Thirtieth Anniversary «6611
Lee de Forest and William Shockley Discuss Electronics 612

Network Synthesis Using Tchebycheff Polynomial Series
SIDNEY DARLINGTON 613

A Carrier Telegraph System for Short-Haul Applications
J. L. HYSKO, W. T. REA AND L. C. ROBERTS 666

fficient Coding B. M. OLIVER 724
Statistics of Television Signals E. R. KRETZMER 751
\"xperiments with Linear Prediction in Television Cc. W. HARRISON 764

Photoelectric Properties of Ionically Bombarded Silicon
EDWIN F. KINGSBURY AND RUSSELL S. OHL 802

THE BELL SYSTEM TECHNICAL JOURNAL
PUBLISHED SIX TIMES A YEAR BY THE
AMERICAN TELEPHONE AND TELEGRAPH GOMPANY

FR. KAPPEL O. E. BUCKLEY
H.S. OSBORNE M. J. KELLY
J.J. PILLIOD A. B. CLARK
rR. BOWN D. A. QUARLES

Thirty years ago this month Tor Bretu System TECHNICAL JOURNAL
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under discussion for some years. Dr. R. W. King, who had been one
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JOURNAL was established. Except for a six-year period following 1928,
while he was in England, Dr. King continued as editor until he retired
in 1949.

By July, 1922, when No. 1, Vol. 1 of the JourNAL appeared, research
and development was a long established practice in the Bell System.
The high-vacuum electronic tube, which had already begun to revolu-
tionize electrical communication, was itself a product of Bell System
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will prove increasingly useful to a growing circle of readers.

Dr. de Forest is the inventor of the Audion from which the modern vacuum tube in
its many forms and types has sprung. Dr. Shockley is the leader of the research
group at Bell Telephone Laboratories whose members invented the transistor. Stand-
ing side by side these two men seem to epitomize the basic change in the pattern of
our technical life which has taken place during the first half of the present century—
the change from the struggling individual inventor to the great industrial scientific
laboratory as the source of much of our technological advance.

A general method is developed for finding functions of frequency which
approximate assigned gain or phase characteristics, within the special class
of functions which can be realized exactly as the gain or phase of finite
networks of linear lumped elements. The method is based upon mantpula-
tions of two Tchebycheff polynomial series, one of which represents the
assigned characteristic, and the other the approximating network function.
The wide range of applicability is illustrated with a number of examples.

Network synthesis is the opposite of network analysis—namely, the
design of a network to have assigned characteristics, as opposed to the
evaluation of the characteristics of an assigned network. In general,
there are specifications on the internal constitution of the network, as
well as requirements relating to its external performance. A common
form of the general problem is the design of a finite network of linear
lumped elements, to produce an assigned gain or phase characteristic
over a prescribed interval of useful frequencies. The present paper re-
lates to this particular form.

In general, the restrictions on the network are such that the assigned
performance cannot be matched exactly. This gives rise to an approxi-
mation or interpolation problem. For present purposes, the problem is:
to choose a function of frequency which matches the assigned gain or
phase to a satisfactory accuracy, from that special class of functions
which can be realized exactly with physical finite networks of linear
lumped elements. The function of frequency may be defined in terms
of network singularities (natural modes and infinite loss points). The
~~ + Presented orally, in briefer form, at the 1951 Western Convention of the
Institute of Radio Engineers, and at the Symposium on Modern Network Syn-

thesis sponsored by the Polytechnic Institute of Brooklyn and The Office of
Naval Research, New York City, April, 1952.

interpolation problem may then be regarded as solved when a suitable
set of network singularities has been obtained; for quite different tech-
niques are used to design actual networks with these singularities.

The interpolation problem may be attacked in a number of different
ways; and a variety of different techniques are, in fact, needed to cover
the wide diversity of practical applications. The present topic is a fairly
general way of attacking the problem, based upon manipulations of two
series of Tchebycheff polynomials. The two series represent expansions
of two functions of frequency—one, the ideal assigned gain or phase,
the other, the network approximation to the ideal. The interpolation
problem may be solved in this way because it is feasible, as will be
shown, to determine network singularities from arbitrarily assigned
values of coefficients in the corresponding Tchebycheff polynomial series.

The techniques to be described were derived originally from studies
of the so-called potential analogy; but they can now be developed most
easily without reference thereto.{ In a sense they may be regarded as
extensions of familiar filter theory, using Tchebycheff polynomials, to
more general gain and phase functions. The extensions, however, depend
upon a number of new principles. Extensions of the filter theory applied
to more general problems have been noted in published papers; but
those noted have not used the specific general approach employed here.{
The wide applicability of this general approach will be illustrated by
specific examples.

It will be sufficient for our present purposes to limit the discussion to
the general 4-pole shown in Fig. 1. The 4-pole may be active or passive,
but it must be a stable finite network of linear lumped elements. HE and
V are steady state ac voltages, H the driving voltage and V the response.
The gain a and phase @ are here defined as the real and imaginary parts
of log V/E.

t For the most part, they have used the potential analogy, in such a way that.
Tchebycheff polynomials do not appear at all in general applications. For ex-
amples, see methods of Matthaei?, Bashkow’, and Kuh‘.

The “frequency variable” p represents, of course, tw. The zeros De of
the rational fraction are those values of p at which there is infinite loss.
‘The poles p, are the so-called natural modes, or values of p at which
response V can exist in the absence of driving voltage 7. The scale
factor K determines the average level of transmission. The zeros, poles,
and scale factor together determine the gain and phase completely.

For a physical stable network, the zeros and poles must meet certain
well known restrictions, which are commonly stated in terms of loca-
tions in the complex plane for frequency variable p. Within these re-
strictions, the zeros and poles can be subject to arbitrary choice, say
for purposes of network synthesis.

The symmetries required by the physical restrictions permit a and 8
to be represented separately as follows:T

These expressions hold at all real frequencies, but only at real fre-
quencies. .

Actually, the definitions must vary with the nature of the useful fre-
quency interval. For the present, however, it will be assumed that the
useful interval extends from w = 0 to w; or more precisely, from
w = —w, to +w, (in accordance with the symmetries of gain and phase
functions). Useful intervals which do not include w = 0 require changes
in the definitions, which will be taken up in Section 28.

For our present purposes, Tchebycheff polynomials 7; may be defined
as follows:

The first equation defines an auxiliary angle variable, ¢, in terms of
which 7; is especially simple. The imaginary scale factor 7, associated
with polynomials of odd order, simplifies later analysis. In addition, it

is especially appropriate for general network applications, because the
odd ordered polynomials contribute to the imaginary parts of complex
~ network functions—such as 78 in a + 7b.

It is apparent from (3) that the Tchebycheff polynomials become
simply Fourier harmonics, if they are plotted against a distorted fre-
quency scale—that is, against ¢. This means that they must be ortho-
gonal, over that particular range of frequencies which corresponds to
real values of ¢. From the relation between ¢ and », it is clear that real
values of ¢ cover the frequency interval between —w, and +w,, which
is our useful interval. In other words, the interval of orthogonality coin-
cides with the useful frequency interval. The corresponding interval of
p is of course p = —1a, to +tu..

If a given function is plotted against ¢, instead of w, it may be ex-
panded in a Fourier series. Each term in the series may be replaced by
a Tchebycheff polynomial, to obtain an expansion of a given function
in terms of polynomials, for the specific useful interval w = —w, to
+w,. Established techniques are available for expanding experimental,
or other numerical data, in Fourier series, as well as actual analytic
functions.

In Fig. 2, some of the Tchebycheff polynomials are plotted against w.
The frequencies —w, and +a, are also indicated. Frequencies between
these limits correspond to real values of the angle variable ¢. If this
part of the frequency scale is stretched, in the proper non-uniform way,
the various “loops” not only have the same maximum values, but also
the same shapes. In other words, they become periodic. More specifically,
a stretch which changes the frequency scale into a @ scale changes the
plots into sin k¢ or cos k¢.

¢ by .
z= e* (4)
Substituting z in the exponential equivalent of sin ¢, in the first equa-
tion of (8), gives an alternative definition of z directly in terms of p,

+ A small change in the definition of ¢ would bring the definitions closer to
convention, by replacing both sines by cosines (without altering 7, as a function
of p). This however, would complicate our later analysis.

Substitutions in the exponential equivalents of the other sine and cosine
in (3) give:
Dal
(< + a k even
2

Network applications depend upon the nature of the relationship be-
tween the variable p, and the variable z. The relationship is illustrated
in Fig. 3, which indicates corresponding contours in the p and z planes.

Since angle ¢ is real in the useful interval, z, as defined by (4), has
unit magnitude. In equivalent conformal mapping terms, the unit circle
in the z plane maps onto a segment of the axis of real frequencies in
the p plane—namely the segment extending from p = —1w, to +m, .
Hereafter, we shall say merely that the useful interval in the z plane is
the unit circle, or |z| = 1. The real frequency intervals outside the
useful interval map onto the imaginary axis in the z-plane.

z-plane circles with radii other than unity map onto p-plane ellipses, all
with foci at p = +2, . This is reminiscent of filter theory using Tcheby-
cheff polynomials, and in fact such a filter may be obtained by spacing
z-plane mappings of natural modes uniformly around such a circle. f

t The filter theory is developed in detail in a monograph by Wheeler’, which
also includes an extensive bibliography.

z-plane mappings of network singularities are also an essential part
of synthesis applications. The mapping 2, of a typical zero or pole 7, is
illustrated in Fig. 4. From (5), the analytic relation must be:

By its quadratic nature, there must be exactly two values of z, , corre-
sponding to one p,. The relation is such that replacing z, by —1/z.
leaves p, unchanged; and hence the two values of z, must be negative
reciprocals, each of the other. Thus, one mapping of 7p, falls outside the
unit z-plane circle, and the other inside.

A unique definition of z, may be obtained by requiring that z, must
be the mapping outside the unit circle. Then | z, | > 1 by definition, and
the complete definition of z, may be:

This definition is unique provided network singularities p, are excluded
from that very special line segment of the real frequency axis which
corresponds to the useful frequency interval, —w. << w < +w, (where
| z. | would be exactly 1).

We are going to solve the interpolation problem by choosing the z,
first, instead of the p-plane singularities p, , after formulating the inter-
polation problem in suitable z-plane terms. For this, however, we must

know what further conditions must be imposed upon the z, , so that
the corresponding p, will meet the special conditions necessary for
physical networks. A simple analysis of the definition (8) of z,, and of
the well known restrictions on the p, , leads to the following assertion; |

It is obvious, for example, that conjugate complex z, are necessary
for conjugate complex p, . Also, because | z,| > 1, 2, dominates —1/z, .
Then the sign of Re p, is the same as that of the Re z, , and p, with
negative real parts require z, with negative real parts, and so on.

Thus the direct choice of z, is restricted in exactly the same way as
the choice of p,., except for the additional general requirement | z, | > 1.
The last condition imposes no important restriction on the corresponding
p,. Initially, it was adopted to make z, unique for any p, (not at a
useful real frequency); but this condition does also play an essential
role in the z-plane formulation of the interpolation problem.

A first step in the z-plane formulation of the interpolation problem is
the formulation of the network gain and phase functions, (1) and (2), in
terms of z. This is most usefully examined as a transformation of func-
tional form, rather than as a conformal mapping.

The gain and phase function (1) transforms as follows: The analytic
relation between p and z is regular in the neighborhood of the singular-
ities p, of the network function. Therefore, there will be similar singu-
larities of the transformed function at the z-plane mappings of p, ,
which are z, and —1/z, . These singularities, and also suitable behavior
at infinity, are exhibited by the following expression for a + 76 as a

[] is used here to designate a product of factors of the type following it.f
+ The expression is readily confirmed in the following very elementary man-
ner: For every factor (a _ 2) , in (9), there is also a factor (1+ = . The

Equation (13) holds at all values of p and z, while (14) holds at all
real frequencies. Simplifications of (14) should be noted, good for the
useful interval only. When |z| = 1, 1/z = 2*. Recalling also that log
| x |’ is 2 log | x |, and similar elementary relations, one obtains from (14):

Our applications to network synthesis depend upon a correspondence
which may be shown to exist between certain functions of z and certain
power series in z. The functions of :z may be formulated in terms of
network singularities. The power series in zg may be derived from the
-Tchebycheff polynomial series in p representing the corresponding gain
and phase.

The Tchebycheff polynomial expansion of a gain and phase function
may be written:

If a + 78 corresponds to a finite network, it may be represented by the
function of z in (13). At the same time, 7, may be represented by the
function of z in (6). With these changes, (16) becomes:

The logarithm of the product of the two rational functions, in z and
—1/z respectively, may be written as the sum of two logarithms. The
series in sums of 2” and (—1/z)* may be written as the sum of two series.
Then

The above expression equates the sum of two similar functions, in z
and —1/z respectively, to the sum of two power series, also respectively
in zg and —1/z. The theorem on which the synthesis methods are based
asserts that the functions and power series in z and —1/z may be equated
separately, throughout the useful interval. That is:

The transformation from variable p to variable z
converts an expansion in Tchebycheff polynomials
im p into an expansion in a power series in 2.

Thus, by working with the z, , in place of the p,, one may use a
power series sort of analysis in calculating, or in choosing, the coeffi-
cients C;, in the Tchebycheff polynomial series.

The relations (20) refer to the combined gain and phase function.
Exactly similar relations can readily be obtained, however, for gain and

1 As defined in (8), | #e) > 1. In the useful interval, | z| = 1. Hence | 2/z,| < 1.
It follows that log (1 — z/z,) has a power (MacClauren) series expansion in posi-
tive powers of z, convergent on and within the circle | z | = 1. Finally the first
logarithm in (19) may be expressed as a sum of logarithms of this simple type,
each of which may be expanded separately. Substituting —1/z for z maps the
unit circle onto itself. It follows that the second logarithm in (19) has an expan-
sion in positive powers of —1/z, in the useful interval, provided the first loga-
rithm has an expansion in positive powers of z; and the coefficients in the two
series will be the same.

(The absence of factors } in >, Cyz", as compared with (20), reflects the
factors 2 associated with a and @ in (14).)

Let & + 78 be any function of p which has the following properties:
It must be analytic throughout the useful interval. Further, there are
to be no singularities within a (p-plane) distance e of the useful interval,
where e¢ is finite (but may be small). Finally, at real frequencies, & and
iB are to equal respectively the even and odd parts of & + 7.

Under the conditions stated, a + 78 may always be expanded in
terms of our Tchebycheff polynomials 7. Let >. C,7', be the expansion.
To obtain a parallel to (20), we may form (arbitrarily) a power series
> 40,2". Then we may define a function R(z) by identifying log R(z)
with the power series. All this adds up to the following, comparable to
(20):

The functions of z have the following properties: Because of the mild
restrictions, which we have imposed on the singularities of & + 78, the
series >, C,z" defines a function which is analytic within, and on the
circle |z| = 1. Then A(z), also, is analytic within, and on the circle.
Further, 2(z) has no zeros anywhere in the same region. (R(z), however,
may be more general than the rational fraction in (20).) Finally, because
of the even and odd symmetries, required of & and 7B, (22) may be
broken into the following parallels of the equations (21):

In some applications, it is possible to express R(z) in closed form. In
all applications, it is possible to expand #(z) as a power series, convergent
in the region |z| S 1. The same is true of 1/R(z), since there are no
zeros in the region. Coefficients of either series (#(z) or 1/R(z)) may
readily be calculated by means which we shall examine a little later. For
the present we shall say merely that R(z) is a known function, corre-
sponding to an assigned & + 76.

When the gain and phase function, a + 78, is to approximate @ + 78,
the error in the approximation is (a — &) + i(8 — @). The error may
be expressed in terms of z by taking the difference of corresponding
equations in (20), (22). The difference of the logarithms may be ex-
pressed as a single logarithm of a ratio. Alternatively, and also more
conveniently for our later purposes, it may be expressed as the negative
of the logarithm of the reciprocal ratio. When this is done,

Consider the following arbitrary requirement, as a design criterion:
The series >> C,T;, is to match exactly the series >) C.7;, through

terms of order m. If both series have converged to small remainders:
when k = m, this criterion will surely make a + 78 a good approxima-
tion to a + 78.+ In terms of the coefficients, the criterion requires:

If (25) is applied to the second equation of (24), the power series is
zero through terms of order m. In other words, the logarithm, equated
to the series, will approximate zero in the power series, or “maximally
flat”? manner, to order m. The logarithm is zero when the expression in.
brackets is unity. Further, the logarithm will approximate zero in the
maximally flat manner when, and only when the bracket approximates
unity in the maximally flat manner. Thus a condition which is equivalent.
to (25) is the following:

Zz
, u (1 Tg ) .
ee ie a eee? (27)
K, 2
II l- 7
be
where = is used to indicate equality through power series terms of
order m.

When (27) is applied to network synthesis, the singularities z, , and
scale factor K, are the unknowns, while R(z) is known. If m is equal to
the total number of z, , (27) will determine the network function com-
pletely. When m is smaller, (27) will furnish m + 1 relations between
the network parameters (including the zero order condition), which
may be combined with specifications of other sorts. Since (27) is equiv-
alent to (25), this procedure amounts to the determination of network
parameters which will yield assigned values of the coefficients, C. = C;. ,
k < m, in the Tchebycheff polynomial expansion of a + 78.

Equation (27) applies when both gain and phase are to be approx-
imated. For approximation to gain only, or to phase only, similar rela-
tions may be derived from (21) and (23). Only even ordered Tchebycheff
~~ + When both residues are relatively large, the approximation may still be

good, for the remainders may be quite similar, and the error will be their differ-
ence. In practical applications, this is a not uncommon situation.

polynomials contribute to gain. The following condition turns out to be
the equivalent of Cyn = Cx,k < m:

where = means approximation in accordance with a power series of
even ordered terms, through order 2m. Correspondingly, only odd
ordered Tchebycheff polynomials contribute to phase. The following
condition is equivalent to Co.1 = Cx1,k = 1 tom:

In all cases, unity is approximated with one of the functions appear-
ing in (27), (28), (29). It will be convenient to use H(z) to represent the
error in the approximation, or departure from unity. When gain only is
of interest, the function in (28) is used, and H(z) is defined by:

The example which has been chosen for detailed discussion is the
equalization of the gain distortion produced by two resistance-capacity

type cut-offs. The equalization is to be accomplished with a network
which has n natural modes, but no finite frequencies of infinite loss.
(This is simply one of the arbitrary specifications which define this
problem.)

The two natural modes would be cancelled completely by two infinite
loss points at the same location in the p plane. A network with two
infinite loss points, however, is not physically possible unless it has also
at least two natural modes; and the natural modes will have to introduce
distortion of their own. Thus no finite network will give perfect equali-
zation of unwanted natural modes. Sometimes it is desirable, in practice,
to use an equalizer configuration which produces no finite frequencies of
infinite loss, the entire equalization being accomplished by a suitable
choice of its n modes. Configurations of this sort are illustrated in Fig.
6. Thus, our simple illustrative problem, though chosen to introduce
principles, is also of some practical interest.

The exclusion of finite frequencies of infinite loss simplifies the repre-

Fig. 6—Configurations which produce no finite frequencies of infinite loss.

(For convenience, z, has been written for z, , and K, has been redefined
to avoid the 1/K, required if it is defined as in (21).)

The assigned gain & is even more special. In this particular problem,
a has most of the properties of a network gain a. Specifically, it is the
negative of the gain to be equalized, which in fact corresponds toa
finite network. As a result, R(z) of (22) may be expressed in closed
(rational) form. (Later on, we shall modify the methods appropriate for
this very special situation, so that R(z) need be representable only by
series. )

The specific representation of our present & may be very similar to
the representation of a in (31), as follows:

The constant % is the z-plane mapping of the assigned unwanted
natural modes at p = jo, and may be calculated therefrom by (8). In
(32), Z determines the Cy, , which in turn determine &. The constants
2¢, in (31), are the z-plane mappings of the arbitrary natural modes of
the equalizer. They are to be adjusted to make a approximate a. Then
the network natural modes p, may be calculated from them, by means
of (8).

Taking the difference of corresponding equations in om and (82)
gives the following, analogous to (24):

without regard to phase. It reflects the more specific functional forms of
our present a and @.

The formulas show that the coefficients in the Tchebycheff poly-
nomial expansion of our present a — & are fixed by the logarithm of a
polynomial in z’, of degree n + 2. Since the Tchebycheff polynomial
series is simply one representation of the function or &, this means
that a — & itself is determined by the polynomial in z’. Out of the n + 2
zeros, in terms of z’, n are subject to arbitrary choice, but the other two
are required to be at z= %.

To arrive at a useful choice of the zeros, one may start with the ex-
panded form of the polynomial, which replaces the second equation of
(33) by:

All but two of the coefficients K;, may be assigned arbitrary values, pro-
vided the remaining two are then adjusted to give the required two
zeros at 2 = 2). The corresponding zeros 2; may then be found by
ordinary root extraction methods.

The coefficients may be chosen in such a way that the complex poly-
nomial approximates unity, when | z| = 1. Then the logarithm approx-
imates zero, the coefficients in the power series (34) are small, and since
these are also the coefficients in the Tchebycheff polynomial series in
(33), a — wis small.

A special choice of coefficients, which meets these requirements fairly
well, is the choice determined by (28), with m = n. The function on the
left side of (28) is here the polynomial in (34). For our present purposes,
therefore, (28) becomes:

{Ko + Kz +. Kysae } = 1 (35)
This requires K, = 1, and K, = 0 fork = 1 ton. Then Kasi and Kn4.
must be adjusted to give the two required zeros at z’ = 2) . This gives:

In accordance with Section 9, this special choice of coefficients corre-
sponds to a match of Tchebycheff polynomial series, a to a, through
terms of order 2n:

A sample plot of a — & is shown in Fig. 7. This corresponds to an
equalizer of four natural modes, compensating for an initial loss which
rises to about 8 db at the top useful frequency, or a distortion of +4 db
about the median loss. Residual errors are order of --0.06 db.

A little later we shall return to the question of accuracy, to take up .
methods of estimating what can be done with other numbers of arbi-
trary natural modes, and other values of the assigned modes. First,
however, we should investigate whether the network singularities z,
determined by (36) meet the other necessary conditions.

In the first place, | z,| must be >1. Otherwise, the function of z in
(31) will have no power series expansion over the useful interval | z | = 1;
and (31) will not, in fact, determine the gain a over the useful interval.
It turns out, however, that the condition does not give trouble in the
synthesis of natural modes, when there are no arbitrary frequencies of
infinite loss. This may be demonstrated by the argument outlined below.

The z, are zeros of the polynomial in (84), which we have given the
special form (36), by applying (35). A function theoretic test for | z,| <1

makes use of the contour in the complex plane for the polynomial, corre-
sponding to the z-plane circle | z| = 1. (This is like a Nyquist diagram
except that the contour for the variable, z, is different.) There will be
|z,| < 1 if and only if the contour for the polynomial encloses the
origin.

