ELECTRIC COMMUNICATIONS (ELEKTROSVYAZ) NO. 6

• Document Type: CREST
• Collection: General CIA Records
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• Release Decision: RIPPUB
• Original Classification: K
• Document Page Count: 135
• Document Creation Date: December 23, 2016
• Document Release Date: August 16, 2013
• Sequence Number: 1
• Publication Date: June 1, 1957
• Content Type: REPORT

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ninri
InmanT' iu
RUNIC COMMUNICATIONS
(ELEKTROSVYAZ)
NO. 6
June 1957
Pages 3 - 80
STAT
STAT
STAT
PREPARED BY
TECHNICAL DOCUMENTS I tAt SON OFFICE
MCLTD
STAT
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Table of Contents
,Pjga
Some Geometrical Properties of the uptimum Code)
by N.K.Ignatlyev
Fluctuation Overshoots and their Correlativity
by V.I0Tikhonov
Oa* ?41411 ? 00.440400
saoraaapooe000+1000
Reutive Trigger Circuits: their Operational Analysis
and Computing Methods by A.S.Vladimirov 20
Propagation of Meter and Decimeter Radiowaves over Rough
Surface of the Earth, by N.D.Dymovich 304000i0400.0 40
Calculation of Multistage Amplifiers with Junction
Transistor, by I.N.Migulin oftooloos00000soroosororoiews* 49
Calculation of the Broad Band Grid Transformer According
to a Fixed Input Resistance Characteristic, by
Ya01..A1lterman
60 ..........................................
Calculation of Quench Circuits, by F F.7.hdanov ii00,011114$60040e00 67
An Electronic Telegraph Apparatus, by B.P.Terent'yev .........s.. 74
Obtaining High Harmonic Numbers by Means of a Magnetic
Harmonic Oscillator, by L.T.Kiw
1 000ssiosesefosewegtookies 84
Channel Load during On-Duty Talk by the Telephone
Operator, by V.M.Belous . 0000.411000110?00600064W40040O410. 89
The International Telephone and Telegraph Consultative
Committee (CCITT), by P.A.Frolov .............. . 94
Letter to the Editor 000#60000$06,01: ? 011eili0O00#4006410000OOOtO011if a 106
Books in 1957 .......................................,....... . 111
From the Foreign Press - Brief Notes .......... **** ?OCO?04.0080 ? 114
Authors' Certificates121
.......................................... .
Foreign Patents i0eWite..00.0?0000444.0400000140G011004301,00 0 124
Bibliographical and Abstract MateriAl An mectrocommilnications .. 129
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E
OME GEOMETR CAL PROPERT S OF THE OPTIMUMHOPPE
0.? P
N.K.Ii
y v
8
10
12..
14 _
16 ?
18
20
22
24
26
29
30
32 --
34-
36
3 a-
40_.
42_,
44-
46
48
The problewis discussed o selecting the best configuration
of signal space for distributin
code signal (i.e., a code ensur
signal at all other conditions
investigated when the dots of a
an n-dimensional sphere, on the
within an nadimensional cube.
therein the dots of the optimum
ng the maximum entropy of the
maining equal). Instances are
signal become distributed within
surface of that sphere, and
It can be concluded from the proof
maximum channel capacity that the ideal
characterized by the diutribution of sign
sphere with a radius of Vg-(where P is
Shannon's theorem (Bib1.1) concerning
e ensuring such capacity is geometrically
1 dots on the surface of an n-dimensional
he signal power), and statistically char-
acterized by the normal distribution of the signal and the uniform probability of
all code combinations.
But these characteristics pertain on,y to the ideal code whore the number of
elementary sendings n in every code combi
tion tends toward infinity and the total
amplitude of the signal is uncurtailed by anything.
As for the analogous geometrical and statistical characteristics of the opti-
mum real code (hereafter referred to as (*timum code), these remain unclarified.
The preseut paper makes an attempt at de4ning these characte.ristics.
Hereafter the optimum code will be construed as referring to a uniform n.ivaluel
ode which, at all other conditions (such as equality of signal power or signal
scope and equality of noiseproof feature) being equal, has the maximum entropy,
req.
" 56L
jage the equality ot the notseproof feature of codes by the Vi5lume?
STAT'
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?
?
?
