), representing the perturbing term a,a*
in (1), the maximum value of ¢ is

Equation (34) is not valid for u = 2; for the later case the impulse
response is
4096(1 — 6)r/4mt/2.,

Letting wu = 2 in the equations of the preceding section yields the
near-parabolic index distribution. As shown by (35), the impulse
response has zero width when a, = 0 in the approximation made here.
In the cylindrical fiber there is no index distribution which gives zero
delay distortion among all the various modes. There is a distribution
of index which minimizes the delay distortion, and we now evaluate
this condition. We can use the above theory to evaluate the cylindrical
waveguide by noting the two limiting conditions already known for
low dispersion. Take the form

in which R is the radial transverse coordinate. In two earlier papers*8
it has been pointed out that the value of b; must be different to produce
no dispersion for meridional rays versus skew rays; the difference in
bs was found to be $. Kawakami and Nishizawa® found that b, must
be 3 to give no dispersion for skew rays and must be 3 to give no dis-
persion for meridional rays. We can visualize minimizing the dis -
persion for one ray type, and thus experiencing a maximum dispersion
corresponding to a change in b, of 4. We do this in the approximation
used here by setting u = 2, r = 4, and making

(31), and (41) we find for this “ideal” index distribution in the round
fiber the specific group delay

This agrees reasonably well with the value —n,A?/8c arrived at by
Gloge and Marcatili!® using an entirely different analysis for an
optimum index distribution defined differently.* The rms width of
the impulse response corresponding to (41) and (42) is

In contrast with this, the simple step-index fiber [represented by
a, = 0 and n © in (1) ] has an rms impulse response width of
1 Ni
¢=—=—A., 45
ae (45)
Thus the ‘ideal,’ corresponding to (42) and (43), is smaller by a
factor of 0.264. Since A ~ 0.01 the ideal graded-index fiber has an
impulse response narrower by a factor of about 400 than that for the
conventional fiber.
The fourth-order term represented by (41) corresponds to

This is very little different from the simple parabola—not enough to
see on Fig. 1 where curve @ represents both 6 = 0 and (46) for
A ~ 0.01. More importantly, (46) and (41) imply that the fourth-
order term decreases in size relative to the second-order term as A
decreases. The inaccuracies of material processing are likely to prevent
this as A becomes small. A more probable limit is a fixed value of 6
in (25) as A changes. This results in a specific group delay

* If instead of minimizing the delay distortion for either the skew rays or meridional
rays we had minimized the delay distortion for an index equation (40) at the mean
of the index values giving minimum distortion for the skew and meridional rays—i.e.,
at b = 5/12—then the maximum departure in b for any mode corresponds to a
change in b = +1/12. This corresponds to a, = +1/6a?, leading to

in place of (42). The coefficient 0.13 corresponds almost exactly to the result of
Gloge and Marcatili,° but the present analysis indicates twice the total delay spread
predicted by Gloge and Marcatili due to the + sign.

Fig. 1—Normalized index of refraction versus transverse coordinate (x/a) for the
following parameters in eqs. (1) and (25): curve @, u = 2, 6 = 0; curve @, u = 2,
r = 4,6 = 0.05; curve ©, u = 4, 6 = 0; curve @, u = 4,7 = 2,5 = — 1; curve ©,
u = 8,6 = 0; curve © u = ~,5 =0.

For 6 = 0.05, « becomes 0.00591 niA/c which is narrower than for the
step-index fiber by a factor of about 50 independent of A.

We note from (39) that the impulse response for the ideal index
distribution perturbed by a fourth-order term (r = 4) is a rectangular
pulse, shown as curve () in Fig. 2. However, if the perturbation were
sixth order, r = 6, the impulse response would vary as t—?. Other values
of r give other impulse-response shapes, which we discuss further in
the next section.

Finally we note from (47) that the impulse response due to the
fourth-order perturbation of the ideal distribution may either lead or
lag the impulse at +r = ni/c, depending on whether 6 is positive or
negative [see (1) and (25) ]. Similar effects due to the sign of 6 are

Fig. 2—Impulse response P(t) versus t/tmax where t = +r — ni/c and 7 is given by
(31) and (33). The conditions are the same as defined under Fig. 1.

found for perturbations of the nonideal index distributions and may
be seen in (31).

In Section VI there is a discussion of Fig. 8 which shows the effects
of perturbing the parabolic index distribution in several ways.

In this section we discuss the distributions obtained by setting
a, = 0 in (1) which mean 6 = 0 in (24) and (25).

The total spread in specific group delay for all modes tmax,u 1S given
by (87), which gives a null value when u = 2. As already discussed,
this is a simplification in the vicinity of the ‘‘ideal’’ distribution.
However, (37) gives a valid representation as u departs significantly
from the value 2. Figure 3 shows the variation in fmax,. Versus Uw.
The behavior in the vicinity of u = 2 is a form of singularity. For
5-percent error in uw from the ideal, tmax,u 2% 0.025n,A/c. This com-
pares with 0.0244n,;A/c for the modal delay spread from (47) for
5-percent (at « = a) fourth-order perturbation of the ideal. We con-
clude that it is not particularly important how the ideal is perturbed.

The value of tmax,. from (37) and plotted in Fig. 3 is not very
sensitive to the value of u at values removed from u = 2. The reduction
in rms width of the impulse response is illustrated in Fig. 4. For wu = ©
(the conventional step-index fiber) the value of o/(n1A/c) is 1/V¥12 or
0.289. This is reduced by a factor of 2 for u near 6, and by a factor
of 4 for u near 3.5. These results are identical to those of Gloge and
Marcatili.°

The shape of the impulse response, given by (39) for wu = 2 and
by (34) for u away from 2, is plotted in Fig. 2 for several cases of
interest.

The number of modes which can propagate, given by (28) for the
square fiber, is plotted as a function of uw in Fig. 5. For comparison
the corresponding quantity for the round fiber from Ref. 10 is also
plotted. The ratio is 4/7 at wu = ©, and near 2 for u = 2. The approxi-
mations made here are seen to be good, though not perfect.

We discuss now some of the results for the perturbed index distribu-
tions, eq. (1). We recall the solutions have been obtained assuming
|a,| Ka,or |6| <1.

The solutions for the specific group delay, given in (31) and (83),
contain the quantity Q as a factor in the perturbing term. The factor

Q, given by (82), contains the principal effect of the exponents wu and
r on specific group delay. Figure 6 shows how Q varies with r for the
special case u = 2. The region very near r = 2 is in question since we
know the ‘‘ideal’”’ index distribution differs slightly from u = 2. Else-
where, the results should be significant and may be used in (34),
(31), and (33).

More general curves for Q are given in Fig. 7. We observe that
when r > 0 the maximum value of Q is not very dependent on u but
the most sensitive region of r (giving largest Q) does depend somewhat
on u. An intuitive feel for the changes in the index distribution which
correspond to some of the curves in Fig. 7 can be obtained by examina-
tion of Fig. 8. Figure 8 shows the normalized index n versus transverse
coordinate (a/a). The curve labeled r = 2 corresponds to the pure
parabolic distribution. The other curves correspond to 6 = 0.05 with
various values of r in the perturbing term and u always equal to 2.

We note in I'ig. 7 that, at wu = 2, the value of Q at r = 10 is much
larger than at r = ©. This may seem surprising, since a step-index
change occurs at (z/a) = 1.0 when r=. Below (2/a) = 1 the
r = © curve in Fig. 8 corresponds to a pure parabolic gradient between
the ordinate equal zero and —0.95A.

We also note in Figs. 8 and 7 that dips in the index distribution
near (x/a) equal zero (curves for r = —0.4 and —0.1) yield values of
|Q| comparable to those for r in the range 4 to 20.

The above analysis provides an approximate solution for the delay
distortion to be expected in a wide variety of graded-index fibers,
representable by (1) with |a,| « du.

In general, gradual tapering of the index between the center of the
fiber and the outer support provides reduced delay distortion. Only
in the vicinity of the near-parabolic distribution is the performance
highly sensitive to the exact index distribution. The ‘ideal’ near-
parabolic distribution provides a potential reduction in delay distortion
of several hundred times compared to the step-index distribution of
the conventional clad fiber. With an accuracy of the order of 5 percent
in achieving the “‘ideal’’ distribution, the reduction in delay distortion
is on the order of 50.

Fig. 8—Normalized index versus transverse coordinate (z/a) for several per-
turbations of the parabolic index distribution.

It is a pleasure to acknowledge fruitful discussions with E. A. J.
Marcatili and D. Gloge.

1. S. E. Miller, U. S. Patent No. 3,434,774, “Waveguide for Millimeter and Optical
Waves,” issued March 25, 1969.
2. Nippon Selfoc Co., Ltd., “Light-Conducting Glass Fibers or Fiber Structures
ang ereue on Thereof,” Japanese Patent No. 1,266,521, issued March 8,
3. T. Uchida, M. Furukawa, I. Kitano, K. Koizumi, and H. Matsumura, “A Light
Focusing Fiber,’’ IEEE J. Quan. Elec. (Digest of Technical Papers), QE-5,
June 1969, pp. 25.
E. A. J. Marcatili, ‘“Modes in a Sequence of Thick Astigmatic Lens-Like
Focusers,”’ B.S.T.J., 43, No. 6 (November 1964), pp. 2887-2904.

. S. Kawakami and T. Nishizawa, “An Optical Waveguide with the Optimum
Distribution of the Refractive Index with Reference to Waveform Distortion,”
ee Trans. Microwave Theory and Tech., MTT-16, October 1968, pp.

. D. Gloge, EB. L. Chinnock, and K. Koizumi, “Study of Pulse Distortion in Selfoc
Fibers,’’ Elec. Letters, 8, October 1972, pp. 526-527.

. C. A. Burrus, E. L. Chinnock, D. Gloge, W. S. Holden, T. Li, R. D. Standley,
and D. B. Keck, “Pulse Dispersion and Refractive-Index Profiles of Some

. BE. L. Chinnock, L. G. Cohen, W. S. Holden, R. D. Standley, and D. B. Keck,
“The Length Dependence of Pulse Spreading in the CGW-Bell-10 Optical
Fiber,’”’ Proc. IEEE, 61, October 1973, pp. 1499-1500.

. D. Gloge and E. A. J. Marcatili, “Multimode Theory of Graded-Core Fibers,”’
BS.T.J., 52, No. 9 (November 1973), pp. 1563-1578.

. D. Marcuse, ‘“The Impulse Response of an Optical Fiber With Parabolic Index
Profile,” B.S.T.J., 62, No. 7 (September 1973), pp. 1169-1174.

. S. D. Personick, ‘Receiver Design for Digital Fiber Optic Communication
Systems,” B.S.T.J., 52, No. 6 (July-August 1973), pp. 843-886.

. E. G. Rawson, D. R. Herriott, and J. McKenna, “‘Analysis of Refractive-Index
Distributions in Cylindrical Graded-Index Glass Rods Used as Image Relays,”’
Appl. Opt., 9, March 1970, pp. 753-759.

Copyright © 1974 American Telephone and Telegraph Company
Tue BELL SysTeM TECHNICAL JOURNAL
Vol. 53, No. 2, February 1974
Printed in U.S.A.

Losses and Impulse Response in Parabolic
Index Fibers With Square Cross Section

Mode coupling is studied in a parabolic index fiber with a lossy boundary
and square cross section. Statistical deviations of the fiber axis from perfect
straightness and random changes of its width are considered as causing
mode coupling. The excess loss caused by these mode coupling mechanisms
and the loss penalty incurred for a certain degree of narrowing of the
umpulse response are estimated.

Multimode optical fibers whose cores have parabolic distributions
of the refractive index,!°

are of great practical interest for light transmission over long distances,
since their delay distortion is much less serious than that of con-
ventional clad fibers.

Since no optical fiber can ever be produced free of random imper-
fections, it is important to know how statistical irregularities of the
fiber affect its performance. Random irregularities of the fiber axis
and random changes of the effective width of the fiber cause coupling
among its modes. The mode losses are functions of the mode number.
Absorption losses tend to affect all modes in the same way. However,
if we assume that the fiber boundary either consists of an absorptive
material or is a rough surface that scatters light, we must expect that
higher order modes, whose fields reach strongly into the neighborhood
of the fiber boundary, suffer much higher losses than lower order
modes that are confined to the vicinity of the fiber axis. Coupling of
the low-order modes to the high-loss, high-order modes increases the

overall waveguide losses. One objective of our study of the effect of
waveguide irregularities is thus the determinabion of the excess losses
caused by mode coupling.

The second objective of this study of waveguide irregularities con-
sists in determining the impulse response of the fiber. In the absence
of coupling, each mode transports a fraction of the total power at its
characteristic group velocity. Since the group velocities of different
mode groups are not identical, pulse distortion results.2* Mode
coupling has the beneficial effect of improving the impulse response of
the fiber. It is thus of interest to determine how much reduction of
multimode pulse distortion can be achieved by random bends and
random width changes of the fiber.

The effect of random bends on parabolic index fibers with circular
cross section has been estimated in an earlier paper.* Pure diameter
changes of a fiber with circular cross section leave modes with different
circumferential symmetries uncoupled. Statistical irregularities are
unlikely to result in pure diameter changes without distorting the
circular fiber cross section. However, an analysis of more general
distortions of a fiber with nominally circular cross section is difficult
to perform. For this reason we discuss a fiber with parabolic index
distribution (1) but with square cross section. It seems reasonable to
expect that the performance of a fiber with square cross section is
similar to that of a fiber with circular cross section. We expect to find
the correct order of magnitude of the losses and impulse response of
the round fiber by examining its close relative, the fiber with square
cross section. In particular, it should be possible to assess the relative
effect of random axis deformations as compared to random width
changes. In a square fiber, changes of only one set of opposing walls
leave groups of modes uncoupled from each other. This situation
corresponds to the circular fiber with pure diameter changes that
leave modes of different azimuthal symmetry uncoupled. By allowing
both sets of opposing walls to change their separation randomly, we
are sure that all modes are coupled to each other. This model corre-
sponds to a nominally round fiber whose cross section is deformed in
an arbitrary way that does not conserve the circular symmetry.

ll. THE MODES OF THE PERFECT FIBER WITH SQUARE CROSS SECTION
The modes of an infinitely extended medium with the distribution

The parameter a is an arbitrary constant that, in conjunction with A,
determines the transverse dependence of the refractive index distribu-
tion. However, in the round fiber it is convenient to associate a with
the radius of the fiber boundary so that A is the relative difference
between the values of the refractive index on axis and at the fiber
boundary.

The square of the refractive index distribution (2) does not follow
precisely from (1). However, if one equation is regarded as the precise
distribution of the corresponding quantity, the other holds approxi-
mately provided A is small and we limit r to the range r S a. H, and
H, are Hermite polynomials of degree p and q, and P is the power
carried by the mode. The parameter w is defined as® (& = wvVeouo)

nok lA
and determines the radius of the field distribution with p = q = 0. At
r = w the field has decayed to 1/e of its value on axis. Epq represents
the transverse component of the electric field vector. The longitudinal
field components are relatively much weaker and are not being con-

sidered. The field (3) is only an approximate solution of Maxwell’s
equations. The propagation constant of the mode is given as®

The modes of the square-law medium are mutually orthogonal and
satisfy the relation

So far, the fiber boundary has been ignored. The mode field (8) is an
(approximate) solution of the guided-wave problem if we assume that
the distribution (2) extends to infinity. However, each mode decays
very rapidly outside of a certain region. For a given value of p the

field oscillates as a function of x passing through p zero crossings. The
shape of the function

the oscillatory behavior of the function changes to a rapid decay.
If x’ < a the presence of the wall does not interfere appreciably with
the field distribution. However, if x’ >a the field distribution is
severely altered by the presence of the wall. Since we are assuming
that the interaction of the field with the wall causes power dissipation
either by absorption or by radiation, we consider those modes whose
fields reach the vicinity of the wall with high field intensity as being
effectively cut off. By replacing x’ in (9) with a we obtain the condition
for the maximum value of p that can be allowed for low-loss modes.

The coupling coefficients between two modes are defined by the
general expression®7

We assume that f and g are both random functions of z and that
f/a and g/a are small quantities. The deflection of the optical axis of
the square-law medium has a far more important effect on the modes
than the corresponding deviation of the fiber boundary. The deflection
of the fiber boundary that results from the random bends of its axis
is neglected.

All other coupling coefficients vanish. The two choices (1 or 0) that
are indicated under the square-root sign in (14) and (15) belong to
the corresponding upper or lower sign of the subscript on the left-hand
side. Random deformations of the fiber axis couple only neighboring
modes.

For small values of f/a and g/a we can write this expression approxi-
mately as follows:

All other coupling coefficients vanish. There are nonvanishing diagonal
elements in this case. However, diagonal elements couple each mode
only to itself. This self-coupling is of no importance if f(z) and g(z)
have Fourier spectra with no zero (spatial) frequency component.

P, is the average power carried by the mode labeled », », is its group
velocity, and a, its power loss coefficient. The mode label » is used as
an abbreviation for the set of labels p, g. The power coupling coefficient
hy, is defined as follows :®

The coefficient K,, is the factor of the function f(z) or g(z) appearing
in eqs. (14), (15), (18), and (19). The spatial power spectrum of the
function f(z) [or g(z) ] is defined as

Since the random processes considered here tend to couple each
mode only to one of its neighbors on either side (in mode label space)
the equation system (20) can be converted to a partial differential
equation whose variables are not only the length coordinate z and
time ¢ but in addition the two mode labels p and q.? If the number of
modes below the effective cutoff value is very large, the set of discrete
modes can be regarded as a quasicontinuum. We write

= hog. prdv.a(P prapia =e P 5q) ar hog. p—dn.a(P p_an.a a P 5q)
Sid hog.piatda(Pp,atAq —- P pq) ae hing, p.a—ha(P p,g—ag — P 5q)

The last step follows by considering the discrete mode labels as con-
tinuous variables and replacing differences by differentials. The nota-
tion h(p) and h(q) serves as a reminder that, according to (14) and
(15), the coupling coefficients depend only on p if q is held fixed, and
(18) and (19) show that they depend only on q if p is held fixed. We
thus obtain the approximate partial differential equation

The average mode power P is now regarded as a continuous function
of z, t, p, and g. The group velocity v is a function of p and g. We have
omitted the loss term. We consider the modes as lossless if the vari-
ables p and g remain below the cutoff values (10) and (11) and as
having infinitely high loss if cutoff is exceeded. This fact can be incor-
porated into the theory as a boundary condition by requiring

It has been shown in Ref. 8 how the pulse propagation problem can be
solved by means of a perturbation method if the solutions of (24) for
the time-independent case are known. We thus consider the trial
solution

We separate this equation into two ordinary differential equations by
introducing the separation constant x?:

The equation system (28) and (29) together with the boundary
condition (25) (and an additional one to be discussed later) defines
an eigenvalue problem. The lowest order eigenvalue oi; has the
physical meaning of the steady-state loss of the statistical power
distribution.* This quantity is of interest since it determines the
additional losses that are caused by the statistical irregularities of
the fiber.

For random deformations of the fiber axis we obtain the power
coupling coefficient h(p) from (14) and (21).

We assume that f(z) and g(z) have identical power spectra so that
h(q) follows from h(p) by replacing p with g. According to (6) the
difference of the propagation constants of adjacent modes can be
approximated as

This approximation is independent of the mode numbers. This means
that only one spatial frequency (or actually a very narrow range of
spatial frequencies) of the power spectrum F(Q) is responsible for
mode coupling. For random axis deformations we have

The choice of the Bessel function instead of a Neumann function,
that would also solve (84), is dictated by an additional boundary
condition. Since the partial differential equation (24) can be regarded
as a diffusion process, we must require that no power diffuses into the
lowest order mode p = 0 from negative values of p. This requirement
means that dP/dp = 0 at p = 0. The solution (85) satisfies this
condition. The solution of (29) is similarly

Since the eigenvalues depend on the labels v and yu, we attach these
labels to « and obtain from (37) and (388) (note, pe = qe)

The steady-state power loss, the lowest order eigenvalue o1, follows
from (5), (10) (neglecting the term 4), (31), and wi = 2.405

For random diameter changes we obtain from (18), (21), and (31),
considering that the spacing (in f-space) between adjacent coupled
modes is now twice as large,

d { ,dU a _
dp E = | sd uw KM) U=0. (43)
The solution of this differential equation is
1
U = — cos (p, np+ ¢, 44
a5 (op, In p + ¢) (44)
with
a x 1
P= Alok KOM) 4 =)
The solution of (29) is correspondingly
1
v= A (o, Ing + ¢,) (46)

These solutions have a singular behavior at p = 0 or q = 0. However,
we must keep in mind that p and q are really discrete quantities.
Considering them as continuous variables is an approximate procedure
that can work only for very large values of p or q where the relative
difference between adjacent discrete values becomes small. Since
In p = 0 for p = 1, we allow p and q to vary only between 1 and p..
The requirement that no power diffuses across the lower limit of the
range of the variables imposes the conditions

p, as well as p, are solutions of this equation with integer values of »v.
Since the values of p, are now known, we obtain the eigenvalue g,,
from (45) and (47)

For the lowest order eigenvalue, that is, for the steady-state power
loss coefficient, we obtain with the help of (5) and (81)

The solution of (51) is not a constant. It depends on the waveguide
parameters through its dependence on p, = (a/w)?. A plot of pi as a
function of p, is shown in Fig. 2.

The width of the impulse response of a multimode fiber that is
long enough for the steady-state distribution to establish itself is
given by the formula :®

and V is by definition the difference between the inverse group velocity
of the modes minus the inverse of the maximum group velocity.‘

(p + q). (56)
The expression in parenthesis is an abbreviated way of writing
Gu, VG.) = fap [dgGuVG,, (57)

The integrals extend over the entire range of p and q variables from
either 0 or 1 to the cutoff value p, = q. Requiring the normalization

2 2 —4
Aig =| Inve + ea [|e + ea || (61)
The integrals (57) have the following solutions. For axis deforma-
tions (u # 1):
(Gu, VG@1,) = (Gu, V@u1)
_—- 82Apuu, | 2ui—1  6(ui + uf)

Evaluation of (54) with the help of (62) and (63) yields, for random
deformations of the fiber axis,

Equation (66) determines the pulse width of an impulse after it has
traveled a distance L (LZ must be large enough so that the pulse has
settled down to steady state) in the presence of random deformations
of the fiber axis. The pulse width for uncoupled modes is obtained
from (56)

The relative improvement of the width of the impulse response caused
by mode coupling is characterized by the ratio®

a ten
AT VLK(Q) w
Mode coupling not only shortens the width of the impulse response,
but it also leads to excess loss. In order to find out how much excess

loss is associated with a given improvement of the width of the impulse
response, we form the product of (41) with the square of (68)

For random axis deformations, the product of the square of the im-
provement factor with the excess loss is independent of the waveguide
parameters and the statistics of the random axis deformations.

We have derived expressions for the steady-state loss and the loss
penalty of graded index fibers with square cross section for the case of
random axis deformation and random width changes. The most

conspicuous difference between these two types of fiber imperfections
is the fact that, whereas the Fourier components of the function f(z)
(describing the fiber axis) at the spatial frequency Q are instrumental
in the mode mixing process, the Fourier components at twice the
spatial frequency, 20, determine the mode mixing process in case of
random changes of the width of the fiber. This behavior can easily be
understood. Consider a Gaussian beam of arbitrary width that is
injected into the fiber off axis. The beam undulates periodically
around the fiber axis and also changes its width periodically. The
undulations around the fiber axis have a period!

while the width changes repeat themselves with half that period or at
twice the spatial frequency.!! Random displacements of the fiber axis
couple to the deflections of the beam from its on-axis position. This
deflection is driven by a Fourier component at the spatial frequency

(74)
The beam width changes are correspondingly driven by changes in
the gradient (width) of the parabolic index medium. It is thus clear
that they respond to twice the spatial frequency.

In order to be able to associate specific rms deviations of the fiber
axis or rms width changes with fiber loss we have to consider a par-
ticular statistical model. We choose (arbitrarily) a Gaussian correlation
function

& is the rms deviation of the function f(z) and D its correlation length.
The power spectrum of f(z) is known to be!

At \ = 1 yum wavelength we have a/w = 6.3 or p. = 39.7. The
difference between the propagation constant of adjacent modes is,
according to (32), 2 = 34.5 cm7!. With 6 = Q we calculate the excess
loss values at the peak of the power spectrum at the value of the
correlation length given by (78), Dn = 0.041 cm. For random devia-
tions of the waveguide axis we obtain from (41), (77), and (79)
(¢ in cm)

In order to keep the excess loss below 10 dB/km = 2.3 10-> cm™ we
must keep the rms deviation of the fiber axis below ¢ = 2 X 107° cm.
However, this stringent tolerance requirement results from our as-
sumption that the correlation length of the random irregularities of
the fiber axis assumes its worst possible value (78). If, for example,
the correlation length happens to be D = 0.5 cm we obtain instead
of (80)

so that we can now tolerate ¢ = 6 X 10° cm in order to keep the excess
loss below 10 dB/km. This example shows that it is impossible to
predict the excess loss to be expected from a practical square-law fiber
unless the statistics of its irregularities are known precisely.

For reasons of comparison we state the corresponding value for
random width changes. In this case the spatial frequency that is
instrumental in providing mode coupling is 20 = 69 em~. The worst
possible correlation length is now Dm = 0.02 cm. From (53) and Fig. 2
with p. = 40 we obtain (¢ in em)

The tolerance requirements of random width changes appear a little
less stringent than those of random axis deformations. However, we
have already pointed out that the excess loss value depends critically
on the actual statistics of the fiber. Since the excess loss caused by
random axis deviations depends on a different spatial frequency than
the excess loss caused by random width changes, a loss comparison
of the two effects is not possible.

Next we discuss the loss penalty that is incurred for a given im-
provement of the width of the impulse response of coupled mode
operation compared to uncoupled mode operation. The equations (69)
and (72) show that the loss penalty is independent of the statistics of
the fiber irregularities. This feature makes the loss penalty a useful
quantity. In case of random variations of the fiber axis, the loss penalty
is even independent of the fiber parameters and is simply a dimension-
less number. Let us assume that we want to achieve a ten-fold improve-
ment of the width of the impulse response compared to the impulse
response of uncoupled multimode operation. In this case we have
R = 0.1 and obtain from (69) for random deviations of the fiber axis

The length L needed to incur this loss and at the same time to achieve
R = 0.1 depends on the statistics of the irregularities. However, eq.
(83) tells us that it costs 14 dB in excess loss to achieve a ten-fold
relative pulse width improvement.

For random width changes, the situation is slightly different. Here
the loss penalty depends somewhat on the fiber parameters. For the
values used earlier we find from Fig. 4 with p. = 40,

Fiber irregularities can be introduced intentionally in order to
improve the impulse response. In the conventional fiber with a round
core of constant refractive index that is surrounded by a cladding
with constant index, the loss penalty for pulse distortion improvement
can be reduced (in principle avoided) by tailoring the shape of the
power spectrum carefully. The reason that the shape of the power
spectrum has an influence on the loss penalty is explained by the
observation that the spacing (in 6-space) between adjacent modes of
the conventional fiber is dependent on the mode number, so that a
band of spatial frequencies of the power spectrum takes part in the
mode coupling process.

