THE THEORY OF OPTIMUM NOISE IMMUNITY

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• Document Creation Date: January 4, 2017
• Document Release Date: October 28, 2013
• Sequence Number: 5
• Publication Date: January 1, 1947
• Content Type: REPORT

1C-?????
STAT
STAT
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THE THEORY OF mum( NOISE Imam
by
V. A. KOTEVNIKOV
STAT
STAT
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Methods of combating noise
Classification of noise
Messages and signals
The contents of this book
2-3. Normal fluctuation noise
2-4. Representation of normal fluctuation noise s.s &
Fourierseries
2-5. Linear functions of independent normal random 'variables
2-6. The probability that normal fluctuation noise falls
PART II
TRANSMISSION OF DISCRETE MESSAGES
CHAPTER 3
3-2. The ideal receiver
3-3. Geometric interpretation of the material of chapter
CHAPTER 4
'NOISE IMMUNITY FOR SIGNALS WITH TO DISCRETE VALUES
4-1. Probability of error for the ideal receiver
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4-3. Optimum noise immunity for transmission, with a passive space 31
4-4. Optimum noise immunity for the classical telegraph signal 32
4-5. Noise immunity for the classical telegraph signal and
reception with a synchronous detector 33
AL-G- Noise immunity for the classical telegraph signal and
reception with an ordinary detector 37
4-7. Results on the noise immunity of systems with a passive space 39
4-8. Tho optimum communication system with an active space 40
4-9. Noise immunity for frequency shift keying 41
4-10.0ptimum noise immunity for normal fluctuation noise
with frequency-dependent intensity 45
4-11. Geometric interpretation of the material of chapter 4 46
CHAPTER 5
NOISE IMMUNITY FOR SIGNALS WITH MANY DISCRETE VALUES
5-1. General statement of the problem 47
5-2. Optimum noise immunity for orthogonal equiprobable
signals with the same energy 48
5-3. Example of telegraphy using 32 orthogonal signals 49
5-4. Optimum noise Immunity for compound signals 51
5-5. Example of a five-valued code 53
5-6. The optimum system for signals with many discrete values 54
5-7. Approximate evaluation of optimum noise immunity 58
5-8. Example of the transmission of numerals by Morse code 59
PART III
TRANSMISSION OF SEPARATE PARANETER VALUES
CHAPTER 6
GENERAL THEORY OF THE INFLUENCE OF NOISE ON THE TRANSMISSION OF SEPARATE
PARAMETER VALUES
6-1. General considerations.. 63
5-2. Determination of the probability of the transmitted parameter 64
6-3. The function P(X) near the most probable value X 66
6-4. Error and optimum noise immunity in the presence of
law intensity noise 67
6-5. Second method of determining the error and optimum noise
immunity in the presence of low intensity noise 69
6-6. Summary of chapter 6 72
6-7. Geometric interpretation of the material of chapter 6 73
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CHAPTER 7
THE OPTIMUM NOISE IMMUNITY OF VARIOUS SYSTEMS FOR TRANSMITTING
SEPARATE PARAMETER VALUES IN THE PRESENCE OF LOW INTENSITY NOISE
7-1. Amplitude modulation 74
7-2. Linear modulation 75
7-3. General case of pulse time modulation 76
7-4. Spoolal case of pulse time modulation (optimum noise immunity) 77
7-5. Special case of pulse time modulation (noise Immunity
for the first method of detection) 79
7-6. Special case of pulse time modulation (Wise immunity
for the second method of detection) 83
7-7. Frequency modulation (general case) 85
7-8. Frequency modulation (special case) 87
7-9. Raising the noise immunity without increasing the energy,
length, or bandwidth of the signal 88
CHAPTER 8
NOISE PartilTY FOR TRANSMISSION OF SEPARATE PARAMETER
VALUES IN THE PRESENCE OF STRONG NOISE
8-1. Derivation of the general formula for evaluating the
effect of high intensity noise 91
8-2. Comparison of the formulas for weak and strong noise 93
8-3. Pulse time modulation 94
8-4. Frequency modulation 97
8-5. The system for raising tho noise immunity without increasing
the energy, length, or bandwidth of the signal 99
8-6. Geometric interpretation of the results of chapter 8 100
PART IV
TRAUSMISSION OF WAVEFORMS
CHAPTER 9
GENERAL THEORY OF THE INFLUENCE OF WEAK
NOISE ON THE TRANSMISSION OF WAVEFORMS
9-1. General considerations
103
9-2. Tho influence of weak noise on the transmitted waveforms
104
9-3. Conditions for tho ideal receiver
106
9-4. Means of realizing the ideal receiver
107
9-5. The error for ideal reception
108
9-6. Brief summary of chapter 9
109
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CHAPTER 10
DIRECT MODULATION SYSTEM
10-1. Definition
110
10-2. Dorivation of basic formu3as
110
10-3. Optimum noise immunity for amplitude and linear modulation.