?
space corresponding to every signal dot, afld the entropy of codes by the volume of
the entire space of signal. At such an approach toward the problem posed, there is
no need for exploring the geometry of the distribution of signal dote within the
given volume of space*. Thus, we need merely be concerned with the best configura-
tion of that volume. At such an approach the results will be very generalised but
will still apply only for codes with a high entropy, in which the manner of the mu-
tual distribution of signal dots uithit a given volume of spade (at the si.p.roortie
occu tion of vo ume) exerts virtually no influence on signal power.
It is assumed that a priori selected probabilities of appearanmi cannot be imp.
parted to code combinations and that, most generally speaking, these probabilities
should be regarded as uniform.
Volume, Power Anci Entro))yf Signals
? Let us establish certain initial correlations:
Let xl, x2, xn be the coordinates of a signal dot in the n-dimensional
space which also express the successive voltages of the elementary sendinge of a
given signal.
To establish some simplest relationships between the geometrical and energy
characteristics of the signal let us assume that it is emitted at a resistance
of 1 ohm and that it consists of elementary pulses with a duration of 1 sec.
Thereupon the value
will express the square of the distance between a signal dot i from the origin ot
coordinates, and at the same time it also expresses signal energy, while the value
esidowlemsommorimmessammummo
*It is known that tha optimum distribution of signal dots inside a given volume of
space is one at which the signal dots coincide with the centers of the most densely
aligned n-dimensional spheres.
STAT
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0
2
4,414Yo.
4
10
12-
14
16--
18 If the volume of signal space is fil ed uniformly with signal dots, and if a
2'0 sufficiently great number of signal dots s distributed along every direction of
22 signal space (i.e., if the entropy of the signal is sufficiently high), the summa-
Pit ?11.00,11,...t..T.
expresses the power of the given signal.
The mean signal power for all M code
form probability):
?
combinations is (if they all have a uni-
P=?p
7cf k'
?
24
26
?
78
31J__
3 2
tion of individual dots can be replaced b integration of the volume of signal
pace, V:
P c?n? dv,
1 th
For simplicity, let us assume that eery signal dot is corresponded by a space
34f volume equal to the unit. Then the volof signal space V will express the number
4 0
f signal dots, i.e., V.= M. Oonsequentl7
, the signal entropy is
,
h =?n-loz2V.
In instances when sigpal dots become distributed on the surface of a certain
44 figure and not within its volume, volume V in eqs.(1) and (2) should be substituted
46 by surface S.
48_ We shall establish the relationships between signal power and enthropy on the
50?basis of expressions (1) and (2).
4'32,
.04.0.44
Norinal,??Slawal Distribution
1
? t A T".4 vim .now.eim4wle. 44.1ft hnn4A nenevirstmwdrACke Af. a signal with normal distribe
40141g1,---44114101.161he 4witrika taidi~ 01thr4rW116661
56probability_damity_is expressed ati_folla;s1 STAT
1
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? " ? ?
? .
.1 414( ?-,1?04py
The entropy of the signal with such a distribution reaches, as is known, its
maximum possible value equal to
On considering distribution (3) as a function of a dot in an n-dimensional
space it can be easily concluded that pN is the function of distance r between tl'ie
signal dot and the beginning of the coordinates, i.e.,
This, the probability of appearance of various signal dots depends only on the
value of the radius of r on which they are located.
From the geometrical viewpoint, the distribution pN(xl, x2, xn) can be
regarded as the volume density of the signal. Therefore, upon multiplying it by
the distribution density of the spele volume along the radius of r, we obtain the
radial density of the signal p(r). 'Considering that the volume of an n-dimensional
sphere with radius r equals:
the density of distribution of space volume along the radius is
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This distribution, known under the name of x2-d15tr1but1on, reaches its maximum
at the following value of radius;
and, in proportion with the increase in n, it approximates a distribution of the
following form:
which is illustrated in Fig.l.
Fig.1
From the above formula it can be con-
cluded that at n.--(1-; all signal dots will lie on a single radius equal to Va
Here they will all have a uniform probability.
These characteristics of normal distribution, which appears to be optimum from
the viewpoint of utilization of the signal power, serve as a foundation for the be-
lief that all dots of a signal have to be distributed on the surface of an
n-dimensional sphere in order to obtain the optimum code (i.e., that the "surface-
spherical" signal distribution is to be used).