Now the polynomial in (34), and (85), is merely a special case of the
function on the left in (28), and (80). For this special case (30) becomes

The polynomial cannot enclose the origin without passing through some
negative real value. But this requires an | H(z) | > 1, at some point on
the contour in question, | z| = 1, which happens to be also our uscful
interval. On the other hand, « — a = 0 when >, Riz” = 1, and H(z)
is in the nature of a correction term, which is small in the useful interval
when a — & is small.

The conclusion is: There will be no | z,| < 1 unless the approxima-
tion, a to &, is so poor that a — & exceeds several db in the useful inter-
val. .

Besides the requirement | z,| > 1, the z, must meet physical restric-
tions, which we found to be the same as those limiting the natural
modes p, . The z, may be calculated as follows: The z; are roots of the
polynomial in (35), in terms of z’. All the roots in terms of 2 are 2; ,
except the two required roots at 2) , which correspond to assigned gain @.
Each z, is a square root of a 2;. There are two possible square roots,
however, differing only as to sign. As far as gain @ is concerned, either
choice of sign is permissible; for a depends only on z?. For a physical
network, however, the choice must be such that Re z, < 0. This choice
is possible if, and only if +~/z? has a finite real part. A pure imaginary z,
corresponds to a negative real z; , and thus negative real roots in terms
of 2” are excluded by physical considerations.

Table I lists both z2 and z, for a number of values of n. When 7 is
even, all roots are physical. On the other hand, when n is odd, one root
is always non-physical. In a sense, an odd n is not really appropriate
for the present illustrative problem, with any physical design. An odd n
must necessarily bring in a real natural mode, which merely increases
the sort of distortion we are trying to equalize—that is the distortion
due to unwanted real modes.

The following argument substantiates the suggestion, and also illus-
trates manipulations of a sort which are frequently useful in more gen-
eral applications: The highest order coefficient in (34), Kn42, may be
set aside for adjustments to satisfy physical requirements. The rest of

the coefficients may then be chosen to eliminate terms from the series
> (Co, — Co.)T ox , representing a — &, subject to the previous condi-

If n is odd, all the roots z, can be physical only if Ky42 is negative.
On the other hand, any finite negative K,4. leads to a larger error,
a — a, than Kay. = 0. Reducing K,+2 to zero is the same as reducing
the degree of the polynomial by one, which amounts to reducing n by 1,
from an odd to the next smaller even integer. In other words, a physical
_ design with an odd number of natural modes is less effective, for the
present application, than a simpler network, with the next smaller even
number of modes.

Note that the z, in Table I are proportional to Z% . This means that
root extraction methods need be used only once for each value of n,
after which the roots may be quickly adjusted for any value of %,
corresponding to any assigned value of the two identical modes, fin .

The accuracy of a completed design can be checked by calculating a
from the natural modes p, , and comparing a with a. It is important,
however, to have at least some information about accuracy in advance

of the detailed calculation of the p, . Otherwise, it may be necessary to
carry out several designs, in all detail, in order to obtain one satisfactory
design.

The needed information about accuracy can in fact be obtained from
the error function H(z), which we formulated for general gain applica-
tions in (30), and for the present application in (38). The analysis
which yields (15) may be used to obtain a very similar expression for
a — &, in which R(z)R(—z) appears in combination with the rational
function of z from (15). It may be expressed in terms of the error func-
tion H(z) of (30), as follows:

When H(z) is zero, « — wis zero. When H(z) is small, a — & depends
on phase H(z) as much as on | H(z) |. When H(z) is a positive real,
a — ais negative. When H(z) is imaginary, a — @ is very small. When
H(z) is a negative real, a — ais positive. When H(z) is complex, | a — @ |
is always smaller than with a real H(z) of the same magnitude. The last
statement may be expressed as follows:

The left hand relation is an equality when phase H(z) is an even num-
ber of a radians; the right hand side, when it is an odd number of +
radians.

phase of H(z) varies by (n + 1) radians, which means that H(z) is
successively positive real, imaginary, negative real, imaginary, through
n + 1 half cycles. This accounts for the oscillatory nature of the a — @
curve, illustrated in Fig. 7.

The amplitudes of the oscillations are fixed by | H(z) | , which varies
relatively slowly. Specifically, the two logarithms in (41) determine

envelopes, between which the actual error curve oscillates. Theseare
the dashed lines in Fig. 7.

The maximum error, in the useful interval, is determined by the
maximum value of the envelopes, i.e.,

This function is plotted in Fig. 8, for various values of n. The abscissae
“distortion before equalization’? represent distortion relative to the
median loss in the useful interval, or one half the total variation in the
interval. (This is a function of the top useful frequency w, , relative to
the assigned natural mode  ; and (7) makes w,/fo a simple function of
2.) The figure is convenient for estimating the values of n needed for
specific applications.

The various ripples in a — & do not all have the same amplitude, (43).
For some values of » and %, the amplitudes are almost uniform; for
others they are quite variable. A measure of the variability in ripple

Fig. 8—Distortion before and after equalization—n natural modes equalizing
2 identical natural modes.

The above analysis suggests a way of improving the design deter-
mined by (87) (or the equivalent, (36)). An optimum a — & is commonly
one which has the following properties, in the useful interval:

(This usually minimizes the largest departure in the useful interval,
thereby yielding an “approximation in the Tchebycheff Sense.’’) Since
the variation in phase H(z) determines the number of ripples, while
| H(z) | determines the amplitudes of the ripples, the above conditions
will be met if H(z) has the following properties, in the useful interval:

These conditions may be regarded as alternative design criteria, re-
placing Co, = Cx, . They can in fact be applied to our special example,
and also to certain other special problems which will be noted later.
For more general applications, a suitable H(z) can be defined, but no
reasonably simple procedure has yet been found for calculating the re-
quired constants. (The difficulties will be particularized in a later
section.)

For the present example, (88) may be used to replace (84), and hence
also the second equation of (83), by:

(33) requires H(z) to be a polynomial in 2’, of degree n + 2, with two
zeros of [1 + H(z)] at 2 = 2%. The object is to find an H(z) of this
sort, which also satisfies (45), at least to a good approximation.

The function is a polynomial because the factor [1 — J”"/2°"*") is
divisible by (1 — J/z’). The constants J and G are to be chosen to

give the required double zero of [1 + H(z)] at 2 = %. One value of J ;
so determined, is real and of order 1/2) . This is the appropriate solution.
Then | J°**/z2’"™* | is of order 1/25"** , when | z| = 1. This suggests the
following approximation in place of (47):

The approximation is at least as good as 1/%”"**, compared with
unity, both in the useful interval and in the neighborhood of the singu-
larities % , and z,. This means that the approximation can be used: in
estimating the error a — & (in the useful interval), in calculating J and
G, and in finding the roots 2, .

In the useful interval, |z| = 1, and therefore 1/2 = 2*. Then
(1 — J/z’) is (1 — Jz’)*; and their ratio has magnitude unity. Thus
| H(z) | = |G@|, in the useful interval, to order of 1/%”** compared
with unityt. With | J | <1, phase H (2) varies over the useful interval
to the same extent as the phase of 2’”*”.{ Fig. 9 illustrates the difference
in a — &, as determined by (42) and (48). These curves, however, are
for single values of n and % ; and the improvement obtained by using
(48) would be different with different values of n or % .

The values of J and G, determined from (48), and the requirement
that [1 + H(z)] must have two zeros at 2’ = % , turn out to be:

t This is the most we can expect, when we have n singularities, which can pre-
vent the dominance of only lower order terms, through 2”.

Fig. 9—Comparison of design procedures—four natural modes equalizing two

approximating network gain is generalized, by introducing arbitrary
frequencies of infinite loss, in addition to the arbitrary natural modes.
The methods are also modified for approximation to an assigned phase,
instead of gain, or to phase and gain simultaneously. Finally, the anal-
ysis is modified to permit useful intervals of the “high-pass” type, or
(in the case of gain simulation) of the “band-pass” type.

If we now permit the assigned gain @ to be general, in the sense of
Section 8, we must return to the formulation:

If we retain simulation with a network which has 7 natural modes, and
no frequencies of infinite loss, we must retain the formulation:

a=
: (51)
E Cus" = —tog K211(1 - 5), o=I1,-+-,n
The corresponding formulation of the error is (in place of (83)):
a-a= >) (Cx — Cx)To
(52)

Now the reciprocal of &(z)R(—z) has a power series expansion, in the
region of interest. (Recall Section 8.)

It follows that (53) may be multiplied by this quantity, without
damaging the equality of power series coefficients. In other words (53)
is equivalent to:

Let Ky, be the coefficient of z”* in the polynomial expansion of the left
hand side; and let K;, be the coefficient of 2” in the infinite series ex-
pansion of the right hand side. Then,

These relations are directly applicable to network synthesis, provided

640 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1952
the coefficients K;, can be calculated. Formulae for their calculation
may be derived in the following way:
If (55) is substituted in (50), the result is:t
= 7 . > Co.T ox (58)
> Gio = log > K,2""
When z = 0, the second equation reduces to:

If the functions of z are differentiated, a simple rearrangement gives:
D> kK? = [—D kCae” [D> Kie™] (60)

The right hand side may be expanded as a single power series, and then
like powers on the two sides may be equated separately. The result is:

Synthesis calculations may now be carried out in the following stages.
The assigned gain & is expanded as a Tchebycheff polynomial series, to
determine coefficients Cx, , say through order k = n. The equations (61)
are then used to calculate coefficients K; , also through order n. Each
successive coefficient is computed in terms of those previously deter-
mined. Note that the K,, k < n, are fixed by the same number of
Co,—that is, orders k S n.

Equation (57) is now applied to identify K, with K,, k < m. If all
the network degrees of freedom are to be used to get Cx = Cy, , index
m = n, and (57) determines the polynomial in (55) completely. Other-
wise, m < n, and coefficients K 41 to K, are to be adjusted in accord-
ance with specifications of other kinds. When all the K; have been
determined, the singularities z, are found by root extraction methods,
applied to the right hand side of the first equation of (55).

The previous example might have been carried out in these terms,
but happened to be simpler in the terms used. If (32) is regarded as a
special case of (50), and if (32) is simplified (for purposes illustration)
by using K, = 1, the corresponding R(z)R(—z) becomes simply

| We may think of these equations as defining an infinite network, with na-
tural modes only, which would match the assigned gain & exactly.

If these K; are used to evaluate the polynomial on the left hand side of
(53), in accordance with (55), and if the polynomial is then multiplied
by the above special R(z)R(—z), the result is exactly (36). :

If (64) is used to express (53) in terms of H(z), a “1” may be sub-
tracted from each side of the relation, to get

When m = n, this requires an H(z) of the following form, in terms of
the coefficients K;, derived from R(z)R(—2z):

As in the previous example, | z,| > 1 when the approximation, a to
, is at all reasonable. The z, are again zeros of 1 + H(z); but now
H(z) is defined by (64). If | H(z) | < 1, when |z| = 1, there will be
the same number of zeros of 1 + H(z) as poles, in the region | z| < 1.
Any poles would have to be poles of R(z)R(—z). In Section 8, we noted
that this function is regular in the region | z| < 1. Hence, there will be
no poles, and there will be no | z,| < 1, under ordinary accuracy con-
ditions.

ditions, provided none are pure imaginaries. Since an imaginary 2, Is a
: 2
negative real z, :

There will be no negative real 2} if the polynomial >> Kz” in (55) is
non-zero at all negative real z’. If the requirement is violated, initially,
one or more K;, of the highest orders, must be modified. Graphical
methods are likely to be useful for this, combining plots of the original
polynomial, and proposed changes. An approximation of the form
Co, = Cx, k S m, will still be realized, but with m <n. |

The accuracy of match again may be estimated by the means of (41),
using H(z) of (67) or (68). H(z), however, may not be so easily cal-
culated as for the previous example.

A simpler but less reliable estimate of accuracy is furnished by the
error in the first unmatched coefficient in the Tchebycheff polynomial
series. If K, = K;, through k = m, the leading terms in various series
are as follows. First, from (64) and (68),

Kai = Ba:
Ko
lopes of the ripples in a — a. The variability of the envelopes, across
the useful interval, depends upon higher order coefficients, in compari-
son with the leading term. Calculation of higher order coefficients is

The criteria (45) carry over to general assigned gains, as conditions
on H(z) which, if realized, are usually sufficient to establish approxima-
tion in the Tchebycheff sense. For this purpose we must use the H(z) of
(64), rather than (67) or (68) (which correspond explicitly to Co, = Co ,
k < m). In terms of the polynomial and series representations of (55),
the H(z) of (64) becomes:

The following somewhat special problem is easily solved, in these
terms, and has a direct bearing on various quite different synthesis
techniques: A network is to be designed which combines the functions
of an equalizer or simulator, with those of a filter, or selective network.
In the useful interval, an assigned gain variation @ is to be approximated
in the Tchebycheff sense. At higher frequencies, there is to be a rapidly
increasing loss, or ‘‘sharp filter cut-off.”” The number of natural modes,
n, is to be more than sufficient to match & to the required accuracy, in
the absence of a selectivity requirement, the latitude being used to pro-
duce the required sharp cut-off. In particular, is to be large enough so
that an n term match of Tchebycheff coefficients produces errors that
are negligible compared with those accepted as a price of the sharp
cut-off.

On the above assumption of an ample n, the infinite series }) Kuz”
in (73) may be truncated after the term of order n, and the errors due to
the truncation may be neglected in calculating the design error a — a.t
Then (73) becomes

} The truncated series is merely the polynomial on the left side of (56) which
would be obtained if the filter selectivity were ignored, and m were given the
maximum value, n.

If a sharp cut-off were not required, this approximate H(z) could be
made exactly zero, by using K, = K;, for all coefficients. Then the actual
design error would be determined by the approximation inherent in the
use of (74) in place of (73). For high selectivity, however, K, should be
much larger than K,, as large as possible within assigned limits on
a — Gin the useful range. (It is readily established that K,z” will deter-
mine a at asymptotically high frequencies.) The other K; are then to
be adjusted so that a — & exhibits the desired ‘‘equal ripples.”

Multiplying Gz’” into the numerator gives a rational fraction which is
obviously consistent with (74). The coefficients K;, of (74) which corre-
spond to (75) are:

In the useful interval [>> K./z2”] is [D> Kiz™]*. Hence the polynomials
in z and 1/z have identical magnitudes, in the useful interval; and,
since also |z| = 1, | H(z)| = |G@| in (75). The phase variation, over
the useful interval, is the same for H(z) as for 2", which yields the same
number of ripples in a — & as an ordinary Tchebycheff filter of like
degree. t

The constant G is arbitrary, except that its sign must be properly
chosen to avoid non-physical natural modes. Increasing G increases the
filter selectivity, but also increases a — @ in the useful interval. G and n
are to be chosen together, to realize an assigned selectivity within an
assigned limit on distortion.

The above analysis may be related to the following filter problem:
Required to design a filter which has flat gain, in the useful interval,
but which has m assigned frequencies of infinite loss, in addition to n
arbitrary natural modes (m S n). The n arbitrary natural modes may
be regarded as compensating for gain variations due to the assigned
frequencies of infinite loss, in the useful interval, while reinforcing their
effects at other frequencies. Compensation of effects of the infinite loss
points is the same as s¢mulation of the effects of natural modes at the
same (assigned) frequencies. The approximation in the useful interval
+ This assumes an @ with the general characteristics described in Section 8,

which are such that the numerator and denominator of the fraction in (75) will
each have a net phase shift of zero, across the useful interval.

is to be no better than necessary, so that there may be a maximum
reinforcing of losses at other frequencies. In these terms, (58), and
>> Ky2"" in (78), correspond to the assigned natural modes (at the
same locations as the assigned frequencies of infinite loss). Then the
ideal >> Kz” is itself a polynomial, of degree m < n, and (74) is exact,
rather than an approximation to (73). Then (76) determines the n
arbitrary modes in such a way that the net filter gain approximates
zero in. the Tchebycheff sense, over the useful interval.

Equation (75) may be related to work of Bashkow’. The (arbitrary)
amplitude of the (equal) maxima of | a — @|, computed from H(z) of
(75), depends only on | G | . The frequencies at which the maxima occur
correspond to phase H(z) = sz, which is independent of | G|. Thus,
the locations of the maxima are invariant to the arbitrary amplitude,
within the range where (75) applies. (75) applies only when (74) may be
used in place of (73). Generally, (74) only approximates (73), and the
approximation introduces small variations in the maxima of |a — @|
(when « corresponds to (76)). If the maxima themselves are sufficiently
small, the small variations will be large percentage variations; and the
adjustments to compensate for the variations will yield significant
shifts in the location of the maxima. In other words, the locations of
the maxima of |a@ — @|, required for equal amplitudes, are largely
invariant to the magnitude of the equal amplitudes, but only to an
approximation which becomes worse as the amplitudes are decreased.

Bashkow states the invariance of the frequencies of maximum
| a — &|, as a more or less empirical conclusion, based on a quite dif-
ferent approach to the same synthesis problem.

Equation (75) may be related also to work of Kuh.’ The natural
modes 2, are zeros of 1 + A(z). In other words, H(z,) = —1. Using the
H(z) of (75) gives the following:

- Jt must be remembered, however, that this formulation is permissible
only if the approximations, inherent in (75), are justified when z = z,
(as well as in the useful interval). Taking the logarithm of each side
gives:
log {Ko + +++ Knze"}
2 K,\ (78)
= log (—G) + 2n log z, + log, Ko + --:> om
Equation (58) may now be applied, to replace the logarithms by
power series, provided the truncation of the series is again justified, and
provided the convergence of >> C22” is also proper, at both z = z, and
z= 1/z,. This gives

(Summations >> are all with respect to k; and there is one equation for
each o.) This is a suitable rule, for obtaining an |a — &| with cqual
maxima, whenever the approximations are in fact unimportant. It is
not at all clear, however, just when the approximations become sig-
nificant.

Kuh uses the potential analogy approach for the same sort of syn-
thesis problem. He spaces the natural modes along a p-plane contour
defined in fairly complicated potential analogy terms. It can be shown,
however, that mapping his potentials from p plane to z plane leads to
(79).

When the network is to approximate & in the useful interval, but is
not required to supply selectivity at other frequencies, the approxima-
tion (74) is usually untenable. It is generally necessary to retain the
exact formulation (73).

When selectivity is not required, the phase excursion of H(z), in the
useful interval, can usually be increased to that of 2°”* (as in (67),
corresponding to Co, = Cx, k S n). As a step toward meeting the first

design constants are the Q, . They are to be small enough so that they
do not affect the total phase excursion H(z), when |z| = 1. Their
specific values are to make | H(z) | approximately constant, when
|z| = 1. In general the (required) series in 2” in the numerator makes
it extremely difficult to determine the required values for the Q,;. No
reasonably simple general procedure has yet been found.

The preceding sections were devoted to the approximation a to &,
using ” arbitrary natural modes, but no arbitrary frequencies of in-
finite loss. Similar techniques may be used when there are n” arbitrary
natural modes and n’ arbitrary frequencies of infinite loss. As the appli-
cations become more involved, however, routine calculations must be
supplemented increasingly with an element of art.

For simultaneous design of natural modes and frequencies of infinite
loss, we must go back from (81) to the a formulation in (21). This we
shall now write:

The functions N and D are polynomials. The coefficients will be defined
as follows:

By comparison with (21), the zeros of N, in terms of 2’, are the 2,”.
(Note the minus sign in 81.) The zeros of D are then the z,”. For physical
networks, n” = n’.

Equations (50), describing &, may be retained as they stand. Com-
bining (50) and (81) gives, in place of (52):

The function R(z)R(—z) is exactly the same as before. The reciprocal
of the function will still be >) Ki.z”, with the K; related to Cx by (58),
(59), (61). The new rational fraction N/D will appear where previously
we had the polynomial Ky + --: K,2”.

- This condition may be used to determine the coefficients K; : Ky, of
N and D (in combination with conditions of other sorts, when
m <n” +n’). When the coefficients have been calculated, the (z-plane)
natural modes z, may be determined from the roots of N, exactly as
the z, of previous sections. The infinite loss points z, may be calculated
from the roots of D, in exactly the same way except that Re z, need
not be negative. Sione of the z, must be such that complex and imaginary
z, are in conjugate pairs. Note that there can be conjugate imaginary <z,
only if D has a fo eeu erie double negative real zero.

When m = n” + n’, the following method may be used to salouleta
the K; and KE deteenined by (84). The relation is first multiplied by
D, to get:

Then algebraic manipulation is used to evaluate the power series equiv-
alent of the right hand side, through terms of order n” + 1’, using the
known values of the K, , but general values of the K; of D. Each coef-
ficient is a linear function of the unknowns, K; .

Now the polynomial N, in (85), has no terms of order k > n”. There-
fore (85) requires zero coefficients in the expansion of the right hand
side, from order n” + 1 to order n” + n’. Equating these coefficients
to zero gives n’ linear equations in the n’ unknown K; . Solving for the
K;, determines polynomial D. The values calculated for the K; may
then be used in lower order coefficients of the expansion of the right
hand side of (85), which are exactly the coefficients K;, of N.

When n” — n’ = 0 or 1, a continued fraction method is likely to be
preferable. Various established techniquest may be used to convert the
series >) K.2" into a continued fraction of the form:
cor 1
Ss Kz” = a +

{ See, for example, Fry’s applications of continued fractions to network de-
sign.®

If the continued fraction is truncated after the term of order m, and is
rearranged as a rational fraction N/D, it will obey equation (84). The
degrees of N and D will be such that n” + n’ = m, and n” — n’ = 0
or 1. The continued fraction may be associated with the hypothetical
ladder network shown in Fig. 10, with variable impedance shunt branches
proportional to 2’. The impedance of the (truncated) ladder is N/D.

The accuracy of match, a — @, may again be evaluated from the
final network singularities; or by (41), with H(z) as in (80), before the
singularities have been determined from roots of N and D. A rougher
estimate may again be obtained from the error in the first unmatched
term in the Tchebycheff polynomial series. As before, (equation (72)),
this is equal to the leading term in H(z), with 2°”"”” replaced by —Tom4.2 .

H() 7 (a1a2 oes Om) OmyrK a (88)
The corresponding mismatch in Tchebycheff polynomial terms is:
7 Cae as
Compe T Comte = (a1d2 nee Am) Omyrko (89)

When frequencies of infinite loss are to be chosen, as well as natural
modes, the situation in regard to | z, | < 1 is somewhat less favorable.

It is still true that 1 + H(z) will have the same number of zeros as
poles in the region | z| < 1, so long as a — @ is reasonably small in the
interval |z| = 1. In equation (87), however, the poles of 1 + H(z)
include the zeros z, of D (the arbitrary infinite loss points), as well as
the poles of R(z)R(—z). When the coefficients of D are to be chosen as
in the previous section, the contour rule merely says that any z, and z,
in the region | z| < 1 will occur in like numbers.