In case of the parabolic index fiber, only one spatial frequency
(or actually a narrow range of spatial frequencies) is responsible for
mode coupling. The shape of the power spectrum is thus immaterial,
only its value at the spatial frequency © enters into the picture. The
expressions (69) and (72) show that the loss penalty of the parabolic

index fiber is independent of the power spectrum. No loss advantage
is to be gained by using especially shaped power spectra. One might
think that an advantage could be gained by departing from the
square-law index profile in order to change the mode spacing and
sample more of the power spectrum. But as soon as the index distribu-
tion deviates slightly from the parabolic profile the uncoupled impulse
response becomes much broader. The mode coupling mechanism would
now have to work against a far less favorable (uncoupled) impulse re-
sponse so that it seems unlikely that an advantage can be gained
in this way.

Finally, we consider an example of pulse width reduction by random
irregularities. We can introduce intentional deviations of the fiber
axis from perfect straightness in order to cause mode coupling. Since
the coupling process must be random, we could use deformation func-
tions f(z) and g(z) that are sinusoidal in shape but have a random
phase. This introduces a power spectrum centered around the spatial
frequency of the sinusoidal process having a finite width. Instead of
pursuing this idea further, we assume that we have somehow created
an axis deformation whose power spectrum reaches beyond the fre-
quency 2 of (32). For simplicity, and to have a definite case in mind,
we choose

This power spectrum is flat from zero spatial frequencies to a cutoff
value of 6 = 20 and zero for 6 > 20. The rms deviation ¢ of the fiber
axis from a straight line appears in (85). Using the fiber parameters
(79) we obtain from (68) (¢, LZ in cm)

A ten-fold improvement of the width of the impulse response
(compared to the uncoupled case), R = 0.1, over a length of
L = 1 km = 10° cm requires an rms deviation of the fiber axis of
& = 2.2 X 10-* cm. We already know that we pay for this improve-
ment of the impulse response with a loss penalty of 14 dB. Very slight
random deviations from perfect straightness are already very effective
in providing mode coupling and improving the width of the impulse
response.

For random width changes we have to allow for a wider power
spectrum. Letting the power spectrum again extend twice as far as
the effective spatial frequency, 2Q in this case, forces us to divide
(85) by 2. We thus find from Fig. 3 and (71)

Dz
H= n(x). (90)
The quantum mechanical treatment of this problem leads to an ex-
pression for the “energy” EF of the ray that has the form"

E=- ae (91)
We define the ‘turning point’’ of the light rays associated with the
wave field (88) by the condition that p,, which is proportional to the
slope of the light ray, must vanish. That means that the ray trajectory
is tangential to the optical axis as the rays turn back in their path
leading them away from the axis. Using p, = 0 and equating (90)
and (91) we find the following condition for the turning point:

Substitution of (89) and (98) into (92) leads with the help of (5) to
the formula (9) for the turning point.

The physical argument advanced here serves the purpose of defining
the range in which the Hermite polynomial has an oscillatory behavior.
This range is given by

Outside of this range the Hermite polynomial grows monotonically to
infinite values. However, since the Hermite polynomial enters the
mode field (88) only as a product with a Gaussian function, the mode
field decays rapidly without oscillation outside of the range given
by (94).

1. S. E. Miller, U.S. Patent No. 3,434,774, ‘“‘Waveguide for Millimeter and Optical
Waves,” issued March 25, 1969.
2. D. Gloge and E. A. J. Marcatili, ‘‘Multimode Theory of Graded-Core Fibers,”’
B.S.T.J., 52, No. 9 (November 1973), pp. 1563-1578.

3. D. Marcuse, ‘The Impulse Response of an Optical Fiber With Parabolic Index
Profile,” B.S.T.J., 52, No. 7 (September 1973), pp. 1169-1174.

D. Marcuse, “Losses and Impulse Response of a Parabolic Index Fiber with
Random Bends,” B.8.T.J., 62, No. 8 (October 1973), pp. 1423-1437.

. A. W. Snyder, “Coupled Mode Theory for Optical Fibers,” J. Opt. Soc. Am.,
62, No. 11 (November 1972), pp. 1267-1277.

D. Marcuse, “Coupled Mode Theory of Round Optical Fibers,’”’ B.S.T.J., 52,
No. 6 (July-August 1973), pp. 817-842.

D. Marcuse, “Pulse Propagation in Multimode Dielectric Waveguides,’’ B.S.T.J.,
61, No. 6 (July-August 1972), pp. 1199-1232.

Copyright © 1974 American Telephone and Telegraph Company
Tue Be. SysteEM TECHNICAL JOURNAL
Vol. 53, No. 2, February 1974
Printed in U.S.A.

Transverse Coupling in Fiber Optics
Part I: Coupling Between Trapped Modes

Two perturbation formulas have been proposed to evaluate the coupling
between parallel optical waveguides, one involving a line integral and the
other a surface integral. They are shown to be identical. The former
expression 1s preferred because of its greater simplicity. The case of two
parallel lossy dielectric slabs is discussed as an example.

There has been a renewed interest during the last few years in the
evaluation of the transverse* coupling between two parallel open wave-
guides in connection with integrated optics circuitry? and long-
distance optical communication by bundles of glass fibers.

The coupling between two open waveguides can be obtained by
replacing the field of one waveguide by an equivalent current and
evaluating the perturbation caused by this current on the other
waveguide.‘ A more direct and slightly more general (but essentially
equivalent) derivation, based on Lorentz’s reciprocity theorem, is given
in this paper. A related result, applicable only to lossless fibers, has
been used to evaluate the coupling between dielectric rods with circular
cross section.® The perturbation formula derived in this paper involves
an integral along a contour located between the two waveguides. A
seemingly different perturbation formula has been recently proposed
that involves a surface integral over the cross section. The two
formulas are shown to be in fact identical. We will not discuss in
detail other coupling formulas such as the ones proposed in Refs. 7
or 3. In Ref. 7, the coupling is obtained by applying the Rayleigh-Ritz

“The word “transverse” is used here to distinguish the problem of two dielectric
waveguides lying side by side, where the transfer of power takes place in transverse
directions, and the axial coupling between two waveguides placed end to end, where
the transfer of power takes place along the z axis (the later arrangement is discussed,
for instance, in Ref. 1).

optimization technique to a variational expression. The formula ob-
tained by this method involves surface integrals and is rather com-
plicated. In Ref. 3, analytic expressions were obtained for the coupling
between two identical rectangular fibers that agree well with numerical
calculations based on the exact field equations. The approach, how-
ever, is restricted to fibers with a particular geometry.

Let the time dependence of the sources be denoted exp (—xt).
Maxwell’s equations are, in a source-free region with scalar permit-
tivity « and permeability u,,

Vv X E = KUoH, (la)
V XH =—x«eE. (1b)
Any two solutions (E, H) and (E,, H,) of eq. (1) satisfy the relation
V-J =0, (2a)

Integrating over a volume V, Lorentz reciprocity theorem is obtained
[, XH - EX H,)-d8 = 0, (2c)

where S denotes the surface enclosing V, and dS a vector normal to
S pointing outward with magnitude dS. Let the medium be uniform
along z, that is, e be independent of z. If

(E, H) = (E., Ex, H., Ht) exp (yz) (3)
denotes a solution of Maxwell’s equations, then
(E*, H+) = (—E,, Ey, Hz, — Hk) exp (— 72) (4)

Let us now consider two open waveguides a and b uniform along
the z-axis, and let S be the surface S, + Sz; + Cadz shown in Fig. 1.
The field (E,, H.) in eq. (2) is taken as the field of a trapped mode on

waveguide a in the absence of waveguide b. The dependence of (E., H.)
on 2 is denoted exp (y.z). The field (E+, H*) is the adjoint field of a
trapped mode of the two coupled waveguides, with an exp (—TIz)
dependence on z. Letting the spacing dz between S; and S, tend to
zero, eq. (2) becomes

a
where dC, is a vector perpendicular to the contour C., pointing out-
ward. Proceeding similarly for waveguide b we obtain
(»— 1) f (By x HY — BY x Hh) -dS,
b
= = (E, X H+ — E+ X H;)-dCs. (5b)
Co

Because the coupling between the two waveguides is small, we can
assume that the field E, H at plane z = 0 is the sum of the fields of
the two waveguides, that is,

Substituting these expressions, eqs. (6), in eqs. (5a) and (5b) we
observe that the cross terms can be neglected on the left-hand sides
(l.h.s.) because (Es, Hz) is small when (E,, H.) is large, and vice
versa. On the right-hand sides (r-h.s.), on the contrary, only the
cross terms remain, as we can verify by applying Lorentz reciprocity
theorem to each waveguide. Multiplying together the l.h.s. and r.h.s.
of eq. (5a) and (5b) the desired equation for I is obtained.

Because the coupling takes place only if yz ~ ys, the coupling c
(resp. cy) is independent of the choice of the contour C, (resp. Cy) as
long as it surrounds only one waveguide. By choosing the two contours
as coincident in the region where the fields of the two trapped modes
have a significant intensity and using eq. (4), we find that c, is equal
to cy. It is shown in the appendix that our result, eq. (6), can be
expressed in the form given in Ref. 6. The expression, eq. (6), however,
is simpler to evaluate.

Let us now assume that the contours C., Cy coincide with the y
axis and are closed at infinity where the fields vanish. The general
expression, eq. (6), becomes

Let us specialize eq. (7) to symmetrical stratified dielectric wave-
guides such as the slabs shown in Fig. 2. The fields are assumed to
be independent of y. For TE waves the electric field has only one

the fields H, and E, of the uncoupled waveguides being evaluated at the
same point between the two waveguides.

If the waveguides are homogeneous dielectric slabs of thickness 2d
and complex permittivity « we have

within the slabs (obvious changes in the origin of the x axis were
made). In eqs. (10a) and (10b) we have defined

(See, for instance, Ref. 6.) Substituting EH from eqs. (10a) and (10b)
in eq. (9) we obtain, using eq. (12),

where D denotes the spacing between the slabs. This expression
coincides with the result given in Ref. 6 when appropriate changes of
notation are made.

Let us now make a general comment. The coupling formula, eq.
(6), rests on the existence of a divergenceless quantity, the vector J
in eq. (2a). Coupling formulas similar to eq. (6) can be derived from
other wave equations. For the case of the scalar parabolic wave
equation® applicable to the propagation of radio waves in atmospheric
ducts,* the vector J has components

The above expression for J can be obtained by analogy with the
equivalent quantum mechanical problem.°

In conclusion, we have derived a simple coupling formula which is
more general than previous similar expressions‘:> because it is appli-
cable to lossy fibers. In order to evaluate explicitly the coupling, one
needs to know the normalized field of each waveguide, in the absence
of the other, along some line located between the two waveguides.
For slabs and rods with circular cross section, exact solutions are
available. In general, however, we have to resort to numerical tech-
niques or to measurements made at a convenient wavelength on a
scaled version of the open waveguide. In asecond part of this paper,”
we will apply eq. (6) to mode-selecting systems.

* A similar equation is applicable to anisotropic fibers that have small transverse
variation of permittivity.’ Note that, in this approximation, a curvature of the fiber
axis is equivalent to a constant gradient of refractive index.

The purpose of this appendix is to show that for lossless fibers the
coupling formulas given in Refs. 4 and 6 coincide.
Let (E, H) denote a field in free space

1
V X H = —x«e,E, (14)
and (Es, Hs) a field in a dielectric with permittivity e(r)
V X Es = «uoHs,
X Eo = kyuoHs (15)

in any source-free volume V bounded by S. Let now the surface S
be the surface S, + S; + Csdz shown in Fig. 1, (Es, Hs) be the field
of a trapped mode of waveguide 6 with an exp (yz) dependence on a,
and (E, H) be the adjoint field (Ez, Hz) of a trapped mode of wave-
guide a, with an exp (—7az) dependence on z. The field (Ez, Hi)
satisfies eq. (14) inside the surface S that we have just defined. If the
two trapped modes are degenerate, that is, if yz = yo, the contribu-
tions of the two surfaces S; and S; on the l.h.s. of eq. (16) cancel out.
Therefore, letting dz tend to zero, eq. (16) becomes

A similar relation can be obtained for waveguide a. Our coupling
equation, eq. (6), can therefore be written in the form given in Ref. 6,
except for the fact that in eq. (6) E and E, represent fields at the
same frequency. In Ref. 6 the field E7 is defined at the opposite
angular frequency —x, that is, EZ is replaced by Ez”, where the asterisk
denotes complex conjugation. For lossless fibers this difference is
unimportant because E7 can be assumed real.

3. E. A. J. ’ Mavontili “Dielectric Rectangular Waveguide and Aaa a
for Integrated Optics,” B.S.T.J., 48, No. 7 (September 1969), p

4, J. A. Arnaud, “General Properties of Periodic Structures,’ Sections B and E, in
Crossed Field Microwave Devices, Vol. 1, E. Okress ed., New York: Academic
Press, 1961.

.R. Vanclooster and P. Phariseau, “The Coupling of Two Parallel Dielectric
Fibers,’ Physica, 7, June 1970, pp. 485 and 501.

D. Marcuse, ‘The Coupling of Degenerate Modes in Two Parallel Dielectric
Waveguides,” B.S.T.J., 50, No. 6 (July-August 1971), pp. 1791-1816.

. M. Matsuhara and N. Numagai, “Coupling Theory of Open-Type Transmission
Lines and its Application to Optical Circuits,” Electron. Commun. Japan, 564,
April 1972, p. 102.

J. A. Arnaud, tBierehowonality Relations for Bianisotropic Media,” J. Opt. Soc.
Am., 63, February 1973, p. 238.

. Vv. A, Fock, “Theory of Radio-Wave Propagation in an Inhomogeneous Atmo-
sphere for a Raised Source,”’ Bull. Acad. Sciences de l’URSS, sér. phys. 14,
No. 1, p. 70 (1950). In Russian. Translation in V. A. Fock, "Electromagnetic
sean and Propagation Problems, Oxford: Pergamon Press, 1965, Chapter

10. J. at Arnaud, “‘Transverse Coupling in Fiber Optics, Part II,’’ to appear in
BS.T.J., 66, No. 4 (April 1974).

Copyright © 1974 American Telephone and Telegraph Company
Tue Bet System TECHNICAL JOURNAL
Vol. 53, No. 2, February 1974
Printed in U.S.A.

Performance Models of an Experimental
Computer Communication Network

This paper reports the results of a performance study of an experimental
computer communication network. The network is currently being designed
and butlt in order to test concepts and techniques that may find future
application. The network consists of synchronous digital transmission
lines connected in loops to a Central Switch. User traffic enters the system
through multiplexers connected to the synchronous lines. The Central
Switch has the two-fold function of routing and controlling traffic.

Two multiplexing techniques were examined, Demand Multiplexing
(DM) and Synchronous Time Division Multiplexing (STDM). In both
techniques, user messages are blocked into fixed size packets, prior to
multiplexing on the line. The synchronous line can carry these packets at a
maximum rate of 4000 packet slots per second. In STDM each terminal
as assigned a packet slot which recurs periodically. In contrast, for DM,
packets are multiplexed on the line asynchronously into unoccupied packet
slots. Alternative implementations of the DM technique were studied, one
where each terminal transmits and receives at a maximum rate of 4000
packets per second and another where the maximum rate is 2000 packets
per second.

As part of its message-handling function, the Central Switch buffers
messages in transit. This allows User Terminals to transmit and receive
messages with a degree of independence from one another. However, the
terminals’ strategy affects the amount of storage required in the Central
Switch. In order to prevent the loss of information when there is insufficient
buffering, there ts a mechanism to inhibit traffic from User Terminals
when the Central Switch buffer ts near overflow. Due to thts control of
traffic, there ts a relationship between the amount of data that flows through
the switch and the amount of buffering in the switch.

Simulation results showed that there was little difference in delay
performance between the two implementations of DM. However, an analysis

comparing DM and STDM showed a great difference in performance for
all but the very heaviest line loadings. This difference increases as the
number of terminals sharing the T1 line increases.

Our study concentrated on two aspects of buffering in the Central Switch.
We examined the relationship between throughput and the amount of
storage available in the switch. The results of a simulation study showed
that throughput can be quite high for all but minimal storage in the switch.
Moreover, a strategy that dedicates buffers does quite well compared to
common buffering. The second aspect of the study concentrated upon the
User Terminal’s strategy. Since each terminal acts tndependently, there
may be strategies that make particularly high demands upon storage
capacity in the Central Switch. An analysis showed that at the loadings
where the system would be expected to operate, the user strategy in trans-
mitting and receiving messages has little effect.

An experimental computer communication network is currently
being designed and built. The function of this network is to provide a
flexible communication medium between computers, users, and pe-
ripheral devices. The network can accommodate sources with varying
input-output rates and varying activity. Many of the components of
the system employ techniques that are new. In order to gain insight
into the operation of these components and thereby aid in design
decisions, mathematical models were developed. The study of these
models involved both analysis and simulation. The results are pre-
sented in the form of sets of curves.

The system under study consists of several T1 carrier lines,* con-
figured as loops, connected to a Central Switch (see Fig. 1). The
system is accessed through Terminal Interface Units (TIU) connected
between User Terminals and the T1 line. In addition to forming an
interface between the user and the T1 line, the TIU also does signaling.
This signaling plays a role in switching calls and controlling the
traffic flow.

There may be a wide variation of users accessing the system, ranging
from Teletypes' to high-speed computer systems. The switch receives
messages from all terminals and delivers messages to all addressed
terminals so that any terminal in the system may communicate with

“The T1 carrier line is a digital synchronous short-haul transmission system
operating at 1.544-Mb/s rate.

any other terminal. All data pass through the switch even when two
terminals are on the same T1 line.

The T1 line operates at a rate of 1.544 X 10® b/s. For purposes of
synchronization and timing, the bit flow is divided into frames of 193
bits, with a flow of 8000 frames per second. The multiplexing arrange-
ment in the system under study is such that a “network frame’
consists of two adjacent T1 frames. Figure 2 indicates schematically
how the 386 bits of the pair of T1 frames are allocated. The 50 bits
required for framing and timing are part of the operation of the T1
carrier system. The assignment of the remaining 336 bits in the net-
work frame is peculiar to the system under study.

User data are blocked into 256-bit packets and multiplexed on the
line. Twenty-four bits of header information are attached to these
information packets. (In the sequel we shall use the term “‘packet slot”
in referring to this 280-bit block assigned to data and header.)

Twenty-four of the 336 bits are used to carry signal packets. Signal
packets convey control and routing information between the TIU and
the switch. The remaining 32 bits of the frame pair are used for error
detection in order to insure the integrity of the information in the
signal and data packets.

From the foregoing we see that the information-carrying capability
of the T1 loop is 4000 packets per second, each packet bearing 256
data bits, yielding a total information capacity of 1.024 Mb/s. There
are many strategies that can be used to divide this capacity among
the terminals connected to the loop. We shall evaluate the performance
of two strategies, Synchronous Time Division Multiplexing and
Demand Multiplexing. In Synchronous Time Division Multiplexing
(STDM), each User Terminal is assigned a particular packet slot
which recurs periodically. The terminal may multiplex data into its
slot and receive data only in this same slot. For example, if there are
ten terminals on the Tl loop and each terminal receives the same
service, a particular terminal may multiplex packets on the line at a
maximum rate of 400 packets per second. The time between packet
multiplexing for these terminals is a constant 1/400 second.

In Demand Multiplexing (DM), packet slots are not assigned to a
particular terminal. If a terminal has a packet to transmit to the
switch, it inserts the packet into the first slot that is empty. Thus,
unlike the STDM system, the flow of packets into the switch has no
particular ordering as to originating terminal. So that the switch can
sort packets according to their originating terminal, each packet has
an address label in its 24-bit header. Similarly, information packets
going from the switch to the terminal are not ordered and a header
is required for each packet. Furthermore, each TIU must be able to
recognize packets addressed to it. As we shall see, the number of bits
required for addressing is relatively small.

Once a packet has been multiplexed on the loop either from the
switch or from a terminal, it has priority over incoming traffic until

it reaches its destination. A terminal must wait for an empty data
packet slot before it can place a waiting packet onto the line. As in
the STDM, the implementation also allows a terminal to place an
outgoing packet into a slot from which it is removing an incoming
packet.

We consider two implementations of the DM system, which corre-
spond to the maximum speed at which terminals can transmit or
receive. In the adjacent slot seizure implementation, terminals can
transmit and receive at a 4000-packet-per-second rate. We consider
an alternate implementation where the terminal is constrained to
operate at a 2000-packet-per-second rate. In this case, a terminal can
only write into or read from alternate packet slots.

A major component of the system is the Central Switch. The function
of the switch is to route and control the flow of information. All
messages generated at User Terminals pass through the switch where
they are passed on to their destination terminals. Now the operation
of the system (see Section V) is such that, as it may not be possible to
deliver a message to its destination immediately, messages are tempo-
rarily stored in the switch. Also, destination terminals have some
control over the way that these stored messages are read out of the
switch’s buffer.

The storage capability of the switch is not unlimited; therefore,
the flow of information packets into the switch must be controlled.
The switch does this by informing terminal TIU of the amount of
storage currently available in the switch. The terminal does not
transmit information packets when there is no room in the switch,
but holds them until storage is available.

As mentioned earlier, models of the system were studied in order to
gain insight into performance and thereby guide design decisions. The
models studied are approximations to actual system operation. We
felt that the study of more exact, hence more complicated, models
would have involved far more time and effort, without a corresponding
increase in insight.

We study the performance of multiplexing techniques on the loop
as measured by message delay. In the switch we study packet storage
requirements from two points of view, throughput and user strategy.
Since the switch inhibits the flow of information when the storage in
the switch is used up, there is a relationship between throughput and
storage capacity. We also study the effect of different user readout
strategies on switch storage requirements.

As a guide to the reader, we pause to summarize the main body of
the paper before plunging into details. In Section III, analytical and
simulation approaches to the loop multiplexing problem are presented.
Section IV is devoted to a discussion of our results on loop multiplexing.
The relationship between storage capacity in the switch and through-
put is considered in Section V. The results of a simulation study of
capacity and throughput are presented in Section VI. In Section VII
we consider the effect of a user’s strategy on storage requirements in
the switch. The results of this study are presented in Section VIII.

Although the analytical and simulation techniques used in our study
are not restricted to a particular message distribution, we concentrated
on the case where 30 percent of the messages are 32 packets long
(8192 bits) and the remaining messages are one packet in duration
(256 bits). This message distribution was our best guess at the actual
distribution of messages in the system and reflects the fact that most
terminals will, in fact, be computers. In the sequel we use the term
variable message length to designate this distribution. The case where
all messages are one packet in duration was also studied to some
extent. In referring to this latter distribution we use the term constant
message length.

The results of our studies of loop multiplexing are presented in
Section IV. Simulation results for Demand Multiplexing indicate
little difference in performance between alternate and adjacent slot
seizure (see Figs. 6 and 7). The simulation was carried out for the
variable message length distribution which consists of a large propor-
tion of long messages. One would expect this distribution to be es-
pecially sensitive to the minimum time required to transmit and
receive these long messages. In contrast, for the constant message
length distribution, these maximum speeds, 2000 packets per second
(alternate slot seizure) or 4000 packets per second (adjacent slot
seizure), should have much less effect since the time to transmit a
single packet is the same for both.

Analytical results show, not unexpectedly, that Demand Multi-
plexing yields better performance than Synchronous Time Division
Multiplexing (see Fig. 8). Further, as the number of terminals served
by a loop increases so also does the advantage of Demand Multiplexing
(see Fig. 9). However, as the loading for a particular loop configuration
increases, the difference in performance between DM and STDM
decreases (see Fig. 8).

Results on information storage in the switch are presented in
Sections VI and VIII. Our results indicate that, for all but minimum
storage allocation, storage in the switch does not markedly affect
throughput (see Figs. 12 and 13). The study also showed that, for the
message distributions we considered, little is gained by dynamically
allocating storage in the switch, as it is needed, holding nothing in
reserve. Indeed, in certain circumstances, a static storage assignment
does better simply because a static assignment insures reserves in
the switch.

Our study showed that, for the loadings under which the system
may be expected to operate, the effect of user strategy on switch
storage requirements is not pronounced (see Figs. 14 and 15).

Two techniques for multiplexing data on the line are under con-
sideration, Synchronous Time Division Multiplexing and Demand
Multiplexing. In this section we shall present models designed to
evaluate the performance of each of these techniques. These models
are studied using both mathematical analysis and simulation. Simula-
tion is necessary in situations where mathematical analysis is not
possible.

A basic consideration in the design of the system is the response
time to interactive users. An important component in this response
time is message delay. We define message delay to be the time elapsing
between the arrival of a message at a User Terminal and the departure
of the last packet of the message from the terminal. Message delay is
the criterion that we use to evaluate the multiplexing techniques
under study.

We assumed that messages arrive at the User Terminals at a Poisson
rate, and that the entire message arrives instantaneously. One would
encounter this sort of behavior when a computer outputs directly
from its memory to the loop, since the operation of the computer is
at a much higher speed than the operation of the loop.

In both of the system configurations under study, the message delay
may be broken up into two components, queuing delay and multi-
plexing delay. Recall that messages arrive at the station at a Poisson
rate. Since it takes a nonzero amount of time to multiplex each message,
there is a nonzero probability that when a message arrives at the
station it must wait until previously arrived messages have been
multiplexed onto the line. Once all prior messages have been multi-
plexed it takes additional time to multiplex the newly arrived message.

The problem of determining message delay for Synchronous Time
Division Multiplexing and Demand Multiplexing with adjacent slot
seizure is mathematically tractable. In order to study message delay
in the DM case with alternate slot seizure, simulation is required.

A sketch of the loop model is shown in Fig. 3. N terminals are
connected to the loop. The flow of data is unidirectional around the
loop and is shown as counterclockwise in Fig. 3. The first terminal
after the switch is labeled terminal number 1, the second terminal
number 2, and so on. Messages arrive for multiplexing at a terminal
at a rate of \ messages per second. We also assume that messages
flow from the switch to each terminal at a rate of \ messages per
second. The result is that the volume of traffic flow around the loop
is symmetric. The total traffic flow from the switch to the terminals
on the loop is NA messages per second. In the case of Demand Multi-
plexing this flow of return messages affects the operation of the loop.
We shall ignore the interaction between the loop and the rest of the
system by assuming that return message flow is Poisson. This assump-
tion makes analysis possible and considerably simplifies simulation. We
shall return to a consideration of this assumption after we present
our models.

As we have seen, the bit flow on the T1 loop is formated so that
slots, into which information packets can be inserted, flow at a rate of
4000 per second. For Synchronous Time Division Multiplexing, each
of these packet slots are assigned to particular terminals on a periodic
basis. If there are N terminals connected to the T1 loop and each
terminal is accorded equal treatment, then a packet slot is available
every T. = NT, seconds, where T, is the duration of a packet slot.
In the sequel we shall refer to 7’. as the cycle time. For each terminal,
we take the end of one cycle and the beginning of the next to be the
end of the packet slot assigned to that terminal. We assume that a
terminal may always write into its assigned slot even if it is simul-
taneously reading from the slot.