Nonetheless, from the viewpoint of optimal utilization of signal power at a
finite n, it is preferable to fill the entire volume of the n-dimensional sphere
with signal dots (i.e., to employ the "volume-spherical" signal-distribution).
Let us investigate the basic characteristics of these distributions.
Spherical Sigpal Distributions
Let us begin by examining an instance of volume-spherical distribution which
can be expressed as follows:
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askr,',A-vizisr.5" ,(14,;,,R1
where R is radius of the sphere confining the space volume of a signal.
In this case, the total volume of the signal space will be:
The volume of the space limited by the current radius r will be
while its differential will be
Consequently, in accordance with eq.(1) it can be written that
whence we obtain
or
On substituting eq.(12) into eq.(11), we obtain
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..f.1.7414WOVAMM4;
On then substituting eq.(1 into eq(2) we obtain the entropy of the signal
?
l'in(r1-11-2)P
g
(14)
?
We proceed with our examination of the instance of surface-spherical distribu-
tion, which can be thus expressed:
p (XI, x2, 3C1)
(15)
1.1
At xi2 >R2,
1.1
where
npn_i
= ??=.1.111.
(16)
and which expresses the surface of the sphere of radius R on which the signal dots
are distributed.
,Inasmuch as in the given case r = R, in accordance with eq.(1)
R=V
and consequently
11
11-1
itT n (nP)
(17)
whence, in accordance with eq.(2), we obtain the entropy of the signal.
le?
wilP 4
F +
k 2
(1)
?
STAT
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rtmaginat621,47';70; totwow's40..",',
Let us now conduct a comparison of the entropy values Hy and in accordance
with eqs.(14) and (18).
First of all, considering that
we therefore obtain at n -*
(1-1
1
n-peo
Ms both the above-examined distributions become equivalent to each other at
their limit, and then they also become equivalent to normal distribution. This re-
sult can be attributed to the circumstance that at an increase in the number of di.
mensions the entire volume of the n-dimensional sphere becomes concentrated at its
surface.
For comparison of the entropy ovIA n5. at finite values of n, let us
express them reciprocally by eliminating P. Proceeding from eqs.(14) and (18) we
obtain
n--1 IHy 1 ?n og2
ft
?????
n
At high values of entropy the second term of thislermula can be neglected,
and it can be considered that
n
(19)
From the above formula it can be concluded that the transition from volume-
spherical to surface-spherical distribution leads to the loss of .1. part of the
signal entropy, which is equivalent to the loss of one of the n dimensions of
signal space.
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e
Let us investigate the geometrical sense of the loss of one spatial dimension
at the use of a surface-spherical signal distribution.
The position of a signal dot on the surface of an n-dimensional sphere of a
known radius R can be completely determined by the presence of n 1 coordinates of
that dot, since the nth coordinate can be found from the equation of the sphere
x2 x2' '?
1 2 ' n
However, if the radius of the sphere is not known (i.e., if the scale of the
coordinates of the dot lying on the sphere's surface is not known), then it is neces-
sary to know all n of the coordinates of that dot in order to determine its position
completely Thus, knowledge of the nth coordinate, which is redundant from the view-
point of transmission of information at a known signal scale, becomes a prerequisite
when the signal scale is not known.
Accordingly, when a surface-spherical signal distribution is employed, one of
the dimensions of space is "expended" on compensating the loss of the signal scale.
The possibility of effecting reception of a coded signal at any (so long as
sufficiently slow) fluctuations in its level (i.e., despite the loss of its scale)
is of a tremendous practical significance. The core of the matter is that long-
distance transmission of signals invariably causes the level of these signals to be
subject to chaotic fluctuations, owing to the changes in conditions of propagation
and also owing to changes in the amplification factor of the transmission channel.
These fluctuations are, as a rule, so slow compared with the duration of code com
binEtions that the amplitude ratios within every code combination can be regarded
as unchanged, In tho case of surface-spherical distribution such fluctuations in
the signal level, which geometrically correspond only to changes in the length of
the radius-vector of a signal dot, do not incur the appearance of any error.
9
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In instances when the limiting factor in the increase in channel capacity is
not signal power but signal scope, the distribution of signal dots is bound to take
place within an n-dimensional cube.
If the signal scope is denoted by L, then the volume of the signal space
On further proceeding from eq.(1) the formula for mean signal power can be
written as follows:
and the signal entropy becomes
On comparing eqs.(14) and (23) we obtain
From the above formula it can be concluded that the difference in the entropies
of volume-spherical and cubic distributions at an increase in n (from 1 to m)in
--JAT
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creases from zero to log V( Tr (which altogether amounts to about of a binary
digit).