Fortunately, the frequent occurrence of | z,| < 1 is softened by the
following curious circumstance. Almost always, any | 2, | and | 2 | <1
are so nearly identical that factors (¢ — z,) and (z — z,) may be can-
celled out without any important effect on H(z), or a — &. Cancellations
of this sort were encountered a number of times before an explanation
was discovered. Actually the explanation is quite simple.

At any zero of 1 + H(z), H(z) = —1. On the other hand, A(z) is
small when | z| = 1. Generally, it is much smaller in most of the inter-
val |z| < 1. For instance, when Cy, = C2, through k = m, H(z) is pro-
portional to 2°”*’, in the neighborhood of z = 0. As a result, | H(z) |
rarely becomes as large as 1, in the region |z| < 1, except in the very
close proximity of a pole. In other words, in the region | z| < 1, any
zero 2, is usually very close to a pole 2,—usually so close that the
corresponding factors z — z, may be cancleled out without significant
effect on a — &.

The occurrence of non-physical natural modes (Rez, = 0) is the same
as before; but adjustments to correct for these, in an efficient manner,
are much more complicated. In addition, there may be non-physical
infinite loss points, 2. To correct for non-physical singularities, the
simplest procedure would be to change one or both of the highest order
coefficients in N and D of (82), that is Ky, and K‘,,. This would be
inefficient, however, for it would spoil the match of C,,, to Car, or Cn
to C,, . The unmodified design, defined by (84), can match terms through
order n” -++ n’, and it is desirable to change only the highest order terms
in adjusting the design.

More efficient adjustments are in fact feasible. They sometimes re-
quire an increased element of art; but the art may be based on specific
principles. Some particularly useful principles are described in the next
section. These apply to various other corrections besides correction of
non-physical zeros and poles. Examples are reduction in phase to make
two-terminal realization possible, and increase in shunt capacity in
two-terminal designs. In general, they offer a means of making
m <n’ + n” in (84), and using the remaining degrees of freedom to
meet other conditions.

Suppose Ni/D, and N2/D: are two rational fractions, representing
special choices of N and D in (82), such that:

The following combination represents an approximation to >) Kz”,
of the order indicated :f

The left hand side of (92) may be used as N/D in (84), to match
Cox to Cx, through k = m. Adjustment of the function F may be used to
satisfy other requirements, in addition to accuracy specifications.

Frequently, Ni/D,; may be the rational fraction corresponding to
truncation of the continued fraction, (86), after the term in dyr4n-.
Then N2/Dz is likely to be a truncation of order m < n” + n’. The
corresponding F is likely to be proportional to 2, or at most a simple
polynomial in 2’. When F is a constant, and m = n” + n’ — 1, the
use of these particular rational functions in (92), to determine N/D,
corresponds to matching Cy, to Cx through k = n” + n’ — 1, but
leaving Con+4n’) Subject to adjustment. Specificially, Cocn++4n) depends
on the choice of /, which may hinge upon such special conditions as
the elimination of non-physical singularities.

Problems which call for more complicated combinations are by no
means uncommon. Skill may be needed in the choice of specific com-
binations which will solve specific problems. Computations may be

(If only natural modes are to be used, the suitable replacement for the
first equation or (55) is here obtained merely by using D = 1, and K.,
2, 2¢, in place of their squares.)

A comparable expression is needed for the assigned gain and phase
a + 76. In place of (50), we now repeat (22), and redefine the coeffi-
cients K; in accordance with

The definition of K, has been changed in such a way that it is now
related to C;,/2 exactly as it was previously (in 58) related to Cy, .
Equations (59), (61) may be applied to calculating the K;, by merely
substituting therein a C,/2 for every Cx .

} For example, a simple recursion formula may be used to assemble the poly-
nomials N and D which correspond to truncation of the continued fraction (86)

is either N or D and P, corresponds to truncation of the sontmued fraction after
the term in a, . The formula holds for n 2 2.

Equations (84), (85), (86) may now be modified, for the new N, D
and. K,, by merely using z in place of 2’ wherever it occurs (including
z" in place of 2”).| The modifications of equations (84), (85), and the
truncation of (86) after am now lead to C, = C,, k S m, instead of the
previous Cx, = Cy . This means that m must be twice as great to match
coefficients out to the same actual orders. This is to be expected since
now one half of our design parameters are used to approximate phase
B, leaving only half for approximating gain a. Equation (89) must be
changed not only in regard to the orders of C,, C; , but also in regard
to the factors $ in (94), (96). This gives

The most important change is in regard to the zeros and poles z, .
The polynomials N and D now determine z, and z, directly, instead of
their squares. There is no opportunity to adjust the sign of Re z, by
choosing the correct sign of +~/2/2. When non-physical singularities ap-
pear, adjustments of high order coefficients may be tried. Section 23
applies provided z’ is replaced by z. If the specification of the problem
permits added delay, linear phase may be added to @ + 78 to increase
the probability of physical singularities{t. (Addition of linear phase
changes only C,, in >> O.7;. A negative change in C, increases the
delay.)

Sometimes it is required to approximate an assigned phase, without
regard to gain. More commonly, it is required to approximate an as-
signed phase, using an “‘all-pass” network, which has a theoretically
zero gain. These two problems, however, are very nearly identical, due
to circumstances explained at the end of this section.

For approximation to phase only, we go back to the 6 equation in
(21). As before, products of factors (¢ — 2.) are replaced by polynomial

ft The well known relation between the gain and phase of any physical network
(See for instance Bode) may give some information regarding the reasonable-
ness of 8. It must be remembered, however, that departures of network gain a,
from the assigned gain &, outside the useful interval, may affect the permissible
phase 8, within the interval.

§ Up to the present, applications to phase problems have not been developed
to the same extent as for gain. Techniques have been explored, however, to deter-
mine how such applications may in fact be carried out.

Using n to represent n” + n’, the total number of network singularities,
we may write N and D as follows, in place of (82) or (95):

To arrive at a design procedure most easily (but not the simplest
design procedure), one may express the assigned gain 8 in the following
way (comparable to (58) and (96)):

>> Cra® = —log > Biz"
Coefficients K;, may again be calculated by a modification of (61). This
time Cx, is replaced by C,, wherever it appears in (61), and then all
even ordered C; are made zero (since only odd terms appears in >, C,2"
of (101)). Note that even ordered K; are not usually zero, even though
even ordered C;, are.
The degrees of N and D, in (99), are such that we can make C;, = C;
through terms of order k = 2n. This requires merely:

As stated, the condition applies to C; of both even and odd orders.
Since even ordered C; are zero, it means that at least n even ordered
C;, will be zero, in addition to the match between n odd orders. (102)
is sufficient to determine an N and a D without reference to (100). If
the (equal) degrees n of (99) are assumed, however, the N and D deter-
mined by (102) will be found to obey (100) automatically (provided
>> Kiz* corresponds to an odd series >) C2", as here assumed).

of the known relation (100), connecting N and D. Let EF and O be re-
spectively the sums of even and odd terms in N. Then N is L + O and
(100) requires D to be EF — O. The ratio O/E may be related to >> C,2"
of (98) as follows:

Let coefficients K; be defined by:
E+0= >> Ke | (105)
Then H and O are respectively the sums of the even and odd terms.
The complete series may now be related to the (odd) series >) C,z" as
follows:
> 2Cyz = —log > Kye" (106)
This fixes the K; of Z + O in terms of the C;.
To make C, = C;, through m odd orders, (102) is now replaced by
oO” O .

mo
The symbol = designates equality of power series through m odd orders.
(All even terms are now zero on both sides.) The right hand side may be
expressed as a continued fraction of the following form, comparable to
(86):

The coefficients of O and EH may be calculated by an appropriate
modification of (61). (Calculate like K, of (101), after dividing all C,
by 2). After O and EF have been evaluated, by truncating the continued
fraction (108), their sum gives polynomial N of (99).

The natural modes and frequencies of infinite loss are determined
from the zeros of the polynomial N. By (21), each zero is either a (z-
plane) natural mode, z, , or the negative of an infinite loss point = ae
If gain variations are inconsequential, there is likely to be some latitude
in designating each zero as az, , or asa —Z,.

A zero of N with a positive real part would make a non-physical natural
mode, and hence it must be a —z,, corresponding to an infinite loss
point. A zero of N with a negative real part can be a natural mode z; ,
but this may not be required. It may be either a z, or a —2, , provided
conjugate zeros are assigned in the same way, and provided the total
number of —z, does not exceed the total number of 2, . The latter con-
dition requires:

The continued fraction (108) shows how many zeros will have nega-’
tive real parts, before any zeros have been calculated. The following
theorem makes this easy:

The number of zeros of N which have negative real parts
is equal to the number of positive coefficients in the trun-
cation of the continued fraction (108) which determined N.

When gain is not to be disregarded, but is to be exactly zero, the
synthesis technique needs few changes. The phase of an ‘‘all-pass’’ net-
work is related to the natural modes z, as follows:

Cuz I ¢ 7 *)
Dy, —— = —log ——_*4 (109)
(+2)
This may be regarded as a special case of (20) for a = 0 (which makes
C,, = O for k even, and also happens to require z, = —z,). In functional
form however, it is more like 78 of (21). It differs in only two regards.
In the power series in z, each C;, is divided by two. In the rational frac-
tion in 2, all the zeros correspond to natural modes, and the poles cor-
respond to frequencies of infinite loss; but the poles are also exactly the
negatives of the zeros, as in the 78 equations of (21).

Accordingly, the phase synthesis technique which ignores gain varia-
tions may be applied to the zero gain form of the problem by cutting

every C; in two. All zeros of N = E + O must be construed as natural
modes z, . Finally, the network must have as many infinite loss points
as natural modes, such that 24, = —z, . (Integer 7 is now the number
of natural modes, rather than the total number of singularities.)

For physical networks, all the first n terms of the continued fraction
(108) must now be positive. To meet this condition it may be necessary
to add linear phase to the assigned phase (by adding a negative cor-
rection to C,). It appears that sufficient linear phase will always lead
to a physical design, provided the number of modes n is increased to
retain a reasonable accuracy.

When the assigned phase £ is linear, the calculations are relatively
simple.

If a delay D is to be approximated over a frequency interval extending
tow = a,

If delay D is to be realized without regard to gain variations, the ap-
propriate O/E is

A known continued fraction expansion of tanh X may be applied to
(111), to obtain the coefficients of (108) without bothering with (105).t
The result may be arranged as follows:

Truncation of the continued fraction gives O/E, and then O + EH. The
zeros 2, turn out to be proportional to 5 , and therefore root extraction
techniques are required only for one D, for each n. The zeros are tabu-
lated for sample n’s, in Table IT.

+ For the expansion of tanh X, reference may be made to a text on continued
fractions by Wall!"!, page 349, equation 91.6.

The error in the first mismatched Tchebycheff coefficient is a rough
measure of accuracy. It may be shown to be

Conga ct Cons =
This measure of accuracy is plotted in Fig. 11, for various numbers of
natural modes n. A sample detailed curve of 8 — 8 is shown in Fig. 12,
with dotted lines corresponding to the estimated error (113).

If delay D is such that the error is reasonable, all the zeros may be
natural modes. If these are combined with a like number of infinite
loss points, such that <4, = —z,, , an all-pass network will be obtained,
instead of one which approximates D without regard to gain. The all
pass network will produce twice the delay, and twice the nonlinearity
of phase. In other words, for an all pass network, both coérdinates in
Fig. 11 must be doubled.

10 20 30 40 5060 80 100 200 300 400 600 1000
PHASE ANGLE IN DEGREES AT TOP USEFUL FREQUENCY
(MULTIPLY BY 2 FOR ALL-PASS NETWORKS)

Fig. 12—Error when six natural modes approximate a linear phase, with a
slope giving 402° at top useful frequency.

The singularities Z, , % correspond, of course, to the network singu-
larities which are to be boiled down. If only @, or only £, is to be ap-
proximated, suitable modifications are readily derived from (21).
The boiling down is accomplished by requiring
ae al 115)
DD | (
where N/D is of lower total degree than N/D. If m = n+ n’, and
n” — n’ = 0 or 1, the continued fraction method can again be used.
This requires expansion of N/D in continued fraction form, instead of
DS Kiz".

All the previous analysis applies to a useful frequency interval —w, <
w < +w.. Its important characteristics are as follows: It is a single
continuous interval, with » = 0 at its center. Useful intervals with other

With this transformation, the whole of the previous z-plane analysis
may be applied at once to high-pass useful intervals, except where
linear phase is involved.

The situation is more complicated in regard to “‘band-pass”’ intervals.
If the useful interval includes the frequencies between w.. and w., the
complete useful interval (p-plane mapping of |z|= 1) must include
also the “image” frequencies, between —w. and —w.. Otherwise,
conjugate complex z-plane singularities z, will not lead to conjugate
network singularities p, . When there are two disjoint parts of the use-
ful interval, the appropriate relation between p and z is relatively com-
plicated. Up to the present, no corresponding technique has been dis-
covered for approximating assigned phases over band-pass intervals,
in Tchebycheff polynomial terms. Gain approximations can be handled,
however, and for a quite simple reason. Gain functions are even func-
tions, and behave in the p’ plane much as gain-and-phase functions
behave in the p plane. In the p’ plane, —w and +w are identical, and a
band-pass useful interval is a single segment of the w” axis.

The three coefficients, a, b, c are subject to two conditions, stemming
from the requirement that the interval |z| = 1 must map onto the
interval w?1 < w’ < we. This leaves one arbitrary degree of freedom.
Its choice may be related to ordinary least squares approximations in
the following way:

If a = >> CxT'x, , the first n terms approximate a in the least squares
sense. In other words, the integrated square of the error is a minimum,
relative to all possible choices of the first n coefficients Cs, , provided
the integration extends over the useful frequency interval,.and in-
cludes an appropriate “weight function”. When (117) relates z to p,
the arbitrary degree of freedom in the choice of the constants a, b, ¢
permits selection of any one of a family of weight functions. Conventional

In applying least squares analysis, it must be borne in mind that the
network gain a does not approximate the assigned gain & in the simple
least squares sense. When Cy, = Cy, for k S n, a — & depends upon
two least squares approximations. The first n terms of > CxT'x, represent
a least squares approximation to a, and are made identical with the
first n terms of >> Cx,7'x, , Which represent a least squares approximation
to &.

An additional singularity, 2} , is also needed, corresponding to the finite
poles of (117). It may be defined as follows:

When p; — p’, in a of (2), is expressed in terms of z and z, , (117)
introduces denominator factors (1 — 2’/z3) and (1 — 1/z0z”). As a re-
sult, a of (21) must be changed to the following, for band-pass intervals:

When definite values have been chosen for a, b, ¢ of (117) (in order
that the C, may be calculated), (1 — 2’/z5) in (120) is not subject to
arbitrary adjustment. This situation can be handled by defining N/D
as the rational fraction in the a equations of (21), as before, but re-

+ For general discussions of orthogonal functions and least squares approxi-
mations, see Courant and Hilbert®, and also a short text by Jackson.??

Fig. 14 illustrates an application of the technique to the simulation of
a coaxial cable attenuation (which is nearly proportional to +~/w).

Fig. 14—Simulation of a coaxial cable attenuation—Attenuation at top useful
frequency = 46 db; Network = four constant-resistance sections.

network singularities. This makes it possible to apply power series ap-
proximation methods, in terms of z, to obtain approximations based on
Tchebycheff polynomial series, in terms of frequency.

“Maximally flat’? approximations in terms of z may be used to match
the first m terms in the Tchebycheff polynomial series representing net-
work gain or phase to the corresponding terms in the series representing
assigned gain or phase. In this way, a Tchebycheff polynomial type of
least squares approximation to the network function is made identical
to the corresponding least squares approximation to the ideal function.
The overall error, network function minus ideal function, is then the
difference between the two least squares errors.

The z-plane analysis may also be manipulated, in a quite different
way, to approach an equal ripple type of approximation (which usually
represents approximation in the Tchebycheff sense). The complications
are such that applications have been limited to problems of certain
quite special types. On the other hand, analysis of this sort has been
found useful in clarifying various other ways of seeking equal ripple
approximations.

1. S. Darlington, ‘‘The Potential Analogue Method of Network Synthesis,’’ Beil
System Tech. J., 30, pp. 315-365, Apr. 1951.

2. G. L. Matthaei, ‘‘A General Method for Synthesis of Filter Transfer Func- -
tions as Applied to L-C and R-C Filter Examples,” Stanford University
Electronics Laboratory Technical Report No. 39, Aug. 31, 1951 (for Office of
Naval Research, NR-078-360).

3. T. R. Bashkow, ‘“‘A Contribution to Network Synthesis by Potential Anal-
ogy,’’ Stanford University Electronics Laboratory Technical Report No. 26,
June 30, 1950 (for Office of Naval Research, NR-078-360).

. B.S. Kuh, ‘A Study of the Network-Synthesis Approximation Problem for
Arbitrary Loss Functions,’’ Stanford University Electronics Laboratory Tech-
nical Report No. 44, Feb. 14, 1952 (for Office of Naval Research, NR-078-360).

. R. Courant and D. Hilbert, Methoden der Mathematischen Physik, Vol. I,
Chap. 2, Julius Springer, Berlin, 1931.

. C. Lanczos, ‘“Trigonometric Interpolation of Empirical and Analytic Func-
tions,’”’ J. of Math. and Phys., 17, pp. 123-199, 1938-39.

H. A. Wheeler, ‘‘Potential Analog for Frequency Selectors with Oscillating
acre Wheeler Monograph No. 15, Wheeler Laboratories, Great Neck,
. Y., 1951.

. T. C. Fry, “Use of Continued Fractions in Design of Electrical Networks,”
American Math. Soc. Bulletin, 36, pp. 463-498, July-Aug., 1929.

trand Co., New York, 1945.
11. H. 8. Wall, Analytic Theory of Continued Fractions, D. Van Nostrand Co.,
New York, 1948.

designed for application in large groups at telegraph central offices and
for trunk-service operation over toll telephone circuits employing stand-
ard levels. It has proved very economical in this field. However, the
very features which make for economy in large installations (such as
amplitude modulation, common carrier supply and testing equipment,
and standardized operating conditions) cause this equipment to be
costly when it is applied a few channels at a time in outlying offices;
these may not be equipped with either telegraph battery supplies or
telegraph boards. Moreover, the 40Cl equipment, being a carrier-on-
for-mark and carrier-off-for-space system, does not lend itself to the
provision of TWX toll subscriber line supervision identical to that of
local stations without the addition of rather complex and expensive
supervisory applique circuits. Where TWX supervision is involved these
supervisory circuits are required to generate and recognize supervisory
signal patterns capable of being distinguished from transmission space
signals and communication breaks, which are long space signals.

Consequently, it was decided to develop a new carrier telegraph
system especially aimed at the needs of fringe areas. One of the problems
to which much thought was given concerned the choice between ampli-
tude-modulation and frequency-shift operation. A frequency-shift sys-
tem provides some reduction in the effect of noise and other interference
on transmission and it is also less affected by rapid level changes. Al-
though these advantages were attractive, it was not clear that they were
sufficient to justify the added complexity and cost entailed by the
adoption of this type of transmission, in view of the quiet and stable cir-
cuits encountered in the Bell System plant. What finally swung the bal-
ance to a frequency-shift system was its advantage in handling TWX
supervisory signals. With transmission accomplished by shifting the car-
rier frequency, supervisory signals could be sent by turning the carrier
on and off. A cheap and simple circuit might then be used to distinguish
between transmission and supervision.

From the foregoing discussion it will be evident that during the twelve
years since the 40Cl system was developed the needs of the Bell System
have changed. Fortunately, the designer’s art has concurrently made
great strides in making available new miniature apparatus and elec-
tronic techniques such as have been exploited so successfully in the
143A type electronic telegraph regenerative repeater,’ the V3 telephone
repeater’ and the N-1 carrier telephone system. As a result, the
channel terminal of the new 43A1 carrier telegraph system, being small,
inexpensive, self-contained and all-electronic with no electro-mechanical

The 43A1 system provides two groups of channel-frequency alloca-
tions, as follows:

a—A three-channel high-frequency allocation, using frequencies be-
tween the upper edge of the voice-frequency band and the lower edge of
the type-C carrier telephone band. This allocation is primarily for opera-
tion on open-wire lines but can also be operated on cable circuits where
the loading provides a suitably high cut-off.

b—A voice-frequency allocation capable of providing six channels
on two-wire circuits or twelve channels on four-wire circuits. The chan-
nels of this allocation are for operation over telephone speech channels
on any of the standard facilities, including broad-band carrier and cable
or open-wire physical circuits.

The present frequency allocations are shown in Fig. 1. The voice-
frequency system is based on twelve nominal midband frequencies
spaced 170 cycles apart from 595 cycles to 2635 cycles, omitting 1615
cycles. The carrier frequency is shifted +35 cycles about midband, and
either the higher or the lower frequency may be used for marking sig-

iii 9 ee. ee ars
= re) 9 o oO iy)
VOICE FREQUENcy .2@ © @ 2 & Y o @ XS wu t+ ©
£ 35VSHIFT + as et ee ee ae
SMSMSMSMSMSM SMSMSMSMSMS5M
CHANNEL NUMBER 2 3 4 5 6 7 9 10 1 12 13 14
e 2 @ 2@ 2@ @ Sa a
2-WIRE n 2 2 mH 4 9 fo oo 8 9
VOICE FREQUENCY 4 XL o@ 2 ¥ r gx VN YT ©
— | | | | | | Tae
SMSMSMSMSMSM  SMSMSMSMSM SM
CHANNEL NUMBER 2 3 4 5 6 7 a3 4°58 & 7
R58 e ee
HIGH FREQUENCY 8 2 8 &® 2 8
o _ ” 9) © °o
a) vt v vt vt Wy)

nals. The high-frequency system is based on six midband frequencies
spaced from 200 to 240 cycles apart. The frequency shift ranges from
+40 cycles in the lowest channel to -+-50 cycles in the highest. These
wider spacings and shifts were adopted to ease the problem of designing
inexpensive filters and oscillators for the higher frequencies.

Channel 1 of the high-frequency system employs adjacent frequency
assignments for the two directions of transmission. The lower frequency
path employs a downward shift for marking signals and the higher-
frequency path an upward shift for marking signals. In half-duplex
operation this minimizes interference from the strong signals at the
transmitter output to the weak signals at the receiver input. The steady
marking frequency, which is being sent against the flow of traffic, is
shifted away from the band over which the message is passing.

ror)
RECEIVE SUPERVISORY SIGNALS =
(IN SWITCHBOARD TERMINATION ONLY): ehariniy ree
it LOOP tid
“+t
i a P+130V I Ee
MODULATOR ru . Lt
SUPERVISORY | o--—-* 1 ie
TRIODE | I ~T
REVERSING POR Ste ae
SWITCH SENDING
TRIODE 5

downward transformation (7500:600) from the impedance of the buffer
amplifier to that of the line so that no amplifier output transformer
is required.