In order to develop an expression for the delay encountered by a
message, let us assume that a message consisting of mzi1 packets
arrives at a terminal whose buffer is empty, i.e., all previously arriving
messages have been transmitted. If the message arrives w seconds
before the end of a cycle, then a total of w + (mz41 — 1)T. seconds
elapse before the entire message is transmitted. Now if previous
messages have not been transmitted, a newly arrived message suffers
queuing delay as well as this multiplexing delay. For the purposes of
analysis we categorize the packets of previously arrived messages into
two classes: packets held over from previous cycles and packets that
have arrived during the present cycle in the time interval T, — w.
We may write the total delay queuing and multiplexing as:

In eq. (1), gis the number of packets remaining from previous cycles,
Lis the number of messages arriving in the interval T, — w, and m;
is the number of packets in the ith of these L messages. The mean
value of d; is shown in the appendix to be

d= T.(m 5) tag ar (2)
where m is the average message length in packets. Higher moments of
d; can be found since, in eq. (1), the terms g7., Te )o#41 mi + w, and
(mr41 —1)T. are independent random variables whose moment-
generating functions can be calculated (see the appendix). Expressions

for the moment-generating function of di and for the mean-square
value of d; are given in the appendix.*

We now consider two implementations of Demand Multiplexing,
adjacent slot seizure and alternate slot seizure. With adjacent slot
seizure, a typical sequence of information slots leaving the switch
might look as shown in Fig. 4. For purposes of explanation in Fig. 4
we assume messages are either three packets long or one packet long.
The numbers in the slots correspond to the destination of the packet.
No number in a slot indicates that it is empty. When the first three
slots shown in Fig. 4 pass terminal 1, it is blocked. Terminal 1 may
insert an information packet into slot 4 and into successive slots up to
slot 10 when it is again blocked until slot 11. Terminal 1 also removes
packets from slots 7 and 9. Terminal 2 removes the packets from slots
1, 2, and 3 and may insert information packets into these slots.
Terminal 2 will be blocked when slots 10, 13, 15, 16, and 17 pass.
Terminal 2 will also be blocked by terminal 1, if terminal 1 multiplexes
packets onto the lines. The same rules apply to all of the other termi-
nals on the loop. A terminal is free to insert data into empty slots and
slots from which it removes data. Once an information packet has
been inserted into an empty slot, it has priority over any other incom-
ing packets.

Alternate slot seizure is similar except in one important respect.
Since the terminal cannot receive nor transmit on adjacent information
slots, it is limited to a maximum rate of 2000 packets per second. A
typical flow of information is indicated in Fig. 5. Terminal 1 is blocked
in slots 1, 3, and 5 but may transmit in slots 2 and 4. If terminal 1
begins by inserting a packet in slot 8, the next slot that is available to
it is slot 11. These same rules apply to all of the other terminals in
the loop.

The problem of message delay for Demand Multiplexing with
adjacent slot seizure has received considerable attention recently.?~+
Expressions for message delay that are relevant to our study can be
found in Ref. 5. In order to make clear the assumptions that led to
these expressions we shall sketch the analysis.

Basic assumptions of our study are that each terminal on the
loop receives as much traffic as it transmits (see Fig. 3) and may
write into slots containing packets addressed to it. Therefore, each

terminal “sees” the same traffic volume, (N — 1)\ messages per
second. The flow of traffic past each terminal consists of alternate
busy and idle periods. Messages are multiplexed into line idle periods
and are blocked by busy periods.

The probability distribution of message delay depends upon the
distributions of the line busy and idle periods. In order to find these
distributions, a basic assumption about the nature of the traffic flow
out of the switch is necessary. We assume that the line busy and idle
periods out of the switch are caused by the Poisson arrival of messages
at the switch. This is not difficult to justify when there is light loading
on the loops connected to the switch. Under light loading, messages
arriving at User Terminals encounter little blocking and are conveyed
immediately to the switch. Message arrival at the terminals is Poisson.
For the message lengths and loadings we shall consider, the time
between message arrivals is large compared to a slot time so that the
discretization of message flow on the loop has little effect. As loading
increases, the situation is less clear. However, messages arrive from all
of the loops connected to the switch, tending to randomize
message flow.

A line busy period at the output of the switch is initiated by a
message arrival, which under the Poisson assumption is instantaneous.
The busy period is lengthened if another message arrives while the
previous message is being transmitted. The duration of a busy period
is the same as the duration of the busy period of an M/G/1 queue

for which results are well known.* Furthermore, under the Poisson
assumption, the interval between message arrivals has a negative
exponential distribution. Since each terminal sees (NV — 1)\ messages
per second, the duration of a line idle period, as seen by a TIU, is
negative exponential with mean [(N — 1)A]}7.

We assume that the statistics of line busy and idle periods are the
same for all terminals on the loop. The removal of messages from the
data stream may affect this assumption. Also, strictly speaking, these
statistics also depend upon the multiplexing strategy of the switch.
Further work involving simulation must be done to verify this
assumption.

Messages are multiplexed packet-by-packet into empty slots. The
multiplexing is interrupted by the advent of a busy period and is
resumed when the busy period is over. Armed with the statistics of
the line busy and idle periods, the message delay can be found. In
the language of Queuing Theory, the model is that of a server that
suffers periodic breakdowns. The results of the analysis appear as sets
of curves that will be discussed presently.

From the foregoing we have an analytical approach for the calcu-
lation of delay in the case of Demand Multiplexing with adjacent
slot seizure. There are inherent difficulties that preclude an analytical
solution in the case of alternate slot seizure. The basic difficulty lies in
calculating the durations of line busy and idle periods. For alternate
slot seizure a terminal is blocked only if there are two or more inter-
fering terminals. Thus a line idle period is terminated when a terminal
begins transmitting, if there is at least one other terminal already
transmitting.

Message delay in the case of alternate slot seizure was studied by
means of simulation. The simulation program also could be used to
study adjacent slot seizure. Although adjacent slot seizure can be
analyzed, it was simulated primarily as a check.

In order to insure that the basic assumptions that underlie our
model are understood, we outline the basic structure of the simulation
program. The number of terminals on the loop is an input variable
to the program. Input variables also determine the rate of message
arrival, the length of long messages in packets, and the mixture of
long and short messages. The simulation was carried out for the
variable length message distribution. The basic time unit of the
program corresponds to the duration of a packet slot, 1/4000 second.
During each time unit a message may arrive to be multiplexed on the

line. The random message arrival is simulated by comparing the
output of a pseudorandom number generator to a threshold. If the
test indicates that a message has arrived, the length of the message
and the terminal to which it arrived are chosen randomly. A basic
assumption here is that during a packet slot time no more than one
message arrives at all N terminals. For the loadings of interest, this is
not a restrictive assumption.

For each of the N terminals in the loop simulation, numbers are
stored indicating the current number of packets and messages in the
terminal buffer. In each basic time unit, terminal buffers 1, 2, --- , N
are examined in succession until a nonempty buffer is found. If adjacent
slot seizure is being simulated, a packet is removed from the first
nonempty buffer. In the simulation of alternate slot seizure, a packet
is removed from the first nonempty buffer from which a packet was
not removed in the previous time unit. After either a packet has
been removed from a buffer or all buffers have been examined, the
program shifts to the next basic time unit and the cycle repeats
beginning with message arrival.

Our interest is in the Nth terminal as it sees traffic from N — 1
other terminals. The line busy and idle periods seen by this terminal
are measured. The program also measures the number of messages
remaining in terminal N’s buffer immediately after an entire message
has been removed from the buffer. This measurement was not made
every time a message departs, but periodically. The length of the
period between measurements was varied from run to run. The reason
for this is to guard against high correlation between measurements,
thereby insuring independent samples.

Results of the simulation and the analysis of the previous section
are shown as curves which show the mean and the standard deviation
of delay as a function of loading. From these curves we draw con-
clusions about the relative merits of the systems under study.

On Fig. 6 are shown plots of average delay measured in the simula-
tion as a function of loading for 5- and 20-terminal loops. The line
loading is the portion of the time that the line is occupied. For com-
parison the results of the theoretical calculation of average delay is
also shown in Fig. 6.

The simulation also yielded estimates of the standard deviation of
message delay. Results are shown on Fig. 7, where the standard devia-
tion of delay is shown as a function of loading. As in Fig. 6, the results
of analysis for adjacent slot seizure calculations are shown.

In all of the curves the message distribution is the variable length
message distribution defined earlier. The resulting average message
length is 10.3 packets per message. The message arrival rate at each
station in the loop in terms of loading, p, is p/(10.3N) messages per
slot time. (Each slot time is 1/4000 second.) Thus for 0.103 loading
on a 20-terminal loop, messages arrive at a rate of 0.0005 message
per slot time or 2 messages per second at each terminal.

O ADJACENT SLOT SEIZURE — 5 TERMINALS
4 ALTERNATE SLOT SEIZURE — 5 TERMINALS
X ADJACENT SLOT SEIZURE — 20 TERMINALS
© ALTERNATE SLOT SEIZURE — 20 TERMINALS

4 ALTERNATE SLOT SEIZURE — 5 TERMINALS /
X ADJACENT SLOT SEIZURE — 20 TERMINALS / oO
© ALTERNATE SLOT SEIZURE — 20 TERMINALS

equilibrium had been reached. The duration of runs and the random
sequences used in the simulation were varied. The standard deviation
of the estimates of the mean values of delay shown on Fig. 6 can be
estimated. We assume that the measured standard deviations are the
true standard deviations. The standard deviation of the mean is then
the measured standard deviation divided by the square root of the
number of samples. The results indicate that the standard deviation
of the mean is small compared to the mean value. The standard
deviation is largest relative to the mean at light loadings on the
20-terminal loop where it is approximately 5 percent of the mean.

There is a basic difficulty in making measurements at light loadings
on a 20-terminal loop. Due to the relatively low departure rate, fewer
independent samples can be gathered. Except for the lighter values
of line loading on the 20-terminal loop there is good correspondence
between the results of simulation and theory. Even at these lighter
loadings the results of simulation are not so far from theory as to cast
doubt on the simulation.

The simulation results show that adjacent slot seizure yields some-
what better delay performance than alternate slot seizure. For almost
all values of line loading, adjacent slot seizure gives lower values of

mean delay and standard deviation of delay. Moreover, measurements
made on loops with 2, 10, and 64 terminals, not shown here, yield much
the same result.

The reader will notice, however, that for most values of line loading,
the difference between alternate and adjacent slot seizure is not large.
The difference is small enough so that ease of implementation should
probably determine the choice between the two.

Because of pressures of time we did not compute message delay
distributions. However, estimates of the distribution of message delay
can be calculated from the simulation values of mean and standard
deviation. From the Tchebychev inequality we have

where y is the mean and o the standard deviation of the delay. On a
20-terminal loop with alternate slot seizure this inequality shows that
at 0.515 loading 90 percent of the messages suffer delays of less than
83 milliseconds. However, the Tchebychev inequality often gives a
rather loose bound. Under a rather tenuous assumption about the
distribution of delay, one obtains a more optimistic result. If we
compare means and standard deviations of delay we see that, for the
same line loadings, the means and the standard deviations are roughly
the same. For an exponentially distributed random variable the mean
and the standard deviation are equal. If we assume that delay is
exponentially distributed we find

where 1/a is the standard deviation. For 0.515 loading on a 20-terminal
loop, 90 percent of the messages have delay less than 50.5 milliseconds.

A second phase of our work on loop multiplexing was devoted to a
comparison of Synchronous Time Division Multiplexing and Demand
Multiplexing. In order to simplify analysis we have ignored the fact
that for DM information packets must contain the address of the
transmitting terminal. No such addressing is required for STDM.
However, such addressing information is a negligible part of an
information packet. For example, for a 64-terminal loop, only 6 bits
are necessary to specify the address of a terminal. We feel that the
small improvement in accuracy that could be attained by considering
addressing did not justify the complications introduced into the
analysis.

Fig. 8—Average delay versus loading in DM and STDM (30 percent of messages
are 32 packets long).

Typical results are shown on Fig. 8 where delay in both packet
times and milliseconds is shown as a function of line loading for the
variable length message case. As the curves show, DM is clearly
superior to STDM. This superiority is more pronounced at the lighter
loading, where the multiplexing time is the strongest component of
delay. In the absence of interfering traffic, the time required to multi-
plex a message in DM is an average of 10.3 slot times. In contrast,
for an STDM system with N terminals, the average time required to
multiplex a message is 10.3 X N slot times. As the loading increases,
the difference between the two systems decreases. Line traffic in the
DM system interferes with message multiplexing and as the load
increases so does the interference.

Similar results have been obtained for the constant length message
distribution. Computations of the standard deviation of delay for
both constant and variable length message distributions also show the
same basic pattern.

Another view of the performance is indicated on Fig. 9 where average
delay is shown as a function of the number of terminals in the loop

Fig. 9—Message delay versus number of stations (30 percent of messages are
32 packets long).

for fixed values of loading. In Fig. 9 the dependence of delay in the
STDM system on the number of terminals in the loop is marked.
There is little of this dependence in the case of DM. However, DM
is more sensitive to changes in load than STDM. Notice the large
jump in delay from 0.5 loading to 0.9 loading in the case of DM.
Although we have not shown them, similar results obtain for the
constant length message distribution.

The second phase of our work involved a study of buffering in the
switch. Streams of data enter the switch from the loops connected to
it. In the DM implementation entering packets are labeled as to
originating terminal. In the STDM implementation each terminal has
an assigned packet slot recurring periodically. System operation is
such that, at any given time, each terminal in the system transmits to
and receives from only one other terminal in the system. Information
on which pairs of terminals are linked together is stored in the switch.

Therefore, given the origin of an information packet, the switch
determines its destination by looking in a table.

A terminal can rapidly change the destination of the packets that
it transmits. Stored in the Central Switch is a list of up to 64 possible
correspondent terminals for each terminal. A terminal that is trans-
mitting to terminal A, for example, may select a new destination, say
terminal B. By means of signal packets (see Section I) the Central
Switch is notified of this change in destination. After the change all
information packets transmitted from the originating terminal are
routed to terminal B. A terminal can select only from the list of its
64 correspondent terminals stored in the switch. However, this list
can be altered by the originating terminal when it wishes to make
connection with a new terminal or drop connection with an old. Again,
signal packets are used to communicate between the terminal and
the switch. The process of altering the list requires much more time
then switching between terminals already on the list.

At a given instant of time, a terminal transmits to and receives from
the same terminal. Further, each terminal in the system acts indepen-
dently in selecting the correspondent terminal that is the destination
of its packets. Thus a terminal may select a destination terminal that
is, at that point in time, corresponding with a third terminal. In this
event the packets that are transmitted are stored temporarily in the
Central Switch. The Central Switch, again using signal packets,
notifies the destination terminal that packets from a particular originat-
ing terminal are waiting to be delivered. It may happen that, for a
particularly busy terminal, there may be messages from several
different originating terminals stored in the switch waiting to be
delivered. The receiving terminal is free to choose the order in which
these messages are read out of the switch buffer.

In connection with this routing and selection procedure we use the
term virtual channel as a notational shorthand. As we have seen, each
User Terminal has stored in the switch a list of as many as 64 corre-
spondent terminals. When a terminal selects the 7th correspondent on
this list, we say that the terminal selects the zth virtual channel.
When we say that a terminal transmits and receives over virtual
channel 7 we mean that the terminal transmits to and receives from
the zth correspondent terminal on the list stored in the Central Switch.

As we have indicated, it may be necessary to store information
packets in the switch before they can be delivered. As a practical
necessity, the amount of storage in the switch is finite and under
heavy loading conditions storage may be used up. In this situation the

switch sends signal packets to User Terminals which inhibit transmis-
sion until storage is available in the switch.

Our study of packet storage capacity in the switch focused on two
aspects of the problem, throughput and user strategy. Given the
random nature of the message flow in the system, there will be occa-
sions when all of the storage assigned to a channel is used up and the
transmitting terminal is inhibited. If this condition occurs often
enough, there will be a significant effect on the total throughput of
data. Secondly, the user through his virtual channel selection strategy
can affect the amount of storage that is required in the Central Switch.
As we have noted earlier, a certain amount of time is required for a
User Terminal to switch from one virtual channel to another. During
this switching time, the terminal cannot read packets out of the
switch. If User Terminals pursue a strategy calling for frequent
switches, demands on switch storage may be too large.

In order to study throughput and the effect of user strategy on
buffer requirements, a simplified model was constructed. The model is
shown in Fig. 10. N independent data streams carrying \ messages per
second flow into N buffers. These data streams represent traffic from
correspondent terminals flowing over different virtual channels to the
same destination terminal. The destination terminal’s changing of
virtual channels is represented by the switch in Fig. 10 moving from
buffer to buffer. In the model the time required to switch buffers is
taken to be either zero or eight packet times (1/4000 second).

A good deal of previous work on buffer occupancy has been based
on a Poisson arrival model for messages in the input data stream. In
the Poisson model, messages arrive instantaneously with an ex-

ponentially distributed interval of time between messages. A more
realistic model, for our study of throughput, is one in which messages
arrive over a time interval proportional to the message length with
the time between the beginning of one message and the end of the
previous message being exponentially distributed. This latter model is
more appropriate to buffering in the switch where the arrival and
departure of messages is over T1 lines and the read-in and write-out
rate of messages is the same. In Section VIII, results based on the
continuous arrival model will be compared with the Poisson arrival
model.

The model used in our study is indeed something of a simplification.
Because terminals share the same T1 loop, messages are likely,
especially in heavy loading, to be broken up when they are multiplexed.
In our model we do not take this effect into account. For example, in
our model a message with 32 packets would occupy 32 successive
packet slots on the input line. In the actual system, there may well
be gaps in these messages. Similarly, we assume that messages going to
a particular destination terminal have sole access to the T1 line, when
in fact the line is shared. Thus we assume that messages can be read
out of buffers at will. Fortunately, these two effects tend to cancel out.
In our model we read in faster and read out faster than reality. Also,
we are not looking at absolute measures of performance, but are

comparing different implementations. We felt that a more complicated
input output model would not improve this comparison significantly.

Even this simplified model was not amenable to analysis and a
Monte Carlo simulation program was written. The basic functions of
the program is to measure the throughput as a function of storage
capacity and to measure the average occupancy of the buffers. Input
variables to the program determine the amount of storage available,
the number of input lines and buffers, the time required to switch
between buffers, and the probability of message arrival.

The program is easily changed to handle either common or dedicated
buffering. For common buffering, an input variable is the total storage
available. When a packet is inserted in a buffer, this number is reduced
by one. For dedicated storage, the storage for each line’s buffer is an
input variable. As a packet is inserted in a buffer, the amount of
storage available for that buffer is reduced by one.

There is a simple relationship between the probability of message
arrival and load. Let p be the probability that a message is generated
in a slot. It can be shown that the portion of the slots on each input
line that are busy is given by the expression

| ee 16 aa
pm +1 —p’
where 7m is the mean length of a message in packets. The assumption
here is that there is no limitation on the content of the buffer into
which the input line feeds. N input lines feed into the buffers, conse-

Fig. 11—Line load as a function of the probability of message arrival (80 percent
of messages are 32 packets long).

quently the maximum occupancy of the line carrying messages out of
the buffers is
_ pNm

= DET p ”)
The output line will attain this maximum occupancy if no time is
required to switch between buffers. The relationship between loading,
L, and the quantity pN, which we designate as the probability of
message arrival, given in eq. (3) is plotted in Fig. 11.

required by a User Terminal to select a new virtual channel. When a
packet is removed from a buffer, the amount of storage available is
increased by one up to some fixed amount.

In successive simulation runs the amount of packet storage available
was varied with all other parameters held constant. For very large
amounts of storage, there is always room in the buffers. As the amount
of storage is decreased, it is increasingly likely that packet flow from
an input line is inhibited. For relatively low amounts of storage, it will
often happen that there is no room in the buffer. In this case, the flow
of messages will be halted frequently and the number of packets flow-
ing into the buffers per unit time will be reduced.

In the measurement portion of the program, the main focus was on
throughput. Programs were run for 20,000 cycles, and the total number
of packets that were fed into buffers were measured. By varying the
total amount of storage available, with all other parameters fixed, one
obtains the relationship between throughput and storage. Simulation
runs were made for the constant and variable length message dis-
tributions. Measurements were also made of the total number of
messages in the buffers. The results of these latter measurements will
be considered in a later section dealing with user strategy.

Typical results of simulation are shown on Figs. 12 and 13 for 5
and 20 input lines, respectively. In obtaining the results shown on both
figures the variable length message distribution was used. The switch-
ing time is 8 packet slots. If the line rate is 4000 packets per second,
the time required to switch is 2 milliseconds. The curves show nor-
malized throughput as a function of the total packet storage with
message arrival probability as a parameter. For each loading the
throughput is normalized to the throughput measured at very large
storage capacity.

The basic configuration of the curves is as one might expect. As the
storage capacity decreases, the throughput decreases. Further, the
normalized throughput decreases faster for the larger values of loading.

The results show that, even for a limited amount of storage, the
throughput is high. For example, if there are two packet slots for each
buffer, the throughput is over 70 percent even for high loading. Results
(not shown) for the case of zero switching time show that for this
same amount of storage the throughput is over 90 percent.

A surprising result shown on Figs. 12 and 13 is that dedicated
storage shows better performance than common storage when there is

Fig. 12—Throughput versus packet storage for 5 input lines (switching time is
equal to 8 slot times).

a limited amount of buffering available. A combination of factors
produces this result. First of all, even though storage may be held in
common, it is committed to input lines (virtual channels) in a specific

Fig. 13—Throughput versus packet storage for 20 input lines (switching time is
equal to 8 slot times).

way that may be far from optimum. The preponderance of traffic
is contained in messages that are 32 packets long. If the amount of
storage held in common is limited, one channel may absorb all of the
storage that is available in the switch. We have also simulated models
where all messages are one packet long. In this case common storage
is superior to dedicated. However, even in this case the difference
between common and dedicated storage is not great.

In the foregoing, messages are generated regardless of whether there
is room in the switch or not. We have also examined an implementation
where messages are not generated until there is buffer space available.
The results of a simulation study of this implementation are essentially
the same as the results presented here.

Before concluding this section let us consider the reliability of the
foregoing results. First of all, a good many simulation runs were made
whose results are not shown here. In these runs, the number of chan-
nels, the starting points of the random sequence used in the Monte
Carlo technique, and the running time of the simulation were varied.
All of the results were in conformity with the results presented in
this paper.

Recall that an input parameter to the program was the probability
of message arrival. In Fig. 11 the theoretical relationship between
loading and this quantity is shown. Also shown on Fig. 11 is the
loading obtained in the simulation. As seen on Fig. 11, the results of
simulation are within 5 percent of the theoretical values. This is
additional indication that the simulation runs were long enough to
obtain representative data sequences.

Estimates of the standard deviation of the throughput were made.
This was done on each simulation run by measuring the throughput
every 1000 cycles, obtaining a sequence of 20 points. (Recall that the
simulation runs were 20,000 cycles long.) The mean, yu, and the vari-
ance, o”, of these points was calculated, giving an estimate of the
mean and standard deviation of the throughput in 1000-cycle intervals.
The sum of these points is the total throughput for the simulation
run. If we assume that the throughputs for successive 1000-cycle
intervals form a sequence of independent, identically distributed
random variables, the variance of the throughput for a 20,000-cycle
run is 2002. The coefficient of variation or the ratio of the standard
deviation to the mean for a 20,000-cycle run is

Our measurements show that in all cases the coefficient of variation is
less than 0.1 and in many cases it is less than 0.05. This result means
that with different starting points in the random sequences we would
expect a relatively small variation around the points we have plotted
on Figs. 12 and 13.

At any given point in time, a User Terminal knows which virtual
channels have messages waiting to be delivered. A terminal is free to
select these virtual channels in any order. We assume that a terminal
will not interrupt the reading out of a message in order to switch to a
new channel. Since the channel selection procedure is entirely in the
hands of the User Terminal, we studied the effect of different strategies
on system storage requirements. Accordingly, a calculation of buffer
occupancy statistics for different user strategies was performed.

As in the study of throughput we use the model shown on Fig. 10.
Again, input lines correspond to virtual channels and switching
between buffers corresponds to the selection of new virtual channels.
In order to make the analysis tractable, we assume that messages
arrive at a Poisson rate of ) messages per second over each input line.
Further, we assume that eight packet slot times are required to switch
buffers. In order to calculate bounds, we also consider the case where
no time is required to switch.

We analyze the buffer occupancy of the 1-by-1 and the random
strategies by using the theory of the M/G/1 queue. Messages arrive
at all buffers at a Poisson rate NA messages per second. If the time
required to switch between buffers is zero, we can take the service

time in both strategies to be either 1 slot time or 32 slot times, depend-
ing on whether a message is long or short. An upper bound on storage
requirement for the 1-by-1 strategy can be found by assuming that
after each message is read out one always switches to a nonempty
buffer. We can analyze this situation by adding the switch time to the
time required to read out each message. Thus we have read out times
of 9 slot times for short messages and 40 slot times for long messages.
This is an upper bound because there is a nonzero probability that
all of the messages are in the same buffer and there is no reason for
the station to select a new virtual channel.

For the random strategy the service time is slightly different than
1-by-1. With probability 1/N no switching takes place and the service
time is simply the time required to multiplex a message.

Let b be a random variable denoting the number of slots required
to read a message out of a buffer, including switching time. We write
b = (m+ w)T,, where w is the time required to switch in slot times.
The random variable w is independent of m. If, in the case of random
switching, 8 slot times are required to switch between buffers, the
probability that w = 8 is 1-1/N and the probability that w = 0 is
1/N. From the analysis of the M/G/1 queue,® it can be shown that
the mean number of messages in all N buffers is

(4)
where b? is the 7th moment of b and ) is the average message arrival
rate in messages per second. The mean-square number of messages in
the buffer is

Our primary interest is in the number of data packets in switch
buffers rather than in the number of messages. It can be shown that
the mean and the mean-square number of packets in all N buffers is
given respectively by

The foregoing considers packet storage requirements in all of the
buffers in the switch. We shall focus our attention on the number of
packets in individual buffers assigned to virtual channels. The mean

In the sequel we shall consider the time to switch between buffers,
w, as fixed; therefore, © = w and w*? = w*. The quantities mT, and
mT? denote the first two moments of the time required to read a
message out of a buffer. The mean number of messages in the buffer is

An expression for the mean-square number of packets in each buffer
can be derived from the work presented in Ref. 6. This expression is
rather lengthy and provides little insight; therefore, we shall omit it.

Results of computation using this expression will be presented in
the sequel.

The results of computations using the formula derived in the fore-
going are shown in Figs. 14 and 15 for 5 and 20 buffers, respectively.
In these figures the average occupancy of each buffer is shown as a
function of load, which is the product of the message arrival rate and
the average time required to read out a message, NAm. As expected,
the lowest buffer occupancy occurs in the case where no time is re-
quired to select a new virtual channel. When an 8-slot-time channel
select time is required, the technique with the lower occupancy
depends upon the loading. At light loading, the “empty before switch”’
strategy shows poorer performance because time is wasted stopping at
empty buffers. It must of course be remembered that this is only an
upper bound for the ““empty before switch” that selects only nonempty
buffers. As the loading increases, there are fewer empty buffers and
the performance of the “empty before switch” strategy improves
relative to the l1-by-1 strategy.