A code with cubic (or more exactly, with volume-cubic) signal distribution dis-
plays the same shortcoming as the code with volume-spherical distribution: it re-
quires strict maintenance of the constancy of signal level.
To eliminate this shortcoming it is possible - just as in the case of spherical
distribution - to eliminate the signal dots from the interior of the volume and to
pass over to "surface-cubic" distribution, which will likewise lead to the loss of
1
approximately--7-1 part of the entropy of the signal.
Conclusions
The above analysis demonstrated that the projection of the geometric properties
of the ideal code onto a real code is not the best solution of the problem posed.
At a limited number of sendings in a code combination, the maximum entropy is
ensured by volume-spherical distribution.
At a limited signal scope, the maximum entropy is ensured by volume-cubic dis-
tribution. Such an entropy proves to be slightly smaller than the entropy of
volume-spherical distribution, but the realization of the optimum code can then be-
come much simpler. For instance, a pulse code with a volume-cubic signal distribu-
tion is materialized in the form of the pulse expression of the number of code com-
bination in the calculating system based on b (where b is number of code levels).
Volume distributions can also be employed at a variable signal scale. For this
purpose, scale pulses have to be introduced into the signal. The slower the possi-
ble fluctuations in signal scale, the rarer can be the distribution of these pulses,
and the smaller will be the reduction in entropy of the signal awing to these
pulses.
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1. Shannon, C. gm Statistical Theory of the;fransmission of Electrical Signals*
Collection an: The Theory of thelYansmission of Electrical Signals 44 the
Presence of Noise. Edited by N.A Zhelesnovi IL (1953)
Article received by the Editors 20 December 1956.
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vi.ovm:P
FLUCTUATION OVERSHOOTS AND MR CCEIRELATION
by
V. I . Tikhonov
This article cites the results of an experimental investigation
of the distribution of the overshoots of normal and Rayleigh
fluctuations, in terms of length, and it evaluates their correlation.
46.
Electronic relays and lock-on circuits are widely used in various coding sys-
tems. For an analysis of the effect of random noise on an electronic relay with a
single stable equilibrium position the following relay model may be used in a number
of cases: it operate; whenever the noise voltage (t) exceeds the relay operating
level a during a time of At > To, where To is the time of relay operation. Here it
is possible to compute with sufficient ease the mean number of spurious operations
Fig . 1
Fig.2
14
of the relay and to determine also the other statistical characteristics, if the
values of noise on the relay operating level, spaced by the-duration of the relay
pules, are not correlated (Bib1.1). Such a postulate holds true for the pulse
noises subject to Poisson's law. However, it may be doubtful whether this postulate
is also valid for fluctuation noise.
It will be demonstrated below that fluctuation overshoots can be considered as
even at a relay operating level of a 0, where 02 is the
STAT
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scattering of fluctuations of ,(t). Here an overshoot is cbfistrued as referring to
an event where the fluctuation voltage becomes F(t) > a during a time T termed the
pulse duration (Fig.1). The interval T between
Fig.3
durations T and intervals T between
two successive overshoots is analogously defined.
A special experiment has been devised to
solve the above problem. The normal fluctuation
noise at the outputs of three amplifiers was
photographed from the screen of an OK-17 oscillo-
graph. The relationship between the square of
the amplification factor and the frequency for
each amplifier is illustrated in Fig.2.
The numerous oscillograms taken were pro-
cessed and used in plotting the functions of the
probability density p(I) and q(T) for overshoot
them on the level a = a (Fig.3).
We will demonstrate that a nominal division of T and T corresponds to a time
of 0.17 oec for the first amplifier, 0.12 p.sec
40 for the second amplifier, and 0.087 sec for the
Fig.4
third amplifier.
Of the characteristic, of the three ampli-
fiers only the frequency characteristic of the
second amplifier is sufficiently well approxi-
mated by the following function:
K2(f)=exp =---20,5 td sec
1
Correspondingly, the correlation factor for
normal fluctuations at the output of the second amplifier will equal
)t
(2 STAT
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Let us now examine the fluctuations n(t) obtainable from normal fluctuations
of t) by the following transform:
j(t) 6,f E(/)?a
at t(t)