Hither unity-ratio line coils or a hybrid coil may be used to connect
the unbalanced sending and receiving filters to a balanced line. The
hybrid coil is used with a two-wire line when the send and receive
frequencies occupy adjacent bands.

When the channel is used in TWX service as a toll subscriber line,
the subscriber calls the operator to initiate a call by closing the power
switch on his teletypewriter. This connects power to the teletypewriter
motor, closes the transmission circuit to the teletypewriter and applies
plate battery voltage to the transmitting oscillator in the channel ter-
minal, resulting in the transmission of carrier current over the line.
At the distant (switchboard) terminal the receipt of carrier current
energizes a supervisory signal receiving circuit which is responsive to
carrier-on and carrier-off conditions in the receive band. In this circuit,
carrier voltage appearing at the plate of the limiter tube is rectified
and applied to the grid of the supervisory triode. The operation of a
relay in the plate circuit of this tube causes a line lamp at the switch-
board to light.

A disconnect signal is sent by the subscriber at the end of a call by
opening the teletypewriter power switch. This removes the oscillator
plate voltage. At the central office, the receipt of the resulting no-carrier
signal de-energizes the supervisory receiving circuit and causes the super-
visory lamp in the operator’s cord circuit to light steadily. To recall
the operator during a call the subscriber opens and recloses his power
switch. This causes the cord lamp at the switchboard to flash.

An Rc circuit slows the rise of current in the supervisory receiving
tube to guard against false operation of the switchboard line lamp due
to noise impulses during the carrier-off, that is, the idle condition.

On the de side of the channel terminal, provision is made for optional
wiring arrangements to connect to the circuits of the various telegraph
test boards, service boards and TWX switchboards, as well as to local
- teletypewriter loops, using telegraph voltages of either 130 or 48 volts.
In offices where a negative 130-volt battery is not provided, operation
with a single positive 130-volt battery is possible.

(a) shows a conventional open-and-close circuit and the wave shapes
which it produces at the central office end and at the far end of a capaci-

In half-duplex operation, one de loop at each channel terminal serves
for both sending and receiving. The central office end of this loop is
connected to the grid of the sending triode and to the plates of the re-
ceiving tetrodes. If a marking signal is being received from the carrier
line while the teletypewriter contacts in the loop are closed, the receiving
tubes conduct, current flows in the loop and the teletypewriter in the
loop receives a marking signal. Under this condition the office end of
the loop is positive with respect to the cathode of the sending triode;
hence this tube passes a marking signal toward the carrier line. When a
spacing signal is received from the carrier line the tetrodes are cut off,
the loop current is reduced practically to zero, the teletypewriter re-
ceives a spacing signal and the voltage at the office end of the loop be-
comes more positive. Hence a marking signal continues to be transmitted
to the line during the receipt of either mark or space signals from the
line.

When the subscriber opens the loop at the teletypewriter to break
transmission coming from the distant terminal, a clean-cut space should
be transmitted to the line regardless of any incoming signals. The re-
sistor shunted between the plates and cathodes of the receiving tubes
causes the central office end of the open loop to assume the same potential
as the tetrode cathodes. This insures that a steady spacing potential
will be applied to the send tube even though the tetrodes are cut off
by an incoming space. This provides a rapid, clean break. However, if
a large leakage exists across the loop conductors, the resistor will not be
able to keep the sending tube in a cut-off condition and a break by the
subscriber will result in the incoming transmission being reflected in an
inverted condition to the distant carrier terminal. In such a case the
distant sending subscriber would be broken by a “bust-up” of local
copy or by operation of the keyboard break lock. This would normally
be caused only by a trouble condition in cable loops.

On quiet circuits, total distortion per section averages 1 to 2 per cent
at 60 words per minute and about 5 per cent at 100 words per minute.
Plots of received signal distortion versus level of received carrier are
shown on Fig. 6 for both signaling speeds.

The channel terminal employs a formed sheet-metal framework and
occupies a space 103 inches high, 5} inches wide and 7? inches deep over-
all. Fig. 7 shows a 43A1 channel terminal. It is plug-terminated, and
hence removable for maintenance or repair at a bench.

The basic portion of the channel terminal is common to all frequency
allocations. The oscillator network and send filter, which constitute the
elements determining the transmitted frequency, form a plug-termi-
nated sub-assembly 7% inches high, 53 inches wide, and 13 inches deep.
The receive filter and discriminator, which select the received frequency,
form a plug-terminated sub-assembly of the same size.

A rear view of the channel terminal with the send frequency unit
removed is shown in Fig. 8. With both frequency units in place, the
rear of the channel terminal is almost completely enclosed. When they
are removed, the wiring and apparatus terminals of the basic channel
terminal are readily accessible for test and repair.

Tube sockets, potentiometers, test points, switches and the inductor
of the low-pass filter are mounted on the front panel. Small resistors,
capacitors, and germanium diodes are assembled on a plastic “ladder”
which is mounted vertically in the space between the frequency units.

As shown in Fig. 9, three channel terminals may be mounted abreast
on a welded metal frame which is fastened to any of the standard bay
frameworks designed to accommodate 19-inch mounting plates. The
unit mounting frame carries the multicontact receptacles into which the
channel terminals are plugged. Twenty-four channel terminals may be
mounted on an 114-foot relay rack, with line coils and certain auxiliary
equipment.

Where arrangements for switching between half and full-duplex opera-
tion are required, duplex switches for a number of channel terminals
are mounted on a narrow plate between the channel terminal mounting
frameworks.

Loop rheostats, when required, may be mounted adjacent to the
channel terminals or in a loop pad bay along with other loop rheostats
that may be associated with electronic loop repeaters. The latter arrange-

ment concentrates the heat dissipated. by these rheostats at a place
where it will not be harmful.

of the sending tube at the central office terminal, actuates the station
ringer, which is connected to the local loop whenever the teletypewriter
power switch is in the OFF position. .

A channel terminal dissipates about 25 watts. Tube heaters consume
about half an ampere at 24 volts and the remainder ‘of the channel term-
inal, exclusive of its loop-terminating portion, consumes 50 ma at 130
volts. The loop terminating portion dissipates 20, 30 or 62.5 ma at 80
volts, depending upon the type of local circuit employed.

The 43A1 system is capable of working with a great variety of line
levels. The send level may be adjusted for any value from +6 dbm
downward: The receiving equipment operates satisfactorily with —45
dbm or even —50 dbm. But the levels actually used are controlled by
crosstalk and noise conditions in the line.

Receiving levels are normally limited by lightning interference on
open wire and by noise on cable circuits. The minimum tolerable levels
are about —40 dbm on open wire, —45 dbm on four-wire cable circuits
and —35 dbm on two-wire cable.

In Fig. 11, a comparison is made of the effects of static on the 43A1
system and on the 40C amplitude-modulation system. It gives the
results of simultaneous tests on the 2465-cycle bands of the two systems,
using the static from a record made at Madison, Florida. The 48A1
channel could tolerate about 4 db stronger static than the 40C.

A typical circuit layout of the 43A1 system working in the frequency
band between the voice and type-C carrier on an open-wire line is shown
in Fig. 12. The telegraph circuits extend from 48A1 channel terminals
located in a central office, at the left, to 180B1 sets in subscriber stations,
at the right. In the central office, the send and receive paths of the chan-
nel terminal are combined in a hybrid coil. With the moderate degree of
balance provided by the network of this hybrid coil, the allowable dif-
ference between send and receive levels of the middle channel may be

TIME, AS MEASURED IN A 10 KC
BAND BY A METER HAVING A 10
MILLISECOND INTEGRATION TIME
CIRCUIT.

1
-24 -22 -20 -18 -16 -14 -12 -10 -8 -6 -4 -2
SIGNAL-TO- INTERFERENCE * RATIO IN DECIBELS
Fig. 11—Comparison of amplitude and frequency shift modulation with static

35 db or more. The telegraph channels are next combined. with the voice
frequency circuit by means of a 150A filter, and are connected to the
composite set and line through the low-pass section of the 121A (type
-C) carrier line filter. As a result of the cut-offs of the 150A high-pass
and 121A low-pass filter components, the pass band of the telegraph is
about 3.7 to 5.4kce. At the outlying terminal of the open-wire line, the
telegraph is separated from the voice and type-C carrier circuits by
similar filters and connected to the individual subscriber stations by a
branching network and branch lines.

The typical arrangement on a two-wire circuit in the voice feouisaey
range is shown in Fig. 13. Six channels are available, using six of the
twelve frequency bands for transmission east to west and the other six
bands west to east. As in the high frequency case, a branching network
and branch lines at the outlying end connect the circuits to the sub-
scribers. Fig. 14 shows a layout in which branch lines are connected at
intermediate points in the telephone circuit. At these intermediate points
the impedance of the branching network is made high, in order to keep
the balance at the telephone repeater from being harmed excessively.
Though the network attenuates greatly the signals through it, the tele-
graph level is usually sufficiently high so that this loss can be tolerated.

The branching network at the outlying terminal has low impedance.
Taps on the transformers in the network permit the impedance ratios
to be adjusted to suit the line impedances between which the network
operates. Since several circuits may pass through this network, a short-
circuit on one branch should not be capable of degrading transmission
in the other branches. To prevent this, resistances are inserted in series
with each branch of such a value that a short circuit will not cause more
than 3 db excess loss in other channels. It has been shown by tests that,

patch. Obviously echo suppressors must be disabled when a message
circuit is used for telegraph.

When the telegraph circuit contains one or more branches at inter-
mediate points, it would be difficult and often impractical to use an
ordinary message circuit to replace the telegraph stem in emergencies.
The branching location frequently will not be manned and so no one
will be available to patch the branch line to the message circuit. In such
cases each intermediate branch circuit may be made good over a separate
message circuit which is individual to it. Fig. 16 shows this arrangement.
A patch trunk is provided between the 43A1 channel terminal at the
central office and the telephone switchboard. At the switchboard which
is nearest to the intermediate branch subscriber, the branch line is

SWITCHBOARD SWITCHBOARD
Fig. 16—Emergency circuit for intermediate branch.

It is expected that the field of application of the 48A1 system will be
broadened by further development over the next few years. More fre-
quencies will be provided, both in and above the voice band. Means will
be designed for passing TWX supervisory signals over a direct-current
loop from a subscriber station to a channel terminal installed in a nearby
central office. The built-in supervisory arrangements of the 43A1 equip-
ment will be exploited to gbtain inexpensive straightforward trunks
for use both between TWX switchboards and from switchboards to
Line Concentrating Units. The supervisory feature will also be employed
in private line service to provide an open circuit alarm.

1. R 8. Caruthers, H, R. Huntley, W. E. Kahl, L. Pedersen, ‘‘A New Telephone
Carrier System for Medium-Haul Circuits,’”’ Elec. Eng., 70, pp. 692-697, Aug.
1951.

2. B. P. Hamilton, “Carrier Telegraphy in the Bell System,”’ Bell Labs. Record,
26, pp. 58-62, Feb. 1948.

3. J. A. Duncan, R. D. Parker and R. E. Pierce, ‘‘Telegraphy in the Bell System,”’
A.LE.E. Transactions, 68, pp. 1032-1044, 1944.

4. B. Ostendorf, Jr., ‘“New Electronic Telegraph Regenerative Repeater,” Elec.
Eng., 69, pp. 237-240, March, 1950.

O
NEGRASKA OE
TOWAL DAVENPORT
j CITY xe - ——s
, O D CHICAGO
DENVER . ~\ROCK
(coL) ; /MUSCATINE. ISLAND

MEPREGIONAL CENTER BURLINGTON
O O O MT. PLEASANT
A SECTIONAL CENTER SHENANDOAH CLARINDA CORYDON ,
@ PRIMARY OUTLET — = Ne
o TOLL CENTER
— TOLL CABLE ST. JOSEPH YIKEOKUK
---- FUTURE TOLL CABLE MS a Re Ted
— OPEN WIRE KANSAS CITY

<
yn”
n
(0)
9)
isa}
<
Vv
5
NORTHWESTERN BELL TELEPHONE CIRCUITS
I TOTAL NUMBER OF CIRCUITS = 2720
Y) TOTAL NUMBER OF CIRCUIT MILES = 86,400
> END OF 1949
oO
a
1S)
Ww
re)
=
z
Ww
16)
ry
ao oO 20 40 60 80 100 120 140 160 180 200
OPEN-WIRE LENGTH IN MILES
Fig. 2—Distribution of circuit lengths in Iowa in 1950.
io
90

The type-O carrier system followed the type-N system closely in
time, and, in effect, covers the same range of circuit lengths for open
wire lines that N provides for cables. It was both natural and expedient
that many of the N features were carried over directly into the O design.
It was necessary, however, to make important distinctions as well. These
similarities and differences will be discussed in some detail.

The transmitting and receiving voice frequency subassemblies are
reused with substantially no modification. This provides the O system
with the same compandor and the same 3700 cycle signaling system as
used in N.

An important difference between the two systems is concerned with
the use of single sideband in the O rather than double sideband as in
the N. This choice is an economic one. The double sideband system is
relatively easier to design and less expensive than the single sideband
arrangement. The use of double sideband in cables is practicable in
many cables because of the relative abundance of conductors as com-
pared with open wire pairs. In some cases the use of single sideband in
cables may be attractive as compared with the cost of new outside plant
for certain length ranges.

Another distinction between the two systems is the provision of cir-
cuits in smaller groups in O. In N the basic group is 12, although systems
may in some instances be partially equipped. In O, the desire to furnish
smaller circuits groups resulted in the choice of a basic four-channel
group. The full complement per pair for O, including a channel replacing
the voice circuit, is sixteen channels.

to provide sufficient regulator range to accommodate line variations due
to wet or icy lines. The repeater and receiving group regulator range
common to four channels is in the order of 40 to 50 db, or approximately
four times the regulating range of an N repeater. The range of the
twin-channel regulator is comparable to the N individual channel regu-
lator, but the O regulator is shared by two channels forming adjacent
sidebands of the same carrier.

The use of a single sideband imposes more severe requirements on
channel band filters. The use of a material called ferrite, in combination
with a crystal, affords an efficient channel band filter in a small space
when compared to previous single sideband channel filters employing
air-core coils. Ferrite coils are employed in a coil-condenser type of filter
to provide separation for the various four-channel groups. While the N
system employs only receiving channel band filters, O has filters in
both the transmitting and receiving terminals.

The O system employs the double modulation principle for all groups.
This arrangement. permits the use of only four channel band filter de-
signs for all of the 32 channel frequency allocations. The frequency range
for these basic channel bands has been selected to provide the most
economical overall filter design.

The use of die-castings has been extended in a number of ways. No-
table among these is a die-cast framework, used in both the terminal
and the repeater. The plug-in technique has been expanded to provide
plug-in filters for channel and group band filters.

The O modulation plan is shown on Fig. 5. The single-sideband chan-
nel filters for all groups are in the frequency range from 180 to 196 ke.

By the use of different group carrier frequencies the several four-channel

5, high and low group assignments are used for the two directions of
each four-channel system. A repeater is provided for each four-channel
system and, except in the case of OA, the high and low frequency groups
are ‘frogged”’ at each repeater, as in the N system.

Figure 6 shows a block diagram of a complete O system. On this figure,
and in general on other figures showing filters, a letter code is assigned
to designate the kind of filter, with a subscript letter to indicate the
particular system in which the filter is used. The several filter designa-
tions and number codes are collected for reference in Table I. Much of
the apparent complexity of the system, particularly as regards filters,

arises out of the use of a single pair for both directions of transmission.

| MODULATOR |
OB SYSTEM OB SYSTEM
40-76 KC 40-76 KC
4 Cp, | GROUP INEL
CHANNELS Cc | OSCILLATOR | B CHAN Ss
(180-196 KC) S Ca UNIT | (180-196 KC)
| MODULATOR |
OC SYSTEM OC SYSTEM
80-116 KC 80-116 KC

GROUP
CHANNELS Cc Co pew nnn nm en wn wn a rr nr ee ee Co Cc CHANNELS
(180-196 KC) ° Gg ¢ | UNIT | Wg0-196 Kc)

GROUP GROUP
CHANNELS Cop Cp pee enn en em own a ee -- Cp , CHANNELS
(180-196 Kc) } UNIT UNIT | (1g0-196 KC)

Line filters (G) are provided to separate the OB, OC, and OD Systems
from the OA frequency ranges. Because of the lower attenuation and
slope in the OA frequency range, and the better line coupling factors,
the repeaters do not “frog” the high and low groups.

A block diagram of a typical carrier terminal is shown on Fig. 7,
in this case the OB terminal. The terminal is comprised of four channel
units, two twin-channel units, group transmitting and receiving units,
and a group oscillator. An oscillator in each of the twin channel units
supplies carrier to the transmitting channel modulators. The same oscil-
lator supplies transmitted carrier for two associated sidebands. The
original carrier is balanced out in the transmitting modulator. This
method results in a more accurate control of the transmitted carrier.
level. The group oscillators supply the necessary frequencies to the

TaBLeE I
Filter Codes for Each System
Location and Function Filter Symbol)
Common OA OB oc | OD
TERMINAL ONLY
Transmitting low pass..... None 168F
Receiving low pass........ None 169G
Carrier pickoff (184ke)..... Fi 532A
Carrier pickoff (192kc)..... F2 532B
Signal pickoff............. None 169A
Channel band pass........ None 529A
Channel band pass........ None 529B
Group transmitting........ E. 540A*
Group receiving........... A+ D 530J | 531B | 5380C | 530F
Group receiving........... B+D 531F | 5381C | 530D | 530G
TERMINAL AND REPEATER....
Directional................ C 5380H | 580A | 530B | 5380E
G2 219St
LINC ie ose re ehoaetelox wets G 5Gl 5387At
Gl 538A
REPEATER ONLY
AVRINATY o% (cx Beaiey eee A+B 531A | 531D | 531E
A 530K
B 530L

* Except OA system. f Cut-apart region between 36 and 40 ke. tf Cut-apart
region between 30 and 40 ke. 538A is a 587A filter with housing and protectors
for pole mounting.

;
|
MODU-
LATOR
! I
HYBRID & |
| | NETWORK
| !
l
| BAND FILTER |
1 | (PLUG IN AND!
Hersh Sees e Ses ere em Rena somal Rae See ae | REVERSIBLE |
a ee ar IS a Poca Se ene Pe epee Ieee PEE ee SN ats | 529A OR 529B!
EXPANDOR SUB-ASSEMBLY ! | !
i I | { |
<— _|ExPANDOR I |
[or |
ONTROL ba DEMODU- 1
1 | LATOR | 192-196 KC
1 | ' \(OR 180-184 KC)
VARIO LPF rg | REC <—_
H LOSSER (3100) tf ! BPF a
1 |
] toy ! I
{ toy ! |
, ted ee 1
| MERCURY RECT & BPF
RELAY DELAY a LIMITER < (3700) [1 |
! # !
' i | '
1 , | i
| { | i

plug-in filter reversible in its socket, this grouping can be made to serve
two channels as follows:
eee ees 4 eae oS
192-196 Receiving 1) 180-184. Receiving
A similar paired filter serves
ae —— d eae eas
. (188-192 Receiving 184-188 Receiving
Thus only two basic kinds of paired channel band filters are required,
rather than four kinds. In these filters, as well as the reversible group
filters, the filter designations are so arranged that when the filter is in
place the proper filter designation is in view.

LINE FREQUENCIES IN KILOCYCLES FREQUENCIES IN KILOCYCLES
FOR EACH CHANNEL FOR EACH CHANNEL
44 52 184 192
40 48 56 180 188 196
1 2 3 4
LOW fee 8
GROUP 4 3 2 1 fence
236
GROUP
CARRIER
64 72
60 68 76 256
GROUP
CARRIER
HIGH more S=sTort
GROUP t, “2 3 4 s———.J ">
4 3 2 1
180 188 196
184 192

picked off by a narrow band crystal filter. This same carrier is supplied
to the receiving side of the channel units for demodulating the associated
sidebands.

The OB group transmitting unit is shown on Figure 11. It receives the
four sidebands and two transmitted carriers and places thein in the
proper high or low line frequency assignment. The transmitting group
unit, depending on the optional connection to the group oscillator (Fig.
11), can be either a high group transmitting unit or a low group trans-
mitting unit.

For convenience the noise generator is contained in the group trans-
mitting unit. On very quiet circuits this noise source provides a means of
masking crosstalk. In ordinary usage the noise thus provided is not
noticeable on the circuit, but is sufficient to reduce the chance of hearing
intelligible crosstalk to a small value.

CARRIER SUPPLY OSCILLATOR TRANSMITTED
TO TRANSMITTING <= 184 KC CARRIER
MODULATORS (OR 192 KC)
<_

fier, and operates principally on the two received carriers, although the
sidebands are fed back also.

FROM == REC UNIT
40-56KC
<TUNITS (OR 60-76 KC)
FROM -- [
TWIN- Ii ee
CHANNEL , | H
UNITS => i
amid | 256 236 3700 |
| KC KC CYCLES |
|
| THER LGR
p bese \
ey (eerie eee eee fees: 4
TO GROUP | TO KEYING
; REC UNIT } CIRCUITS

=
REG LINE
AMPLIFIER AMPLIFIER
' TO OC AND
OD SYSTEMS
|
eee [to pati
CONTROL
| AMPLIFIER
|
| RECTI-
! FIER
|
x*
| LINE
i osc
FILT --
(116 Kc) les
]
!
p--------------------}--------------+-+------
LINE REG
| AMPLIFIER
1
|
t
f |
t 1
t 1
| 1
CONTROL TO OA SYSTEM
AMPLIFIER OR TO TYPE C
: | CARRIER AND
! 'VOICE FACILITIES
|
| i]
| '
be beste one eee ae Ae ek Se eee Se ee a eee ee ee

* FILTERS ARE PLUG-IN AND REVERSIBLE ** BASIC AMPLIFIER USED FOR OB, OC AND OD FOR
REPEATERS AND GROUP RECEIVING UNITS

60
-60 -55 -50 -45 -40 -35 -30 -25 -20 -15 +10 -5 O 5 10
COMPRESSOR 1000-CYCLE INPUT IN DBM (AT ZERO LEVEL)
Fig. 16—Typical over-all channel load characteristic.

with repeaters will not result in a substantial change of the net loss
variation, assuming the repeater section losses do not exceed the range
of the regulators.

The line regulator is assisted by the twin channel regulator, for which
a characteristic is shown on Fig. 19. This regulator is similar to the
individual channel regulator of N, and serves two channels having a
common carrier. This fact alone does not materially change the effective-

it controls in any case. There are other important differences, however,
between N and O channel regulators. In N the channel regulator follows
the channel band filter and thus tends to compensate for its flat trans-
mission variations. Also the N regulator is controlled from the demodu-
lator de output and thus compensates in some degree for demodulator
variations. In O, the twin-channel control is ahead of both the channel
band filter and the channel demodulator, and therefore does not make
up their variations.

