SELECTED ARTICLES BY VARIOUS AUTHORS FROM RADIOTEKHNIKA NR. 6

• Document Type: CREST
• Collection: General CIA Records
• Document Number (FOIA) /ESDN (CREST): CIA-RDP81-01043R001900020013-1
• Release Decision: RIPPUB
• Original Classification: K
• Document Page Count: 112
• Document Creation Date: December 27, 2016
• Document Release Date: April 2, 2013
• Sequence Number: 13
• Publication Date: January 1, 1955
• Content Type: REPORT

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Table of !Contents
Page
Passage of Signal and Noise through a Limiter and
Differentiating System, by V.I.Bunimovich .............. 1
Reception of :Pulse Signals by the Method of Mutual
Correlation, by V.I.Chaykovskiy ......................... 16
:.
Build-Up Processes sari Detection of Pulse Signals,'
by A.A.Kulikovskiy ..................................... 23
34
44
with Plane-Parallel Gratings, by A.M.Modelt ............ 68
Triode Frequency Converters for Meter Waves, by I.I.Levenstern
and G.G.Kostandi .......................................
A Method of Studying Transient Processes in Linear Systems,
by ft.D.Leytes and L.N.Gutman ...........................
f
Propagation of a Plane Electromagnetic Wave in Space Filled
Relative Evaluation of Communications Channels using
Different Systems of Modulation, with Respect to
? their Traffic-Carrying Capacity, by A.G.Zyuko .......... 77
Increasing the Efficiency of Reactance Tubes, by A.D.Artym ..... 90
Authorfs Abstract -Band Resonant Systems with Constant Pass
Band,~by I.M.Simontov .................................. lOb
New Books ...................................................... 110
i ;~. ~ ?STAT
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PASSAGE OF SIGNAL AND NOISE THROUGH A LIMITER AND DIFFERIIJTIATING_SYSTEM
- by ,
V.T.Bunimovich,
Full Member of the Society
The mean number of pulses and the law of distribution of pulse height obtained
as a result of the passage of a sinusoidal signal accompanied by noise through a
limiter and a differentiating system, are determined.
1. In this paper the following problem is considered. The voltage at the in-
put of a system consisting of a limiter and a differentiating system is composed cf
a sinusoidal voltage (signal) and a voltage of fluctuating character (noise)s which
obeys the normal law of probability distribution. The spectres of the fluctuations
is concentrated mainly in a relatively narrow frequency band. Within the limits of
? this same band lies the signal frequency.
At the output of the system, as a result of the limiting and differentiation
pulses are formed, whose height, owing to the presence of noise, is distributed
according to a certain law. The mean number of pulses in unit time whose height ex-
ceeds an assigned value is determined, together with the law of distribution of
pulse height. The assigned values are the effective values of the sinusoidal signal.
and noise at the input.
It is assumed in. the calculation that the voltage at the output of the dif-
__ ferentiating system~is~proportional to the derivative of the voltage at its input.
;For this reason the following discussion is applicable directly to the practical
---case, under the condition that the time constant of the actual differentiating sys-
---~tem is sufficiently small.
.,: ~ ~ ~ ,
_ -6 2. The variations with time of the quantities with which we are concerned are
illustrated schematically by the curves of Fig.l. A straight line parallel to the
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__ axis of abscissas, marks the level of limiting._ The curve of U the dashed ine
'represents the time-dependence of_the total voltage at the limiter input.- The curve
~~ of voltage at the limiter output (the heavy continuous line) is formed of segments of
_,the curve U and of rectilinear portions parallel to-the -time axis
-(andlloc`at~ at the
? _distance a away from it, that distance being equal to the level of limiting), and of
U,V
Fig.l
?
portions coinciding with that axis. The same figure shows the pulses formed as a
_ result of the differentiation of the voltage after its limiting. The pulse height is
denoted by H. If there were no limiting (more exactly, if the level of limiting
a -- ~ ), then the voltage at the output of the differentiating system would not be of
pulse character, but would vary by a law represented by the curve V (the dotted line).