Fig. 14—Average number of packets in each buffer for 5 buffers (380 percent of
messages are 32 packets long).

Fig. 15—Average number of packets in each buffer for 20 buffers (30 percent of
messages are 32 packets long).

In the previous section we considered an ‘‘empty before switch”
strategy that skipped empty buffers. As we have mentioned, the
problem of calculating occupancy statistics for this technique is
intractable. However, the results shown on Figs. 14 and 15 form
bounds on the skipping empty technique. The shaded areas in the
figures indicate the areas in which the statistics for this method lie.

If the system is operated below 0.5 loading, the difference between
the different channel switching strategies is not very large. For example,
for 20 channels and 0.4 loading (see Fig. 15) the average occupancy for
1-by-1 rotating strategy is 1.1 packets. For the “‘empty before switch”
strategy skipping empties, the average occupancy is between 0.4 and
1.0 packet. As the load increases beyond 0.5 loading, the 1-by-1
strategy leads to saturation and the cyclic system is clearly superior.

Results for the standard deviations of buffer occupancy have been
obtained. These results support the foregoing conclusions.

The simulation program discussed in the previous section computed
means and standard deviations of buffer occupancy for an “empty
before switch” strategy with skipping of empty buffers. The results
are shown on Figs. 14 and 15. A comparison of analysis and simulation

indicates that for the most part the analysis gives upper bounds to
the simulation. This is not unexpected since, in the simulation pro-
gram, messages arrive over an interval of time, whereas for the Poisson
arrival model used in the analysis, messages arrive instantaneously.

Equation (1) of the text is the following expression for the delay
in Synchronous Time Division Multiplexing:

The delay, d,, is the sum of the three mutually independent random
variables qT... T. Di1 mi + w, and (mz41 — 1)T.. Thus the moment-
generating function for di is the product of the moment-generating
function for these three variables. In this appendix we shall calculate
the moment-generating functions of each of these.

Recall that in STDM dedicated packet slots are available to each
terminal cyclically every 7’, seconds. We take the end of one cycle
and the beginning of the next to be the end of a dedicated packet slot.

Let gq; be the number of packets remaining at the end of the jth
cycle and let a; be the number of packets arriving during the jth
cycle. We can write

where U(zx) is such that U(x) = 1 for xz > 0 and U(x) = O forz < 0.
Taking expectation on both sides of (14) and assuming equilibrium
(i.e., Eqj41 = Eq;) we find that
Elu(q; + aj41)] = ELaj41].
But
Elu(q; + aj+1)] = P-La; + aiz1 > 0].

If we assume equilibrium has been reached, we may define
Q(s) = ELe~*% ]
for all 7. From (16) after some manipulation we have

where A(s) = E[e~**]. Since messages arrive at a Poisson rate i,
it can be shown that
A(s) = e>Pell—M(a)) | (18)

where M(s) is the generating function of the messages.
By successive differentiation it can be shown that the first two
moments of q are

where 4, a’, and a are the first three moments respectively of the
number of packets arriving in a cycle 7',. By successive differentiation

where 7, m2, and m3 are respectively the first three moments of the
number of packets in a message.

We turn now to the second term in (1), f Aw + TT. Diei1m; A
message arrives at random during a cycle, w seconds before the end
of a cycle. In the time interval T.. — w, L messages arrive, all of which
have priority over the newly arrived message. Since message arrival
is random, the quantities L and w are mutually dependent random
variables. Conditioned on w, the probability that L messages arrive
in the interval T, — w is
M(T. — w)t

The random variable w is uniformly distributed in the interval (0, 7’).
Let r(t) be the density function of the random variable T.m;. We may
write

where r)(t) is the Z-fold convolution of r(t). The Laplace-Stieltjes
transform of this can be shown to be

The first two moments of f can be found by successive differentiation
of (23):

generating function of the message is M(s), the generating function of
this term is easily shown to be

The mean value of delay is the sum of the terms q7., f, and g. The
variance of the delay can be calculated by summing the variances of
qT, f, and g.

1. A. G. Kohneim, “Service Epochs in a Loop System,” presented at the 22nd Int.
Symp. Computer-Communications Networks and Teletraffic, Polytech. Inst.
Brooklyn, Brooklyn, N. Y., April 1972.

. J. F. Hayes and D. N. Sherman, “Traffic Analysis of a Ring Switched Data
Transmission System,” B.S.T.J., 50, No. 9 (November 1971), pp. 2947-2978.

. R. R. Anderson, J. F. Hayes, and D. N. Sherman, “Simulated Performance of a
Ring Switched Data Network,’ IEEE Trans. Commun., COM-20, No. 3
(June 1972), pp. 576-591.

. B. Avi-Itzhak, ‘“Heavy Traffic Characteristics of a Circular Data Network,”
B.S.T.J., 50, No. 8 (October 1971), pp. 2521-2549.

. J. F. Hayes and D. N. Sherman, ‘‘A Study of Data Multiplexing Techniques and
Delay Performance,” B.S.T.J., 51, No. 9 (November 1972), pp. 1983-2011.

. A. G. Fraser, “Interconnecting Computers and Digital Equipment,” internal
report available upon request.

Copyright © 1974 American Telephone and Telegraph Company
Tue Bey System TEcHNIcCAL JOURNAL
Vol. 53, No. 2, February 1974
Printed in U.S.A.

Peak-Load Traffic Administration of a Rural
Multiplexer with Concentration

A procedure is proposed for estimating the main-station capacity of an
SLM* (Subscriber Loop Multiplexer) system by observing the traffic load
when the system ts partially filled. The procedure is intended to be usable
in unattended offices, and requires only one measurement per week and
very few calculations. In contrast to the usual practice of measuring load
in a time-consistent busy-hour, we work with weekly peak loads, and so
our method is based upon the statistical theory of extreme values. The
validity and precision of the procedure have been investigated by applying
at to data from a study of rural traffic and by a Monte Carlo study of tts
behavior. Use of this administrative procedure should give the average
SLM _ system a capacity of about 120 rural residential customers, in
contrast to the limit of 80 that would be necessary in the absence of traffic
measurements.

I TN PRODUCTION 6440 ).ccuwesd dy Sa Ge eee sal ee ree ers hss 262
Ho THE ‘BASIC: PROCH DUR s iss hy bee Sire NBs ahaa es oe ee hs 263
IJI. THE DISTRIBUTION OF WEEKLY PEAK TRAFFIC LOADS... 265
IV. THE MATHEMATICAL MODEL..................0...00 000005: 265
V.. THE SERVICE: CRITBRION 54; caccs Oona ibe Fauna ek vate 267
Vi" THE MONTE: CARLO STUDY. oc) oo vine jin 35-4 Seca ents 270
VII. DISCUSSION OF ASSUMPTIONS............... 0.00.00 0 000 e eee 273
VIII. SUMMARY AND CONCLUSIONS............... 0002200002000 5. 275
Dx ACKNOWLEDGMENT sce direitos dike sae ended Sok hia s 276

APPENDIX A—Number of Candidate Busy-Hours and Their Load
DDUSEP ID ULBON os. 2.6 5 2s 5 see RRR Sa Aste Soaps cape odeeed Ba gesclos spl Aba t Wqen pode eh eae 276
APPENDIX B—Rules for Traffic Administration............0.. 000000002 278

The SLM (Subscriber Loop Multiplexer) system is a digital carrier
and switching system that was developed to provide economically for
main-station growth and upgraded service on long rural cable routes.
It is capable of serving 80 lines, all sharing 24 channels. Each of the
80 lines can be used for single- or multi-party service. (For a detailed
description of the SLM system see Ref. 1.) For the purposes of this
paper, which is concerned with traffic, the SLM system may be viewed
as a remote line-concentrator serving 80 lines on 24 full-access channels.

The quality of service given to SLM subscribers should be kept
well above levels that might lead to complaints, and to the need for
hasty rearrangements that would interfere with the orderly growth of
subscriber plant. Hence, service for these subscribers should not be
noticeably different from that for customers served by physical pairs
to the central office. This service objective will be met if blocking
exceeds one-half percent in no more than a few hours per year. (It is
possible to imagine a distinct service, for sparsely populated rural
areas, in which a less stringent service objective would be appropriate.)

The Rural Line Study,?" a study of subscriber line usage in rural
areas, has confirmed that rural residential subscribers like those studied
in the territory of South Central Bell can almost always be served on
one SLM system with essentially no blocking in groups of 80 main
stations or more. (In fact, as shown below, most rural systems should be
able to serve many more than 80 main stations.) However, the load per
main station does vary greatly from place to place and from customer
to customer. Thus even 80 main stations will in a few cases generate
enough load to cause undesirably frequent blocking in excess of one-
half percent. Some means of monitoring the traffic performance of
SLM systems are therefore necessary.

One such means is a register which records the total amount of a
system’s all-channels-busy time since the register was last read and
reset to zero. But such a register, which can indicate by its readings
when a system is overloaded, cannot be used to foresee an overloaded
condition until too many lines have already been assigned to the
system. When the register’s readings exceed a specified threshold, the
administrator’s response must be to remove lines from the system and
to serve them on other facilities.* Because of the long lcad-times
involved in cable planning and installation, a major goal of the work
reported here was to avoid this situation.

Since some additional traffic-monitoring capability was necessary,
it seemed best to provide a measurement which could be used to
predict the ultimate main-station capacity of a system, and thus to
guide the loading of the system and the planning of relief facilities.
The natural quantity to measure is carried load; and the system’s
second traffic register, which was also chosen to minimize the volume
of data to be taken and processed, records peak hourly carried load—
that is, the highest hourly carried load that has occurred since this
register was last read and reset to zero. As shown below, it is sufficient
to take readings once a week. This frequency is compatible with a
normal schedule of visits to unattended offices. Note that the time
that this peak traffic occurred is neither recorded nor of importance
to the procedure to be described, so that the usual time-consistent
busy-hour is not identified.

Weekly peak load measurements can be used whenever 40 or more
main stations are assigned to an SLM system. We begin with N weekly

peaks, 11, --- , ty. (N is normally equal to 4.) We simply calculate
the mean,
Se tide 1
r= ve Liy ( )
and the variance,
1 4 a ;
p= yh wo (2)

Fig. 1—Estimated main-station capacity as determined by mean and variance
of weekly peak loads.

currently working main stations, we find the point defined by the
coordinates £ and v. This point will fall in one of the regions labeled
40, 45, --- , 160. The label of that region is the estimated main-station
capacity of the SLM system.

For example, let us say we have 80 working main stations. We
observe four weekly peaks and calculate a mean of 260 CCS and a
variance of 1400 (CCS)?. Figure 1 is the chart corresponding to 80
working main stations. The point (260, 1400) falls in the region
labeled 120. This is the estimated main-station capacity for the system
in question. Repeating this estimation procedure four times (using
data from 16 weeks of operation) and calculating the weighted mean
of the estimates, using the respective numbers of working main
stations as weights, we obtain the predicted capacity of the system in

main stations. Some precautions which must be observed in drawing
conclusions from this process are summarized below.

which is sometimes referred to as ‘“Gumbel’s first asymptotic distribu-
tion.’’ It has been shown‘ that, for the so-called ‘‘exponential’’ class
of distributions, the largest value of a random sample of size n will be
asymptotically distributed (as n ©) according to (8). The exponen-
tial class includes most well-known distributions with an infinite tail
to the right, such as the normal, lognormal, and gamma.

Gumbel suggests that wu and @ be estimated by replacing E(x) and
V(x) by their sample values, (1) and (2) respectively, and solving
(5) and then (4) for a and u:4

With J main stations served by an SLM system, the weekly peaks
will have an extreme-value distribution with parameters u; and ay.
If we increase the number of main stations from J to K, the weekly
peaks will have a new extreme-value distribution with parameters ux
and ax. In this section we describe a method for estimating ux and
ax when J and K are known and u, and a; have been estimated from
measurements. That is, we want to know what the distribution of
weekly peaks will look like for K main stations when we have observed
this distribution with only J main stations being served.

Suppose that, during any week, there are n hours in which the
weekly peak traffic load may occur. We know from experience that
the weekly peak can occur during almost any waking hour.? However,
for any given week, n will be much smaller than the number of waking
hours. (In Appendix A we show that n = 10 seems to be an appropriate
choice for our purposes.) We call these n hours (whose actual times of
occurrence are not specified) the candidate busy-hours.

We now assume that each main station generates a load with mean
pw and standard deviation o during each of the candidate busy-hours.
If customers behave independently, the load distribution in candidate
busy-hours must have mean Ju and standard deviation ovVJ. Let this
distribution be F, with density f = F’. The weekly peak will then be
the maximum value in a random sample of size n from the distribution
F. If F is in the exponential class of distributions, the distribution of
this maximum can be approximated by the extreme-value distribution
(3). Gumbel! shows that the parameters wu and a@ are given approxi-
mately by

Since the candidate-busy-hour loads are the sums of the loads from
J main stations, it seems reasonable to assume that F is normal,” as
suggested by the central-limit theorem. (As mentioned above, we
take J and K to be at least 40.)

Let ® be the standard unit-normal distribution function and ¢ = ®’
the corresponding density. Define v by the relation

Then » is the 1 — (1/n) quantile of the unit-normal distribution, for
which tables and computer subroutines are available. Then from (8)
and (9) it is readily seen that

If the K — J subscribers to be added come from the same population
as the J subscribers already being served, the candidate-busy-hour
load distribution for K main stations will be normal with mean Ku
and standard deviation oVK. The weekly-peak-load distribution will
have parameters defined by (11) and (12) with J replaced by K. From
the four equations (11), (12), and the corresponding equations for K
main stations, the variables » and o can be algebraically eliminated to
yield these expressions for ux and ax in terms of u,; and a;:

Here C,, = nvd(v), a function of n only. Hence, for a given n, we
can estimate the parameters of the weekly-peak-load distribution for
K main stations by observing the weekly peaks generated by J(<K)
main stations.

To introduce the service criterion, we define a heavy-load hour as
an hour in which the offered load is such that the probability of block-
ing is 0.005 or greater. We first attempted to limit the frequency of
such hours to no more than four times a year, but found, as shown
below, that the statistical characteristics of our administrative pro-
cedure lead to a different formulation of the service criterion.

Jones proposed a model for relating the probability of blocking in a
concentrator to the number of channels, the number of customer lines,
the percentage of intra-system traffic, and the total source load.°
Johnson has written a computer program which calculates load-
service relations based on Jones’s model.® Although Jones’s model
assumes blocked calls cleared, whereas waiting and retrials occur in a
real system, we believe that these effects are compensated by the
lopsided distribution of load per line,’ so that this model is appropriate.
Laue showed from the Rural Line Study that, for 80 main stations,
the expected intra-system traffic (IST) should be about 18 percent,

but because of temporal and customer variation we used the more
conservative value 30 percent.? (In subscriber concentrators which are
not arranged for remote switching of intra-system calls,* such calls
occupy two channels. Thus for a given offered load, IST increases the
variability of the traffic and hence the blocking also.) It is known that
the percentage of IST increases with the number of main stations; but
the value 30 percent is used throughout because the smoothing effect
of party-line interference, which we neglect, grows with the number
of main stations.? Omission of party-line interference from the model
is equivalent to treating a main station as a “‘traffic source.” For 24
channels and an IST of 30 percent, the finite-source effect makes that
load which results in a 0.005 probability of blocking a decreasing
function of the number of main stations. This load varies from 526 CCS
for 40 main stations to 471 CCS for 160 main stations; we call it
L(K), the load that causes a heavy-load hour.

From our model we have an estimate of wx and ax. The probability
of a heavy-load hour in any week for an SLM system with K main
stations assigned is then estimated as

Our goal is to let heavy-load hours occur about once every quarter
of a year (13 weeks). We now invoke Gumbel’s definition of return
pertod as the mean time (in weeks) between heavy-load hours, which is

If we choose K so that the estimated return period is exactly 13 weeks,
the temporal variation in the observed weekly peak loads will cause
a distribution of return periods, among systems, centered around
13 weeks.

(Note that the expected number of heavy-load hours per year—in
other words, the frequency of heavy-load hours—is 52/[R(K)] or
52:P(K), so that a 13-week return-period is equivalent to 4 heavy-
load hours per year.)

*In the SLM system, as in many such systems, the cost of this capability would
not be justified; and it would have the further disadvantage of preventing operator
access to such calls.

This would produce an even chance of having a heavy-load hour in
any 13-week period. But as shown in the next section, this situation
only occurs when u, and a, are estimated from more weeks’ peak-load
data than are conveniently obtainable in practice.

Furthermore, the scaling procedure described above makes K de-
pendent on the value of J at which u,; and a; were measured; that is,
the charts corresponding to J main stations, of which Fig. 1 is the
example for J = 80, differ considerably from each other. And this
method of determining K 1s appreciably affected by using the chart
for J working main stations (MS) when the actual number of MS
served has varied widely during the measurement period, even with
the correct mean of J. Thus the number of working main stations
must be held nearly constant (actually within 10 percent) during the
N weeks of peak-load measurements that yield Z and v from eqs. (1)
and (2). Yet we do not think it practical to impose limits on the
growth of an SLM system’s fill over periods exceeding four weeks in
length, merely so as to obtain usable data, so long as the actual MS fill
is not too high.

We resolve this difficulty by setting N = 4. This results in a very
broad distribution of actual return-periods. We then locate this
distribution so that its right-hand tail (in terms of MS capacities) is
not too large: In particular, we limit the probability of a return period
less than four weeks to a few percent. This is equivalent to setting the
mean return-period for all systems equal to 37 weeks, and to having
at least one heavy-load hour occur in any 13-week period with prob-
ability 0.3. Thus we choose K (to the nearest multiple of 5) so as to
satisfy the equation

The following section shows that this way of estimating K must be
repeated four times, and the results averaged, in order to make the
resulting distribution of return periods acceptably narrow. When this
is done (to obtain what we call the predicted capacities), practically no
systems should end up with return periods less than four weeks—that
is, with more than 1 heavy-load hour per month. This, then, is the
service criterion: No system should have more than 1 heavy-load hour in
the average month. This criterion is extremely conservative: First, it

means that the average system has less than 2 heavy-load hours per
year (return period, 37 weeks). Second, it typically implies about one
originating call with dial-tone delay and one blocked incoming call per
month for the worst systems. Systems with more frequent heavy-load
hours (return periods less than four weeks) would be called overloaded
and would have to have main stations removed. Detection of over-
loaded systems by means of all-channels-busy readings is covered
elsewhere. (Two heavy-load hours can, of course, occur in one month
without implying an overloaded condition.) Proper use of the adminis-
trative procedure should make the occurrence of overloads ex-
tremely rare.

In order to evaluate the statistical variability of the main-station
capacities that would be predicted by the recommended procedure,
we simulated its performance in a Monte Carlo study. For each of
five samples of weekly peak loads actually observed in the Rural
Line Study, we estimated the parameters of their distribution from
(6) and (7) and converted these, by means of (13) and (14), to esti-
mates of the values uy and a, that would have existed with J = 40

jo] owoonownwnd wo ooenowwnonodwW od
re) Serre ase HE BSBLS2RCRAS HSE SB
Kee wr Kw Kr K—MXr Ker eX Ke ee

and J = 80 main stations working. Using these as the parameters of
the extreme-value distribution (3), we repeatedly generated random
samples of size four from that distribution. We calculated the mean
and variance of each such sample and estimated the main-station
capacity. This process was repeated 100 times for each of the five
samples and for both 40 and 80 working main stations, yielding a
sample distribution of the estimated main-station capacity. The spread
of this distribution is attributable to the variability of the weekly
peaks drawn from the distribution (8).

Figure 2 shows an example representing 1000 runs based on the
1FR* data from McComb, Mississippi, described in Ref. 3. Alongside
the estimated-main-station-capacity scale is a scale which gives the
mean return-period of heavy-load hours for the indicated number of
main stations. Note that if the procedure were followed with only four
weeks’ data, some systems would be overloaded (with return periods
of heavy-load hours as short_as three weeks) and others, with much
spare capacity, would be greatly underloaded. Table IA summarizes the
Monte Carlo results for the five samples. The statistics listed are:

(z) The percentage of cases that would have a return period of
less than the desired 13 weeks.

We see from Table IA that the percentage of systems that would be
overloaded ranges from 2 to 14 and the percentage of systems that
would be seriously underloaded ranges from 3 to 12. On the basis of
these results it was decided that four weeks’ data (one measurement
month) are not enough for a final determination of main-station
capacity. We therefore recommend the use of the mean of the main-
station-capacity estimates of four samples of four weeks each. (This
should be a weighted mean, the weights being the average numbers of
working MS in the four measurement months. This weighting accounts
for the greater predictive value of an estimate that is based on the
traffic of a larger fraction of the stations that will ultimately be
served.) A Monte Carlo study was carried out for this procedure and
the results are shown in Table IB. Note that an overloaded or badly

Table |— Percentage of systems that would have particularly
high or low loads—summary of Monte Carlo results

Return Return Underloaded
Period Period by More Than
<13 Weeks <4 Weeks 20 MS
Working Main Stations: 40 80 40 80 40 80

underloaded system would result very infrequently. Figure 3 shows a
histogram of the sample distribution based on the McComb data. We
distinguish the result of the modified procedure, using the mean of
four estimated capacities, by calling it the predicted main-station
capacity; and Fig. 3 is so labeled.

The last two pairs of columns in Table I relate to cases in which
the administrative system may be said to have performed inadequately.
The criterion for this is more stringent on the overload side, as neces-
sitated by the inherent variability of the predicted capacities. This
difficulty could be cured by taking many more measurements—a solu-
tion whose cost, in our judgment, would not be justified.

An alternative to the averaging procedure would be to calculate
the mean and variance of 16 weekly peaks and to use the main-
station-capacity charts only once. The precision of this approach was
shown by a Monte Carlo study to be comparable to that resulting
from averaging the four estimates from groups of four weekly peaks.
But, as mentioned above, if a system is being filled during the measure-
ment period, there is a much better chance of the number of working

) onononmnonononeondcnaoewndognsd
wW OOoRR DWDAMDDODOHKHFE_ HKH_ NNYON TT OW O

The spread of the capacity estimates, as indicated by Table I,
was slightly larger with 80 main stations working than it was with 40.
We have found no convincing explanation for this unexpected result.

Several assumptions used in the foregoing argument are known to
be invalid. Nevertheless, the procedure we recommend worked well
when applied to the Rural Line Study data. [Some confirmation of its
efficacy comes from weekly peak loads observed in an SLM system trial
in Brandon, Mississippi, which closely fit the Gumbel distribution (3). ]

But in order to distinguish the realistic from the idealized features of
our mathematical model, we summarize the latter in this section.

The concept of a candidate busy-hour is fictitious. However, we
know of no better means of projecting the distribution of weekly peaks
from the current number of working main stations to a larger number
of main stations; and it seems intuitively reasonable that not all
waking hours of the week should be equally likely to contain the
week’s highest load. Furthermore, as shown in Appendix A, the pro-
cedure does seem to be quite insensitive to the form of the assumed
distribution of candidate-busy-hour loads and to the number n of
candidate busy-hours. We do not believe that this aspect of our
model will cause problems in practice.

It is also assumed that the loads generated during a candidate
busy-hour by each of the (existing or future) working main stations
are statistically independent and identically distributed. In fact, sub-
scribers in an area do not behave independently, since they are subject
to similar influences from events in their shared environment. More
important is the critical assumption that the load parameters are
identical for all customers. Departures from this assumption among
customers already served may actually offset other errors in the
model, as noted in Section V; but the possibility of adding to an SLM
system a group of subscribers whose traffic characteristics differ
considerably from those of the ones already being served constitutes
the greatest danger in using the predictive method of administration
here proposed. One way of guarding against this danger is to limit the
number of main stations that may be added to a system before addi-
tional measurements are taken; and a precaution of this kind is
included among the rules of administration given in Appendix B. New
subscribers should be added to a system even more cautiously if they
are thought to differ sociologically from those whose loads have been
measured, especially if those to be added have higher incomes. Business
telephones, in particular, may generate several times the loads typical
of residential service.

The problem of seasonal variation has not been mentioned. No
seasonal effects were observed in the Rural Line Study, but some
SLM systems, especially in resort areas, will have highly seasonal
loads. Our administrative procedure contains no internal safeguards
against this source of error. Local knowledge will have to ensure that
only weekly peak loads recorded in the busy season are used as inputs
to the computations, and in some cases this may force planning to be
based on fewer than four 4-week estimates of MS capacity.

In the discussion above, it has been assumed that the number of
working main stations remains constant during a measurement month.
The case in which inward or outward movement occurs during or
between measurement months is covered by some of the rules in
Appendix B.

It is not known whether any error is introduced by applying to
peak loads the load-service relations deduced from Jones’s model.

In fact, most rural systems will be able to serve many more than
the nominal 80 MS. Application of our procedure to the data from the
Rural Line Study led to a sample of predicted capacities whose mean
slightly exceeds 120 MS, but this figure must be viewed with caution
for three reasons: First, this is the mean of the limits imposed by
traffic considerations alone; geographical constraints associated with
transmission criteria may keep it from being attained in practice.
(Lack of demand for multiparty service could also act to limit main-
station fills.) Second, the Rural Line Study was conducted in eight
rural areas in the territory of South Central Bell; loads would cer-
tainly be larger in some suburban applications, and as yet we have no
assurance that our data are representative even of rural areas in other
parts of the country. And third, the predicted capacities have a very
wide spread; a small but significant fraction of systems are predicted
to have a capacity of only 80 MS, confirming the early choice of that
number as appropriate for rural use. Studies of suburban traffic that
are now in progress should lead to an evaluation of SLM main-station
capacities in the suburban environment.

In this study we have viewed the capacity of a telephone system in
terms of its behavior in the presence of peak rather than average
demand. (In this general sense our work has many forerunners, in the
Bell System and elsewhere.) Instead of measuring the load in a time-
consistent busy-hour, we record only the load carried during the
busiest hour of the week. This reduces the measurements required to

only one observation per week by focusing on the hour that is most
important to the quality of service, regardless of when that hour
occurs. It has the corresponding disadvantage of working with a
traffic statistic that is volatile (as compared to the more stable mean,
for example) and therefore difficult to predict with accuracy.

M. M. Buchner, Jr., was deeply involved in the planning which led
to the idea of using a peak-load measurement for the SLM system.

In the body of this paper we assume that the candidate-busy-hour
loads are normally distributed. Although the central-limit theorem
supports this assumption, the gamma distribution has in some situa-
tions been found to be more descriptive of offered loads; and unlike
the normal, it does not imply the existence of negative loads. This
appendix summarizes a study comparing the normal and gamma
distributions as bases for scaling peak loads, and also leading to the
choice of n = 10 as the number of candidate busy-hours.