A statement might be interpolated at this point to emphasize that
the relative advantages of single-sideband and double-sideband trans-
mission are by no means easily listed and evaluated, since the differences
are many and devious, some necessarily and some fortuitously. An ex-
ample worthy of note is that in N it is necessary to be concerned about
relative phase shift of the sidebands and in the instances of longer cir-

cuits, perhaps to equalize this phase shift in order to prevent serious
reduction of signal output, or variation in channel net loss with fre-
quency. No such concern applies to O.

In regard to filter characteristics, it seems obvious that complete
coverage is not feasible in this description. Instead typical curves only
will be shown.

Fig. 20 shows the general characteristics of filters for separating the
wanted sideband from the carrier and unwanted sideband. The trans-
mitting and receiving filters have similar shapes. The carrier pick-off
filter characteristic is shown in the same figure. Fig. 21 shows the filter
characteristics for separating the voice and signaling (3700 cycle) func-
tions.

Another filter case of interest is the line filter for separating, for ex-
ample, the OA system from the OB system, and from the OC and OD
systems, as well, if they are employed. Fig. 22 shows the configuration
and loss characteristics of the G1 (537A or 538A) filter. A Cg directional
filter (580A) characteristic is shown on Figure 23. This filter assembly
includes two filters to accommodate the OB high and low group assign-

4 3 2 1 Oo 4
FREQUENCY IN KILOCYCLES PER SECOND (FROM CARRIER)
Fig. 20—Typical channel band and carrier pick-off filter characteristic.

ments. Similar characteristics apply to the Ag + Bg auxiliary filter
(531A). The A and B characteristics are used in the group receiving
filters Ag + Dg , (5381B) and Bg + Dg, (531C). The D filter is a band-
pass filter with relatively gradual cutoff to pass the 180-196 band for
the channel filters, and has peaks at the group carrier frequencies of
236 ke and 256 ke. These filter characteristics are not shown.

and a crystal with the necessary fixed and adjustable capacitors. The
small size is made possible partly by the high Q ferrite coil, and partly
by the circuit configuration employing it. As compared with filters em-
ploying air-core coils and having comparable cutoffs, the reduction in
size of these filters is very striking.

The group receiving unit is shown with its plug-in filters on Fig. 31.
The filters are held in place by stud screws and nuts. The arrangement
shown on Fig. 31 is also used with different filters for the repeater ampli-

PREGUENEY IN Recs: PER econ
Fig. 22—Typical line filter characteristics (G1, 537A or 538A).

plug-in units. Since there is no interference between plug-in units in
inserting and removing them, can covers have been eliminated. This
fact, and the somewhat wider distribution of units having a high con-
centration of vacuum tubes, result in a relatively low temperature rise
for O as compared with N. Blower facilities are not provided in the
terminal.

plug-in group oscillators, which are also shown in the photograph, to-
gether with fuses, alarm lamps, etc.

Pole mounted repeaters are housed in a cabinet, similar to that used
for N. Such a cabinet, equipped with four repeaters is shown on Fig. 37.

Since a maximum of four repeaters would have to be supplied by one
pair of wires, it is not feasible to transmit power for the repeaters over
line pairs. Instead the cabinet contains rectifiers and a line voltage
regulator for obtaining 130 volts de from commercial ac supply. For
reserve power supply, a cabinet is available containing a 24-volt storage
battery and a dynamotor to supply 130 volts de to two repeaters (or
two dynamotors to supply four repeaters) in case of power failure.

Fig. 38—Tes
their being held during the interval of failure; and second, to make all
circuits busy at both terminals, to prevent false seizure by operators or
automatic switching equipment. Since many O systems may be em-
ployed in situations where one terminal is unattended, facilities are
included whereby, after failure, the system can be tested from either
end, through the use of one of the signaling channels. If it is indicated

that the system is operable it can be placed in service again without the
necessity of a trip to the unattended terminal.

Arrangements are provided by which two O circuits can have their E
and M signaling control leads interconnected without the use of the
signaling converter, which is otherwise required. This feature is employed
when two circuits are connected together on a permanent or semi-per-
manent basis to form a single trunk.

To facilitate testing at terminal points a test stand (Fig. 38) has been
provided which supports an O, or N, channel unit during test and
adjustment. By a patch cord, the channel unit can be connected to its
original framework if desired. Built in pin jacks permit bridging meas-
urements to be made at selected points in the transmission circuit.

The term coding, as applied to electrical communication, has several
meanings. It means the representation of letters as sequences of dots
and dashes. It means the representation of signal sample amplitudes as
groups of pulses having two or more possible amplitudes as in pulse
code modulation. Lately, it has also come to be the generic term for
any process by which a message or message wave is converted into a
signal suitable for a given channel. In this usage single-sideband modula-
tion, frequency modulation and pulse code modulation are examples of
encoding procedures, while microphones, teletypewriters and television
cameras are examples of encoding devices.

This is a nice concept, but it is useful to distinguish between two classes
of encoding processes and devices: those which make no use of the
statistical properties of the signal, and those which do. In the first class,
the encoding operation consists simply of a one-to-one conversion of
the message into a new physical variable, as a microphone converts sound
pressure into a proportional voltage or current, or of the one-to-one
remapping of the message into a new representation without regard to
probabilities, as by ordinary amplitude, frequency or pulse code modula-

tion. In ordinary PCM for example, the message samples are converted
into groups of on-or-off pulses. The particular combination of pulses
in any group depends only upon the amplitude of the particular sample,
not upon any other property of the message, and the same time is allotted
to each group, regardless of the probability of that group or of the
amplitude it represents. Almost all the processes and devices used in
present day communication belong to this first class. In the second class,
the probabilities of the message are taken into account so that short
representations are used for likely messages or likely subsequences, longer
representations for less likely ones. Morse code, for example, uses short
code groups for the common letters, longer code groups for the rare ones.

Processes of the first class we may call non-statistical coding processes,
or simply modulation or remapping processes. The time of transmission
is the same for all messages of the same length, and all messages are
handled by the system with equal facility (or difficulty). These processes
require no memory and have a small and constant delay. They are
inefficient in their use of channel capacity.

Processes of the second class we may call statistical encoding proc-
esses. These processes in general require memory. The time of trans-
mission .of messages of the same length may be different so that if
messages are to be accepted and delivered by the system at constant
rates, variable delays may be necessary at the sending and receiving
ends. They are more efficient in their use of channel capacity. It is with
this second type of process that this paper is concerned, although pro-
cesses of the first type may be used as component steps. Thus we con-
sider systems of the type shown in Fig. 1, with the accent on the word
“efficient”’.

SIGNAL= EFFICIENT DESCRIPTION OF MESSAGE
Fig. 1—Reversible statistical encoding.

static discharges. The best type of signal for one channel may be very
poor for another.

In the following sections it is assumed that the channel transmission
characteristic is flat in amplitude and delay over a definite band and
zero outside. It is also assumed that the channel has a definite peak signal
power limitation, and that the noise is white gaussian noise. Such a
channel is no mere academic ideal. It is in fact quite closely approached
in practice by many circuits. Moreover, the conclusions based on these
assumptions can usually be modified or extended to other actual cases,
such as that of noise with non-uniform spectral distribution (as for
example the coaxial cable).

If the bandwidth of the channel is W, we can (using single sideband
modulation, if necessary) transmit over it without distortion from fre-
quency limitation signals containing frequencies from 0 to W (or —W
to W in the Fourier sense). Such a wave can assume no more than 2W
independent amplitudes per second. Any set of samples of the wave
taken at regular intervals sa serves to specify the wave completely.
The wave may be thought of as a series of (sin x)/2 pulses centered on
the samples and of proportional height, and indeed the wave may be
reconstructed from the samples in this fashion. This is the well-known
sampling theorem’. Thus a message source of bandwidth W can supply
at most 2W independent symbols (samples) per second, and this same
number can be transmitted as overlapping, but independently dis-
tinguishable pulses by a circuit of bandwidth W.

Since, as will appear later, channels which are to transmit signals
resulting from efficient statistical encoding must be relatively invulner-
able to noise, we shall assume that the pulses on the channel are quan-
tized. This allows regenerative repeatering to be used to eliminate the
accumulation of noise’. If there are b quantizing levels, and if the levels
are sufficiently separated so that the probability of noise causing in-
correct readings is negligibly small, then the capacity of the channel in
bits/sec is”

Such a circuit talks in an alphabet of b “letters” and uses a language
in which all combinations of these letters are allowed. There are no for-
bidden or impossible ‘‘words”. The circuit has a vocabulary of b one-
letter words, b” two-letter words, b” n-letter words. The basic ineffi-
ciency in present day electrical communication is that we build circuits
with unrestricted vocabularies and then send signals over them which

in which C is the capacity in bits/sec, W is the bandwidth and P/N the
ratio of average signal power to average noise power. This capacity can
only be approached, never exceeded, and is only reached when the sig-
nal itself has the statistics of a white noise. The expression sets a limit
for practical endeavor, and also gives the theoretical rate of exchange
between W and P/N.

A practical quantized channel, operated so that the loss of informa-
tion due to incorrectly received levels is negligible requires about 20
db more peak signal power than the average signal power of the ideal
channel to attain the same capacity’. However, bandwidth and signal-
to-noise ratio are still exchanged on the same basis. For example, a satis-
factory television picture could be sent over a channel with, say, 100
levels. This would require a (peak) signal to rms noise ratio of some 40 +
20 = 60 db. The bandwidth could be halved by a sort of reverse PCM:
by using one pulse: to represent two picture elements. But there are 10,000
combinations of two samples each of which can have any of 100 values.
Hence the new combination pulse would need 10,000 distinguishable
levels and this would require a signal to noise ratio of 80 + 20 = (2 xX
40) + 20 = 100 db.

It is evident that while bandwidth compression by non-statistical or
straight signal remapping means is not an impossibility, it is neverthe-

less impractical when the signal to noise ratios are already high. What
we should really try to do is make our descriptions of our messages more
efficient so that less channel capacity is required in the first place. The
saving can then be taken either in bandwidth or in signal-to-noise ratio,
whichever fits the requirements of our channels best.

Messages can either be continuous waves like speech, music, or tele-
vision; or they can consist of a succession of discrete characters each with
a finite set of possible values, such as English text. Because a finite band-
width and a small added noise are both permissible, continuous signals
can be converted to discrete signals by the processes of sampling and
quantizing’. This permits us to talk about them as equivalent from the
communication engineering viewpoint. Since many of the principles
which follow are easier to think of with discrete messages and since
quantization of the channel is assumed for reasons already stated, we
shall think of our messages as always being available in discrete form.

Then if all the message samples were independent and if all quantizing
levels were equally likely, the information per sample would be

and the message would use the full capacity of a channel with ¢ quan-
tizing levels, and bandwidth S/2. Or by remapping k message samples

(with the ¢ possible levels) into (85) k samples, a channel with b levels
and bandwidth W = S/2 (75) could be loaded to full capacity.

However, it is not true that the successive samples of typical messages
are independent, nor is it true that the various sample amplitudes are
in general equiprobable. If these things were true, speech and music
would sound like white noise, pictures would look like the snowstorm

where VN = number of message sequences of length n. If the successive
symbols of the message are independent but not equiprobable, then a long
sequence will contain x; symbols of type 1, x2 of type 2, etc. The number
of possible combinations of these symbols will be

n!
N ~ II x;! ,
7
so that log N = log n! — >> log 2;!
7
For large enough n, all the x; will be large also and we may write, by

But since >) x; = n, and since for large n, x, > p(j)n where p(j) is the
probability of the j symbol, we have

which is the expression Shannon derives more rigorously’. H; is a maxi-
mum when all the p(j) are equal to 1/¢. Then Hi = log, ¢ = Hy. The
more unequal the p(j), i.e., the more peaked the probability distribution,
the smaller H, becomes.

If the successive samples are not independent, the message source will
pass through a sequence of states which are determined by the past of
the message*. In each state there will be a set of conditional probabilities
describing the choice of the next symbol. If the state is 7 and the condi-
tional probability (in this state) of the next symbol being the j" is p,(j),
then the information produced by this selection is

The average rate of the source is then found by averaging (6) over all
states with the proper weighting; thus

The greater the correlation between successive symbols or samples of a
message, the more peaked the distributions p,(7) become on the average,
and this results in a lower value for H. As Shannon points out, the in-
formation rate of a source, as given by (7), is simply the average un-
certainty as to the next symbol when all the past is known. But in a
properly operating communication channel the past of the message is
available at both ends, so that it should be possible to signal over the
channel at the rate H bits/message symbol, rather than Hy as we now
do. In present day communication systems we ignore the past and
pretend each sample is a complete surprise.

By completely efficient statistical coding it should be possible to re-
duce the required channel capacity by the factor H/H>. Whether or not
this improvement can be actually reached in practice depends upon the
amount of past required to uniquely specify the state of the message
source. If long range statistical influences exist, then long segments of
the past must be remembered. If there are m symbols in the past which
determine the present state and each symbol has ¢ possible values,
there will be ¢” states possible (although only 2”” of these are at all prob-
able for large m). If m is large the number of possible states becomes fan-

* In a philosophical sense the state of a message source may be dependent on
many other factors besides the past of the message. If the source is a human being,
for example, the state will depend on a large number of intangibles. If these could
really be taken into account the resulting H for the message might be quite low.
If the universe is strictly deterministic one might say that H is ‘‘really”’ always
zero. When we describe the drawing of balls from the urn in terms of probabilities,
we admit our ignorance as to the exact detail of the mixing operation which has
occurred in the urn. Likewise the information rate of a source is a measure of our
ignorance of the exact state of the source. From a communication engineering
standpoint, the knowledge of the state of the source is confined to that given by
the past of the message.

tastically large and complete statistical encoding becomes an economic
impossibility if not a technical one.

Let Bi be a particular combination (the i) of k symbols in the past
of the message. Each of these combinations at least partially determines
the state of the system. Hence we can write an approximation to (7):

IF, > H,ask— o. If only m symbols in the past influence the present
state, then k need only be as great as m, in order that F,, = H. In any
case the sequence Ff, , F,, --- F, is monotone decreasing. Naturally
one should always pick the k symbols in the past which exert the great-
est effect upon the present state, ie. which cause ps*(j) to be as highly

peaked as possible, on the average. In English these would be the im-
mediately previous letters; in television, the picture elements in the im-
mediate space-time vicinity of the present element.

Suppose we break the message up into blocks of length k. Each of these
blocks may be considered to be a character in a new (and huge) alpha-
bet. If we ignore any influences from previous blocks, i.e. if we consider
the blocks to be independent, then the information per block will be
simply

Since there are / symbols per block, the information per symbol, G; is
1 ,
Ge = — 7 2 p(Bi) log p(Bi). (10)

As k > «, G, — H, since the amount of statistical influence ignored
(between blocks) becomes negligible compared with that taken into
account.

If d is the number of binary digits required to specify a message n
symbols long, then as n > «, d/nH — 1. For large n there are thus
2”"" messages which are at all likely out of 2"”° = ¢” possible sequences
(in an ¢ letter alphabet). The probability that a purely random source
will produce a message (i.e., a sequence with all the proper statistics) is
therefore

for large n. Even if Hp — H is small, p — 0 rapidly for large n. This is
why white noise never produces anything resembling a picture on a
-television screen, for instance. For in television signals, Hj) — H > 1

As given by (11), p, also represents the fraction of the possible signals
on a channel of ¢ levels which are likely ever to be used by messages of
length n without statistical encoding. |

Since a sequence of binary digits can be remapped by a non-statistical
process into a channel with b quantizing levels, or indeed into a wide
variety of other signalling alphabets, it suffices to consider statistical
~ coding processes and codes which reduce the message to a sequence of
binary digits. An efficient code is then one for which the average number
of binary digits, H., per message symbol lies between Hp and H. As the
efficiency increases H/H, — 1, so this ratio may be taken as an efficiency
index. With highly efficient processes, the sequences of binary digits
produced will have little residual correlation, i.e., they will be nearly
random sequences. Since the encoding process must be reversible the
receiver must be able to recognize the beginnings and ends of code groups.
Since we have at our disposal only zeros and ones, the divisions between
code groups must either be marked by a special code group reserved for
this purpose, or else the code must have the property that no short code
group is duplicated as the beginning of a longer group.

A code which satisfies this latter requirement and which is capable of
unity efficiency is the so-called Shannon-Fano code, developed inde-
pendently by C. E. Shannon of Bell Telephone Laboratories and R. M.
Fano of the Massachusetts Institute of Technology. This code is con-
structed as follows: One writes down all the possible message sequences
of length & in order of decreasing probability. This list is then divided
into two groups of as nearly equal probability as possible. One then
writes zero as the first digit of the code for all messages in the top half,
one as the first digit for all messages in the bottom half. Each of these
groups is again divided into two subsets of nearly equal probability
and a zero is written as the second digit if the message is in the top
subsets, a one if it is in the bottom. The process is continued until
there is only one message in each subset. Fig. 2a shows the code which
results when this process is applied to a particularly simple probability
distribution p(B) = (1/2)*. Here each code group is a series of ones
followed by a zero. The receiver knows a code group is finished as soon
as a zero appears. Although the longer groups contain mostly ones, their
probability is less and on the average as many zeros are sent as ones.

Fig. 2—Shannon-Fano codes for three different distributions. The successive
bisections are indicated by the dashed lines and the number gives the step at
which that bisection took place.

If the successive message segments are independent, the code will gen-
erate a random sequence of zeros and ones. Fig. 2b shows the code which
results with another distribution. Here the termination of each code
group is more complicated but the non-duplicative property exists so
the receiver can still identify the groups. Fig. 2c shows the code which
results when all the p(Bj) are equal. It is the ordinary binary code.

The length of each code group is equal to log 1/p(B%), for the cases
shown in the figures. This is true in general so long as it is possible to
divide the list into subgroups which are of exactly equal probability.

When this is not possible, some code groups may be one digit longer
as Shannon shows. The average number of digits per message symbol
using this code is therefore given by

For large k, H, — G, — H and the efficiency approaches unity. With
small k, H, increases both because the smaller list of messages cannot be
so accurately divided repeatedly into equal probability subsets (so-
called “‘granularity”’ trouble), and also because more statistics are ig-
nored between the shorter blocks.

The ordinary binary code provides a stabauoal match between mes-
sage source and channel only if the various message blocks Bi have
equal probability p(Bj) = 1/2", and are mutually independent. With
k = 1, p(B) = p(y) and the “blocks” are merely the successive symbols.

The application of the Shannon-Fano code to a block of k symbols of
a message in an ¢ letter alphabet requires that ¢* different codes be used.
The receiver must be able to recognize each of these and to regenerate
the proper message block when a particular code is received. If f is on
the order of 10 to 100 as is typically the case, we very quickly run out of
room to house the receiver and money to build it with. On the other
hand, if k is small, say on the order of 1 to 3, considerable statistical
information between blocks is ignored. These considerations led to the
development of a class of encoders known as n-grammers. The name
stems from the fact that they operate on the n-gram statistics of the

message, to produce a reduced signal having more nearly independent
symbols, but (in return) a highly peaked simple probability distribution
which allows savings with Shannon-Fano coding on a symbol-by-sym-
bol (k = 1) basis. .

The simplest member of this class is the monogrammer. It is basically
merely a re-ordering device. The operation may be best understood by
the following example. Suppose someone supplied us with English text
encoded into a quantized pulse signal as follows:

The device shown in Fig. 4 will accomplish this translation. The orig-
inal signal is applied to the vertical deflecting plates of a cathode ray
tube. The rest position of the spot corresponds to “‘space’’, i.e. no pulse.
A pulse one unit high deflects the spot to A, a pulse two units high de-
flects the spot to B, etc.

Now in front of these spot positions we place a number of light at-
tenuating filters. In front of the ‘“‘space”’ position we place an opaque
mask. Hence when the spot is deflected to ‘‘space”’ the photocell receives
no light and no pulse is sent. In front of the ‘‘E”’ position we place a
mask having one unit of transmission. So although F is received as a
pulse 5 units high, it is sent as a pulse of unit height. In front of the

“T” position we place a mask with two units transmission, and so on.
The signal amplitudes as received are thus re-ordered in the desired
fashion.

The resulting signal has lower average power and this can sometimes
be an advantage, particularly if several such signals are to be sent over
a common channel by frequency division. In this case the extreme rarity
of occurrence of high peak powers on all channels simultaneously means

that the system can be designed to have a lower peak power capacity.
The signal out of the monogrammer can be remapped into binary digits
using a Shannon-Fano code, pulse by pulse. However, this could have
been done equally well with the original signal merely by rearranging the
code groups in the coder tube. It is when we extend the principle to di-
grams and trigrams that the potentialities of the system become evident.

We can easily take account of the influence of the preceding message
symbol. To do this we apply the signal to the vertical plates as before,
and to the horizontal plates we apply the signal delayed by an amount
equal to the time between successive pulses as shown in Fig. 5. Thus the
beam is deflected vertically by the present message symbol, and horizon-
tally by the previous message symbol. Whereas before we used a single
column of optical filters chosen in accordance with the simple probabil-
ities of the letters, we now have 27 columns, one for each letter and one
for the space. The filters in each column are chosen in accordance with
the conditional probabilities which apply when the corresponding letter
was the previous symbol. For example, in the ‘‘Q” column (last letter
Q), and the ‘‘U”’ row (present letter U) the mask would be opaque, since
U is most common after Q. In general, the transmission of cell 77, in the
i column and j" row, is proportional to the rank of the entry for p;(7)
when the entire distribution (conditioned on 7) is ordered in a monotone
decreasing sequence. The amplitude distribution of the output pulses

from the digrammer will be more peaked toward zero amplitude than
that of the monogrammer. This is illustrated by the signals in the figures.
At the receiver the same type of device, but with an inverse mask can be
used to convert the signal back to its original form.

The digrammer can, with a little assistance, supply all the data re-
quired to prepare the encoding mask. If typical signals from the message
source are applied to the cathode ray tube (without mask) for a long
time, and a time exposure is made of the face of the tube, a lattice of
spots will be obtained on the film. These spots will be dense where the
high probability combinations occur and less dense elsewhere. The order
of decreasing density in each column is noted, and the filter transmis-
sions are arranged in the same order.

It is, of course, not necessary to use a phosphor, optical filters, and a
photocell. An array of targets each of which connects to the appropriate
tap on a load resistor might be simpler and more efficient. The cathode
ray tube itself can be replaced with an appropriate diode switching net-
work. Relay networks could be used for low-speed operation.

At the digrammer level we run out of new dimensions to use in the
cathode ray tube. The principle can, however, be extended to trigram-
ming and genera] n-gramming. For example, tetragramming could be
accomplished by using a bank of ¢ digrammers all in parallel, and all
deflected by the present and previous samples. Only one of these tubes
would be turned on at a time however. Which one this was would depend
on the other two previous symbols of the tetragram. These (by addi-
tional delays) would be applied to the deflecting plates of a master switch-
ing tube having an array of target plates in place of a mask. Depending
on the particular combination of signal samples applied to this tube,
the beam would strike a particular target. The target current would then
be used to turn on the beam of a particular digrammer tube, namely the
one with the proper mask for that particular combination of two past
symbols.