111
10-4. Optimum noise immunity for phase modulation
112
306-5. Noise immunity for amplitude modulation and ordinary receptions...]-13
30-6. Noise immunity for phase modulation and ordinary reception
306-7. Noise immunity for single-sideband transmission
CHAPTER 11
115
115
PULSE MODULATION SYSTENS
11-1. Definition
116
11-2. A way of realizing the pulse modulation systom.. .
.117
11-3. Optimum noise immunity for the pulse modulation system
119
11-4. Noise immunity for the receiver analyzed in section 11-2
121
31-5. Optimum noise immunity for pulse amplitude modulation
124
11-6. Optimum noise immunity for pulse time modulation
125
11-7. Optimum noise immunity for pulse frequency modulation
126
CHAPTER 12
INTEGRAL MODULATION SYSTEMS
12-1. Definition 127
32-2. Optimum noise immunity for integral modulation systems 127
12-3. Optimum noise immunity for frequency modulation 129
CHAPTER 13
EVALUATION OF THE INFLUENCE OF STRONG NOISE OK
THE TRANSMISSION OF WAVEFORMS
13-1. General considerations
130
33-2. Maximum discrimination of transmitted waveforms
130
33-3. Maximum discrimination for phase modulation
132
13-4. Maximum discrimination for weak noise
133
33-5. Maximum discrimination for weak noise and phase modulation
135
APPENDICES
Appendix A. The specific enorgy of high-frequency waveforms 137
Appendix B. Representation of normal fluctuation noise by
137
Appendix C. Tho instantaneous value of normal fluctuation noise 139
Appendix D. Normal fluctuation noise made up of arbitrary pulses 139
two amplitude-modulatod waves
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PREFACE
This book is the author's doctoral dissertation, presented in January,
1947, before the academic council of the Molotov Energy Institute in Moscow.
Despite the fact that many works devoted to noise immunity have appeared in
the time that has elapsed since the writing of this dissertation, not all of
the topics considered in it have as yet appeared in print. Considering the
great interest shown in these matters, and also the number of references made
to this work in the literature, the author has deemed it appropriate to publish
it, without introducing any supplementary material. However, in preparing the
manuscript for publication, it was somewhat condensed, at the expense of material
of secondary interest. Moreover, Chapter 2, which contains auxiliary mathema?
tical material, has boon revised somewhat, to make it easier reading, and some
of the material has been relegated to the appendices.
The author
TRANSLATOR'S PREFACE
This translation is as faithful as is consistent with an English style
that is not overly turgid. How this was achieved will be apparent to anyone
familiar with the stylistic peculiarities of scientific Russian. I have
occasionally added footnotes where I thought the text needed some clarifica?
tion. These comments have been indicated by the word 'translator' in parentheses.
I have also corrected numerous typographical errors appearing in the mathematical
expressions of the original text.
R. A. S.
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PART I
AUXILIARY MATERIAL
CRAFTER 1
INTRODUCTION
1-1. Methods of combating noise
Ordinarily, a radio receiver is acted upon not only by disturbances (signals) pro-
duced by the radio transmitter, but also by disturbances (noise) produced by a large
variety of sources. The noise combines with the signals and corrupts them; in the case
of telegraphic reception this leads to errors, and in the case of telephonic reception
to background noise, static, etc. When the signals are too small compared to the noise,
reception becomes impossible.
The following methods of combating noise are used:
1. Deoreasing the strength of the noise by taking action against their sources.
2. Inoreasing the ratio of the strength of the signals to that of the noise by in-
creasing the transmitter power and by using directional antennas.
3. Improving the receivers.
4. Changing the form of the signals while keeping their power fixed. (This is done
with the aim of faoilitating the combating of noise in the receiver.)
Tho first two methods are not considered in this book, which is devoted rather to
the last two methods, and has as its goal to examine whether it is possible to decrease
the effect of noise by improving the receivers, given the existing kinds of signals. In
particular, what can be achieved in combating noise by changing the fora of the signals?
What form of the signals is optimum for this purpose?
1-2. Classification of noise
We can classify the noise which impedes radio reception into the following categories:
A. Sinusoidal noise consisting of one or a finite number (usually small) of sinusoidal
oscillations. This category of noise includes interference from the parasitic radiation
of one or more radio stations operating at frequencies near that of the station being
received.