The gamma distribution function, also a member of the exponential
class, takes the form

normal. It is considerably more difficult to manipulate algebraically

which are simple for the normal, are not available for the gamma.
With a change of variable in (18), eqs. (8) and (9) become

The procedure requires that given n, u, and a we solve (19) and (20)
for 7 and 8. This was done by using a modified regula-falsi method of
iteration. Solving for u and a when 7 and 8 are given is somewhat
easier: Given 7, the value of u/8 can be found from a subroutine for
the inverse incomplete gamma function, and from u/8 the product
ua is easily calculated from (20).

It is well-known that the sum of independent, identically distributed
gamma variables is also gamma. Hence, if the candidate-busy-hour
load distribution for J main stations is gamma with parameters 6
and 7, the individual main-station mean load p = 6y/J with variance
o? = 6y/J. Thus the candidate-busy-hour load distribution for K
main stations is also gamma with scale parameter 8 and shape param-
eter Kn/J.

We now have all that is necessary to arrive at wx and ax, for both
the gamma and normal distributions, if we are given u, and a; and
any value of n.

In choosing the form of the candidate-busy-hour distribution F and
the number n of candidate busy-hours, we used the Rural Line Study
data to evaluate the precision and bias of the estimation procedure.
We chose 13 groups of lines, varying in size from 54 to 254 main
stations, on which we had peak-load data in series whose lengths
varied from 14 to 51 weeks. We divided each group in two in such a
way that the two subgroups were approximately equal in the numbers
of main stations for each class of service. We found the weekly peaks
for each subgroup and, combining the two subgroups, for each whole
group as well. From the subgroups’ weekly peaks we predicted the
distribution of the weekly peaks for each whole group and then com-
pared these predictions with the observed whole-group weekly peaks.
This was done with both normal and gamma candidate-busy-hour
load distributions, for numbers of candidate busy-hours ranging from
6 to 18.

To determine the best model two measures were used: the root-
mean-square deviations of the subgroup predicted values from the
total-group estimated values of the mean peak load and of the 37-week-
return-period load. Table II shows the results of these calculations.
Since the gamma and normal assumptions perform about equally
well, and since calculation of the charts (such as that shown in Fig. 1)
is many times easier, faster, and cheaper with the normal, we chose
it as a model for the distribution of candidate-busy-hour loads. The
results of our procedure are not very sensitive to the value of n; and

Table |! — Variability of predicted peak loads — square-root
of the mean-squared deviation (in CCS)

we took n = 10 because this value gave optimal performance in
predicting the mean weekly peak load (the more stable statistic) and
nearly optimal for the 37-week-return-period load.

Table II also gives us an evaluation of the procedure; and in par-
ticular, it tests the assumption of homogeneity within the groups of
customers represented. The rms error for the predicted mean weekly
peak is 13 CCS, or about four to six main stations. This means that
most such predictions should be accurate to within 10 main stations.
(Since these results come from 14 to 51 weeks’ data, the statistical
variability that would result from using only four weekly peaks is not
represented here; it was investigated in the Monte Carlo study dis-
cussed in the body of this paper.)

In order to ensure that the conditions for validity of our mathe-
matical model are satisfied, or nearly so, and to guard against erroneous

*The proposed procedure, together with these operational rules, is now being
tested in field use. This appendix is included here only to illustrate problems that
arise in reducing the peak-load approach to practice.

Table III — Comparison of predicted and observed peaks (in CCS)
from the 13 Rural Line Study groups

predicted capacities when they are not, we give the following rules ane
guidelines for the use of the administrative procedure.

Readings should be taken (and the WPL register reset to zero) at
the same time every week. When this is not possible, each measure-
ment week must contain no more than six weekdays and no less than
four weekdays. When this condition is violated, the data should be
discarded.

Measurement weeks need not be contiguous. Studies have shown
that there is no serial correlation among weekly peaks. A missing
week should therefore not affect the results, so long as there are four
weekly peaks in each measurement month.

The number J of main stations associated with a measurement
month should be the mean of the numbers of main stations being
served at the times the four weekly peak loads were recorded. This
mean should be rounded to the nearest multiple of ten when one of
the charts such as Fig. 1 is to be chosen.

Inward. or outward movement of main stations served by an SLM
system will tend to increase the variance of the weekly peak loads and
so to decrease the accuracy of capacity estimates. A study has shown
that if the number of main stations varies more than 10 percent during
a measurement month, the peak-load data should not be used to
estimate a main-station capacity.

The MS capacity predicted from several measurement months is
the weighted mean of the monthly estimates. The weights are the
numbers of working main stations for each measurement month.

No measurements should be used when fewer than 40 main stations
are being served. For this reason, the first of the charts for estimating
main-station capacities is for J = 40 working main stations.

No system should be loaded beyond 60 main stations until the
available data (used in accordance with the rules in this appendix, and
consisting of at least an estimate based upon one measurement-
month) indicate that it is safe to do so. (The Rural Line Study shows
that even some rural systems may suffer excessive blocking with 80
main stations.)

No system may serve more than 160 main stations, with this
exception: Unusual configurations in which no concentrator-blocking

is possible, such as 24 lines with 8 MS on each, are perfectly acceptable
with respect to the considerations treated here.

Except for the initial fill, no more than 40 main stations may be
added to an SLM system on the basis of a single predicted capacity. If
a predicted capacity exceeds the present fill by more than 40 MS, the
system should be allowed to grow by 40 MS and a new series of mea-
surements taken. This rule embodies a compromise between the
value of predicting main-station capacities and the danger of adding
customers unlike those already served.

If the current main-station capacity estimate was calculated from
fewer than four measurement months, the number of main stations
may be increased to the lesser of

(2) the current main-station-capacity estimate minus 20, and
(17) the mean of the current main-station-capacity estimate and
the current number of working main stations.

These restrictions are necessary because of the statistical variability
of such estimates.

1. I. M. MeNair, Jr., “Digital Multiplexer for Expanded Rural Service,” Bell
Laboratories Record, 50, No. 3 (March 1972), pp. 80-86.

5. A. W. Jones, ‘‘Measurement of Non-Random Telephone Traffic with Special
Reference to Line Concentrators,’ Third International Teletraffic Congress,
Paper No. 21, Paris, 1961.

. W. S. Hayward, Jr., ‘Traffic Engineering and Administration of Line Concen-
trators,’”” Second International Teletraffic Congress, The Hague, 1958.

Copyright © 1974 American Telephone and Telegraph Company
Tue Bey System TECHNICAL JOURNAL
Vol. 53, No. 2, February 1974
Printed in U.S.A.

The response of notch filters to sudden excitations is analyzed. Unit
step and stepped trigonometric inputs are considered for the class of
filters derived from low-pass networks by a frequency transformation.
It 1s possible in some cases to approximate the transient solutions in terms
of Laguerre functions and deduce general properties of notch filters from
these solutions. The use of phasing sections to modify the transient response
as also examined. It 1s shown that this method can be used to effectively
reduce the overshoot in the response to a stepped trigonometric excitation.

In order to accurately measure noise levels in compandored com-
munication systems, it is necessary to set the compandor charac-
teristics at approximately the values associated with signal trans-
mission. This is done by applying a so-called ‘‘holding tone” which is
subsequently removed by a notch filter incorporated into the measuring
set. This investigation originated in connection with the design of such
a notch filter for an impulse noise counter.! The filter has to meet
both frequency and time domain requirements. The frequency response
requirements could be readily met with existing filter design pro-
cedures. However, the time domain characteristics of notch filters
needed investigation with regard to the suitability for the present
application. The time domain requirement is that the filter when
combined with a C-message weighting filter? and excited with a stepped
trigonometric time function at the notch frequency should, in the
transient state, have only a specified overshoot level. This requirement
is imposed by the necessity to distinguish in the measuring set between
sudden gain and phase variations and impulse noise.

To examine the transient response of notch filters, a class of such
filters derived from low-pass filters by a frequency transformation is

considered. The transient response to a unit step function, and to a
stepped trigonometric function with the notch frequency, is expressed
in terms of the low-pass impulse response. It is shown that the low-
pass and notch filter response are for both excitations related by a
Hankel transform. Some general properties of the transient response
are deduced by considering notch filters for which the response can
be obtained approximately in terms of generalized Laguerre functions.

This investigation shows that notch filters would distort narrow
time pulses of duration less than one-half the notch frequency period.
The amount of distortion is related to the notch depth. The response
of a notch filter to a stepped trigonometric function can be kept to a
low level only after a certain time duration, which depends on the
filter parameters. The transient response of a notch filter followed by
a low-pass filter may still assume large values at relatively short times
from the beginning of the response. A method of decreasing the
transient response at such times by the use of phasing sections is also
presented. The use of phasing sections is of particular importance
where it is necessary to modify the transient response without affecting
the frequency response.

Although this work is primarily concerned with notch filters, the
methods used may also be applied to determine the transient response
of high-pass and bandpass filters, when derived from low-pass filters
by a frequency transformation.

A class of notch filters derived from low-pass filters by a frequency
transformation’ is considered. The transformation corresponds to re-
placing the inductances and capacitances in the low-pass filters with
parallel and series resonant circuits, respectively. Let 7',(s) be the
low-pass transfer function in the complex frequency domain s. The
transformation is given by

= xy (1)
where w, = 2rf., fo = notch frequency, and @ is a constant. (For
low-pass filters normalized such that for s = jw the 3-dB bandwidth
is at w = 1, B is the 3-dB circular bandwidth of the notch filter.) The
transfer function of the notch filter in the complex frequency domain
2, T(z), is related to the low-pass transfer function T',(s) by

To investigate the transient response of notch filters, that response
is related to the impulse response of the low-pass filter. Such a rela-
tionship is obtained by first expressing the transfer function of the
low-pass filter in terms of the Laplace transform of the impulse re-
sponse, fx(t),

then from (2) the transfer function of the notch filter by
Ty(2) = i exp — | os s | fr (t)dt. (4)
The time response, fys(t), of the notch filter to a unit step function

can now be obtained by inversion of the Laplace transformation,
through integration in the complex plane over the contour I, as follows:

where y is a positive constant. The relationship between step response
of the notch filter and the impulse response of the low-pass filter is
obtained by the substitution of (4) into (5) yielding

1 et
fs = 5 fo [ew — (a5) fwauae 6)
In Appendix A it is shown that (6) can be expressed as follows:

where 1 (¢) is the unit step function and J,(y) is a Bessel function of
order n.
In (7) the first term is the undistorted unit step response and the
second term expresses the distortion introduced by the notch filter.
The first term can be simplified by observing that from (8)

[feat = 7,00). (8)
Without loss of generality, 7’1(s) can be normalized to be equal to
unity at zero frequency. The expression in the brackets of the second

The integration in (9) can be performed approximately by using
the method of stationary phase. This is a good approximation when
the Hankel transform is a slowly varying function in comparison to
the Bessel function. It is also shown in Appendix A that with the
stationary phase approximation (9) can be approximated by

An example considered subsequently shows that (10) is indeed a good
approximation for B K ws, i.e., for narrowband notch filters.

lll. TRANSIENT RESPONSE OF NOTCH FILTERS TO A UNIT STEPPED SINE
FUNCTION
The transient response to this function is obtained in a manner
similar to the response to a unit step function. However, a more
general response function, fym(t), is considered,

providing for the possibility of a low-pass notch combination. The
time response of a notch filter to a stepped sine function with the
notch frequency is obtained as a special case, by setting m = 0.
Proceeding in a manner similar to Appendix A, it can be readily
shown that (11) can be expressed as follows:

Equation (12) also contains a Hankel transform but of order m.
Similarly as before, (12) can be approximated by using the method
of stationary phase. It can also be readily shown, from the results in
Appendix A, that with this approximation (12) is given by

To gain some insight into the behavior of the transient response, a
low-pass transfer function is chosen for which the integrals (10) and
(13) can be evaluated. As mentioned before, these integrals are Hankel
transforms. A class of functions for which Hankel transforms can be
evaluated in closed form are the generalized Laguerre functions.** In
fact, for these functions the Hankel transforms are self-reciprocal.®
These functions form orthogonal sets so that, in principle, any passive
filter impulse response can be expanded in terms of these functions.
However, the impulse response of some low-pass filters may be ap-
proximated closely by a single Laguerre function. Such a response can
be used to obtain an estimate of the notch filter response.

The transfer function (14) is normalized such that 7';(0) = 1.0. This
function can be considered as consisting of (m + 1) cascaded low-pass
filter sections and, for ag =—au, of n phasing sections. For physical
realizability of the n cascaded sections, it is necessary and sufficient’ ®
that |a2| 2 a1.

These polynomials are oscillatory for positive arguments and mono-
tonically increasing for negative arguments.

The integral (10) with fz(u) given by (15) can be reduced to a
tabulated integral for the special case n = 0. The applicable integral

oo !
i ettatmtuthy, (Q0VV de = e-¥ VLE (Y). (17)
0
For this special case the low-pass transfer function is
T1(8) = (18)
L ~ (s+ 17?

where without loss of generality a1 is set equal to one.
The transient response of the notch filter is obtained by using (10),
(15), and (17) and is approximately

It is of interest to determine the accuracy of (19) for different
values of m. This is done in Appendix B for m = 0, 1, and 2 where
(19) is compared with the exact solutions. It is shown that the accuracy
of (19) is of order 6/w, when expressions multiplied by both sin wt
and cos wf are considered. However, expressions multiplied by sin wt
only are of order (8/w.)?, so that near the maximum amplitudes of
the second term in (19), expected near cos w,t & 0, the approximation

good approximation for 6/w, <1, i.e., for notch filters in which the
3-dB bandwidth is much smaller than the notch frequency.

In Fig. 1 the function e~*Li,(x) is shown for m = 0, 1, 2. It follows
from this figure, in conjunction with (19), that the distortion of the
unit step function will be particularly pronounced in the time vicinity
T = 1/2w,. In Fig. 2 the values of the second term in (19) are plotted
as a function of B7'/2 for m = 0, 1, 2. This graph gives an estimate of
the distortion of the unit step function, and is of particular. significance
when considering the transient response of notch filters to pulses of
short time duration. This investigation shows that pulses of duration
less than one-half notch frequency period will be considerably dis-
torted by notch filters.

To obtain a numerical estimate of the distortion, a notch filter with
the following requirements is considered: (7) Notch frequency—1010
Hz. (77) Notch depth at least — 30 dB in the frequency range 995-1025
Hz. A notch filter with these requirements is derived from the low-pass
transfer function (18) for m = 0, 1, 2, 3.

It readily follows from the transformation (1) that 8 can be deter-
mined from the following equations.

Table I lists the filter parameters, including the maximum value,
Gm, of the second term in (19).

It is evident from. Table I that the 3-dB bandwidth and the distor-
tion term a, decrease as the number of sections increases. The differ-
ence is particularly pronounced between m = 0 and m = 1. A further
decrease in the distortion term can be obtained by decreasing 8; how-
ever, this also reduces the depth of the notch.

The listed values of a, seem to be representative of what is obtain-
able with notch filters of specified notch depth and 3-dB bandwidth.
For example, computation of a two-section notch filter derived from
a.Tschebyscheff low-pass filter with 0.5-dB ripple gave a value for
am of 0.18. The notch depth of this filter was also —30 dB and the
3-dB bandwidth 170 Hz. The lower value obtained with this filter can
be attributed to the more oscillatory behavior" of the low-pass impulse
response. Equation (10) suggests such an interpretation.

The low-pass transfer function (14) and the corresponding impulse
response (15) are again considered. With that impulse response the
integral (13) is given by

The integral (23) can be evaluated in closed form, again yielding
Laguerre functions®

It is noted that with the condition for physical realizability of the
n cascaded sections in (14), |az| 2 a1, that the argument of the
Laguerre functions is always positive, and hence the functions are
oscillatory.

From (11), (14), and (1), (24) corresponds to the inverse of the
following Laplace transform:

The terms in the brackets in (25) can be interpreted as transforms
of (z) a stepped sine function, (77) an m-section low-pass filter, (777)
a notch filter section, (iv) n phasing sections for ag = —a, or n addi-
tional notch sections for a2 >.

The transient response of a notch filter followed by a low-pass
filter is of interest. However, (25) is restricted to a particular low-pass
filter with a high-frequency cutoff in the vicinity of the notch fre-
quency, and will not be considered further.

The special case ag >, treated previously for the unit step re-
sponse, can be obtained from (26) and is in agreement with the result
obtained by performing the integration directly by using (17).

For az —>* and m = 0, 1, 2, a comparison was made between the
exact solutions and the approximate solution (26). The comparison
showed the same accuracies as for the unit step response.

The time response due to a stepped cosine excitation can be ob-
tained by differentiating (26) and dividing by w,. It readily follows

that, within the accuracy of (26), the same expression is obtained but
with sin w,t replaced by cos wz.

It is noted that (26) gives the correct value for t = 0. This value
can be obtained by using the initial value theorem of Laplace
transforms.

Graphs of e~*L8(x) are shown in Fig. 3 for n = 0, 1, 2. These graphs
give the envelope of the transient response (26) for ag —«. The
effect of finite values of a, can be deduced from the graphs. For
example, for a: negative the arguments of the Laguerre functions
increase reaching maximum values of (6t)/a, for a1 = —a». Therefore,
for the same ft the spacing between the zeros would decrease and
the maximum values increase. For positive a2 the opposite would be
the case.

It has been shown above that the transient response of notch filters
can be obtained from the Hankel transform of the low-pass impulse
response. This property can be used to deduce the qualitative charac-
teristics of notch filters derived from conventional low-pass filters of
the Bessel, Butterworth, and Tschebyscheff type. The impulse re-
sponse of conventional filters, with transfer function polynomials of
the same order and approximately the same bandwidth, has similarities
to the impulse response considered. Therefore, the graphs shown in
Fig. 3 are also representative of the transient response of notch filters
derived from conventional filters.

The transient response of notch filters can be kept arbitrarily low
at large times, such times being defined after the first zero of the
envelope, t,. The time ¢, can be kept small by the choice of the number
of sections m, and/or by choosing 8/a: large. However, for the interval
0 St St, these methods are not effective. In fact, it has been shown
above from the initial value theorem (27) that, at ¢ = 0, the envelope
of the response is unity independent of the filter parameters. A low-
pass filter combined with a notch filter would cause the transient
response to be zero at ¢ = 0, and behave, for small t, as ¢*— if k is the
order with which the transfer function goes to zero as s > ©. However,
with a given low-pass filter the transient response may not be reduced
to a desired level at small t. An additional method of reducing the
response by the use of phasing sections is discussed subsequently.

To illustrate some of the properties of notch filters, numerical
computations for a three-section filter (m = 2), with the parameters
given in Table I, have been performed. Figure 4 shows the filter
response to a step function and Fig. 5 the response to a stepped cosine
function. The computed results are essentially in agreement with
those obtained based on the approximate method. Figure 6 shows the
transient response of this filter followed by a C-message weighting
filter, when excited with a stepped cosine function. A comparison of
Figs. 5 and 6 shows that the C-message filter reduced, as expected,
the first lobe of the response, affected only slightly the second lobe
and increased the subsequent lobes. Increasing 8 and hence the 3-dB
bandwidth of the notch is not very effective in reducing the second
lobe. This led to the investigation of phasing sections as a means of
reducing the transient response.

Fig. 6—Response of notch and C-message weighting filters to stepped cosine
function.

V. TRANSIENT RESPONSE OF PHASING SECTIONS
The transient response of a phasing section is considered when

Pols) = Fes + é o
and the Laplace transform of the damped sine function, Fa(s), by
Fa(s) = ge ye pe Las + 8 (29)

The first term in (82) is the damped sine function and the other two
terms are introduced by the phasing section. In order that the last
two terms be of significance, it is necessary that these terms be com-
parable to the first term. This will be the case for d = b, for which
(32) reduces to

= a—- Cw,
If, in addition, (a/2)? « b? and (c/2)? « d*, (37) simplifies further
and, for (a/2 — c/2)t «1, (37) is approximately given by

Equation (88) contains the damped sine function but modified by
the term (1 — ct). This term can be used to introduce a zero in the
time vicinity where the damped sine function assumes a maximum
value.

To illustrate the above, a phasing section is introduced to modify
the impulse response of a C-message weighting filter. The computed
impulse response of the filter is shown in Fig. 7 and has relatively
large values in the time vicinity of 0.4 ms. The impulse response
modified by a phasing section is shown in Fig. 8. The phasing section
parameters are c = 2-10 and d = 108. These parameters have been
chosen on the basis of the above analysis. It is evident the large values
of the response have been reduced, but the modified response has

appreciable values for a much longer time duration than the initial
response. This behavior can be explained on the basis of Parseval’s
theorem," since the absolute value of the Fourier transform of the
response is the same with and without the phasing section.

Fig. 8—Impulse response of C-message weighting filter with phasing section.

Table !I—Two-section filter with phasing section
(3-dB bandwidth 390 Hz, 30-dB bandwidth 80 Hz)

Phasing sections can be used to reduce the overshoot of the response
of a notch filter followed by a low or bandpass filter and excited with
a stepped trigonometric function at the notch frequency. Such sections
are of particular importance where the transient response has to be
modified without affecting the amplitude of the frequency response or
where a modification is needed at times shortly after the beginning of
the response. However, such sections may also introduce considerable
distortions of the unit step response.

As an example, the performance of a 1010-Hz notch filter with
and without a phasing section is considered. The transfer function,
T(z), of the filter and phasing section can be written as

This filter was derived from a Tschebyscheff low-pass filter, and
an operational amplifier version was synthesized and built. Figures 9a
and 9b show the computed response to a stepped cosine of the filter
combined with a C-message weighting filter without and with the
phasing section. Figures 9c and 9d show photos of the corresponding
oscilloscope displays obtained with the actual filters. Good agreement
was obtained between the computed and measured response. The
effect of the phasing section on the response is evident in this figure.
About a 4-dB reduction in the overshoot was obtained with the
phasing section.

The transient response of a class of notch filters which are derived
from low-pass filters by a frequency transformation was investigated.
General expressions for the transient response due to a unit step

Fig. 9—Computed and measured response of notch filter with C-message weighting
filter to a stepped cosine: (a) and (c) without phasing section, (b) and (d) with
phasing section.

function and a stepped trigonometric function have been obtained in
terms of the low-pass impulse response.

The transient response of certain types of notch filters can be formu-
lated approximately in terms of Laguerre functions. These filters have
been examined in detail and some general properties of the notch
filter response have been deduced from this formulation.

Notch filters may considerably distort short time pulses (time
duration less than one-half notch frequency period). The amount of
distortion depends on the notch depth.

The response of notch filters to stepped trigonometric functions can
be kept at low levels only after a certain time interval from the begin-
ning of the response. The length of the time interval depends on the
filter parameters.

A method of reducing the transient response at short time intervals
by the use of phasing sections was presented. This method may prove

particularly useful in applications where it is necessary to modify the
transient response without affecting the frequency response.

The author wishes to express his gratitude to E. R. Nagelberg for
his critical reading of the manuscript and for his helpful comments
and suggestions, Mrs. A. M. Franz for the computational assistance,
Mrs. E. Y. McBride for the synthesis of the operational amplifier
version of the notch filter and phasing section, and R. R. Redington
who built the filter and measured its charactcristics.

The relationship between the step response of the notch filter and
the impulse response of the low-pass filter from which the notch filter
is derived is given by (6) of the text,

After interchanging the order of the Rance and expanding the
exponential function, (40) can be written

Equation (41) contains a sum of inverse Laplace transforms of a
tabulated form !>16

Equation (42) is of its own interest, since it may be used to obtain
the step response of a bandpass filter derived from a low-pass filter by
a frequency transformation. A direct derivation of (42) follows.

After interchanging the order of integration and expanding the
exponential function in a power series, (43) can be written

The inverse of each term in (44) is zero for a negative exponential
argument. For a positive argument u S é, the inverse is readily ob-
tained ; hence,

~ | gu) Py m! (m + 2v)!
The summation of the terms in (45) gives a Bessel function " of order
2v, and hence (42). Using (42), (41) can be written
n n—1
fast) =f" faotaut f° fay fo bw" er

The series in (46) can be summed yielding a Bessel function of
order one; hence,

The integration with respect to u can be interpreted as a Hankel
transform of the low-pass impulse response. For an impulse response
which is not very oscillatory, and for 6 « w., the Hankel transform
will be a slowly varying function in comparison to the Bessel function.
Under these conditions an approximation to (47) can be obtained by
using the method of stationary phase.

The stationary phase method!* approximates integrals, 7, of the
following type:

where & is large and g(x) is a slowly varying function. The approxi-
mation considers only contributions from the vicinity of stationary

points where (dy)/dx = 0, and is of order (1/k). The approximate
value of the integral (48) is

To bring (47) to a form suitable for evaluation with the stationary
phase method, the Bessel function is expressed in terms of modulus
and phase.!9

(2) = 2— 4 + d0(2), (51)
where 6,(z) and M,(z) are slowly varying functions for large z with
lim, «0 5o(z) = 0 and lim,... M,(z) = V2/ (2).

With z = 2wovix — 22, the integral in (47) has a stationary point at
x = t/2. The approximate value of (47), obtained by using (49), is

For large values of ¢ such that M,(z) and 6,(t) can be approximated
with their asymptotic values,

It is of interest to note that the stationary phase method gives the
correct value for the integral

This integral can be evaluated exactly by using (42) with »v = 0 and
g(u) = 1.0. The left-hand side of (42) is readily inverted yielding (54).

Comparison of Exact and Approximate Solutions
Consider a notch filter transfer function

The Laplace transform of the time response due to a unit step
function is

A comparison of (57) with (19) shows that both are of the same
form but w is replaced by w,. Hence, the approximation is of order

. Members of the Technical Staff, Bell Laboratories, Transmission Systems for

M.E. Van Valkenburg, Introduction to Modern Network Synthesis, New York:
John Wiley, 1960, pp. 479-486.
P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Part I, New York:
McGraw-Hill, 1953, p. 944.
W. T. Howell, “On a Class of Function Which are Self-Reciprocal in the Hankel
Transform,’’ Phil. Mag., 25, Series 7, April 1938, pp. 622-628.
. A. Erdelyi, et al., T'ables of Integral Transforms, Vol. 2, New York: McGraw-Hill,
1954, p. 43.
. E. A. Guillemin, Synthesis of Passive Networks, New York: John Wiley, 1957,
p. 489.
. Ref. 3, p. 348.

. Ref. 6, Vol. 1, p. 175.
. I. S. Gradshteyn and I. M. Ryzhik, Tables of Integral Series and Products, New

. Ref. 4, p. 456.
. Ref. 6, Vol. 1, p. 1383.
. N. W. McLachlan, Modern Operational Calculus, London: The MacMillan Com-

Copyright © 1974 American Telephone and Telegraph Company
Tue BELL SystEM TECHNICAL JOURNAL
Vol. 53, No. 2, February 1974
Printed in U.S.A.

The purpose of this paper 1s to make comparisons between optimum,
linear phase, finite impulse response (FIR) digital filters and infinite
impulse response (IIR) digital filters which meet equivalent frequency
domain specifications. The basis of comparison is, for the most part,
the number of multiplications per sample required in the usual realiza-
tions of these filters—t.e., the cascade form for IIR filters, and the direct
form for FIR filters. Comparisons are also made between group-delay
equalized filters and linear phase FIR filters. Considerations dealing
with finite word-length effects are discussed for both these filter types. A
set of design charts 1s also presented for determining the minimum filter
order required to meet given low-pass filter specifications for both digital
and analog filters.