The complete array of equipment is admittedly rather staggering, but
then, rather efficient coding should result. In practice it would probably
be found that the masks of many of the tubes would be so similar that
little gain resulted from differentiating between them. That is, the state
of the message source might be nearly equivalent for several past com-
binations. In these cases, the group of tubes could be replaced with one
having the best average mask, and the corresponding targets on the
switching tube then tied together. This compromise would be particu-
larly warranted for those tubes which were rarely used anyway. By these

tricks it should be possible to keep the growth of equipment down to
something approaching 2”” rather than ¢”.

The output signal from the n-grammer will be, as we have seen, a series
of pulses with an amplitude distribution very peaked toward zero and
small pulses. If ¢ ,the alphabet, is large, these pulses can be efficiently
encoded into a Shannon-Fano code. For small alphabets, granularity
trouble can be reduced by remapping the output pulses two-by-two
into pulses of base ¢’, and then encoding these into the Shannon-Fano
code.

dictive-subtractive coding. In an actual system the reduced signal
would ordinarily be encoded into Shannon-Fano code groups before
transmission over the channel.

If so is the present sample amplitude, and s; , s: , 83 «++ S, are previous
sample amplitudes we compute a predicted value, s,, for the present
sample which is given by

where 6 < 3 quantizing level. If the conditional probability distribution
for the present sample is p,,...s,(80), then the difference, or output, or

“error” signal, ¢, will have the conditional distribution p,,...s,(€ + sp)
for this particular case. The simple distribution is then the weighted
average over all cases, 1.e.

Predictive-subtractive coding has especial merit when a simple func-
tion can be used for computing s, . This is often the case. When the
function is simply a weighted sum of the past sample amplitudes, i.e.
when

we have what is known as linear prediction. Of course, linear prediction
can always be used, but it may not be good enough with some types of
messages.

As Wiener has shown the coefficients a, b, c --- which minimize
é are readily computed. For simplicity, assume only two message sam-
ples, s; and Ss , from the past are to be used. We then have

where Ao , Ai , and Az are the values of the auto-covariance of the mes-
sage wave at displacements of 0, 1, and 2 sampling periods. Thus

The autocorrelation (normalized auto-covariance) is given by ¢; = a :
0
A» is proportional to the average power in the message wave, so the
=
ratio p = *— is the ratio of the power in the error signal to the power
0

If ¢. = ¢;, then the expressions simplify to
a=, b= 0, p=1- 1.
As can easily be shown, if ¢(x) = e “|, then all the coefficients except

a are zero, and a has the value e *. In other words, if the autocorrela-
tion function is of exponential shape, the previous sample alone is needed

* From the preceding expression for p, we see that p = 0 (i.e., perfect predic-
tion is possible) if:

If ¢. = 1, the message samples alternate between two independent but constant
values. For this case a = 0, b = 1. If $2 = 26, — 1 the autocorrelation is a cosine
wave so the message consists of samples of a sinusoid. In this case a = 2¢,, b =
—1. If ¢ is nearly unity, the sinusoid is of low frequency, and the prediction
approaches ‘‘slope’”’ prediction (i.e. extrapolation of a straight line through the
last two samples).

In any case where perfect prediction is possible the wave is periodic and there-
fore H = 0.

Power reduction is an index of merit when many reduced signals are
to be sent by frequency division over one channel, as we have said.
When the object is to reduce the channel capacity required for a single
message source, then it is the upper bound entropy of the reduced signal
which should be minimized, not the power. That is we want —>.; p(J)
log p(j) to be minimized. For certain types of signals this requires the
modes to be shifted to zero, although this is by no means a general rule.
Shifting the modes to zero may actually increase the entropy of the
“reduced” signal over that of the original message, by adding too many
new symbol levels, as the example in the last section shows.

If the original message has ¢ quantizing levels, the reduced message
after predictive-subtractive coding will in general contain more than
levels since an error of more than : can be made in either direction.
An n-gramming operation, on the other hand, never increases the al-
phabet.

QQ
| | | ] ee
Ne >
\y 2) OF
Ay yd oe
9
as ee pee Vee
FIRST ORDER ROWS THEN NaS
ADD COLUMNS FOR J
DIGRAMMER DISTRIBUTION YP
Vv

Fig. 8—Joint probability distribution (divide all coefficients by the sum over
each array).

Thus the most likely level is that of the previous sample. A camule
differing by one level is 1/a times as likely, one differing by m levels
is a” times as likely. Figure 8 shows a plot of the relative values of
p(t, 7) (neglecting the factor K). For ¢ = 4, the total array would be
the 4 x 4 portion enclosed by the dashed line. This sort of distribution
is:rather similar to those of typical television signals, as shown by pre-
liminary measurements, although typical values of a have yet to be
determined. With no statistical coding, the required channel capacity is

H, may be computed from the array of relative coefficients by adding
the rows to form the sums

Since, with the assumed distribution, the S; are all nearly equal very
little reduction in channel capacity is achieved by this step.

With linear prediction, the modes of the distributions (¢ = 7) could
be centered at zero merely by sending the difference between the present
and previous sample (previous value prediction). This would give a
reduced signal whose distribution may be found by adding the array
along the diagonals. The required channel capacity is then given by:

The distribution of the signal from a digrammer is found by rearrang-
ing each row of the table in order of decreasing probability and then
adding the resulting columns. Call these sums Sz . The digrammer out-
put will thus require a channel capacity:

H = — 7 p(t) D2 pj) log ps(j)
u ]
1 tek
= ——H 2h
loge i+ > me k lag a
Values for the above quantities were computed for a = zg and n =

2, 3, 4, 6, 8, 16, 32, ©. For the case of a = 2, we find that
K = [8-401 — 2°9]°
and that as f—> o,
H,, Hp , H > $ + loge 3 = 2.918 bits.
The results are shown in the Table I and also are plotted in Fig. 9.
While Hy and H; increase without limit as ¢ is increased, H, , Ho,

and H quickly approach a definite limit. This limit exists because we
assumed that the decrease in joint probability as a function of number

(These figures were computed by slide rule so the fourth figure is not very
significant.)

of levels off the diagonal was the same regardless of ¢. In typical signals
this is not true. The decrease is more apt to depend on amplitude differ-
ence and the finer the quantum step, the more levels a given difference
represents. As a result, the probability will fall off less per level off the
diagonal, and doubling ¢ will in general add one bit to H.

On the other hand, doubling the sampling rate will not in general
double the required channel capacity, for the closer spaced samples will

We have seen in the last two sections how it is possible to convert
a message for which H « Hy as a result primarily of intersymbol correla-
tion, into a reduced signal for which H « Hy as a result primarily of a
highly peaked probability distribution in the individual symbols (i.e.
one for which H, — H). Since the operations are reversible, the true
information rate, H, is preserved. In the original signal it was the
conditional distributions which were peaked, while the simple distribu-
tion was relatively flat. In the reduced signal the simple distribution is
peaked.

The result is that whereas a Shannon-Fano code would only have been
effective on the original message if applied to blocks two or more symbols
in length (and then it would ignore correlation between blocks), in the
reduced signal the code will be effective on a symbol to symbol basis.

l=s8
a side a lies Perera (a) Original, (b) After linear modal pre-
diction, (c) After digramming.

The encoding of the reduced signal into binary digits presents no
theoretical difficulties. A PCM type coder tube* with the appropriate
Shannon-Fano groups built into it is all that is needed. The biggest
practical complication arises out of the fact that the code groups are of
different length. Some messages, such as written text, can be fed into
the system as fast as it can handle them. The transmission time will
then vary with the message complexity. Others, such as television are
generated and must be accepted and delivered at a constant rate. One
solution is then to take the binary digits in big and little batches ‘as they
come from the coder and store the surplus in a sort of pulse ‘‘surge tank”
before they are sent over the channel at a regular rate. At the receiver,
a similar sort of storage register is necessary as the pulses arrive over
the channel at a regular rate and are used by the decoder at a varying
rate. Devices which will perform this variable delay function satisfacto-
rily for signals with relatively slow sampling frequency are available,
and as the art progresses there is every reason to believe that high speed
sampled signals like television can be handled also. :

It will be noticed that the digramming or prediction operation, while
it involves memory, does not introduce appreciable transmission delay.
Each symbol of the reduced signal appears the moment the correspond-
ing message sample is applied. The total transmission delay required
for statistical coding thus depends upon how much variation is required
in the variable delay units. This in turn depends upon the degree of
stationarity in the “local information rate” of the message. For example,
in television, if each line could be described (by the n-grammer and
subsequent coder) in the same total number of binary digits, then the
total delay variation and total delay would be less than one line time.
Since this is not true, we either must have enough channel capacity to
send in one line time the number of digits corresponding to the “worst”
line, or enough variable delay to average the existing rate over many
lines.

Probably the most practical solution is to provide sufficient channel
capacity and variable delay to take care of all but a small fraction of
the possible message sequences. Then when an unusual stretch of message
continues long enough for the variable delay to be nearly all used up, the
system should fail in some relatively harmless way. In television, the
sampling rate could be momentarily reduced, for example. This would
degrade the resolution in rare situations, but a small amount of this
could be tolerated in return for transmission savings.

Tf long blocks of the message are efficiently encoded as a group, then
an error in transmission may cause the whole block to be reproduced

1. Oliver, Pierce, and Shannon, ‘‘The Philosophy of PCM,” Proc. Inst. Radio
Engrs., Nov. 1948.

2. C. E. Shannon, ‘‘A Mathematical Theory of Communication,” Bell System
Tech. J., July and Oct. 1948; Shannon and Weaver, ‘‘The Mathematical
Theory of Communication,’’ University of Illinois Press, 1949.

3. C. E. Shannon, ‘‘Prediction and the Entropy of Printed English,’’ Bell System
Tech. J., Jan. 1951.

4. R. W. Sears, ‘‘Electron Beam Deflection Tube for Pulse Code Modulation,”’
Bell System Tech. J., Jan. 1948; W. M. Goodall, ‘“Television by Pulse Code
Modulation,’’ Bell System Tech. J., Jan. 1951.

One of the teachings of information theory is that most communica-
tion signals convey information at a rate well below the capacity of the
channels provided for them. The excess capacity 1s required to accom-
modate the redundancy, or repeated information, which the signals
contain in addition to the actual information. Removal of some of this
redundancy would reduce the channel capacity required for transmission,
thus opening the way for possible bandwidth reduction. In order to
remove redundancy, one must first understand it; the amount and nature
of the redundancy can be completely defined in terms of various statis-
tical parameters characterizing the signal.

It has been pointed out that the existence of redundancy is particularly
evident in the case of television; moreover, its elimination is highly de-
sirable because of the large bandwith presently required for transmis-
sion. Evidence of redundancy is found in the subject matter of televi-
sion—the average scene or picture. Knowing part of a picture, one can
generally draw certain inferences about the remainder; or, knowing a
sequence of frames, one can, on the average, make a good guess or pre-
diction about the next frame. In either case, knowledge of the past re-
moves uncertainty as to the future, leaving less actual information to be
transmitted.

Another way of looking at this is to visualize the picture as an array
of approximately 210,000 dots, 500 vertically, 420 horizontally, cor-
responding, respectively, to the 500 scanning lines and 420 resolvable

picture elements per line of the standard television raster. Each dot can
have, say, 100 distinguishable brightness values in a good-quality pic-
ture. The number of possible combinations is therefore approximately
10077"? or 10° At the usual rate of 30 frames per second it would
take approximately 10°": years to transmit all these “pictures,” which
our present television system is fully prepared to transmit! The vast
majority of these “pictures” will, of course, never be transmitted in this
age because the average picture statistics virtually preclude the pos-
siblity of their occurrence.

If all of the redundancy alluded to in the preceding paragraph were
to be expressed in terms of statistics, the array of data would be stagger-
ing.* Redundancy encompassing even a small part of a single frame
implies statistics of enormously high order because of the large number
of possible past histories. The initial attention should therefore be
focused on local redundancy, encompassing only a few adjoining pic-
ture elements. Accordingly, measurements have been made of the fol-
lowing statistical quantities.

1. Simple probability distribution of signal amplitudes corresponding
to picture brightness. This encompasses only a single picture element,
revealing the relative probabilities of this or any element’s assuming
the various possible brightness values, in the absence of any past-his-
tory information.

2. Simple probability distribution of error amplitudes resulting from
linear prediction of television signals. Only the simplest type of linear
prediction is considered here, so-called previous-value prediction, which
predicts each picture element to have the same brightness value as the
preceding one. The prediction error signal is simply the difference be-
tween the picture signal and a replica delayed by one Nyquist interval
(one-half the reciprocal bandwidth or the time interval corresponding
to the spacing between picture elements). The distribution of this error
signal encompasses tio picture elements (past history of one element)
and therefore is a condensed version of the family of first-order joint
probability distributions.

3. Autocorrelation of typical pictures. This statistical quantity is’ an
even more streamlined version of various families of different-order
joint probability distributions. Each family corresponds to just a single
point on the autocorrelation curve; the ordinates of the curve represent
the average correlation between picture elements spaced by various

* Complete statistics extending, say, over one frame period, would comprise
one conditional probability distribution per picture element for each possible
past history. With the approximate figures cited above, the number of distribution
curves (many of which would be similar) is 210,000 X "10 419,999 op 10420,004.3,

distances. This correlation, say, between horizontally adjoining elements
is simply the average product of the two brightness values of each pair
of neighbors, relative to the average square of all brightness values.

The three quantities enumerated above contain a great deal of sta-
tistics in very compact form, but these statistics are essentially of a local
and linear nature. They do not include the bulk of the large-scale re-
dundancy, which is of a far-flung and nonlinear nature.

AUTOCORRELATION
For a function of time, f(t), the autocorrelation can be expressed as

averaged over all time, for various values of the time shift 7. In the case
of a picture transparency, the optical transmission is a function of two-
dimensional space, expressible in polar coordinates as 7'(s/¢), and the
autocorrelation can be expressed in analogous fashion. The time variable
t is replaced by the space coordinate s/¢, and the correlation time shift 7
is replaced by a space shift As/@, so that the new expression is

Fig. 3—Close-up view of slide holding assembly and shifting mechanism of
picture autocorrelator.

The apparatus used to measure autocorrelation is shown in Figs. 1
and 2. The chamber at the bottom contains a light source of very
constant intensity and a convex lens to collimate the light. The middle
part, made of accurately machined aluminum, holds the two identical
slides of the picture under test, and an aperture exposing a large circular
area of the slides. The top chamber contains a collector lens and a photo-
multiplier tube which (on a microammeter not shown) gives a sensitive
indication of the total light transmitted through the slides. Fig. 3
shows a close-up view of the slide-holding assembly. Two close-fitting
graduated aluminum rings permit accurately determined rotation of
both slides or one slide, and the micrometer drive permits translational
displacements measurable to within one mil (moving the two slides by
equal and opposite amounts); the separation between picture elements
is approximately 7.5 mils horizontally and 5 mils vertically (for the
23” by 34” slide size used).

The light transmission is always a maximum when the two slides are
in precise register (As = 0). For large shifts the transmission fluctuates .
about a nonzero asymptote. The nonzero asymptote results from the
fact that the average transmission is always positive, and the fluctuation
from the fact that large displacements introduce substantial amounts of
new picture material into the aperture. Since these components tend to

obscure the correlation effects, it is useful to make additional measure-
ments which enable us to subtract them out completely. This leaves us
with a ‘pure’ autocorrelation A(As/@), which is then normalized so as
to have a peak value of unity. It is given by

where 7, (a /2) is the transmission through the two cascaded slides
shifted by equal and opposite amounts = at an angle @ with the hori-

SCENE C "SCENE D
Fig. 4—Test pictures whose statistics are included in this article.

STATISTICS OF TELEVISION SIGNALS 757
SHIFT, AS, NUMBER OF VERTICAL PICTURE ELEMENTS
~-100 -80 -60 -40 ~20 ie) 20 40 60 80 100

Fig. 6—Contours of constant autocorrelation for Scene A. In general there are
no preferred directions of correlation.

Fig. 7—Plots of autocorrelation for small shifts. Ai is the autocorrelation for
a shift of one horizontal elemental distance, Azo for two horizontal elemental dis-
tances, and Ao, for one vertical elemental distance. Alternatively Aio may be

A probability distribution of amplitudes is generally shown as a plot
of probability density versus signal amplitude. Probability density, say,
corresponding to amplitude x1, is the probability of finding the signal
amplitude between 21 and x1 + dz, divided by the differential amplitude
increment dx. Conversely, the probability of finding the signal ampli-
tude between 7 and x; + dz is given by p(x,)dx, p(x) being the proba-
bility density corresponding to amplitude 2.

If a cathode-ray spot is deflected, say horizontally, by the signal in
question, its average dwell time at any point is directly proportional to
the corresponding probability density. In the optical system shown in
Fig. 8, a cylindrical lens maps each point into a vertical line which is

UNDEFLECTED DEFLECTED
IMAGE *, /|MAGE
5XP it f \ !
CATHODE LOW Sy
RAY TUBE TRANSMISSION, |
CONVEX \ AE
LENS\, aS |
ao
_UNDEFLECTED LA a
SPOT f “1
_DEFLECTED ee |
SPOT aa
CYLINDRICAL’ re
LENS OPTICAL
DENSITY
1SIGNAL WEDGE SIGNAL

then tapered in intensity by an optical density wedge before reaching a
high-contrast photographic film. Depending on the dwell time at any
amplitude level, the corresponding tapered line has enough average
intensity to blacken the film up to a certain level. This level is pro-
portional to log p(x), since the density wedge is tapered exponentially
so that the intensity of each tapered line of light reaching the film
diminishes, say, by a factor of ten for each inch we travel up the line.
The film in effect traces out a contour of constant exposure.

Two or three iterated photographic printings increase the effective
gamma, sufficiently to yield a contour of ample sharpness. This contour
is then changed to a sharp line by a simple dark room trick: while the
film is in the development tray, already fully developed, it is momentarily
exposed to light. The blackened portion of the film is unaffected, the
clear portion is fully blackened, while the transition contour, being
partly opaque, is not fully blackened. By printing from this film we then

obtain a well-defined black-on-white curve of p(x) versus x on a loga-
rithmic probability scale. The logarithmic scale has the advantage of
making the curve shape independent of exposure length and giving uni-
form relative accuracy over the entire range.

Fig. 9 shows some typical results obtained by means of the “proba-
biloscope.” The two small curves are distributions of two different still
pictures. The left-hand end corresponds to black, the right-hand end to
peak white; the blanking intervals (slightly blacker than black)’ cause
the peaks at the extreme left. (The signals did not contain any synchro-
nizing pulses.) The tall and slender curve at the right of Fig. 9 is the
distribution of errors resulting from previous-value prediction of one
of the pictures in Fig. 4. The peak corresponds to zero error which is
seen to be most probable, as it should be if the prediction criterion is
good. Increasingly larger errors are increasingly improbable or rare.
The six decades of probability density spanned by the curve were ob-
tained in three separate exposures and subsequently joined, since stray

Fig. 9—Typical probability distributions as obtained from the probabiloscope.

Curves at left are for video signals; right-hand curve is for difference between
video signal and delayed replica.

-where p; is the simple probability of the signal’s falling into the zth
level. Since the 64 p,’s are unequal, Hmax is necessarily less than 6 bits.
For all available data the average value of Hmax turns out to be ap-
proximately 5 bits, indicating a one-bit redundancy. The latter figure
is essentially independent of quantization.

The prediction error signal still contains all the useful picture in-
formation. The maximum possible information content per sample (max-
imum in that all samples are assumed to be completely independent)
is still given by (4) but in this case the 64 values of p; are obtained
from the peaked error distribution. The average* result from all available
data turns out to be approximately 3.4 bits below the 6-bit ceiling, show-

* This average was computed by averaging the various redundancy values ob-
tained for the individual pictures, rather than averaging all statistical data and
then finding one corresponding average redundancy. The average computed here

is more favorable and can be realized only if, optimum coding is performed on a
short-term basis rather than on the basis of one set of long-term statistics.

ing that the original signal must have contained at least 3.4 bits of
redundancy.

The autocorrelation can also furnish a lower bound to the redundancy,
as has been pointed out by P. Elias in his Letter to the Editor of the
Proceedings of the I.R.E. for July, 1951. If, for example, the correlation
Ai, between horizontally adjoining picture elements, is high, the cor-
responding lower-bound redundancy is very roughly equal to

Alternatively, taking the Fourier transform of the autocorrelation yields
the power spectrum P(f), from which we can find the lower-bound re-
dundancy through the relation

Using either method, one obtains approximately 2.4 bits for the
average* of the available data. This is an approximate bound, in that .
it applies strictly only to functions having gaussian amplitude distri-
butions.

Suppose, then, that we have exposed an average redundancy of at
least 3 bits per sample. This means a potential 3-bit reduction in the
channel capacity required for television transmission. In a 6-bit system
(64 amplitude levels) this means a 50 per cent reduction, and hence a
potential halving of the bandwidth with the aid of an ideal coding scheme.
It is true that the decorrelated signal is somewhat “frail,” i.e., vulner-
able to interference, so that it might be desirable to use a “rugged”
system of the PCM variety for transmission. Thus, if a Shannon-Fano
code were used, the 3-bit decorrelation should enable us to send tele-
vision by an average of 3 on-off pulses per picture sample rather than
6. This represents a two-to-one saving over the usual PCM bandwidth.
More spectacular reductions are likely to be achievable only by tap-
ping the large-scale redundancies mentioned earlier.

where 7’:2 is the optical transmission of frames 1 and 2 in cascade, 7’; is the average
of the individual transmission of frames 1 and 2, and 71: is the average of the
transmissions of two cascaded slides of frame 1 and two cascaded slides of frame
2, respectively. In all cascade transmission measurements, the two frames must
be in precise register.

The correlation present in a signal makes possible the prediction of the
future of the signal in terms of the past and present. If the method used for
prediction makes full use of the entire pertinent past, then the error signal—
the difference between the actual and the predicted signal—will be a com-
pletely random wave of lower power than the original signal but containing
all the information of the original.

One method of prediction, which does not make full use of the past, but
which 1s nevertheless remarkably effective with certain signals and also
appealing because of tts relative simplicity, is linear prediction. Here the
prediction for the next signal sample is simply the sum of previous signal
samples each multiplied by an appropriate weighting factor. The best values
for the weighting coefficients depend upon the statistics of the signal, but
once they have been determined the prediction may be done with relatively
simple apparatus.

Linear prediction is perhaps the most expedient elementary means of
removing first order correlation in a television message. Before discussing
the advantages and disadvantages of linear prediction, it might be well
to consider what is generally meant by correlation in a television pic-
ture and why it should be removed.