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B. Impulse noire consisting of separate impulses which follow ono another at such
largo time intervals that the transients produced in tho receiver by one impulse have
substantially died out by the time the next is-pulse arrives. This category of noise
includes some kinds of atmospheric noise and noise from electrical apparatus.
.
C. Nornal fluctuation fleas()1 or, as it is sonetirhes called, smonthcd-out noise.
This also consists of separate impulses.occuraing at random time intervals, but the im-
pulses follow one another so rapidly that the transients produced in the receiver by the
individual impulses are superimposed in numbers large enough to narrant tho application
of the laws of large numbers of probability theory. This category of noise includes
vacuum tubo noise, noise due to tho thermal motion of electrons in circuits, and some
kinds of atmospheric noise and noise from electrical apparatus. At very high frequencies
this kind of noise is encountered almost exclusively.
D. Impulse noise of an intermediate type, which occurs when the transients produced
in the receiver by the individual impulses are superimposed, but not in ntmtbers large
enough to warrant the application with sufficient accuracy of the laws of large numbers.
This kind of noise is intermediate between categories B and C.
Methods of studying the action of sinusoidal end impulse noise on radio receivers
are at present quite well developed. The study of impulse noise of the intermediate
typo, when the transients produced by the individual impulses are just beginning to be
superimposed, is much more difficult. Moreover, in this case, we need to know not only
the shapes of the separate impulses, but also the probability of superposition of impulses
which have various shapes, and which obey various time distributions. In nost cases we
do not have this information about the noise, and it seems to be quite difficult to obtain.
For these reasons, and also because noise of category C is often encountered, in what
follows wo shall consider only noise of this latter category; no shall often designate
normal fluctuation noise simply as noise.
3-3. Messages and signals
By a message we shall mc-on that which is tc be transmitted. The messages with which
1. The use of the word "normal" alluaes to the fact that no deal here with one of a
variety of possible fluctuation processes.
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we shall bo concerned can be divided into three categories.
A. Discrete messages.
B. Messages in the form of separate numbers (parameters), which can take on any
values in certain ranges.
C. Messages in the form of wave trains, which can assume a continuous infinity of
different waveforms.
The messages which aro transmitted in telegraphy belong to the category of discrete
messages. In this case, they consist of discrete letters, numerals, and characters,
which can take on a finite number of discrete values. Moreover, in many instances, the
messages transmitted in remote-control systems belong to this category.
In the case of the transmission of individual measurements with the aid of tele-
metering, the messages consist of the values of certain parameters (e.g., temperature,
pressure, etc.) measured at given time intervals. These quantities usually take on
arbitrary values lying within certain ranges. Thus, in this case wo cannot restrict
ourselves to a finite number of possible discrete messages. Messages of this kind belong
to category B.
In the case of telephony, the messages are acoustical vibrations, or the electrical
vibrations taking place in the microphone, which can take on an infinite number of differ-
ent forms. These messages belong to category C. In television, the oscillations acting
on the transmitter can be taken as the message; this message also belongs to the last
category.
re shall assume that some variation in voltage, produced by the operation of the
transmitter, acts upon the receiver input. re have called this variation in voltage a
signal. Clearly, there will be a signal corresponding to each possible transmitted
message. The receiver must use this voltage waveform (i.e., signal) to reproduce the
message to which the signal corresponds.
3-4. The contents of this book
In this book we consider the influence of normal fluctuation noise on the transmission
of messages. The problem which will concern us is the following: re assume that when the
tv
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noise perturbation is not superimposed on the signal, then the receiver will reproduce the
transmitted message exactly. If noise is added to the signal, then tho sum of two voltages
will act upon the receiver input, i.e., the signal voltage plus the noise voltage. In this
case, depending on the sun voltage. the receiver will reproduce some message or other,
which in a given instance ray be different from the one that was transmitted. Clearly,
each sum voltage which acts upon tho receiver produces the particular message which cor-
responds to it. This correspondence may be different for different receivers. Depending
on this correspondence, a receiver will be more or less subject to the influence of noise
for a given kind of transmission. Ve shall find out what this correspondence ought to be
for the message corruption to be the least possible. The receiver which has this optimum
correspondence will be called ideal.