Although a great deal is known about the properties of different
types of digital filters, very little has been done to relate the various
designs as to performance and complexity of realization. Thus the
filter designer must learn the details of several design procedures
before being able to make a wise decision on a suitable filter for his
specific application. It is the purpose of this paper to add insight into
some of the problems that have been encountered by filter designers
by: (2) presenting new and useful design curves for digital and analog
low-pass filters, and (27) making several comparisons between optimum
(quasi-equiripple), linear phase, FIR low-pass filters and equiripple
(elliptic) IIR filters which meet equivalent frequency domain specifica-
tions. Although these results are presented for low-pass filter designs,
they are easily extended to the case of bandpass, bandstop, and high-
pass filters by the well-known frequency band transformations.

The organization of the paper is as follows. After defining the
terminology to be used, the design relationships between the FIR
filter parameters are reviewed. The design relationships for the IIR
filter parameters are developed and a novel graphical interpretation
of these relations is presented in the form of useful filter design charts
(applicable to both digital and analog filters). Using the design rela-
tionships, the filter orders required to achieve equivalent performance
are compared for different ranges of filter parameters.

The design procedures for the two general classes of digital filters,
FIR and IIR, have essentially progressed along independent paths. As
a result, the terminology used in specifying the filter performance is
generally not quite the same. Thus it is instructive to define the most
commonly accepted definitions of the filter parameters for these two
classes of filters. The relations between these parameter sets for
equivalence are then established. Figure la shows the amplitude
response of a typical optimum FIR low-pass filter and Fig. 1b shows
the magnitude response of a typical elliptic low-pass filter. For the
FIR case the amplitude response in the passband (0 S f S F,)*
generally oscillates between 1 + 6; and 1 — 61, where 6; is the passband
ripple. In the stopband (Ff, S f S 0.5) the amplitude response oscil-
lates between +6, and —é_2 where 62 is the stopband ripple. For the
elliptic case the magnitude response is constrained to always be less

* Throughout this paper the frequency scale has been normalized with respect to
the sampling frequency. Thus the normalized sampling frequency is 1.0 and the
frequency range graphed is 0 < f S 0.5 or, equivalently, 0 Sw Sx.

than 1.0. Thus, in the passband (0 S f S F,) the magnitude response
oscillates between 1 and 1 — 6,. In the stopband the magnitude re-
sponse oscillates between 62 and 0. It is straightforward to relate 44,
52, 61, and 4 so the resulting magnitude characteristics are equivalent.
If the FIR amplitude characteristic is scaled by 1/(1 + 6;) and the
magnitude of the resulting amplitude response is taken, then the
following relationships are obtained:

At this point it is convenient to define the additional filter terms £,
ATT, and 7 as

filter. The parameter 7 has been shown to be a basic analog filter
parameter! which will be used in the filter design curves given in a
later section.

The five basic FIR filter parameters are Fy, F;, 61, 52, and N, the
duration of the filter impulse response in samples. For the general
case of optimum, linear phase, low-pass FIR filters, there exist no
simple analytical relationships between these five filter parameters,
except in special cases, e.g., one passband or one stopband ripple.
However, an approximate empirical relationship between the filter
parameters has recently been obtained? which accurately satisfies
known design results for a wide range of values of the filter parameters.
The relationship is of the form:

Equation (10) can generally predict the value of N required to meet
specifications on 61, 62, Fp, and F, to within +2. In the cases where
F’, is very close to 0, or F; is very close to 0.5, eq. (10) tends to over-
estimate the required N. It should be noted that eq. (10) shows the
estimate of N to be independent of specific values of F, or Fy, but
instead is dependent only on the transition width, (Ff, — Fp).

This expression is a modification of the design relationship for FIR
filters designed by windowing techniques (where 6; = 62). See Ref. 3,
pp. 237-238.

One of the most general procedures for designing IIR digital filters
is through the bilinear transformation of an appropriate continuous

filter. There are two equivalent techniques for obtaining the desired
digital filter using the bilinear transformation and these are illustrated
in Fig. 2. The technique of Fig. 2a begins with an analog low-pass
filter with normalized passband cutoff frequency of 1 radian per second,
and analog stopband cutoff frequency of Q, radians per second. This
filter is bilinearly transformed‘ to give a ‘‘normalized’’ digital filter
with passband cutoff frequency 7/2 radians per second (or f = 0.25 on
the normalized scale) and stopband cutoff frequency 4,. The relation
between the frequency variables Q and @ is given by

Thus Q, and 4, are simply related by eq. (15) with Q = Q,, and 6 = 4,.
Finally, a digital all-pass transformation® is used to give the desired
digital low-pass filter with passband cutoff frequency w, and stopband
cutoff frequency w,. The relation between the frequency variables w
and @ is given by

The second technique (shown in Fig. 2b) begins with the identical
analog low-pass filter as in the first technique and immediately per-
forms a low-pass—to-low-pass transformation to give an analog filter
with passband cutoff frequency 2, and stopband cutoff frequency Q,.
The relation between the frequency variables Q and 2 is

The resulting low-pass filter is then transformed to a digital filter using
the bilinear transformation, giving the same end result as in the first
technique (Fig. 2a). The relation between frequency variables w and Q is

In the case where the prototype normalized analog filter is an elliptic
filter, it is relatively easy to derive a design formula relating the various
filter parameters, both in the analog and digital cases. For either the
analog or digital case the order, n, of the elliptic filter is related to
the remaining filter parameters by the equation

DISCRETE
CONTINUOUS NORMALIZED
NORMALIZED BILINEAR LOW-PASS FILTER ALL—PASS pr eal oe
LOW-PASS TRANSFORMATION TRANSFORMATION a
FILTER
FILTER
1 “ 1
1-8, 1-8 1-84
3 s 4
2 2 2
0 a 0 A) 0 w
0  ~ 0 7/2 Gs 7 0 Wy Ws 7
(a)
CONTINUOUS CONTINUOUS
NORMALIZED LOW-PASS-»LOW_PASS LOW-PASS FILTER BILINEAR iG et
LOW-PASS TRANSFORMATION TRANSFORMATION
FILTER
FILTER
1 1 1
& A
1-9) 1-8, 1-6;
A A A
52 52 A 62
0 o 0 - ~ 0 0 a
0 1 0, 0 Op Q, 0 Mp Ws 7

Fig. 2—T wo techniques for transforming a continuous normalized low-pass filter to a digital low-pass filter.

h=n= (22)
Thus eq. (20) relates filter order, n, to the parameters F’,, , [through
eq. (21) ] and 6; and 62 [through eq. (22) ].

For the case when the prototype filter is a Chebyshev filter (either
type I—equiripple passband, monotone stopband, or type I]—maxi-
mally flat passband, equiripple stopband) the design equation becomes

and 7 and k are defined as eqs. (21) and (22). Finally for a prototype
Butterworth filter (maximally-flat magnitude, all pole) the design
equation is

Although eqs. (20) through (25) completely describe the design’
curves for both analog and digital filters, it is generally quite helpful
to see the relationships between filter parameters displayed in a
meaningful way. Since, in general, there are five filter parameters
there is no simple way of presenting these relationships on a single
plot, even in terms of well-known nomograph procedures.* There is,
however, a simple and straightforward way of including all design
relations for both digital and analog filters, for any prototype filter,
using a sequence of three charts.

The first chart(s) relates the filter design parameter n to the pass-
band and stopband ripple specifications 6; and 62 or their equivalents.
The second chart(s) graphs the filter design equation relating filter
order n, design parameter n, and transition ratio k. The third chart(s)
relates transition ratio k to passband cutoff frequency F’, and transition
bandwidth ».

Figures 3a through 3d show four possibilities for Chart No. 1. The
graphs of Figs. 3a and 3b correspond to digital filters with 6, as a param-

Fig. 3—Plots of 7 versus stopband specification, with parameter passband specifi-
cation for low-pass filters.

eter (Fig. 3a) or 20 logio (1 + 61)(dB) as a parameter (Fig. 3b). The
graphs of Figs. 3c and 3d correspond to analog filters with absolute rip-
ple 6; as a parameter (Fig. 3c) or total ripple 20 logio [1/(1 — 6) ](dB)
as a parameter (Fig. 3d).

Chart No. 2 represents the design relations particular to the proto-
type filters, i.e., eq. (20) for elliptic filters, eq. (23) for Chebyshev
filters, and eq. (25) for Butterworth filters. For these graphs the param-
eter 7 is plotted versus transition ratio, k, with filter order, n, as the
parameter. Figures 4a through 4c show the resulting graphs for elliptic
filters, Chebyshev filters, and Butterworth filters, respectively. The
horizontal scale on each of these graphs is a nonuniform scale which
was chosen to provide a reasonably good spacing of the curves for the
various values of n. The actual nonlinear scale used is represented by
the equation

where x is the z-axis coordinate (0 S$ x S 1) and k is the transition
width. Thus, the scale is linear for small values of k and highly non-
linear near k = 1.0.

Chart No. 3 represents the relation between the transition ratio and
the filter cutoff frequencies [eq. (21) ]. For these graphs the passband
cutoff frequency, F’,, is plotted versus transition ratio, k, for various
values of normalized transition width, v, defined as

filters. The scale for transition ratio is identical to the seale used for
Chart No. 2.

To illustrate how to use the set of charts of Figs. 3 through 5, con-
sider the determination of filter order n required to meet the following
specifications:

6; = 0.01 (} +0.086-dB passband ripple)
52 = 0.0001 (80-dB stopband loss)
passband cutoff frequency = 480 Hz
stopband edge frequency = 520 Hz
sampling frequency = 8000 Hz.

For the determination of filter order n for a digital filter of the
elliptic type, the charts of Figs. 3a, 4a, and 5a are used (Fig. 4a special-
izes the design to the elliptic type). To obtain the value of 7 on Fig.
3a, we use the curve 6; = 0.01 and find its intersection with the line
52 = 0.0001 which yields a value of 7 approximately equal to 2 X 10-5.
To obtain the transition ratio, we use Fig. 5a by finding the intersection
of the curve v = F, — F, = 0.005 with line F, = 0.06; this yields a
value of 0.923 for the transition ratio (this agrees nicely with
F,/F, = 0.06/0.065 = 0.923, an alternate way of arriving at the same
result). Finally, the filter order, n, can now be determined from Fig. 4a
by finding the intersection of the lines 7 = 2 X 10-5 and transition
ratio = 0.923; thus the required theoretical elliptic filter order is
11.5. In order to meet specifications on all four parameters, a
12th-order filter must be used.

However, there are several tradeoffs possible for the final filter
specifications. For example, if 7 is held fixed at 2 X 10-5 and the
transition ratio is changed to approximately 0.94 to lie on the n = 12
curve, then either F, or F, can be varied to match this new value of
transition ratio. The tradeoffs here are obtained from Fig. 5a. If the
transition ratio is held fixed, then for n = 12 we find 7 is 1.0 X 107-5;
from Chart No. 1 (Fig. 3a) we can observe the tradeoff as 6, and 6, are
varied for this new value of 7. Finally, both transition ratio and 7
can be varied, e.g., to 0.93 for transition ratio and 1.5 X 10-5 for ,
so as to make their intersection remain on the n = 12 curve; now all
four filter parameters can be varied to match the new values of 7
and transition ratio.

It is interesting to note that if a Chebyshev or Butterworth filter
type is specified in place of the elliptic, the designer need only substi-
tute Figs. 4b or 4c for Fig. 4a as Chart No. 2 and proceed as before.
In both cases of the example given, the required filter order consider-
ably exceeds the maximum limit of 20 of the curves; thus the “effi-
ciency”’ of the elliptic design is clearly seen.

Clearly, this design procedure presents a tremendous amount of
flexibility to the designer—more so than is generally available in most

Fig. 4—Plots of 7 versus transition ratio as a function of filter order n for elliptic,
Chebyshev, and Butterworth low-pass filters.

programs for filter order determination. Furthermore, the insight into
the design problem afforded by this graphical technique allows the designer
to get a feeling for the way in which small changes in filter specification
affect the required filter order. Quite often the designer is willing to
changes his ideas on “required”’ specifications, especially if it reduces
the filter order necessary to meet his specifications.

Based on the design formulas of the preceding sections, it is possible
to make some comparisons between optimum FIR low-pass filters and
equivalent elliptic filters. The main basis of comparison will be the
number of multiplications per input sample* required in the most
standard realization of each filter type, i.e., the direct form for the
FIR case and the cascade form for the elliptic case.’ Direct realization
of an N-point impulse response filter with linear phase requires

*The number of multiplications per input sample is a useful measure of the com-
putational complexity of the filtering operations as it represents the number of
multiply-add operations required for a software implementation of the algorithm as
well as for a general hardware implementation.

Fig. 5—Plots of passband cutoff frequency versus transition ratio as a function of
transition width for discrete and continuous low-pass filters.

[(N + 1)/2] multiplications per sample, whereas cascade realization
of an nth-order elliptic filter (all zeros on the unit circle) requires
[(8n + 3)/2]* multiplications per sample where [- ] denotes “integer
part of.”

Thus, one basis of comparison between equivalent filter designs (i.e.,
both meeting the same specifications on 61, 52, F,, and F,) is in terms
of the efficiency of the respective realizations, i.e., which structure
requires fewer multiplications per sample. Equivalence between struc-
tures is attained when the condition

Using the appropriate filter design formulas, we have measured the
quantity N/n as a function of n for a large range of values of Fp, 41,
and 62. Figure 6 shows two typical sets of curves which were obtained.
Figure 6a shows data for the case F, = 0.15, 6; = 0.1, 62 = 0.1, 0.01,
0.001, 0.0001, and Fig. 6b shows data for F, = 0.35, 6: = 0.00001, and
the same range of 62 as in Fig. 6a. Also shown in these plots is the line
N/n = 3 for showing where the data lie with respect to the fixed
portion of eq. (29). As seen in this figure, for certain values of F’y, 41,
and 62, the ratio of N/n falls below the equivalence level of eq. (29),
i.e., the FIR filter is more efficient than the elliptic filter. However, in
general, the elliptic filter is more efficient than the optimum FIR
filter, and, in the case of high-order elliptic designs, the ratio of N/n
is often in the hundreds or thousands.

Based on our examination of large amounts of data, the following
general observation can be made: the most favorable conditions for
the FIR design are large values of 6;, small values of 62, and large
transition widths (i.e., small transition ratios). One also observes the
following behavior:

(c) For values of F, = 0.3, the ratio N/n always exceeded 3 + 1/n
for all values of 61, 52, and n.
(22) For values of n 2 7, the ratio N/n always exceeded 3 + 1/n
for all values of 61, 62, and Fp.

“This number of multiplications per sample for the IIR filter assumes that any
scaling between sections is an integer power of 2 and is performed entirely by shifts
of the data. If finer scaling multipliers are included between each cascade section,
the realization requires [(4n + 3)/2] multiplications per sample.

Fig. 6—Plots of the ratio N/n as a function of n for optimum FIR filters and elliptic
filters meeting identical specifications on 61, 52, Fy, and F,.

(it) The smaller the value of F’,, the larger the range of 61, 62, and
n for which N/n was less than 3 + 1/n.

Since the design formula for N for the optimum FIR case is not
exact but only an estimate, measurements were also made of the
required theoretical value of n (elliptical filter order) to meet the
specifications of optimum FIR filters which had already been designed.
Typical results of these measurements are shown in Fig. 7. Figure 7a
shows the theoretical order n (n need not be an integer) required to
match specifications on F,, F,, for 6; = 0.1, 52 = 0.1, 0.01, 0.001,
0.0001, and 0.00001, as a function of F, for a set of optimum FIR
filters with N = 21. (It should be noted that as F’, varies, F; also
varies so as to achieve the desired specifications on 6; and 62.) Figure
7b shows similar measurements for N = 41. In Fig. 7a the theoretical
point of equivalence is n = 6.3, whereas in Fig. 7b it is n = 13. From
this figure it is seen that for these cases the elliptic filter is always
more efficient than the equivalent FIR filter, as anticipated by the
discussion in the preceding paragraphs. :

In summary, elliptic filters are generally more efficient in achieving
given specifications on the frequency response than optimum FIR
filters. However, the FIR filters have the additional useful property
that their phase is exactly linear, i.e., there is no group delay distortion.
For the elliptic filter, however, there is generally a large amount of
group delay distortion (concentrated primarily near the band edge). A
question of both theoretical and practical importance is whether, in
cases when the additional requirement of a flat delay is specified, it is
more desirable to equalize the delay of an elliptic filter or to use the
equivalent optimum FIR filter (with its constant group delay). In the
next section we discuss various aspects of this question. It should be
noted that the above alternatives are not the only possibilities for
obtaining a digital filter which meets frequency domain specifications
on both magnitude and group delay responses. For example, a filter
can be designed, using modern optimization procedures, where the
number of poles and zeros are unequal. In such cases, the comparisons
between FIR and IIR filters are quite distinct from those to be dis-
cussed in the next section.

Vil. COMPARISONS OF OPTIMUM FIR FILTERS AND DELAY-EQUALIZED
ELLIPTIC FILTERS

Recently developed optimization procedures’ make it possible to
design an all-pass equalizer which can equalize the group delay of any

Fig. 7—Theoretical order of elliptic filters required to meet given specification on
51, 52, F,, and F’, as a function of F’, for various values of 52. Optimum FIR filters
with N = 21 meet the specifications for all filters of (a), whereas N = 41 is re-
quired for all filters of (b).

digital filter to any desired accuracy over a restricted band of fre-
quencies. As an example of the use of this procedure, Fig. 8 shows
plots of the group delay of a 6th-order (unequalized) elliptic filter
(with parameters 6; = 0.01, 6: = 0.0001, F, = 0.24163, F, = 0.34842)
and the equalized group delay using a 10th-order all-pass filter. The
relative error in the equalized delay curve is 3.6 percent of the average
delay in the passband. In this case the equalized elliptic filter requires
20 multiplications per sample, whereas an optimum FIR filter which
achieves the same specifications requires only 11 multiplications per
sample.

The difficulty with trying to equalize the group delay of a filter
lies in the fact that the equalized filter must have a total delay greater
than the largest delay in the unequalized filter which always occurs
near the passband cutoff frequency. Thus, in the example of Fig. 8,
even though the delay throughout most of the passband is between
2 and 6 samples, the delay at the edge of the band is about 15 samples.
It can be shown that an all-pass equalizer of degree n, has the

Fig. 8—The group delay of an unequalized and an equalized elliptic filter. The
equalizer is of 10th degree and the elliptic filter is of 6th degree.

where 7,(w) is the equalizer group delay and the integral is taken over
half the sampling interval (0 S w S zm). Since 7,(w) = 0, i.e., group
delays add, to justify eq. (30) it is sufficient to show that a first-degree
all-pass equalizer has the required property. The z-transform of a first-
degree all-pass equalizer is
kee /e

H(z) = i ae (31)
where a is the pole position and 1/a is the zero position in the z-plane.
The group delay is commonly defined as

for the first-degree equalizer. Integrating eq. (33) from 0 to w and
normalizing by 27 gives

The significance of eq. (30) is that one can estimate the minimum-
order equalizer required to equalize a given group delay characteristic
by determining the area between the line 7 = 7max and the curve
7,() and dividing by 7, where tmax is the maximum value of 7,(w)
in the passband. Thus in the example of Fig. 8, the estimated order of
the equalizer is approximately (13 X w/2)/m = 6.5. Of course, the
required order of the equalizer must be greater than the estimate given
above, since this estimate assumes the delay of the equalizer exactly
compensates the delay of the unequalized filter. As the degree of the
equalizer is increased over the estimate, the peak error of approxima-
tion decreases monotonically.

We have used the above algorithm, along with initial estimates of
equalizer order, to equalize three sets of elliptic filters. The data for

Table |— Comparisons between optimum FIR and equalized
elliptic digital filters

_ * Ni is the number of multiplications per sample for the optimum FIR filter; N2
is the number of multiplications per sample for the equalized elliptic filter.

these three sets of filters are given in Tables I through III. Included
in the table are the filter specifications (61, 62, Fp, F,); the required
elliptic order n; the required FIR filter duration N; the equalizer
order n.; the average passband delay, 7, (in samples), of the equalized
filter; the percentage ripple, r, in the passband group delay of the
equalized filter; and a comparison between the number of multiplica-
tions per sample required in both the optimum FIR filter and the
equalized elliptic filters. The data in these tables indicate that to
achieve equalization to within about a 3-percent error requires on the
order of 30 percent more multiplications per sample for the equalized
filter than for the optimum FIR design, although in most cases the
unequalized elliptic filter was more efficient than the optimum FIR
designs. Thus it would appear that, at least for these restricted results,
if constant group delay is required in addition to the equiripple magni-

Table I|— Comparisons between optimum FIR and equalized
elliptic digital filters

* N, is the number of multiplications per sample for the optimum FIR filter; N2
is the number of multiplications per sample for the equalized elliptic filter.

tude characteristics, then the optimum FIR filter is always more
efficient than an equalized elliptic filter. It should also be noted that
the delay of the optimum FIR filter [(N — 1)/2 samples] was always
less than the delay of the equalized elliptic filter.

The examples of Tables I through III considered filters where the
order of the unequalized elliptic filter was six or less. It can be argued
that, for higher-order elliptic designs, the relative efficiency of the
elliptic filter over the optimum FIR filter is far greater than for lower-
order designs; hence in these cases perhaps the equalized filter may
still be more efficient than the optimum FIR design. This conjecture
turns out to be untestable because high-order elliptic filters have a
peak passband delay timax which is much larger than for low-order
filters, hence the order required for the equalizer becomes extremely
large and thus is not even practical to consider if equalization over
the entire passband is required. To illustrate this point, Fig. 9 shows
the group delay of a 10th-order elliptic low-pass filter with F, = 0.25.
Using eq. (30) to get an estimate of n. we arrive at a value of n, = 45.
Since this value of n, is only an underbound on the actual order of the

Table II! —- Comparisons between optimum FIR and equalized
elliptic digital filters

* N; is the number of multiplications per sample for the optimum FIR filter; N2
is the number of multiplications per sample for the equalized elliptic filter.

equalizer, it is clear that it is not practical to try to obtain such a
high-degree equalizer.

Another interesting question which arises when one considers the
idea of equalizing an IIR filter is how does the cascade combination
of an elliptic filter and an all-pass equalizer compare to the optimum
IIR filter which best approximates both the desired magnitude and
group delay characteristics? It is clear that the optimum IIR filter can
be no worse than the cascade; the question remains as to how much

better it can be. There is no clear-cut answer to this question. However,
based on our experience with equalized elliptic filters, several observa-
tions can be made. (We shall use the z-plane pole-zero plot of a typical
equalizer filter, shown in Fig. 10, to aid in understanding the nature of
the equalized filter.)

(17) The zeros of the equalizer lie outside the unit circle to give
positive delay.

(iz) The poles of the elliptic filter are constrained by the transition
width requirements of the low-pass filter.

(iv) The poles of the equalizer lie approximately on a circle of fixed
radius, and are approximately equally spaced in the passband.

If the zeros of the optimum filter are not constrained to lie on the
unit circle, then each second-order section will require four multipli-
cations per sample, rather than the three multiplications for each
second-order section of the elliptic design and the two multiplications
for each second-order section of the all-pass equalizer. Based on the

above observations, it seems unlikely that there is much to gain by
using the optimum IIR filter over the equalized filter.

In this paper we have considered only one basis for comparison
between optimum FIR filters and equivalent IIR designs, that mea-
sure being the number of multiplications per sample required in the
standard method of realization for each of these filter types. The
justification for this measure is that in hardware (and generally in
software) the number of multiplications per sample is an excellent
measure of the complexity required in the implementation as well as
the factor which determines the maximum throughput rate of the
system.® However, there are many other ways for comparing these
filter types when one takes into consideration the various finite word-
length effects which occur in a practical design situation. In this section
we review several of these design issues.

Among the various finite word-length effects are roundoff noise,
both uncorrelated and correlated (e.g., limit cycles), and coefficient
quantization sensitivity. For direct-form FIR realization, the peak
roundoff noise can easily be made to be less than 4 of the least signifi-
cant bit by accumulating partial sums in an extended length register
and then rounding the final result. For cascade IIR filters realized
with fixed-point arithmetic, the roundoff noise problem is inherently
related to the dynamic range problem,’ and involves the concepts of
pole-zero pairing and section ordering. Jackson” has shown that with
reasonable pairing and ordering the uncorrelated roundoff noise vari-
ance can be minimized. However, even in the best of situations, the
roundoff noise is equivalent to several bits. In terms of correlated
roundoff noise, i.e., limit cycles, the direct-form FIR realization has no
zero-input limit cycles (because no feedback is present), whereas the
cascade IIR realization will generally exhibit zero-input limit cycles.
Kaiser! has extensively studied these limit cycles and has developed
bounds and estimates for their amplitude and frequency.

The coefficient quantization problem is one of the most difficult
finite word length effects to treat analytically. Rounding of infinite
precision filter coefficients to a fixed number of bits alters the overall
frequency response of the filter in a complicated manner. Avenhaus”
has shown that straight rounding of the infinite precision filter co-
efficients is generally inferior to optimizing the filter performance over
the finite set of fixed precision filter coefficients. However, there are

no general procedures for performing this optimization, nor are there
any guarantees of convergence of the existing methods. Furthermore,
in many cases the advantage of optimizing finite precision coefficients
over straight rounding of the infinite precision coefficients is small.
Thus for the case of coefficient quantization neither direct-form
realization of FIR filters nor cascade realization of IIR filters seems
to offer a relative advantage here.

Thus it is difficult, if not impossible, to be quantitative in comparing
FIR and IIR filters based on anything other than number of multipli-
cations per sample. This is why we have used this measure throughout
this paper.

In this paper some comparisons were made between equivalent FIR
and IIR digital filters based on the number of multiplications per
sample required to realize these filters. In the case of low-pass filters
with quasi-equiripple magnitude characteristics, IIR elliptic filters
could generally be realized more efficiently than equivalent linear phase
FIR filters. When the additional requirement of constant group delay
in the passband was added to the specifications, comparisons showed
the linear phase FIR filters to be more efficient than group-delay-
equalized elliptic IIR filters.

Additionally, a novel set of design charts for determining the
minimum filter order required to meet given filter specifications for
both digital and analog elliptic, Chebyshev, and Butterworth low-
pass filters was presented. Explanation of how to use these charts to
gain insight into the various filter parameter tradeoffs was also given.

2. O. Herrmann, L. R. Rabiner, and D. 8S. K. Chan, ‘‘Practical Design Rules for
Optimum Finite Impulse Response Low-Pass Digital Filters,’ B.S.T.J., 52,
No. 6 (July-August 1973), pp. 769-799.

. J. F. Kaiser, “Digital Filters,’’ in System Analysis by Digital Computer, edited
by F. F. Kuo and J. F. Kaiser, New York: John Wiley and Sons, 1966.

. R. M. Golden and J. F. Kaiser, ‘‘Design of Wideband Sampled Data Filters,”
B.S.T.J., 43, No. 4, Pt. 2 (July 1964), pp. 1533-1546.