Almost every. picture that has recognizable features contains both
linear and non-linear correlation. Each type of correlation helps in
identifying one picture from another; however, linear prediction is only
effective in removing linear correlation, and for this reason, future ref-
erences to correlation will refer only to its linear properties. With tele-
vision, a signal is obtained as the result of scanning; hence, the cor-

relation is evident in both space and time. Briefly, correlation is that
relation which the ‘next’? elemental part of the signal has with its past.

To leave correlation in a message is to be redundant, and this effec-
tively loads the transmission medium with a lot of excess ‘words’ not
necessary to the description of the picture at the receiving end. It is’
then more “efficient” to send only the information necessary to identify
the picture, and to restore the redundancy at the receiver.

The more efficient we are in sending pictures over a, given transmission
line, the more alarmed we become at the increasing amount of equipment
that is required at the transmitting and receiving terminals. Certainly
the design will be a compromise between the complexity of apparatus and
the efficiency achieved. The ingenuity of engineers will be taxed along
these lines for years to come; however basically, the general form of
these systems will be similar to that shown in Fig. 1. Although not
always separable, four essential operations are required-namely, decor-
relating, encoding, decoding and correlating. The transmitting decor-
relator and the encoder encompass the principal design problems, since
the decoder and correlator at the receiving end perform the reverse
operations which interpret the code and add in the redundancy that was
removed.

Decorrelation involves prediction, and as the predictors are more
nearly made to predict the future of the signal, the more the output
signal from the decorrelator resembles random noise. The essential
picture information is still present, which means that our original picture
signal can be obtained at the receiving end without theoretical degrada-
tion. The basic job of the encoder is to match the picture information
out of the decorrelator to the channel over which it is to be transmitted.
There are several encoding operations. The first concerns the rate of
information into the encoder, and that required out of it. In the case
of television, there are flat, highly correlated areas as well as areas
containing more concentrated detail. This means that the information

TRANSMISSION
MEDIUM
Fig. 1—Block diagram of an efficient transmission system employing reversible
decorrelating and encoding means.

rate varies when the picture is scanned at the conventional uniform scan-
ning rate. The output of the encoder feeds a transmission line that has
a definite channel capacity, and if maximum efficiency is to be obtained
from this transmission medium, then the rate of information into it
must be held relatively uniform at a value near the channel capacity.
It is the job of the encoder to take the varying rate of information from
the decorrelator and feed it to the channel at a constant rate. At the
receiving end, the decoder must take the constant rate of information
and deliver it to the correlator at the variable rate as originally fed into
the transmitter’s encoder. Thus, to perform this task, a variable or
elastic delay to run ahead or behind, depending on the information con-
tent of the picture being scanned, is an important part of the encoder.
Over a long period of time, the variable delay would average out to
some fixed value. This variable delay must never run out, even when the
detail is concentrated. There are instances when this condition could
not be met, such as an extended reproduction of a snow storm; however,
with good design the system should fail ‘‘safe’’—a slight degradation of
picture quality. This condition can be made infrequent enough to cause
little concern.

The encoder design must also account for noise as well as bandwidth
of the channel and must consider the ultimate effect of an error that
may be introduced by noise along the transmission line. As more re-
dundancy is removed to get at the “essence” of the picture signal, the
more important it is to guard this ‘“‘essence,”’ as mistakes presented to
the receiver will propagate themselves longer in the absence of correla-
tion. Errors can be minimized by rugged systems of modulation such
as PCM, where the signal-to-noise ratio of the transmission line deter-
mines the base of the PCM system selected. In any event, the encoder
must send the information so that the effect of errors will not appreciably
disturb the picture.

Fig. 2 illustrates, in a general way, a means of decorrelating the sig-
nal, S,(é). For purposes of explanation, the encoder and decoder have
been omitted, and the transmission between the receiving and sending
terminals, idealized. The predictors, P, are identical, and base their
prediction, S,(t), on the signal’s past history. In this way, the output
of the computer represents the discrepancy between the actual value of
the signal sample and the predictor’s prediction. By this means we are
sending only our mistakes—the amount by which the next picture ele-
ment surprises us. For example, if the computer is so designed that it

Fig. 2—Decorrelator and correlator showing reversible nature of this method
of removing redundancy.

bases its prediction on the ‘‘previous frame,’”’ and we are transmitting
a “still,” there will be no surprises after the first frame and consequently
no output signal. Certainly it is redundant to send the same picture
more than once.

Linear prediction provides an easily instrumented means of removing
redundancy. With linear prediction the next signal sample is simply
the sum of the previous signal samples, each multiplied by an appropriate
weighting factor. The best values for these weighting coefficients de-
pend on the statistics of the signal.

Fig. 3 is a block diagram of a decorrelator employing linear predic-
tion. The delayed versions of the input signal can be obtained from
taps along the delay line. The weighting coefficients for each of the
delayed signals are selected by loss in their respective paths as shown
by the amplitude controls. The polarity of each signal can be determined
by the switches. The output is simply the sum of these weighted signals.

If we consider the signal on a continuous basis (not quantized or
sampled), linear predictors can be characterized as ordinary linear
filters used to predistort the frequency spectrum of the signal. As such,
they can be designed in the frequency or time domain. However as will

Fig. 3—General block diagram of decorrelator employing linear prediction.
Linear prediction bases its prediction on the weighted sum of previous signal
samples.

be shown, it is much easier to recognize circuit configurations that
reduce ‘redundancy in the time domain. To this end, and for purposes
of encoding, the signal is thought of as signal samples uniformly spaced
at Nyquist intervals. Thus, amplitude values obtained by sampling
a 4.0 me picture signal at $ microsecond intervals serve to specify the
signal completely. Fig. 4 shows a small portion of a television raster where
the signal is represented by signal values spaced at Nyquist intervals,
+. The coordinates shown are designated with respect to the “present
value” of the signal, Spo . The positive coordinate directions are shown
by the arrows. The past is represented by positive coordinates—the future
by negative coordinates. In this way, the previous value of the signal

x
_ Fig. 4—A small portion of a television raster showing geometrical location of
signal samples with relation to the “present value” of the signal, So,o.

taken one Nuquist interval before Soo is designated by S15 — the previous
line samples by So,1, etc.

As previously stated, with linear prediction the next signal sample,
S,(t), is simply the sum of the previous signal samples each multiplied
by an appropriate weighting factor. Thus,

represents the weighted sum of all the previous signal values. The error
signal, e, as shown in Fig. 2, is represented by the difference between the
present vaue of the signal, So and the predictor’s prediction.

ther explanation —namehs “previous value, previous line,”’
“planar” and “circular.”

TIME
Fig. 5—Example of ‘‘previous value’’ prediction, where the error signal is the
difference between the actual value of the signal and the previous value.

It is of interest to mention that “slope” prediction is equivalent to
two “previous value”’ predictors in tandem. Three or more ‘previous
value” predictors in tandem are equivalent to a binomial weighting
of the previous values of the signal to form a predicted signal.

For example, the prediction for three ‘‘previous value” predictors in
tandem is given by

TIME
Fig. 6—Example of “‘slope’’ prediction. Here the next signal value is assumed
to lie on a straight line that intersects the two previous signal values.

“Planar” prediction, shown in Fig. 8, is effectively tandem operation
of “previous value” and ‘previous line” prediction. Planar prediction
may also be thought of as the value represented by a plane above the
present value of the signal when passed through three adjacent signal

Se = Soy
€ = Soo -Soy
Fig. 7—Example of ‘‘previous line’’ prediction. Here the error signal is the

difference between the actual value of the signal and the value of the signal on
the line directly above.

Fig. 8—An example of ‘ planar’’ prediction. Here the prediction is represented
by a plane that has been passed through three adjacent signal values.

values, namely Si, So,1 and S:i,1. The predicted signal is given by
. Sp = S10 + Sor — Sia.
The filter characteristic is given by

2 2
The peak error amplitude for “Planar” prediction can be four times
that of the input signal.

“Planar” prediction has several good characteristics. For example,
if S11, and Si, were white and So,1 black, then So would be predicted
to be black. Thus a change horizontally from white to black would
produce no errors. Similary, if Si,1and So,, were white and Sj black,
then So, would be predicted to be black. This indicates that a change
vertically from white to black would be predicted correctly. In this
manner, all vertical and horizontal contours in a picture are deleted. This
philosophy can be extended to include other directions as well.

“Circular” prediction, illustrated in Fig. 9, is an extension of planar,
since it deletes horizontal, vertical and 52° contours as well. A total of
190.5 microseconds of delay is required, making the required equipment
more elaborate. Also, as more delay is required, more noise is added.

Fig. 9—Past signal samples required for “circular prediction’’—a type of
prediction which removes horizontal, vertical, and +52° straight line picture
contours.

Therefore, indefinite extension of this straight line contour deleting
philosophy is not a paying means of prediction, at least not at the present
state of the art of wide band delay lines. Furthermore, the increasing
diameter of the circle for extension of circular prediction would decrease
its accuracy for finely concentrated detail.

Fig. 10 shows the relative position of picture elements nearest So,
if a wide band field delay were available. The methods of prediction dis-

Fig. 10—Small portion of television raster showing signal samples, including
those of previous field, which would enable time extrapolation-space interpolation
as a method of prediction.
cussed have been essentially an extrapolation in space; however, with a
field delay, interpolation in space, and extrapolation in time would also
be possible.

Experimentally, those types of predictors that involve only a few
Nyquist intervals of delay are easiest to mechanize. Fig. 11 shows a
simplified schematic of a decorrelator that enables an evaluation of
linear prediction schemes having error signals given by @¢ =. do,oSo,o +
@1,091,0 -— Q2,0Se0. This enables an evaluation of ‘‘previous value” and
“slope” prediction. The signal is fed into a terminated delay line having
taps at Nyquist intervals. Each of these signals is individually attenuated
by the potentiometers in the cathode circuit of the cathode followers.
Each output is then fed to its respective polarity switch. The D.P.D.T.
switch determines to which side of the differential amplifier, V,, the
particular signal is sent. Since more than one signal may require the
same polarity, the signals are combined through ‘‘L” type resistance
attenuators to prevent interaction between signals. The D.P.D.T.
switches are so arranged that the other signals are unaffected when a
polarity switch is reversed. The differential amplifier, V4, is a cathode
coupled circuit having the advantage of two identical grids which pro-
duce opposing effects in the output. The output is then matched to the
line by the cathode follower, V;. In this way we can transmit (1) the

Fig. 12—Block diagram of experimental apparatus used to investigate methods
of prediction involving combinations of previous signal values along a line with
those on the line directly above.
frequency of 54.0 mc. The over-all video bandwidth is flat (- 0.1 db) to
5.0 mc. Nonlinear distortion is approximately one percent when the
peak-to-peak signal to r.m.s. noise is 58 db. To give an idea of its com-
plexity, two such units with their associated power supplies require a
seven-foot relay rack for housing. The signal at J1 represents the input
picture signal, even though it may be delayed by a small fraction of a
Nyquist interval from the actual input signal. The signal at Jo is the
same signal as found at Ji but delayed by one line time. Each of these
signals is fed into terminated delay lines to enable additional signal
samples to be obtained. The geometrical location of these signal sam-
ples is illustrated in the upper right section of Fig. 12. Here, six signals
are obtained instead of three as were required for ‘“‘previous value” and
“slope” prediction. These signals are weighted and polarized in the same
manner as the three signals shown in Fig. 11. The output is the sum of
these weighted signals and is given by

€ = ,0S0,0 + 41,0S1,0 + 2,0S2,0 + A180 + 41,1811 + 2821.
The coefficients may assume positive or negative values.
MEASUREMENTS

It is obvious that if we are able to predict the value of most signal
samples closely (which we will be able to do if there is a large amount of
correlation in the picture), then the average amplitude of our mistakes

will be much less than the average amplitude of the original signal.
Thus, by using the decorrelator alone, we can send a message over a
channel with the same bandwidth as before but with less average power.
At first, this might sound like a worthwhile saving; however this lower
average power is accompanied by an even higher peak amplitude which
makes any direct saving less attractive. Furthermore, the low frequency
attentuation of the decorrelator makes the signal vulnerable to low
frequency disturbances, since the correlator must restore (emphasize)
these low frequency components.

A proper but noé entirely adequate method of evaluating the effective-
ness of a predictor is by measuring the ratio of signal power to error
power. This is called ‘““Power Reduction” and is generally expressed in
db. Power reduction simply provides a scale by which we can weigh a
linear predictor’s capabilities. The “not entirely adequate” refers to
the fact that minimum error power may not provide simultaneously the
lowest amount of redundancy for that given type of prediction.

As an example, Fig. 18 shows the power reduction for the relative
weighting of the previous horizontal signal sample as compared to the
present value of the signal, for three pictures-later to be described as
Scene A, Band C. The top-most curve is for Scene B, which is a
simple, soft picture that contains very little detail. For this picture,
the minimum error power coincides (within measurable limits) with the
minimum redundancy. For Scene A and particularly Scene C, mini-
mum error power is considerably different than that for minimum re-
dundancy. This difference between minimum error power and minimum
redundancy also applies to decorrelators using other types of predictors
as well. Minimum redundancy may also be a misleading criterion of a
predictor’s performance, since the. value of the prediction must depend
on the particular type of encoder used, and some types of encoding will
require certain types of redundant information to be retained.

The following pictures are representations of the error signal as photo-
graphed from a 10-inch laboratory monitor. The signals were band
_ limited to 4.3 me. Fig. 14 represents the ‘‘original”’ for three scenes called
A, B and C. These pictures represent, to a first approximation, the
gamut of pictures normally expected to be transmitted. They are by
no means the best or the worst pictures than can be imagined; how-
ever any system should be able to reproduce these pictures without
appreciable distortion. For example, Scene C should be capable of
being sent continuously without the elastic delay running out, etc. _

Fig. 15 shows how the error signal appears for “previous value’”’ pre-
diction. ‘“‘Previous value” prediction is excellent for flat white or dark

Fig. 13—Power reduction for various weighting coefficients for the previous
horizontal sample.

areas as can be observed in the background of Scene A. Where the
separation of a white to black area is made, the error signal is large. It
is this large error signal that informs the receiver of this change in bright-
ness, and until another change occurs, the error output is again zero. This
type of performance produced the flat grey appearance of the back-
ground. In this way, the picture represents only changes in brightness—
a first difference type of picture.

It may be noted that horizontal contour lines have vanished leaving
only vertical contours which pertain to the brightness changes that .
have occurred. This effect is especially evident in Scene C. The
power reductions given in the lower left hand corner of these pictures
are consistent with their complexity.

Fig. 16 shows the error signal appearance for ‘‘slope’’ prediction.
When compared to the error signal for “previous value” prediction
a finer vertical granularity is observed, and this is attributed to sudden

Fig. 14—Three pictures as photographed from the face of a kinescope. Scene
‘fA’’ is a picture of average complexity. Scene ‘‘B”’ is a simple, rather soft picture.
Scene ‘‘C’’ is a complex, highly detailed picture. Roughly, these pictures represent
the gamut of pictures normally expected to be transmitted.

Fig. 15—Three pictures showing the appearance of the error signal when using
“previous value”’ prediction. Note the absence of horizontal contours.

Fig. 16—Three pictures showing the appearance of the error signal when using
‘slope’’ prediction.

Fig. 17—Three pictures showing the appearance of the error signal when using
“previous line’’ prediction. Note the absence of vertical contours.

Fig. 18—Three pictures showing the appearance of the error signal when using
“planar” prediction. Note the absence of horizontal and vertical contours.

changes in brightness. In the case of ‘‘previous value” prediction a sudden
change in brightness produces only one error, where for “slope,” two
errors result. This, to some extent, accounts for the lesser amount of
power reduction measured for these scenes.

Fig. 17 shows the appearance of the error signal for ‘previous line”
prediction in the three scenes. Where vertical contour lines are pre-
dominately left after ‘previous value” prediction shown in Fig. 15,
horizontal contours, are more prevalent now. It can be noted that the
power reduction for ‘previous line’’ prediction is less than that for “pre-
vious value” prediction. This is due principally to the increased distance
of the previous line sample from So,o. If the closest horizontal sample
was taken at the same distance from the present value of the signal as
the previous line sample, then the power reduction using these signal
values individually for prediction would be essentially the same for most
pictures.

Fig. 18 shows the error signal appearance for “‘planar’’ prediction.
Here, vertical as well as horizontal contours are deleted. In Scene A
the tree trunk has almost completely vanished. In Scene B the picture
has an extremely flat appearance. Scene C exhibits the lack of hori-
zontal and vertical contours best, since only sloping contours are left.
The power reduction figures at the lower left hand corner also show values
for minimum error power. For most pictures, the error power can be
reduced by a factor of one-half again over the planar coefficients by
modifying the weighting coefficients. The coefficients for this modified
planar case are given by

These coefficients generally produce an error signal with less power than
the coefficients used for “planar” prediction.

While all pictures contain redundancy, the error signals from these
simple linear predictors shown in Figs. 15, 16, 17 and 18 can visually
be noted still to contain large amounts of redundancy. The contours of
the models and of the various objects are readily identifiable. Were all
redundancy removed, the picture would be completely chaotic and would
appear very much like random noise, although greater efficiency in
transmission would be achieved. For richer rewards, more sophisticated

The author wishes to acknowledge with grateful appreciation the in-
valuable guidance of Dr. B. M. Oliver. It was principally through his ef- ©
orts that this study was made possible.

where M and J are the densities of magnetic and electric polarization
currents.”

and yp, € are constants (not necessarily those of vacuum). If M and J
were given, they would act as sources exciting various modes of propa-
gation in a homogeneous, isotropic waveguide. If M and J are functions
of H and E, they can still be considered as the sources, acting on power
borrowed from the wave, of the various modes. Thus M and J will
provide the coupling between the modes existing in a homogeneous,
isotropic waveguide.

where x is the separation constant and e;, @ are the scale factors in
the expression for the elementary distance

In the case of TM waves the 7’-function must vanish on the boundary
of zero impedance. This boundary condition restricts x to a doubly in-
finite set of values xmn with the corresponding functions 7m, . In the
case of TE waves the normal derivative of the 7-function must vanish
on the boundary of zero impedance. Since we have to consider both
types of waves simultaneously, we shall distinguish between them by
enclosing the subscripts in parentheses for TM waves and in brackets
for TE waves. The double subscript designation of various modes has
been standardized only for rectangular and circular waveguides. For
waveguides of other shapes the standard is to use a single subscript by
arranging the modes in the order of their cutoff frequencies. For con-
venience, we shall use this convention in the general case and denote
TM modes by 7'ta)(u, v), and TE modes by 7'n(u, v). The correspond-
ing cutoff constants will be xq) and xn). In what follows it is under-
stood that whenever the 7-functions should be designated by double
subscripts, our single letter subscripts should be conics as symbols
for ordered double subscripts.

The transverse field components may be derived from the potential
and stream functions,’ V and Il for TM waves and U and W for TE
waves. Thus

The 7-functions corresponding to the various modes of the same va-
riety are orthogonal; that is, the following integrals over the cross-section
vanish,

Similarly the gradients of the 7-functions of the same variety as well as
the fluxes, are orthogonal,

/ i (grad Tey) +(grad Tem) dS = i if (flux Toy): (flux Tom) dS
: (9)
= i / (grad Tm) + (grad Tym) dS = i (flux Tiny) « (flux Tm) dS = 0,

if m ¥% n. The following gradients and fluxes of the 7-functions are ortho-
gonal for all m and n,

On the other hand, grad T'tm; and flux 7;,; are not, in general, orthogonal.
If all modes are present, the potential and stream functions are ©

where the tensor summation convention is used: whenever the same
letter subscript is used in a product, it should receive all values in a
given set and the resulting products should be added. The negative
signs have been inserted in order to avoid a preponderance of negative

signs in later equations. Substituting in (9), we have,
i; = Vin) grad Tees + Vin flux Tn] ; (12)
H, = —T flux T (n) + LItnj grad T tn) .

The 7-functions for the various modes are determined by equation
(4) and the boundary conditions except for arbitrary factors related to
the power levels of the modes. If we choose these constants in such a
way that

then the complex power carried by the wave is given by an expression
similar to that in an ordinary transmission line,

Hence, the V’s and J’s correspond to the voltages and currents in coupled
transmission lines.
In an expanded form equations (12) are

To these we add the following expansions for the longitudinal compo-
nents of H and H

Equations of this form satisfy automatically the boundary conditions
on EF, and H, . The multipliers x, have been inserted arbitrarily in order
to make the physical dimensions of the second factors to correspond to
those of voltage and current.

Multiplying (18) by [—9T (m/e2 dv] dS, (19) by [AT (m/e, du] dS, adding,
and integrating over the cross-section, we obtain

In the first term the summation convention should be ignored. Multiply-
ing (18) by [07 {m/e: du] dS, (19) by [0T{m/e2 dv] dS, adding, and in-
tegrating we find

Subjecting the right column of (17) to a similar treatment, we obtain
three additional equations. Summarizing, we have

In the last terms of equations (25) and (28) the summation convention
should be ignored.

If the components of B and D are linear functions of the components
of H and E respectively, then with the aid of (15) and (16) they can be
expressed as linear functions of Vay, Vin, Ia, Lt, Ven, Lem -
Solving (29) for Vz,¢n) and J.,:.; and making the appropriate substitu-
tions in (25), (26), (27), (28), we obtain the generalized telegraphist’s

dV, ) ¥ ig
Fg Zemomlign — SomimZin — “Teme Very — Lome V in
dl ) I -
ae Tx Ym) (n) Vay — YG) [n] Vinl —e Dm) inl (ny = Dm) tml ta] ,
dV
us 7
ae = —Zimitnyl tn) — Zimmliny — Tmt Viny — Timmy
dL tm) q I
Di = —VimimVing — YemaVoy — Ttmintiny — TT pmimlon -

The transfer impedances Z, the transfer admittances Y, the voltage
transfer coefficients "7, and the current transfer coefficients "T between
various modes are in general functions of z. They are constants if the
properties of the waveguide are independent of the distance along it;
in this case the problem of solving the generalized telegraphist’s equa-
tions reduces to solving an infinite system of linear algebraic equations
and the corresponding characteristic equation.

Similar equations may be derived for spherical waves either in an un-
limited medium or in a medium bounded by a perfectly conducting coni-
cal surface of arbitrary cross-section. If the latter is circular and if the
flare angle is 180°, we have a plane boundary. Hence, the case of spheri-
cal waves in a non-homogeneous medium is included. In the spherical
case we shall use the general orthogonal system of coordinates (r, u, v)
where r is the distance from the center and (u, v) are orthogonal angular
coordinates. In this system the elements of length ds and area dS are
given by . ‘

The transverse field components may be expressed in a form similar
to that for waveguides

where grad and flux of a typical scalar function are defined by equations
(10). Instead of (11) we have

where the T-functions satisfy equation (4) and appropriate boundary
conditions. These functions, their gradients and fluxes are orthogonal.