Next we shall determine the message perturbation which results when noise is added to
the signals, and when the reception is with an ideal receiver; tho message perturbation
obtained in this way will be the least possible under the given conditions, i.e., for real
receivers under the same conditions, the message perturbation cannot bo less. The noise
immunity characterised by this least possible message perturbation will be called the
optimum noise immunity. This noise immunity can be approached in real receivers if the
receiver is close to being ideal, but it cannot be exceeded. By comparing tho optimum
noise immunity with the noise immunity conferred by real receivers, we can judge how close
tho latter are to perfection, and how much the noise immunity can be increased by improving
them, i.e., to what extent it is adyisable to work on further increasing the noise immunity
for a given means of comnunication. Knowledge of the optimum noise immunity makes it easy
to discover and reject methods of connunication for which this noise irnunity is low com-
pared with other methods. This can be done without reference to tho method of reception,
since real receivers cannot achieve noise immunity greater than the optimum. By comparing
the optimum noiso immunity for different means of communication, we can easily explain (as
will be seen subsequently) the basic factors on which tho inmumity depends, and thereby
increase the immunity by changing tho means of communication. In the book, these matters
are illustrated by a whole series of examples which have practical interest. However,
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tho examples considered are far from exhausting all possible cases in which one can apply
tho methods of studying noise immunity developed here.
In tho book, all questions are discussed in connection with radio reception, in the
interest of greater clarity; however, all that is said is directly applicable to other
fields, like, for example, cable communication, acoustical and hydroacoustical signaling,
etc. remover, in the book, all signal and noise disturbances aro considered to be oscil?
lations of voltage; however, nothing is changed if we consider instead oscillations of
current, acoustical pressure, or of any other quantity which characterizes the disturbance
acting on the receiver.
This book does not consider certain irredular perturbations of the signals, which
can strongly affect both the operation of radio receivers and their noise Immunity.
Examples of such pertJrbations are fading, echo phenomena, etc:. remover, it should be
kept in mind that in this book the word noise is henceforth (for brevity) understood to
refer to normal fluctuation noise; indeed, this is the only kind of noise which will be
considered.
CHAPTM 2
AUXILIARY MATHEMATICAL lATERIAL
2-1. Sono definitions
te now introduce some definitions which simplify the subsequent analysis. re assume
that all waveforms under consideration lie in the interval ?T/2,+TA, which is obviously
always the case for sufficiently largo T.
The mean value of a waveform A(t) over the interval T is designated by
(2-1) A(t) =
A(t) at
By the scram- product of two functions A(t) and B(t), we understand tho mean value of
their product over the interval ?TA,+TA. Thus, the scalar product is
+TA
t (2-2) A(t)B(t) = 1 I
Y A(t)B(t) dt .
?TA
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It is clear from the definition that
(2-3) A(t)B(t) = B(t)A(t)
Furthermore
(2-4) AW[B(t) + C(t)] = A(t)B(t) + A(t)C(t)
and
(2-5) [aA(0][6B(t)] = ab A(t) B(t)
'hero a and b aro arbitrary constants. Thus, the scalar product of functions has tho same
properties as tho scalar product of vectors; instead of scalars we havo constants, and
instead of vectors we have functions.
We write
41/t
1
(2-6) A2(t) AWA(t) = a; j. A2(t) dt
In what follows, we shall often encounter the quantity
+TA
(2-7) T A2(t) = f A2(t) dt .
-TA
This quantity will be called the specific energy of the waveform A(t). It equals the
energy expanded in a resistance of 1 ohm acted upon by tho voltage A(t). The quantity
will be called the effoctivo value of the wavoform A(t). A function with effective value
is said to be normalized.
If two functions differ only by a constant, they aro said to coincide in diroction.
The normalized function which coincides in direction with a givon function A(t) is
obviously
We shall say that the functions Al(t), A2(t) An(t) are (mutually) orthogonal, if
(2-10) Ai (t)AI (t) = 0
for all 1 ?i,.Q n, except when i .
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2-2. Representation of a function as a linear combination of orthonornal functions
If the system of functions
(2-11) C1(t),C2(t) c(t)
satisfies the equations
(2-12) C2 (t) = 1
(2-13) Ck(t)C.1(t) = 0 ,
where 1..Sk,Itcn and kit we say that it is a system of az-thermal functions. An example
of such a system of functions is the system
1(t) = sin 4t
I2(t) = VIE cos 4rt
Zn
I3(t) = vr sin 2
I4(t) = VE cos 2 31/A
12n,1(t) = Na. sin Tact
I2m(t) = \PI COS M ;111t
since for this system the relations
(2-15) 1 Ik(t)12,(t) = 0 (k yfit)
are valid. We shall say that a function A(t) can be represented as a linear combination
of a system of functions
(2-16) C1(t).C2(t),...,Cn(t)
if we can write
(2-17) A(t) =? akyt)
k=1
whore some of the ak may vanish.