. A. G. Constantinides, “Spectral Transformations for Digital Filters,’’ Proc. IEE,
117, No. 8, 1970, pp. 1585-1590.

. E. Christian and E. Eisenmann, Filter Design Tables and Graphs, New York:
John Wiley and Sons, 1966.

. L. B. Jackson, “On the Interaction of Roundoff Noise and Dynamic Range in
Digital Filters,” B.S.T.J., 49, No. 2 (February 1970), pp. 159-184.

. A. G. Deczky, “Synthesis of Recursive Digital Filters Using the Minimum
p-Error Criterion,” IEEE Trans. Audio and Electroacoustics, AU-20, No. 4
(October 1972), pp. 257-263.

B. Jackson, J. F. Kaiser, and H. S. McDonald, ‘‘An Approach to the Imple-
mentation of Digital Filters,’ IEEE Trans. Audio and Electroacoustics,
AU-16, No. 3 (September 1968), pp. 413-421.

B. Jackson, ‘‘Roundoff-Noise Analysis for Fixed-Point Digital Filters Realized
in Cascade or Parallel Form,’? [EEE Trans. Audio and Electroacoustics,
AU-18, No. 2 (June 1970), pp. 107-122.

F. Kaiser, “An Overview on Digital Filters,’’ Newsletter of IEEE Circuit
Theory Group, 6, No. 1 (March 1972).

Avenhaus, ‘On the Design of Digital Filters with Coefficients of Limited
Word Length,’ IEEE Trans. Audio and Electroacoustics, AU-20, No. 3
(August 1972), pp. 206-212.

Copyright © 1974 American Telephone and Telegraph Company
Tue BELL SysTeEM TECHNICAL JOURNAL
Vol. 53, No. 2, February 1974
Printed in U.S.A.

On the Behavior of Minimax Relative Error
FIR Digital Differentiators

Optimum (in a minimax relative error sense) linear phase FIR digital
differentiators can be designed in an efficient manner using a Remez opti-
mization procedure. This paper presents data on wideband differentiators
designed with even and odd values of N, the filter impulse response dura-
tion in samples. Based on these data, several interesting observations can
be made, including:

(i) Differentiators with even values of N have peak relative errors which
are approximately one to two orders of magnitude smaller than identical
bandwidth differentiators with odd values of N, and with the same number
of multiplications per sample in a direct convolution realization.

(22) The smaller the bandwidth of the differentiator, the faster the de-
crease of the peak relative error with increasing N.

(iz) The larger the value of N, the faster the decrease of the peak relative
error with decreasing bandwidth.

These observations lead to the conclusions that the bandwidth of a
differentiator should be made as small as possible, and that even values
of N should be used whenever possible. Complete tables of values of the
impulse response coefficients are included for several wideband dtffer-
entiators for even and odd values of N.

part of many practical systems,‘ it is the purpose of this paper to
present new data on the characteristics of optimum FIR differentiators,
as an aid in making informed decisions concerning their use.

A differentiator is a system whose output is the derivative of its
input. The frequency response of a differentiator is purely imaginary
and is proportional to frequency. A sequence of samples of the deriva-
tive of a band-limited signal can be obtained by filtering a sequence
of samples of the signal with a digital filter that approximates the
ideal frequency response of a differentiator over the bandwidth of the
signal. Therefore, digital filters having this type of frequency response
are also called differentiators.

The frequency response of the ideal digital differentiator with a
delay of 7 samples is

The impulse response corresponding to eq. (1) is obtained as the
inverse Fourier transform of eq. (1) and is given by

The impulse response of eq. (5), which corresponds to an ideal differ-
entiator with zero delay, is of infinite duration and obeys the symmetry

The impulse response of eq. (6), which corresponds to an ideal differ-
entiator with one-half-sample advance, is of infinite duration and obeys
the symmetry condition

In Ref. 2 it was shown that the frequency response of an ideal
differentiator with zero delay had a discontinuity at half the sampling
frequency [i.e., w = 7 in eq. (1) with 7 = 0], whereas the frequency
response of an ideal differentiator with a one-half-sample advance had
no discontinuity at w = 7. The frequency response of a half-sample-
advance differentiator has a slope discontinuity at w = 7 but, as we
will see, slope discontinuities are much easier to approximate than
function discontinuities. It should be noted that the output of a one-
half-sample-advance differentiator is the derivative of the input signal
evaluated midway between input samples. For numerical analysis appli-
cations where one desires the derivative at the sample point rather
than midway between samples, the use of differentiators with zero
delay is required. For most signal processing applications, either type
of differentiator is generally appropriate.

We have only considered 7 = 0 and r = — 3 as possible delays for
the ideal differentiator. It can be seen from eq. (4) that these are the
only values of (— 1 < 7 S$ 0) such that the impulse response has desir-
able symmetry properties.

In order to obtain a causal approximation to the ideal differentiator
which has no phase distortion (other than that corresponding to
delay), it can be shown that an FIR approximation is required.°®
Therefore, consider a causal FIR filter with impulse response h(n),
0 <n SN —1, and frequency response

For this system to have exactly linear phase, the impulse response
sequence must satisfy the condition

For N an odd integer, this means that h(n) has odd symmetry about
the sample at n = (N — 1)/2. [This case corresponds to the 7 = 0
case above with an additional delay of (N — 1)/2 samples. ] For N

even, h(n) has odd symmetry about a point halfway between the
samples at n = N/2 and n = (N/2) +1. (This case corresponds to
the r=—4 case above with an additional delay of N/2 samples.)
This implies that the frequency response of a filter satisfying eq. (10)
can be expressed as

H (eis) = e-##N-VPL jHT*(e8)), (11)
where H*(e%) is a purely real function of w. In particular, for N odd,
H* (e#) is
(N-=1) /2
H*(e#) = YY a(n) sin (wn), (12a)
n=1
where

The factor e~J#\N-)/?2 in eq. (11) represents a delay of (N — 1)/2
samples, which may be accounted for when necessary. Therefore, in
approximating the differentiator, the coefficients a(n) and b(n) must
be chosen so that jH*(e”) approximates the ideal differentiator
frequency response of eq. (1) (with 7 either 0 or —4). In many cases
it is not required, and often it is not desirable, that the approximation
be carried out over the entire band 0 S w S wm. Thus, in general, the
real function H*(e”) must approximate

where F’, is the highest frequency where a good approximation to a
differentiator is desired. In the interval 27F, < w < 2r(1 — F,), the
frequency response of the differentiator is generally left unconstrained
—although it certainly could be constrained to approximate zero over

Fig. 1—The frequency response and error curves of an N = 16 wideband
differentiator.

this interval. The frequency F’,, is called the bandwidth of the differen-
tiator. When F, = 0.5, eqs. (1) and (14) are identical (to within the
delay 7). This case is called a full-band differentiator.

The iterative Remez algorithm of McClellan, Parks, and Rabiner can
be used to choose the values of a(n) or b(n) that minimize the peak rela-
tive error of approximation

D(e%) — H*(e#) 7
D(e%*)
Our present concern is the general properties of the resulting approxi-

mations rather than the details of the approximation algorithm which
are available in Ref. 1. The general properties of differentiators de-

signed by the optimization procedure are illustrated by examples given
in the next section.

The approximation to the ideal differentiator is characterized by
a relative error function that is equiripple over the approximation
band 0 S w S 2zxF,. This is illustrated by Figs. 1 through 4 which
are plots of the frequency responses of several wideband differen-
tiators. Figure 1 shows the magnitude response, the approximation
error [D(e%*) — H*(e#)], and the relative approximation error
[D(e*) — H*(e%) ]/[D(e%*) ] as a function of frequency for N = 16
and F, = 0.5. The relative error curve can be seen to be equiripple

Fig. 2—The frequency response and error curves of an N = 32 wideband
differentiator.

with a peak relative error of 0.0136. That is, the maximum error is
1.36 percent of the desired frequency response over the entire band
0 Sw Sm. Figure 2 shows the same curves for N = 32 and F, = 0.5.
By doubling N, the peak relative error is reduced to 0.0062. Figure 3
shows the magnitude response for N = 31 and F, = 0.5. In this case,
the relative error at f = 0.5 is 1.0 since the desired value is 7 and the
approximation is zero. The reason for the undesirable behavior of the
approximation in this case is apparent from eqs. (12a) and (18a).
When JA is odd, A*(e”) is exactly zero at w = 0 and w = 7, inde-
pendent of the choice of a(n) in eq. (12a). When WN is even, H*(e*) is
exactly zero only at w = 0, independent of the choice of b(n) in eq.
(13a). It is quite desirable to have a zero of H*(e#”) at w = 0 since
this is the desired response; however, a zero of H*(e%) at w = 7 is at
odds with the desired response at w = 7. Thus, the inherent zero at
w = «for N odd is a fundamental limitation to the design of extremely
wideband differentiators.? However, if F, is less than 0.5, the resulting

Fig. 4—The frequency response and error curve of an N = 31, F, = 0.4 wideband
difercotiator.

approximation can be quite acceptable as seen in Fig. 4a for the case
N = 31 and F, = 0.4. Figure 4b shows the magnitude response
plotted from f =0 to f = 0.4, the cutoff frequency, and Fig. 4c
shows the error function over this same range. The peak relative error
is 0.000088 over the approximation interval.

As is clear from these examples, the basic parameters that charac-
terize these differentiator approximations are N, F,, and 6, the peak
relative error of approximation. The examples suggest that 6 can be
reduced by increasing N, and by decreasing F',. Also, there seems to
be a distinct advantage in choosing even values of N provided the
half-sample delay is not undesirable. To substantiate these observa-
tions, a large set of measurements of 6 as a function of F, and N were

made. The results are shown in Figs. 5 through 8* and in Tables I
through VIII. Figure 5 shows the dependence of 20 logio 6 upon N for
F, equal to 0.5, 0.45, and 0.40 and for even and odd values of N in
the range 3 S N S 128. The curve for N odd, F, = 0.5, is not in-
cluded since 6 = 1.0 independent of N. Figure 6 shows the same data
as Fig. 5 with a logarithmic horizontal scale. From Figs. 5 and 6 it
is seen that for the same value of F’, the values of 6 for even values
of N are approximately 1 to 2 orders of magnitude (20—40 dB) smaller
than the values of 6 for comparable odd values of N. This difference
between even and odd values of N is due to the frequency response
discontinuity for odd N, which is considerably more difficult to approxi-
mate than the slope discontinuity in the frequency response for even
N. To substantiate this claim, the curve for F, = 0.5, N even, was
subtracted (on a log scale) from the F, = 0.45, N even, and the

“The curves in Figs. 5 through 8 are straight-line connections of measured data
points and do not represent any smoothing or fitting of the data.

Fig. 6—The curves of 20 logio 5 versus logio N for F, = 0.5, 0.45, and 0.40 for
even and odd values of N.

F, = 0.40, N even, curves and the resulting curves were replotted on
the same scales as Fig. 5. It was then seen that the difference between
the N even and N odd curves for fixed values of F', was small (on the
order of 2 dB) and essentially independent of N. Thus the 1 to 2 order-
of-magnitude difference in the deltas is almost entirely accounted for
by the frequency response discontinuity for odd values of N.

Another observation from Figs. 5 and 6 is that the smaller the
bandwidth (F,) of the differentiator, the faster the peak relative error
decreases with increasing N. Thus for F, = 0.5, the value of 20 logio 6
decreases by only about 30 dB as N varies from 4 to 128; whereas for
F, = 0.45, the value of 20 logio 6 decreases by about 98 dB as N
varies from 4 to 52. For the cases when N is odd, the relation

appears to work for 6 smaller than approximately 0.01, where a —0.5.
Thus, for F, = 0.45, a value of N = 41 gives a 6 of 0.000764, whereas
for F, = 0.40, a value of N = 21 gives a value of 5 of 0.000878. This

inverse proportionality between transition bandwidth and filter order
for fixed value of 6 was originally noted by Kaiser.®

Figures 7 and 8 show the dependence of 20 logio 6 upon F’, for even
values of N (N = 4, 8, 16, 32, 64) and odd values of N (N = 5, 9,
17, 33, 65) respectively. Note that since h((N — 1)/2) = 0 when N is
odd, there are the same number of nonzero impulse response coefficients
for differentiators of length 4 and 5, 8 and 9, etc. The data for even
and odd values of N are presented on different figures because of
the different nature of the solution in the two cases. As seen in Fig. 8,
where WN is odd, as F, approaches 0.5, 20 logio 6 approaches 0, inde-
pendent of N. As previously discussed, this is because of the con-
strained zero of H*(e#) at w = 7 when N is odd. For even values of
N, the curves are spaced apart for all values of F,. The main observa-
tion from these figures is that the larger the value of N, the faster the
peak relative error decreases with decreasing differentiator bandwidth.

The curves in Figs. 5 through 8 can be used to determine the length
of impulse response required to meet a given specification of approxi-

mation error. For example, to obtain a peak relative error less than
1 percent (—40 dB) requires the following values of N (as a function
of F,):

These examples indicate the substantial reductions in N that result
when F,, is reduced and when N is changed from odd to even.

The data presented above indicate that the most efficient FIR
differentiators (i.e., having the smallest value of N for a given value
of peak relative approximation error) are obtained when the band-
width, F',, is as small as possible and when N is even. These design
considerations must be viewed in the light of the intended application.

In processing sequences obtained by sampling an analog signal, it is
generally true that a fullband differentiator is not required, since the
sampling rate is generally set somewhat higher than twice the Nyquist
frequency of the analog signal. Thus, differentiator bandwidths on the
order of 0.40 to 0.49 are quite reasonable for most applications.

Since even values of N result in better approximations than com-
parable odd values, it is generally desirable to choose N even. However,
even values of N are sometimes undesirable because the delay is a
nonintegral number of samples. In situations when the differentiator
is part of a larger signal processing system it may be important to
be able to equalize signal delays in different parts of the system. This
may be more difficult to accomplish when the delay of the differen-
tiator is not an integral number of samples. When N is large and the
differentiator is to be realized as a general-purpose computer program,
it may be desirable to use an FFT realization instead of a direct
discrete convolution realization. In such cases, the odd symmetry of
the impulse response about (NV — 1)/2 permits the frequency response
of the differentiator to be purely imaginary only when N is odd, thus
reducing the storage requirements for the transform of the impulse
response and the number of intermediate multiplications by a factor
of two.

A subset of the differentiators designed in this study is given in
Tables I through VIII. Included in these tables are wideband differen-
tiators (F, = 0.5, 0.49, 0.48, 0.45, and 0.40) for both even and odd
values from N = 3 to N = 50.* The value of peak relative error,

denoted D, is given for each case as well as the impulse response
coefficients for the filter. Note that only the first half of the impulse
response is given in the table; i.e., h(n) forn = 0,1, --- , (N/2) — 1.
The remainder of the impulse response can be obtained using eq. (10).
These data should be adequate for most design applications.

The authors would like to acknowledge the valuable comments and
criticisms provided by J. F. Kaiser and H. 8. McDonald.

1. J. H. McClellan, T. W. Parks, and L. R. Rabiner, ‘‘A Computer Program for
Designing Optimum FIR Linear Phase Digital Filters,” IEEE Trans. on Audio
and Electroacoustics, AU-21, No. 6 (December 1973), pp. 506-526.

2. L. R. Rabiner and K. Steiglitz, ‘“The Design of Wide-Band Recursive and Non-
recursive Digital Differentiators,’’ IEEE Trans. Audio and Elect., AU-18,
June 1970, pp. 204-209.

J. L. Hall, “Simulations of Schroeder’s Integrable Model for Motion of the
Basilar Membrane,” unpublished work.

. J. F. Kaiser, “Digital Filters,’ Chapter 7 in System Analysis by Digital Computer,

Copyright © 1974 American Telephone and Telegraph Company
Tue Beit System TECHNICAL JOURNAL
Vol. 53, No. 2, February 1974
Printed in U.S.A.

Optimum (in a minimax sense) linear phase FIR Hilbert transformers
can be designed efficiently using a Remez optimization procedure. This
paper presents useful design data on wideband Hilbert transformers with
even and odd values of N, the impulse response duration (in samples) of
the filter. Based on these data, the following observations can be made:

(2) In the case of equal lower and upper transition regions, Hilbert
transformers with odd values of N can be realized more efficiently than
those with even values of N, assuming the same peak errors of approxima-
tion for both cases. This 1s because every other impulse response sample is
exactly zero for odd values of N.

(22) The peak approximation error for Hilbert transformers with odd
values of N is determined primarily by the minimum of the values of the
lower and upper transition widths.

(222) The peak approximation error for Hilbert transformers with even
values of N is determined primarily by the lower transition width of the
filter.

(cv) The smaller the bandwidth of the Hilbert transformer, the faster
the decrease of peak error of approximation with decreasing bandwidth of
the Hilbert transformer.

(v) The larger the value of N, the faster the decrease of peak approxima-
tion error with decreasing bandwidth of the Hilbert transformer.

These observations lead to the conclusion that the bandwidth of the
Hilbert transformer should be made as small as possible, and that odd
values of N should be used, whenever possible, for efficient direct form
realizations. A set of tables of values of the impulse response coefficients
ts included for several different bandwidth Hilbert transformers and for
both even and odd values of N.

Hilbert transformers have been used in a variety of applications
including radar systems, speech processing systems, modulation
systems, and schemes for efficient sampling of a real bandpass signal.!.
Although the theory of discrete Hilbert transformer systems is well
understood, it is only recently that design techniques have become
available for obtaining optimum (in a minimax or Chebyshev sense)
finite-duration impulse response (FIR) approximations to the ideal
Hilbert transformer.® It is the purpose of this paper to present new
design data on minimax FIR Hilbert transformers to aid in selecting
the most efficient network to meet given design specifications. Herr-
mann’ has already given a small table of data on FIR Hilbert trans-
formers; however, the generality and widespread applicability of the
data presented here justify this more complete elaboration of the
properties of the minimax Hilbert transformer approximations.

Most applications of Hilbert transformers involve the representation
of a real bandpass signal in terms of a complex signal. For continuous-
time signals, such complex signals are analytic functions of time and
thus are called analytic signals. Although the concept of analyticity is
meaningless for discrete-time signals, complex representation of real,
discrete-time, bandpass signals can be used in a similar manner to
analytic signals. Therefore, consider a real sequence x(n) with Fourier
transform X (e”). Corresponding to this sequence, we can construct
the complex sequence

From eq. (1), the Fourier transform of %(n) is
X (e) = X (ei) + 7X (eX)
and from eq. (2) it follows that
X (ei) + jX(e*) =O of

Thus £(n) can be obtained by linear filtering the sequence x(n) with
a system having a frequency response, Ha(e%), as in eq. (4). Such a
system is called an ideal Hilbert transformer and £(n) is the Hilbert
transform of x(n). If one allows the addition of a linear phase term in
the definition of the frequency response of the ‘‘ideal’”’ Hilbert trans-
former, eq. (4) becomes

where 7 is the delay in samples. The impulse response of the ideal
Hilbert transformer of eq. (5) can be shown to be

The frequency response [eq. (5) ] corresponding to rt = 0 has dis-
continuities at both w = 0 and w = 7, whereas the frequency response
corresponding to +r =—2 only has a discontinuity at w = 0. The
impulse response of eq. (7), corresponding to 7 = 0, has odd symmetry,
is noncausal, and is of infinite duration. In addition, all even-numbered
samples of the impulse response are exactly zero—i.e., ha(2n) = 0,

n = 0, +1, +2, ---. The impulse response of eq. (8), corresponding
to r =—4, is also noncausal, is of infinite duration, and obeys the
symmetry condition

This impulse response does not have the property that even-numbered
(or odd-numbered) impulse response samples are zero.

We have only considered + = 0 and +r =—4 for determining the
impulse response of the ideal Hilbert transformer. All other values of
(—1 <r <0) can be shown to yield impulse responses without de-
sirable symmetry properties and hence are not suitable candidates for
approximation or implementation.

To obtain a causal approximation to the ideal Hilbert transformers
with no phase distortion (except for a delay), it can be shown that
an FIR approximation is required.® Since there are a number of subtle
distinctions between FIR systems of even-length and of odd-length,
the next section deals with some of the general properties of FIR
Hilbert transformer approximations.

Consider a causal FIR system with impulse response h(n),
0 <n SN —1, and frequency response

Since the desired frequency response H4(e) is purely imaginary and
h(n) is real, a linear phase approximation to the ideal frequency
response is obtained only when h(n) satisfies the symmetry condition

For N an odd integer, this means that h(n) has odd symmetry about
the sample at n = (N — 1)/2. (This case corresponds to the 7 = 0
case above with an additional delay of (N — 1)/2 samples.) For N
even, h(n) has odd symmetry about a point halfway between the
samples at n = N/2 and n = (N/2)+ 1. (This case corresponds to
the r =—43 case above with an additional delay of N/2 samples.)
This implies that the frequency response of a filter satisfying eq. (11)
can be expressed as

The factor e~#¢\V-D?2 in eq. (12) represents a delay of (N — 1)/2
samples. When WN is odd, this delay is an integer number of sample
intervals, but when A is even, it is an integer plus one-half of a sample
interval. In approximating the Hilbert transformer, the coefficients
a(n) and b(n) must be chosen so that jH*(e%) approximates the ideal
frequency response of eq. (5). In many cases it is not required, and
often it is not desirable, that the approximation be carried out over
the entire band 0 S w S 27. Thus, in general, the real function
H*(e%*) must approximate

where F;, and Fy define the lower and upper cutoff frequencies respec-
tively. Alternatively, PF, and 0.5 — Fy define the lower and upper
transition bandwidths respectively. It can be seen from eqs. (18a)
and (14a) that H*(e%) is constrained to be zero at w = 0, and 7 when
N is odd, and when JN is even, H*(e%”) is constrained to be zero at
w = 0. Thus, fullband approximations are impossible since F'; cannot
be zero, and Fy can be 0.5 only when N is even. In the transition
regions the desired frequency response is left undefined; hence, in
these regions, H*(e%) is unconstrained except at the end points. In
the next section it will be shown by examples that leaving these
regions unconstrained can lead to unsatisfactory FIR approximations.

The impulse response of the ideal Hilbert transformer (with 7 = 0)
has the property

Fig. 1—The impulse response, magnitude repo and error function of an N = 31
Hilbert transformer with Fz, = 0.04 and Fy = 0.46.

and the frequency response is an imaginary, odd, periodic function and
H,(e%*) = Hale), (16)

Similar properties can be achieved for the FIR Hilbert transformers
if N is odd and F, = 0.5 — Fy. To see this, assume that

A*(e) = H*(e*-»)), (17)
Clearly this requires that F, = 0.5 — Fy. Substituting eq. (13a) into
eq. (17), it follows that

Fig. 2—The impulse response, magnitude response, and error function of an N = 32
Hilbert transformer with F;, = 0.04 and Fy = 0.46.

Fig. 3—The magnitude responses of three N = 15 Hilbert transformers with
different upper and lower cutoff frequencies.

Therefore, from eqs. (11), (13b), and (18c), it can be shown that when
(N — 1)/2 1s even, h(n) = 0 for n = 0, 2, 4, --- and when (N — 1)/2
is odd, h(n) = 0 for n = 1, 3, 5, ---. When WN is odd and (N — 1)/2
is even, h(0) and h(N — 1) are both zero so that the actual length of
the impulse response is N — 2 samples. Thus, for symmetrical approxi-
mations [i.e., satisfying eq. (17)], it is only necessary to consider

Fig. 4—The magnitude responses of three N = 16 Hilbert transformers with
different upper and lower cutoff frequencies.

values of N such that (N — 1)/2 is odd, since the FIR approximations
to be discussed in the next section are unique.

In a manner similar to above, it can be shown that when N is even,
alternate coefficients are not zero even if F, = 0.5 — Fu.

This distinction between even- and odd-length impulse responses is
important in direct convolutional realizations of such systems. For
this realization, the convolution summation

requires N/2 multiplications per output sample for N even, whereas
only (N + 1)/4 multiplications per output sample are required for N
odd when alternate samples of h(n) are exactly zero. Thus an odd-
length filter requires about half the computation required by an even-
length filter one sample shorter.

The iterative Remez algorithm of McClellan, Parks, and Rabiner‘
can be used to choose the values of a(n) or b(n) that minimize the
peak approximation error.

The resulting approximation is the unique best approximation (in the
minimax sense) to D(e”) for a given choice of N, Fr, and Fy. The

details of the approximation algorithm are given in Ref. 4. Our present
concern is the general properties of the minimax IIR Hilbert trans-
former approximations obtained using the McClellan, et al., algorithm.

The minimax approximation to the ideal Hilbert transformer is
characterized by an error function that is equiripple over the approxi-
mation band 27F;, S w S 27Fy. This is illustrated in Figs. 1 through
4 which show the responses of several optimum Hilbert transformer
approximations. Figure 1 shows the impulse response, magnitude
response, and the error function of an optimum Hilbert transformer
having N = 31, Fr, = 0.04, and Fy = 0.46. The peak approximation
error is 0.008094. and the error curve is seen to be equiripple. It can be
seen that H(e*) has the symmetry property of eq. (17) and therefore
alternate samples of the impulse response are zero. To see that the
minimax solution must satisfy eq. (17), assume that

Fig. 7—The curves of 20 logio 5 versus N for three sets of upper and lower cutoff
frequencies with N odd.

then because H*(e%*) = —H*(e-%) and because F, = 0.5 — Fy, both
H*(e*) and H*(e*-*)) would satisfy the conditions for optimality,
thus contradicting the uniqueness of the optimum approximation.

Figure 2 shows the same responses as in Fig. 1 for the case N = 82,
F, = 0.04, and Fy = 0.46. As previously noted, even though the
upper and lower transition widths are equal, all the impulse response
coefficients are nonzero because the frequency response cannot have
the required symmetry when N is even. As seen in Fig. 2b, the magni-
tude response is not constrained to be zero at f = 0.5. Figure 2c
shows the error curve to again be equiripple over the band of approxi-
mation. The peak approximation error for this case is 0.007175.

Figures 3 and 4 show the effects of making the upper and lower
transition bandwidths unequal. Figure 3 shows three sets of conditions
for N = 15;1.¢., Fr = 0.02, Fy = 0.48 in Fig. 3a, Fr, = 0.10, Fy = 0.48
in Fig. 3b, and F, = 0.02, Fy = 0.40 in Fig. 3c. The peak approxi-

Fig. 8—The curves of 20 logio 5 versus N for three sets of upper and lower cutoff
frequencies with N even.

mation errors are 0.267 for 3a, 0.261 for 3b, and 0.261 for 3c. Thus,
even though one of the transition widths changes by a factor of 5:1
(i.e., from 0.02 to 0.10), the change in peak error is on the order of
2 percent. Furthermore, as seen in Figs. 2b and 2c, the magnitude
response of the filter peaks up significantly in the wide transition
band—a generally undesirable result. Also, when the transition band-
widths are unequal, the symmetry property of the frequency response
no longer holds and all the impulse response coefficients are nonzero.
In conclusion, the negligible decrease in peak error obtained by using
unequal transition bandwidths is more than offset by the undesirable
effects in the magnitude and impulse responses. Furthermore, based
on this and other similar examples, it is seen that the peak error of
approximation is determined primarily by the smaller of the two
transition widths when the filter impulse response duration is odd.