They are assumed to be normalized as follows
[[(erad 7)- (grad 7) do = 32 [[ravet, (34)
where dQ is an elementary solid angle. Hence, equation (14) will again

represent the complex power flow in the direction of propagation.
The various field components may then be expressed as follows

It should be noted that the physical dimensions of V;,¢,) and I,,tnj are
not those of voltage and current. Substituting in Maxwell’s equations
and using transformations similar to those in the case of plane waves,
we find

Returning to the plane wave case and assuming the following general
linear relations

If we solve the last two equations for I,,tm) and V.,.n) and substitute
in the preceding four equations, we shall obtain the telegraphist’s equa-
tions in their final form (30). Thus, let

“Zm\tn] = / / Jomeex tn T'tnj T'tm BS,
(45)
“Yom (ny = | jooesx(nyT (my Tom AS.
From these coefficients we obtain another set
*Zinitm) = normalized co-factor of *Zpmjtnj ,
*Zcn)(m) = normalized co-factor of *Y (myn -
Then,

Before substituting in equations (39) to (42), the summation index m
in (47) should be changed to avoid conflict with m in the former equa-
tions. It does not seem necessary to make these final substitutions in
their most general form. The results are’ very complicated and in prac-
tice the various coefficients are not independent. Some coefficients may

vanish; others may be small. In isotropic media, pun = boo = be =
By €uu = €» = €z = e€ and the mutual coefficients vanish. In gyromag-
netic media subjected to a strong magnetic field in the z-direction, the
permeability coefficients of superposed ac fields are®

If the entire waveguide is filled with such a medium, assumed to be
homogeneous, equations (48) and (44) become

In view of the orthogonality of the T-functions and the normalization
conditions (13), we have

where the summation convention is waived. In this case all the transfer
coefficients in equations (30) vanish,

8 C. L. Hogan, ‘‘The Ferromagnetic Faraday Effect at Microwave Frequencies
and Its Applications—The Microwave Gyrator, Bell System Tech. J., Jan. 1952,

In rectangular waveguides we choose cartesian coordinates;
thene:=@=1, w= 2,0 = yand

Some of the mutual impedances vanish; thus
Zooey = O, (55)
if either p + sorg +t is even. If p + 8s, as well as q + ¢, is odd,

. 2 2
Se
Similarly
oe Ztpatst) = 9, (57)
if either p + s org + tis even, provided p ~ sandq ¥ t. If p + s, as
well as q + ¢, is odd,
A-Ipgla(pt? — q’s)jupay (58)

Consider now the set of modes which includes TEj10}.. This set in-
cludes TE, modes and all the other modes which are coupled to either
of these modes. Noting that there are no TM») and TM @,) modes, we
obtain the following table in which those modes which do not belong
to the set are marked with a bar:

The principal effect of the gyromagnetic medium on the TEj19) and TEjoy
modes may be understood by taking into account their mutual coupling
but ignoring their coupling to other modes. The equations of propaga-
tion become

dV : i
—— = —Jjoplyo — jopey(8/r')Lio 5
al r
pl = — (joe + saicat) Yom
Z JOM
dV op
ra = jops,(8/ p10) — jou on ,
aI 01} - © a)
ao Jue as V tor} -
For exponentially propagated waves we have
Vio = Vane, Vion = Vine™, 62)

When the mutual permeability is zero, we have two independent modes
whose phase constants are

(8° — Bor)L oy = FhBOF 110) « i
Multiplying term by term, we obtain the characteristic equation
B* — (Bio + Ba)8° + (L — K’)Bio801 = 0. (67)
Solving, we have
B° = 4(Bio + Bor) = 4[(Bi0 —Bor)” + 4KBio801)"”. (68)

The effect of coupling is to increase the larger phase constant and de-
crease the smaller one; that is, to make the slower wave slower, and the
faster wave faster.

Let us assume a > 6; then 61 > 8. Taking the upper sign in (68)
and substituting in the second equation of the set (66), we have

V 101 Bor Tt10) p+ (p? + ky)? °
Hence, the ratio of the power carried in the TEjoy mode to that in the
TE} mode is

If the phase constants of the uncoupled modes are equal, then p = 0
and Po. = Pi for all values of the coupling coefficient. In this case (68)
becomes

The cutoff frequencies of both normal modes are seen to be independent
of either the transverse permeability or the mutual permeability. Since

Hzy 1S a pure imaginary, it effectively increases the transverse permeabil-
ity for one mode and decreases it for the other.

To evaluate the effect of higher order TE and TM modes on wave
propagation we may substitute from (68) in all terms of the character-
istic equation for telegraphist’s equations except the first two diagonal
terms and recalculate the 6’s. Alternatively we may replace TE; and
TE,o1 modes by the normal modes just obtained, recalculate the cou-
pling coefficients, and evaluate the effect of the mode with the greatest
coupling to the modes under consideration.

The approximate location of the natural barrier found in early melts
is shown in Fig. 1. The barrier was generally located in the melt per-
pendicular to the axis of the melting crucible or more accurately to the
direction of the temperature gradient. Plates and rods containing sec-
tions of the photoactive barrier, Fig. la, were cut from the melt and
mounted on convenient supports for laboratory tests. Fig. le shows a
magnified section of one of these barriers.”

Fig. 1—(a) Drawing of melt showing position of photovoltaic barrier and photo
cells with natural barrier. (b) Drawing of melt showing surface type photo cell
made from natural barrier. (c) Magnified, etched section of slowly cooled silicon
showing the transition between p and n silicon forming the barrier which consists
of intermeshed striae of these two varieties.

-6 -5 -4 -3 2 - oO 14
VOLTS (Cc)
Fig. 2a—Spectral response of internal barrier in silicon.
Fig. 2b—Voltage and current photosensitivity of internal barrier in silicon.
Fig. 2c-—Rectification characteristic of internal barrier, dark and illuminated.

That ion velocity has a profound effect on the voltage current char-
acteristic of bombarded surfaces is shown in Fig. 5. These characteristics
were obtained by placing a tungsten point contact under 10 gm of force,

Fhe 5—Liffect of bombardment voltage on the rectification of the intermediate
cells.

Fig. 6—Photocurrent at a constant illuminance versus the bombarding voltage.

on the photo active surfaces of the medium size cells whose spectral
response js given in Fig. 8. However, in order to show the rectifying prop-
erty of the barrier without the complication of a point contact, a disc of
hyper-purity silicon 13” in diameter and about 0.025” thick was given
an optical polish on both faces. One face was bombarded with 30-kv ions.

Fig. 7—Photovoltage at a constant illumination versus temperature of the
bombarded silicon surface.

Fig. 8—Spectral response of the intermediate size cells at various bombarding
voltages.

Electrodes 14” in diameter of evaporated rhodium metal were applied in
like manner to each surface. Contact was made to the collector electrodes
by means of tin discs. Fig. 4 gives the forward and backward log voltage-
log current relation of this large cell. Without bombardment such an
arrangement shows ohmic conductivity so it is evident that the treat-
ment is responsible for the development of a potential barrier beneath
the surface. It is believed from the high dark resistivity of the bom-
barded layer that the intrinsic properties of the silicon are developed
therein. Thus an intrinsic -p type potential barrier is produced similar
to a degree to the n-p junction. One would expect the depth of this bar-
rier to be related to the velocity of the ions. Consequently a study has
been made of the effect of ion velocity on the photoelectric properties.

The photoelectric current at constant illuminance for a series of cells
prepared by bombardment with ions of different energies is shown in
Fig. 6. It is remarkable how quickly and completely the current sensi-
tivity saturates at approximately 500 volts.

The photoresponsiveness improves as the total bombarding charge is
increased until it has reached about 600 microcoulombs per sq. em.
Further bombardment produces no appreciable improvement. In certain
exploratory tests a total charge of about 9000 microcoulombs at 30 kv
has been applied. Under these severe conditions the surface may show
small areas having a slightly etched appearance.

No extensive tests have been made to determine the effect of the rate
of application of the bombarding charge. The apparatus was designed
for use at a rate of about 5 microamperes per sq. cm. It is known how-
ever, that between the limits of about 2.5 and 10 microamperes per sq.
cm. the effects are subject only to the total charge or the total number
of ions which strike the silicon surface.

Six spectral curves are shown in Fig. 8 which illustrate the result ob-
tained with the intermediate size cells over the bombardment voltage

range previously mentioned. The peak of the lowest voltage cell is
definitely toward the blue compared with the other five whose maximum
is constant at about 0.725 pu.

One objective in this study was to obtain evidence relating to the
depth of the barrier below the silicon surface as a function of the energy
of the bombarding particles. The higher the velocity of the particles the
further beneath the surface one would expect the barrier to be located
and as a result there might be a shift in the spectral characteristic toward
the red with increasing depth of the barrier due to the relatively greater
absorption at the blue end. There is however, a selective or secondary
maximum at the peak which sharpens it and nullifies the effect of the
warping of the entire curve. The blue to red shift can be shown as in
Fig. 9 by plotting the ratio of the responses in Fig. 8 at 0.50 » and 1.0 pz.
Thus at low voltage the blue to red ratio is high and decreases as the
bombarding potential is raised.

In the spectral curves it will be noted that there are a number of
secondary humps located near the top of the curves and extending down
on the‘blue side. There is a strong tendency for them to occur at definite
wavelengths and to beevenly spaced regardless of the bombarding voltage.

SPECTRAL MEASUREMENTS ON THE LARGE CELLS AND THE EFFECT OF
MATERIAL COMPOSITION

For the large cells, two grades of silicon were used peu niente by
pyrolytic reduction of SiCl, and called “hyper-pure”. These will be

LOGig—VOLTS =
Fig. 9—Ratio of blue to near infra-red response versus bombarding voltage.

designated B and C, the former being from the same source as “‘silicon
B” referred to in the paper by Scaff et al.” A typical analysis is given
therein. The C silicon was from another source and a spectroscopic analy-
sis indicated it was somewhat more impure than B thus agreeing with
observed differences in its electrical and optical characteristics. An opti-
cal variation of considerable interest is shown in Fig. 10 where the
spectral transmittance of the two grades of silicon is compared in the
infra-red for polished plates each 0.0195” thick.

The transmittance of B decreases a little with increasing wavelength
but C goes down much more. Both however, start to get transparent at
about the same point, 1.14 and also show corresponding absorption
bands superimposed on the main curve. Briggs’ has compared the trans-

WAVELENGTH IN MICRONS
Fig. 10—Spectral transmittance of B and C grades of silicon, polished plates,
0.0195” thick.

mittance from 2 » to 12 uw of the A and B silicons in Scaff’s paper where
the former was much more impure than the C grade. The absorption
of the A silicon increased so rapidly out in the infra-red that a much
thinner sample was used for the measurements than for the B material.
If this difference in thickness is allowed for, the effect of impurity is
very striking.

The spectral response of large area cells made of the B and C materials
and bombarded with 1000-volt helium positive ions is shown in Fig. 11.
The two curves are similar in shape except the one for C silicon is some-
what narrower and in addition is shifted toward the blue. Both have
some of the secondary humps noted previously.

All the cells shown in this paper have indicated a long wave limit of
about 1.2 u. Actually some response can usually be detected out to about
1.3 ». Measurements made some years ago on the internal barrier units
also gave a limit around 1.3 » but relatively more response at 1.2 » with
peaks close to 1.10 4. This difference is reasonable because light was

projected along the barrier plane and not normal to it as in the latest
units, so that with the rapidly increasing transparency in this region,
less infra-red radiation was lost. However, the blue was rapidly at-
tenuated.

When illuminated by tungsten light of 2848°K color temperature, the
large B cells gave 2160 microamps per lumen and the C unit 638. Cor-
recting for a surface reflectance of 0.385, the net sensitivities would be
3510 and 1040. These measurements were made with between 4- and
5-footcandles illuminance on the cells, a region in which the response is
proportional to the intensity. At much higher values of illuminance there
was some falling off of response so that the effective sensitivity was a
little lower. The above measurements were made on a ten ohm microam-
meter which is too low a resistance to affect the linearity. The inter-
mediate cells ran approximately 3000 microamps per lumen in the most
sensitive region of bombardment without correction for surface reflec-
tion and at 10- to 20-footcandles for the same tungsten lamp using a
meter of 76 ohms.

These experiments have served not only to introduce us to some of the
phenomena involved in semiconductor barriers but have also yielded
photo cells having desirable properties. These cells have a high degree of
stability and will stand treatment ruinous to most other cells. They have
a very high current sensitivity to tungsten light and daylight. They re-
quire no associated battery and can be made in large areas. Unlike the
material used in many types of photo cells, silicon does not have the
disadvantage of scarcity. All tests to date indicate that an indefinitely
long life may be expected even under extreme illumination. Fig. 11 sug-
gests that it may be possible to control to some extent the spectral re-
sponse in the region from the deep blue into the infra-red. The long wave
limit is set by the edge of the absorption characteristic.

1. U. S. Patent No. 2,402,839, Filed Mar. 27, 1941.
U. S. Patent No. 2,402,662, Filed May 27, 1941.
U. S. Patent No. 2,443,542, Filed May 27, 1941.
2. J. H. Seaff, H. C. Theuerer and E. E. Schumacher; also W. G. Pfann and J. H.
Seaff, Trans. A. I. M. E., 185, pp. 383-392, 1949.
3. R. S. Ohl, Bell System Tech. J., Jan., 1952. Also see this paper for more details
regarding the method of preparing silicon.
4. H. B. Briggs, Phys. Rev., 77, pp. 727-728, Mar. 1, 1950.

Abstracts of Bell System Technical Papers*
Not Published in This Journal

Mechanical Properties of Discrete Polymer Molecules. W. O. Baxer’,
W. P. Mason’ and J. H. Huss’. J. Polymer Sci., 8, pp. 129-155, Feb.,
1952. (Monograph 1937).

Post-War Achievements of Bell Laboratories: IJ. O. E. Bucxury’.
Bell Tel. Mag., 30, No. 4, pp. 224-237, 1951-1952.

A Portable, Direct-Reading Microwave Noise Generator. Kh. L. CxHin-
nock’. Proc. Inst. Radio Engrs., 40, pp. 160-164, Feb., 1952. (Mono-
graph 1939).

Concerning the early years of this fundamental concept of modern physics—
how Max Planck formulated it at the turn of the century and how others en-
larged it up to 1923.

Results of tests at 10 and 60 me with resistive terminations of 75 to 1000
ohms. Low terminating impedance values yield wide bands but involve higher
insertion losses.

* Certain of these papers are available as Bell System Monographs and may
be obtained on request to the Publication Department, Bell Telephone Labora-
tories, Inc., 463 West Street, New York 14, N. Y. For papers available in this
form, the monograph number is given in parentheses following the date of pub-
lication, and this number should be given in all requests.

Echo Distortion in the FM Transmission of Frequency-Division Mul-
tiplec. W. J. ALBERsHEIM’ and J. P. Scuarer’. Proc. Inst. Radio Engrs.,

The composite multiplex signals generated by frequency-division methods
long standard in telephone communication can be transmitted by the new trans-
continental broad-band FM radio relays. Signal intermodulation by echoes must
be minimized. Such intermodulation is investigated in this paper experimentally
and analytically. Two types of echoes are considered: (1) weak echoes with de-
lays exceeding 0.1 microseconds, caused mainly by mismatched long lines; and
(2) powerful echoes with delays shorter than 0.01 microseconds, caused by multi-
path transmission, and leading to selective fading. By use of random noise sig-
nals, the distortion is evaluated as a function of various parameters of the echo,
the base-band, and the rf modulation.

Experiments have been made on a sample of FeO, cut from a single crystal
in such a way that its ferromagnetic domain pattern includes an individual
domain wall whose motion can be studied. This sample has a permeability which
is high (about 5000) at low frequencies and drops off rapidly above 1000 cycles.
A hysteresis loop and data on wall velocity vs applied field were also taken.
The data are discussed in terms of recent developments in the theory of the
ferromagnetic domain wall. It appears that this theory explains our data satis-
factorily, and that in using it to explain our data we determine some of the
fundamental magnetic constants of Fe;O4. We are also able to gain some insight
into domain wall motion in ferrites generally in this way.

The Drift Mobility of Electrons in Silicon. J. R. Haynes’ and W. C.
Westpnar. Phys Rev., 85, p. 680, Feb. 15, 1952.

Formulas for the Group Sequential Sampling of Attributes. H. L.
Jones’, Ann. Math. Statistics, 23, pp. 72-87, March, 1952.

Some Fundamental Properties of Transmission Systems. F. B.
Lumwettyn. Proc. Inst. Radio Engrs., 40, pp. 271-283, March,
1952. ;

The problem of the minimum loss in relation to the singing point is investi-
gated for generalized transmission systems that must be stable for any combina-

tion of passive terminating impedances. It is concluded that the loss may ap-
proach zero db only in those cases where the image impedances seen at the ends
of the system are purely resistive. Moreover, in such cases, the method of over-
coming the transmission loss, whether by conventional repeaters or by series and
shunt negative impedance loading, or otherwise, is. quite immaterial to the ex-
ternal behavior of the system as long as the image impedances are not changed.
The use of impedance- -correcting networks provides one means of insuring that
the:-phase of the image impedance of the over-all system approaches zero. Gen-
eral relations are derived which connect the image impedance and the image
gain of an active system with its over-all performance properties.

A Recurrence Relation for Three-Line Latin Rectangles. J. RIORDAN’.
Am. Math. Monthly, 59, pp. 159-162, March, 1952.

Capacitors and Communications. Inductive Coordination of Lines. A.
R. WarHNER’ and W. E. Broecker’. Elec. Light and Power, 30, pp. 105-

ANTENNAS: THEORY AND Practice. By Sergei A. Schelkunoff and
Harald T. Friis, 689 + xxii pages, John Wiley and Sons, Inc., New York

Srpney Daruineton, B.S., Harvard University, 1928; B.S. in E.E.,
Massachusetts Institute of Technology, 1929; Ph.D., Columbia Uni-
versity, 1940. Bell Telephone Laboratories, 1929-. Dr. Darlington has
been engaged in research in applied mathematics with emphasis on
network theory.

Paut G. Epwarps, B.E.E., Ohio State University, 1924; E.E., Ohio
State University, 1929. Western Union Telegraph Company, 1919-22;
American Telephone and Telegraph Company, 1922-34; Bell Telephone
Laboratories, 1934-. His main concern in the Laboratories has been
with toll transmission problems, including voice frequency and carrier
systems. Member of the I.R.E., A.I.E.E., Sigma Xi, Tau Beta Pi, and
Kta Kappa Nu.

C. W. Harrison, BS. in E.E., Purdue University, 1938; M.S., Lehigh
University, 1940. Bamberger Broadacasting Company, 1939-41. Bell
Telephone Laboratories, 1941~-. Mr. Harrison is a member or the tele-
vison research group. He formerly designed radio receivers and, later,
microwave relay repeaters. Member of the I.R.E.

JoHn L. Hysxo, B.S. in E.E., Cooper Union, 1921. U. S. Army,
1918-19. Bell Telephone Laboratories, 1919-. Mr. Hysko’s principal
activities in the Laboratories have been in the development of amplitude-
modulation and frequency-shift carrier telegraph systems for land line,
radio teletypewriter and submarine cable applications.

Epwin F. Kinessury, B.S., Colgate University, 1910. United Gas
Improvement Company, 1910-18. U.S. Army, 1918-19. Eastman Kodak
Company, 1919-20. Bell Telephone Laboratories, 1920-51. Mr. Kings-
bury retired in 1951 after a career which was primarily concerned with
television research and development, especially that part dealing with
photoelectric and electrooptical problems. Member of the Franklin In-
stitute, the Optical Society of America, and Phi Beta Kappa; Fellow of
the American Physical Society and the American Association for the
Advancement of Science.

Ernest R. Kretzmer, B.S., Worcester Polytechnic Institute, 1944;
M.S., Massachusetts Institute of Technology, 1946; Sc.D., Massachu-
setts Institute of Technology, 1949. As a member of the Electrical En-
gineering Department at Massachusetts Institute of Technology, Dr.
Kretzmer taught from 1944-46 and conducted research there from
1946-49. Bell Telephone Laboratories, 1949-. He works in the televison
research group, where he has been principally concerned with decor-
relation of television signals. Member of I.R.E. and Sigma Xi.

L. R. Montrort, E.E., University of Virginia, 1926; American Tele-
phone and Telegraph Company 1926-34; Bell Telephone Laboratories,
1934—. Mr. Montfort has been concerned with the engineering of carrier
systems. This has included field work with new systems and field tests
prior to the design of new systems. During the end of World War II
and for a short time thereafter, he assisted in the engineering and test-
ing of microwave radio systems. Member of A.I.E.E., Tau Beta Pi,
Theta Tau, and Sigma Phi Epsilon.

Russet 8. Out, B.S. in Electro-Chemical Engineering, Pennsylvania
State College, 1918; U.S. Army, 1918 (2nd Lieutenant, Signal Corps);
Vacuum tube development, Westinghouse:-Lamp Company, 1919-21;
Instructor in Physics, University of Colorado, 1921-1922. Department of
Development and Research, American Telephone and Telegraph Com-
pany, 1922-27; Bell Telephone Laboratories, 1927-. Mr. Ohl has been
engaged in various exploratory phases of radio research, the results of
which have led to numerous patents. For the past ten or more years he
has been working on some of the problems encountered in the use of
millimeter radio waves. Member of American Physical Society and Alpha
Chi Sigma and Senior Member of the I.R.E.

B. M. Ottver, B.A., Stanford University, 1935; M.S., California
Institute of Technology, 1936; Ph.D., California Institute of Technology,
1939. Bell Telephone Laboratories, 1939-52. During World War II,
Dr. Oliver was engaged in radar research and the rest of his employ-

ment before leaving the laboratories was in the television research group.
Member of I.R.E. and Phi Beta Kappa.

Witton T. Rua, B.S., Princeton University, 1926; American Tele-
phone and Telegraph Company, 1926-34; Bell Telephone Laboratories,
1934—. Except for the years 1941-45, when he worked on military pro-
jects. Mr. Rea has been principally concerned with telegraphy. As Tele-

graph Development Engineer, he is in charge of the development of tele-
graph and telephotograph systems. Senior member of I.R.E. and member
of A.I.E.E. and Phi Beta Kappa.

L. C. Rosrrts, A.B., Harvard University, 1916; B.S. in E.E., Harvard
University, 1919; B.S. in E.E., Massachusetts Institute of Technology,
1918; American Telephone and. Telegraph Company, 1917-34; Bell
Telephone Laboratories, 1934—. Mr. Roberts has been primarily concerned
with the development of de and carrier telegraph except during World
War II when he worked on multichannel and single-channel radio tele-
typewriter developments. Member of A.I.E.E.

S. A. ScHeLKunorr, B.A., M.A. in Mathematics, The State College
of Washington, 1923; Ph.D. in Mathematics, Columbia University,
1928. Engineering Department, Western Electric Company, 1923-25;
Bell Telephone Laboratories, 1925-26. Department of Mathematics,
State College of Washington, 1926-29. Bell Telephone Laboratories,
1929-. Dr. Schelkunoff has been engaged in mathematical research,
especially in the field of electromagnetic theory.

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