If we assume that the functions (2-16) are orthonormal, then, taking the scalar
product of both sides of Eq. (2-17) with Cl(t) and expanding, we obtain, with the use of
Eqs. (2-12) and (2-13)
(2-16) At)C1(t1= .
We call the coefficients ak the coordinates of the function AM in the system (2-16).
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Obviously, the function A(t) is completely characterized by the n coordinates
if the system (2-16) is specified. In particular, if we take as the system of ortho-
normal functions the system (2-14), we obtain
(2-19) A(t) I(t) ,
where
(2-20)
al = A(t)I(t) .
The series (2-19) is the familiar expansion of the function A(t) as a Fourier series in
the interval -TA,+Tit. According to !,2-14), the amplitude of the cosino term of fre-
quency milf is VT a2m, and the amplitude of the corresponding sine term is
VIE a2m-1.
If the oscillation A(t) is a signal, then we usually only consider a finite number
of terns of the sum (2-19), with indices from 2 to c2, say, since the components of the
1
signal are as a rule so small outside a certain frequency range that they are masked by
noise or by the components of other signals being transmitted on neighboring frequencies.
In this case
(2-21)
A(t)
api t(t ) .
1
Let a1 an be the coordinates of the function A(t) in the system (2-6) and let
b1 bn be the coordinates r.f 4-1,n function B(t) in the same system. Then
(2-22) A(t)B(t) = [EakCk(t) lEbkCk(01=
k=1 .k.1 k=1 k k
which follows easily by expanding and using Eqs. (2-12) and (2-13). As a special case,
we have
n
(2-23) A2(
(t) = A(t)A(t) =
'
k
k=1
If C(t) is a normalized function with coordinatesel cn, then
n
(2-24) ;E: c2
k = 1 .
k=1
Furthermore, if the functions A(t) and B(t) arc orthogonal, then according to the formula
(2-22) and the orthogonality condition (2-10), we have
(2-25) :Eakbk = A(t)B(t) = 0 .
k=1
The expressions (2-22), (2-23), and (2-25) are the analogs of the corresponding expressions
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of vector analysis.
Finally, we show that if two functions
A(t)
al (t)
km0
xt
B(t) bkIk(t)
have no components with identical frequencies, i.e., if for all indices k jI 0, either one
of tho ak or one of the bk is zero, then
(2-26)
A(t)E1(t) A(t) B (t )
Indeed, under these conditions
and furthermore
-itic)1-317-7t ? abo
ao, B(t) bo
whence B. (2-20) follows at once.
2-3. Normal fluctuation noise
Ve shall consider noise consisting of a large number of short pulses, randomly distri?
buted in time. Such noise will bo called normal fluctuation noise. This kind of noise
includes thermal noise in conductors, shot noise in vacuum tubes, and, in many cases, atmo?
spheric and man?made noise as viol). Such a noise process can be represented by the expres?
sion
(2-27) W(t) Fk(t?tk) .
k=1
where Fk(t?tk) is the koth pulse occurring in the interval ?Tit,+TA. Te assume that the
pulses are short and begin at the times tk. Thus
(2-28) Fk(t.t,) = 0 for ttk:t6 .
Bore the pulses are to be numbered by the indices k not in the order of their occurrence
in time, but (say) in order of decreasing amplitude.
Suppose that the probability of tk falling in it subinterval of length dt is dt/tr, and
that it does not depend on the location of the subinterval within the interval ?T/t,+TA
nor on the other pulses. Aorcover suppose that A(t) = 0. Then we find that
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(2-29) v(t)A(t)
10
41t
2-- EFk(t-t) A(t) dt
-T k=1 k=1
/z
where
+TA
(2-30) r . 2.j F(t-t) A(t, k k) dt.
k T
-T/2
Assuming that 6 is so small that A(t) changes negligibly in the time 6, wo obtain
A(tk) t k(+6 A(t k)
(2-31) lr =
tJ Fk(t-tk) d T t = qk .
k T
k
where
6
(2-32) qk = F(t) dt
0
is the area of the kith pulse.
The summands rk are mutually independent random variables. If they are bounded, and
if the sum of their variances increases without limit as the number of summands is
in-
creased, then, according to probability theory, we obtain in the limit of infinite n
(2-33)
lin
- Er,
k=1
V42: DlEk
k=1
SC
43A
whereis the mean value, and D rk E(gk - E rk)2 is tho variance of the quantityck;
E E.k
Q'A is the random variable with distribution law
2
(2-34) F(
e dx
x