N = 16. The lower and upper cutoff frequencies for these examples
are: F, = 0.02, Fy = 0.48 for Fig. 4a, Fr, = 0.02, Fy = 0.40 for Fig.
4b, and F; = 0.02, Fy = 0.50 for Fig. 4c. The resulting peak approxi-
mation errors are 0.247920 for 4a, 0.232594 for 4b, and 0.248561 for
4c. For the data of Fig. 4b (where the upper transition width is 5
times the lower transition width) the magnitude response becomes

very large in the upper transition band. For this case, the peak approxi-
mation error is about 6 percent smaller than the peak error for equal
transition widths. Thus the slight decrease in peak error does not
compensate for the undesirable peaking in the transition region of the
frequency response. On the other hand, Fig. 4c shows that setting
the upper transition width to 0, i.e., letting Py = 0.50 produces a
negligible change in the peak error. Based on these and other examples,
it has been found that for even values of N, the peak approximation
error depends almost entirely on the lower transition width (because
of the zero at w = 0). Thus the upper transition width should be
made less than or equal to the lower transition width to minimize the
peaking in the upper transition band.

The basic Hilbert transformer parameters are N, F1, Fy, and 6, the
peak approximation error (or the ripple) of the filter. It is assumed
that Fy = 0.5 — Fz, ie., the upper and lower transition widths are
equal, for the data to be presented in this section. Thus there are only
three parameters: N, 6, and AF = F, = 0.5 — Fy. A large set of

measurements of 6 as a function of AF and N were made, and the
results are shown in Figs. 5 and 6. Figure 5 shows a plot of 20 logio 6
as a function of N for values of AF of 0.01, 0.02, 0.05, and 0.10, and for
even and odd values of N in the range 3 S N S 128, The curves for
even and odd values of N, for fixed transition widths, are almost
indistinguishable on the scales of Fig. 5, and hence a single curve is

given for both.* Based on these curves, it is seen that the larger the
transition bandwidth of the Hilbert transformer, the faster the de-
crease of peak error with increasing N. Thus for AF = 0.01, the value of
20 logio 6 decreases by only about 42 dB as N varies from 3 to 128;

Figure 6 shows plots of 20 logio 6 as a function of AF for even and
odd values of N. The actual values used were N = 3, 4, 7, 8, 15, 16,
31, 32, 63, and 64.7 As seen in this figure, as AF tends to 0, 20 logio 6
tends to 0 dB or a peak error of 1.0, independent of N. It is also seen
from Fig. 6 that the larger the value of N, the faster the decrease of
peak error with increasing transition width.

From Figs. 5 and 6 it is seen that for a fixed value of 6 the product
of N and AF is approximately a constant. Thus a simple relation of

t The values N = 3, 7, 15, 31, 63 were chosen since they satisfy the condition that
(N — 1)/2 be an even integer.

has been suggested by Kaiser as a reasonable approximation to most
cases of interest in Figs. 5 and 6. Similar inverse proportionality be-
tween filter order N and transition with AF was originally noted by
Kaiser.®

As discussed earlier, it has been found that for odd values of N,
the peak error is determined primarily by the smaller transition width
of the filter; whereas for even values of N, the peak error is determined
primarily by the lower transition width. Figures 7 and 8 present data
which essentially verify these claims. Figure 7 shows curves of 20 logio 6
as a function of N for three sets of conditions: (7) Fr = 0.02, Fx = 0.48,
(7) Fy = 0.10, Fy = 0.48, (227) Fz, = 0.02, Fy = 0.40 for filters where
N is odd. The curves for cases (77) and (771) are indistinguishable on
these scales. This figure shows that the maximum differences between
the peak errors for these cases is about 3.4 dB for N = 127, 1.5 dB for
N = 68, and 0.7 dB for N = 31. Figure 8 shows similar results for
even values of N. For this figure the three cases were: (7) Fz = 0.02,
Fy = 0.48, (a) Fy, = 0.02, Fx = 0.50, (222) Fy = 0.02, Fx = 0.40. In
this case the curves for cases (7) and (i) are indistinguishable. The
maximum differences between peak errors are almost identical to the
errors for comparable cases when N is odd. These figures substantiate
the conclusions stated previously—that the peak ripple is determined

primarily by the smaller transition width for N odd, and by the lower
transition width for N even.

The above discussion suggests that, for direct realizations, the most
efficient FIR Hilbert transformer (i.e., using the smallest number of
multiplications per sample to obtain a desired value of peak approxi-
mation error) has as large a transition bandwidth as possible, and an
odd number of impulse response samples. As an example, to obtain a
peak error of less than 1 percent (6 S 0.01) requires the following
values of N (as a function of AF):

The above tables indicate the substantial processing advantages of
odd-length Hilbert transformers with symmetrical frequency responses.
Further discussion of the relative merits of even and odd values
of N for signal processing applications is given in Ref. 7.
A subset of the Hilbert transformers designed in this study is given
in Tables I through VII. These are symmetrical approximations
(F, = 0.5 — Fy), with Fy, = 0.01, 0.02, 0.05, and 0.10 and both even

The authors would like to acknowledge the valuable comments and
criticisms provided by J. F. Kaiser.

1. B. Gold, A. V. Oppenheim, and C. M. Rader, ‘“‘Theory and Implementation of
the Discrete Hilbert Transform,” Proc. Symp. Comput. Process. Comm.,
Polytechnic Press, 1970, pp. 235-250.

3. O. Tena: “Transversal Filters for Hilbert Transformation,’’ Arch. Elek.
Ubertr., 23, No. 12 (December 1969), pp. 581-587.

4, J. H. McClellan, T. W. Parks, and L. R. Rabiner, “A Computer Program for
Designing Optimum FIR Linear Phase Digital Filters,’ IEEE Trans. on Audio
and Electroacoustics, AU-21, No. 6 (December 1973), pp. 506-526.

5. A. J. Gibbs, “On the Frequency-Domain Responses of Causal Digital Filters,”
Ph.D. Thesis, Dept. of Electrical Engineering, University of Wisconsin, 1969.

6. J. F. Kaiser, ‘Digital Filters,’ chapter 7 in System Analysts by Digital Computer,
edited by F. F. Kuo and J. F. Kaiser, New York: John Wiley and Sons, 1966.

7. L. R. Rabiner and R. W. Schafer, ‘On the Behavior of Minimax Relative Error
FIR Digital Differentiators,”’ B.S.T.J., this issue, pp. 333-361.

Jacques A. Arnaud, Dipl. Ing., 1953, Ecole Supérieure d’Electricité,
Paris, France; Docteur Ing., 1963, University of Paris; Docteur es
Science, 1972, University of Paris; Assistant at E.S.E., 1953-1955;
CSF., Centre de Recherche de Corbeville, Orsay, France, 1955-1966 ;
Warnecke Elec. Tubes, Des Plaines, Illinois, 1966-1967; Bell Labora-
tories, 1967—. At CSF., Mr. Arnaud was engaged in research on high-
power traveling-wave tubes and supervised a group working on noise
generators. He is currently a supervisor studying microwave quasi-
optical devices and the theory of optical wave propagation. Senior
Member, IEEE; Member, Optical Society of America.

Marie T. Dolan, B.A., 1954, Montclair State Teachers College;
M.S., 1966, Stevens Institute of Technology ; Bell Laboratories, 1954—.
Ms. Dolan has been a programmer in general mathematical research
and in recent years has been concerned with computerized design of
digital filters. Member, Kappa Mu Epsilon.

Jeremiah F. Hayes, B.E.E., 1956, Manhattan College; M.S., 1961,
New York University ; Ph.D., 1966, University of California, Berkeley ;
Faculty Member, Purdue University, 1966-1969; Bell Laboratories,
1969—. Mr. Hayes’ current research interests are in the area of signal
processing. Member, IEEE, Sigma Xi, Eta Kappa Nu.

Otto Herrmann, Dipl.-Ing. (Electrical Engineering), 1956, and
Dr.-Ing. (Electrical Engineering), 1965, University of Aachen, Ger-
many; venia legendi, 1971, University of Erlangen, Nuremberg, Ger-
many. Mr. Herrmann has worked on problems concerning approxima-
tion theory as applied to analog and digital filter design. From 1959

to 1971 he was a Teaching and Research Assistant at the University
of Aachen, University of Karlsruhe, and University of Erlangen. He
was at Bell Laboratories during the summer of 1972 on leave from the
Technical Faculty at the University of Erlangen. Presently, he teaches
courses in communications, analog computation, and digital signal
processing at the University of Erlangen. Member, Nachrichtentech-
nische Gesellschaft.

James F. Kaiser, E.E., 1952, University of Cincinnati, 8.M., 1954,
and Sc.D., 1959, Massachusetts Institute of Technology; faculty of
the Massachusetts Institute of Technology, 1956-1960; Bell Labora-
tories, 1959—. Mr. Kaiser has been concerned with problems of data
processing, digital filter design, system simulation, and computer
graphics. He is coauthor of two books, Analytical Design of Linear
Feedback Controls with G. C. Newton and L. A. Gould and System
Analysis by Digital Computer with F. Kuo. Fellow, IEEE; member,
Association for Computing Machinery, Society for Industrial and
Applied Mathematics, Eta Kappa Nu, Sigma Xi, Tau Beta Pi.

Richard V. Laue, B.A. (Mathematics), 1953, Hofstra College; M.S.,
1958, and Ph.D. (Applied and Mathematical Statistics), 1966, Rutgers,
The State University; Bell Laboratories, 1959—. From 1959 to
1964, Mr. Laue was involved in statistical studies associated with
the manufacture and selection of electronic components. Since then,
he has been primarily concerned with various statistical aspects of
telephone traffic systems. Member, American Statistical Association,
Institute of Mathematical Statistics, Sigma Xi.

Dietrich Marcuse, Diplom Vorpruefung, 1952, Dipl. Phys., 1954,
Berlin Free University; D.E.E., 1962, Technische Hochschule, Karls-
ruhe, Germany; Siemens and Halske (Germany), 1954-57; Bell
Laboratories, 1957—. At Siemens and Halske, Mr. Marcuse was en-
gaged in transmission research, studying coaxial cable and circular
waveguide transmission. At Bell Laboratories, he has been engaged in
studies of circular electric waveguides and work on gaseous masers. He
spent one year (1966-1967) on leave of absence from Bell Laboratories
at the University of Utah. He is presently working on the transmission
aspect of a light communications system. Mr. Marcuse is the author of
three books. Fellow, IEEE; member, Optical Society of America.

Stewart E. Miller, S.B. and §.M. (Electrical Engineering), 1941,
Massachusetts Institute of Technology ; Bell Laboratories, 1941—. Mr.
Miller was concerned with microwave radar design from 1941 to
1945. At the conclusion of World War II he resumed design work on
coaxial-cable carrier systems until 1949, when he joined the Radio
Research Department. His work was concerned with circular electric
waveguide communication, microwave ferrite devices, and other com-
ponents for microwave radio systems. As Director, Guided Wave
Research, he headed a group which did work leading to a current
millimeter-wave system development. More recently, his interest and
that of the Guided Wave Research group has shifted to the optical
region and to exploration of the use of lasers and associated devices in
transmission. Member, Tau Beta Pi and Eta Kappa Nu; associate
member, Sigma Xi; Fellow of IEEE; and member, National Academy
of Engineering. Recipient, 1972 IEEE Morris N. Liebmann Award.

Lawrence R. Rabiner, 8.B., S.M., 1964, Ph.D., 1967, Massa-
chusetts Institute of Technology; Bell Laboratories, 1962—. Mr.
Rabiner has worked on digital circuitry, military communications
problems, and problems in binaural hearing. Presently he is engaged
in research on speech communications and digital signal processing
techniques. Member, Eta Kappa Nu, Sigma Xi, Tau Beta Pi; Fellow,
Acoustical Society of America; Chairman of the IEEE G-AU Tech-
nical Committee on Digital Signal Processing; vice-president of the
G-AU AdCom, associate editor of the G-AU Transactions; member
of the technical committees on speech communication of both the
IEEE and Acoustical Society.

Ronald W. Schafer, B.S. (E.E.), 1961, and M.S. (E.E.), 1962, Uni-
versity of Nebraska; Ph.D., 1968, Massachusetts Institute of Tech-
nology; Bell Laboratories, 1968—. Mr. Schafer has been engaged in
research on digital waveform processing techniques and speech com-
munication. Member, Phi Eta Sigma, Eta Kappa Nu, Sigma Xi, IEEE,
Acoustical Society of America, and the IEEE G-AU Technical Com-
mittees on Digital Signal Processing and Speech Communication.

Eric Wolman, A.B., 1953, A.M., 1954, and Ph.D. (Applied Mathe-
matics), 1957, Harvard University; Bell Laboratories, 1957—. Mr.
Wolman has worked on various aspects of traffic flow in communication

systems, and now heads the Network Analysis Department. He is a
member of the Evaluation Panel for the Fire Technology Division,
National Bureau of Standards, and served as visiting lecturer on
applied mathematics at Harvard in 1964. Member, AMS, IEEE,
ORSA, Phi Beta Kappa, Sigma Xi, SIAM; Fellow, AAAS.

H. Zucker, Dipl. Ing., 1950, Technische Hochschule, Munich, Ger-
many; M.S.E.E., 1954, Ph.D., 1959, Illinois Institute of Technology;
Bell Laboratories, 1964—. Mr. Zucker was concerned with satellite
communication antennas, optical resonators, and problems in the
areas of electromagnetic theory and optics. More recently, he has
been engaged in analytical studies related to transmission systems.
Member, IEEE, Commission 6 of URSI, Eta Kappa Nu, Sigma Xi.

Copyright © 1974 American Telephone and Telegraph Company
Tue Bewtu System TECHNICAL JOURNAL
Vol. 58, No. 2, February 1974
Printed in U.S.A.

Interframe Picturephone® Coding Using
Unconditional Vertical and Temporal
Subsampling Techniques

A number of articles'~* have described the use of horizontal re-
dundancy removal techniques to reduce the transmission rate required
for coded Picturephone® signals to 6.312 Mb/s. Here an extension of
this work is given which uses unconditional vertical and temporal
subsampling techniques to reduce the required transmission capacity
to 3 Mb/s. This type of processing is unique in that it does not employ
the complex conditional replenishment techniques which typically have
been used to reduce the digital transmission rate to either 2 Mb/s
or 1.5 Mb/s.§

The analog Picturephone signal has an interlaced frame format’
as illustrated in Fig. 1. Two adjacent fields, each containing 125}
alternating lines, combine to form a complete 251-line frame of video
information. In the experimental system being described, two adjacent
fields are ‘‘averaged”’ together at the coder; this averaged 1253 line
field is processed using variable-length coding techniques*® and sent
to the decoder at a 30-Hz rate. The decoder processes the received
averaged field to form a two-field, 251-line frame for transmission to
the station set receiver. Hence, unconditional vertical and temporal
subsampling is being used to reduce the digital transmission rate from
6 Mb/s to 3 Mb/s; vertical subsampling because only 125% lines per
frame are transmitted and temporal subsampling because averaged

fields are only transmitted at a 30-Hz rate instead of the original 60-Hz
rate. The basis for this unconditional vertical subsampling is that
the vertical spatial dimension of a Picturephone signal does not con-
tain 251 lines of detail. There is no vertical aperture correction cir-
cuitry in the station set and the actual resolution is much closer to
the 1253 lines contained in one field of the interlace pattern; the
vertical amplitude response of the station set transmitter varies from
8 to 27 dB down at half the field sampling rate (1253 video lines/2),
depending on the control unit settings at the transmitter. This is not
a station set design failure but a requirement to prevent objectionable
vertical aliasing effects in the displayed scene. In the temporal dimen-
sion the required signal properties are harder to define because the
postfiltering action of the human eye dominates system parameters.
It is known that a 60-Hz field repetition rate was chosen on the basis
of flicker sensitivity, not on the basis of motion rendition, but the
application of general eye temporal sensitivity studies to the present
Picturephone format is of questionable value and leads to inconclusive
results. Therefore, an experimental 3-Mb/s codec was built both to
verify the vertical processing predictions and to see if 30-Hz trans-
mission is subjectively acceptable in the present Picturephone system.

A block diagram of the experimental system is given in Fig. 2.
Prior to A/D conversion the analog video signal is deemphasized,
equalized, clamped, and filtered with a crispened Gaussian filter that
is 20 dB down at 1 MHz. The A/D converter uses a synchronized
1.536-M Hz clock to produce an 8-bit PCM word at each sample time.
The input to the variable-length coder (VLC) is either the unfiltered
(no vertical or temporal filtering) output of the A/D converter or a

filtered version of the same signal. This is done so that an easy AB
test can be made to determine the effects of additional vertical and
temporal prefiltering. In the unfiltered case the input to the VLC
during time 7 is field 7 (f;); in the filtered case the input is the average
of field z (f;) and an estimate of field 7 based on the information con-
tained in field i — 1 (fi-1). If line j is in field 7, then the estimate of
line j from field z — 1 is given by:

l; is an estimate of even fields from odd fields and vice versa. This
particular estimator gives adequate filtering performance and is easy
to implement. The variable-length coder and decoder are the same as
described in Ref. 3 and process either every other field or every other
average field. The vertical and temporal postfilter uses the received
30-Hz fields to reconstruct the proper 60-Hz even-odd field pattern
for the station set receiver. Again two options were designed for the
postfilter. In the vertical dimension both postfilters are the same and
estimate an adjacent field using the relationship given in (1); in the

MANUAL
// SWITCH
i
ANALOG ANALOG 0 ¥
nace | eigethe | fowoeoy 4 [Te oo
AND O
HORIZONTAL] | CONVERTER CODER
‘PREFILTER
\
¥ 44
Pa oan
Fe¢keij) = ESTIMATE OF FIELD fe tet
k-1 OR k+
BASED ON THE
DATA IN FIELD k FIELD isto 3—Mb/s
DELAY SOLATOR CHANNEL
A A
y fj. fi. fisae Fiegreee
/
ANALOG Y FIELD
DIGITAL-TO- VERTICAL DELAY f, | VARIABLE—
OUTPUT , AND ley
CONVERTER Te DECODER
N POSTFILTER
MANUAL ‘e
SWITCH ee A os iat
fit Mey See eas

time domain the postfilter using the estimate f; is a sample-and-hold
postfilter, whereas the one using the estimate (f; + fi:2)/2 is an
interpolating postfilter. The D/A converter turns the 8-bit 1.536-MHz
PCM samples into an analog format.

When the station set transmitter is operating in its highest resolution
condition, with no zoom, and when the coder prefilter is not used, then
vertical aliasing is easily seen on eyeglass rims and around the lips.
This aliasing can be eliminated cither by using the coder vertical
prefilter or by placing the station set in a partial-zoom mode. Both
changes have the effect of introducing additional vertical prefiltering ;
the coder vertical prefilter adds 6 dB of vertical filtering at half the
sampling rate or 623 lines per frame and the use of the zoom mode adds
filtering proportional to the setting, somewhere between 8 and 27 dB
at 62? lines per frame. The effect of the temporal prefiltering associated
with the coder field filtering is insignificant.

The use of the sample-and-hold postfilter (8 dB down at 15 Hz)
results in a slight jerkiness of motion (temporal aliasing) which is
completely removed by using the interpolating postfilter which is 6 dB
down at 15 Hz.

In order to evaluate this 3-Mb/s codee a series of subjective tests
were developed. Fifteen people were given an AB test between the
analog and the coded video (using field filtering at the coder and the
interpolating postfilter at the decoder) and were asked if the impair-
ment added by the codec was:

In this test cach observer was given 15 seconds of analog followed by
15 seconds of the coded scene shown in Fig. 3. During each 15-second
interval, Bonnie carried on a normal conversation for the first 5 seconds,
moved from side to side for the next 5 seconds, and again carried on a
normal conversation for the last 5 seconds. With the station set in its
high-resolution mode, no zoom, the average scale rating was 2.8 in-
dicating that for this scene the impairment added by the 3-Mb/s
codec was noticeable but not objectionable; with the station set in a
partial-zoom mode the average scale rating was 2.3 indicating an

impairment that is just noticeable. These comment scale ratings can
be compared with a more complete series of tests carried out using the
same horizontal processing without any vertical or temporal processing
on three different scenes. This resulted in an overall average scale
rating of 2.1 for the corresponding 6-Mb/s codec.

An experimental system has been built showing that unconditional
vertical and temporal subsampling techniques can be used on a 6-Mb/s
intraframe codec to result in a 8-Mb/s interframe codec. The impair-
ment resulting from this 3-Mb/s codec is rated as being between just
noticeable and noticeable but not objectionable. This unconditional
alternate field transmission technique can also be used as a higher

activity mode in a conditional replenishment type codec as shown in
Ref. 6.

1. J. B. Millard and H. I. Maunsell, “The Picturephone® System: Digital Encoding
of the Video Signal,” B.S.T.J., 50, No. 2 (February 1971), pp. 459-479.

2. R. P. Abbott, “A Differential Pulse-Code-Modulation Coder for Videotelephony
Using Four Bits Per Sample,’ IEEE Trans. Commun. Tech., COM-19, No. 6,
Part 1 (December 1971), pp. 907-912.

M. C. Chow, “Variable Length Redundancy Removal Coders for Differentially
Coded Video Telephone Signals,’ IEEE Trans. Commun. Tech., COM-19,
No. 6, Part 1 (December 1971), pp. 923-926.

4. R. T. Bobilin, ‘“Prefilters, Sampling, and Transmission Rates for Intraframe

. J.C. Candy, Mrs. M. A. Franke, B. G. Haskell, and F. W. Mounts, “Transmitting
Television as Clusters of Frame-to-Frame Differences,” B.S.T.J., 50, No. 6
(July-August 1971), pp. 1889-1917.

6. Y. C. Ching, B. Gotz, D. M. Henderson, and J. B. Millard, “Video Processing
System (VPS)—A Conditional Replenishment Interframe Codec for the
Dee Transmission of Picturephone® Signals at the T1 Rate,’”’ unpublished
work.

. W. B. Cagle, R. R. Stokes, and B. A. Wright, “The Picturephone® System:
aC Video Telephone Station Set,’’ B.S.T.J., 50, No. 2 (February 1971), pp.
271-312.

Simultaneous Measurements of Depolarization by Rain
Using Linear and Circular Polarizations at 18 GHz

Limitations imposed by attenuation during heavy rain on the reli-
ability of microwave systems are well known! and a recent paper?
discussed observations of depolarization of circular polarization by
rain at 18 GHz; it was concluded that depolarization by oblate rain-
drops poses a serious problem for the use of circular polarization.
However, it is desirable that a direct comparison be made by simul-
taneous measurements of linear and circular polarizations on the same
propagation path. Continuous measurements have been made during
the period June 1972 through April 1973 (a total of 35 rain showers) ;
a discussion of these follows a few remarks on the experimental system.

The 2.6-km propagation path used in the experiment is shown by the
dashed line in Fig. 1. Two frequency-swept solid-state transmitters
(with separate antennas) are located at the lower right of the path.
The circularly polarized transmitter operates at a frequency of 18.65
GHz while the linear horizontally polarized transmitter is at a fre-
quency of 18.35 GHz.

The two receivers (with separate antennas) share a common building
at Crawford Hill, Holmdel, New Jersey (upper left of propagation

path, Fig. 1). Each system has a ferrite switch which looks sequentially
at the received fields; e.g., in the circularly polarized system, the de-
sired circular polarization and then at the depolarized component.
Switching rates are of the order of 17 Hz and occur much faster than
the changes in attenuation produced by rain. Strip-chart recordings
are made of all four components.

The clear-day discrimination for both the circular and linear systems
is more than 32 dB, but none of the depolarization data below —32
dB are included.

The extremes of attenuation in linear (horizontal) and circular
polarizations are shown in Fig. 2 for the fades induced by the 35 rain
showers that occurred during the period of June 1972 through April
1973; they have similar magnitudes but the attenuation in linear
(horizontal) polarization is slightly higher than that for circular polari-
zation:* This is as it should be, for it is known‘ that most storms
consist of oblate drops whose major axes are predominantly hori-
zontally aligned, resulting in less attenuation for vertical than for
horizontal polarization, whereas in circular polarization the attenua-
tion is something like the average of vertical and horizontal.

Comparison of the linear cross-polarization discrimination (XPD)
with the simultaneously measured circular polarization discrimination
(CPD) is made in Fig. 3; the curve is the median of the data. The
depolarization is much stronger in circular than in linear polarization;
for example, the former is —15 dB when the latter is —25 dB.

We pursue the comparison further by examining a particular fade
associated with a rain shower that occurred April 1, 1973, as presented
in Fig. 4. The set of two curves on the left pertains to circular polari-
zation, the set of two curves on the right to linear polarization. The
upper curve in each set is a plot of the rain-induced attenuations. As
in Fig. 2 the circular polarization shows a slightly smaller attenuation
than that for horizontal linear polarization. For example, the maximum
attenuation for circular polarizations was 36 dB while that for linear
was 38.5 dB. At the same time, from the lower curve of each set, we
see that the circular polarization discrimination was only 8 dB while
the cross-polarization discrimination for the linear case was 13.5 dB.
Let us examine the measurements at time 19:57; for the circular case,

* The sparseness of attenuations greater than 20 dB is, of course, due to the limited
number of heavy rain fades that occur during a year. Figure 3 of Ref. 3 shows that
this path has rain-induced attenuations that exceed 20 dB about 20 minutes a year.

Fig. 2—Data on linear and circular rain-induced attenuation from June 1972
through April 1973.

Fig. 3—Comparison of simultaneous measurements of linear polarization dis-
crimination (XPD) and circular polarization discrimination (CPD).

Fig. 4—Rain shower of April 1, 1973. Upper curves are rain-induced attenuation.
The lower curves show the level of the depolarized component.

the desired signal was attenuated to a level of —18.5 dB and the de-
polarized component was only 18.5 dB below that (—37 dB). In the
linear case at this same point in time, the rain-induced fade was 19 dB
and the XPD was 28 dB. The linear polarization is depolarized signifi-
cantly less than the circular polarization; therefore, in radio relay sys-
tems at frequencies of the order 20 GHz, linear (vertical or horizontal)
polarization is believed to be preferable in systems relying on polari-
zation discrimination.

1. D. C. Hogg, “Statistics of Attenuation of Microwaves by Intense Rain,” B.S.T.J.,
48, No. 9 (November 1969), pp. 2949-2962.

2. R.A. Semplak, “The Effect of Rain on Circular Polarization at 18 GHz,” B.S.T.J.,
52, No. 6 (July-August 1973), pp. 1029-1031.

3. R. A. Semplak, ‘Dual Frequency Measurements of Rain-Induced Microwave
Attenuation on a 2.6-Kilometer Propagation Path,” B.S.T.J., 50, No. 8
(October 1971), pp. 2599-2606.

4. R. A. Semplak, “Effect of Oblate Raindrops on Attenuation at 30.9 GHz,’’ Radio
Science, 5, No. 3 (March 1970), pp. 559-564.

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