The vertexes of the pulses obtained in the presence of the limiter are formed by seg-
ments of the curve ~.
3. From here cri we shall assume that the spectrum of fluctuation at the input
__..is concentrated, in the main, in a relatively narrow band of frequencies, that is,
cif p ~ is the effective width and it0 the mean frequency of the spectrum at input,
then ~~ ? 1 (in practice, ~~ is equal: to the effective pass-band width of the
_? o
_ ',narrow band filter cor_nected at the input 'of the device and separating a definite
--'spectral band of the fluctuation voltage ). The curve of time-dependence. of such
.voltage, having a narrow spectral band, is approximately sinusoidal in character,
2
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that is, it has?the form of a distinct sinusoid which frequency is equal to the mean
` frequency of the spectrum. This fact is known from experiment (cf., for example,
reference 1), and i:s easily justified theoretically. The curve of total voltage at
the input, U, has the same character, and is made up of the fluctuation voltage with
? a narrow spectral band, and the pure sinusoidal signal voltage. The same thing may
be asserted with respect to the character of the time dependence of the voltage V
(cf..Fig.l) obtained as a result of differentiating the quantity.
Thus the pulse heights H, which would be produced in the presence of limitation
(a being finite)-are equal to the maximum values (amplitudes) of the approximately
sinusoidal quantity V which would be observed at the output of a differentiating net-
work in the absence of limitation (a = ~ ).
Thus the problem of determining the law of distribution of pulse height H re-
duces to the linear problem of the distribution of the maxima (amplitudes) of the
approximately sinusoidal quantity V, representing the result of the differentiation
. of the approximately sinusoidal quantity U. Thus, in solving this problem, the limi-
ter is completely eliminated from the discussion. We remark merely that the value of
the level at which a is limited obviously affects only the pulse duration.
l~. To determine the mean number of maxima of H (that is, the mean number of
pulses) exceeding-a-certain value, let us use at first a method (Bibl.2) directly
based on the fact that the curve V has the form of a "sinusoid" of frequency equal
. to S2 p, and of 'slowly varying amplitude and phase.
'Let us~write the expression for the probability density of the quantity H, in
other words, for the probability that the value of the maximum of the curve V lies
between H and H,+ dH. If the signal were absent, then the probability sought would
be equal to -
n
2 cl
lY/ c H1 dH = t~ e ,~ dH,
(1)
since the latter expression represents, as is commonly known, the distribution law
for the envelope of the quantity V (Bibl.3) if V obeys the normal law. The quantity
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Qv in the formula (Bibl.l) represents the mean-square value of V, ov ~. In the
presence of the signal, the quantity V is the sum of the fluctuation and sinusoidal
voltages, because such a sum is the full voltage U at the input of the differentia-
ting network. We recall that the problem has been reduced to a linear problem. The
limiter has been f~eliminatedt~ (cf. above, Section 3). That is, if at the input
_ 11 = U? -~- ES sin ~'u 1, (2 )
where Un is the voltage of the fluctuating component and Es is the signal amplitude,
then at the output we shall have
V ^ U, V~ = Un and ~Js - `~U LS
l 1+ )
(the dot on top denotes differentiation with respec~ to tune).
In the presence of a signal, the larr of distribution for the amplitude (envelope)
of the value will have the form (Bib1.3, pages 296(21) and 339(50): 4
I/~ ~1- /1S
'l o~'
ii~?p,,,,(H) dN = rV c ~ ]~ ( 1 ~V Nl dH (5 )
o` ` c1 J
where I~ (x) is the Bessel function of zero order of the imaginary argument.
Equation (1) is a special case of eq.(5) for Hs =0.
According to what has been said earlier, eq. (5) represents the law of pulse dis-
tribution according to ?eight (cf. Fi~.~).
For simplicity, we now introduce the notation: `
h_ ii hs_ HS (6)
.. ~y oV
Then the probability of~the quantity h lying between h and h + dh will be
(cf.eq.5) equal to =
It?,, (It) dh = h ~ s I~ (hs h) dh.
(Sa)
The transition from eq.(5) to eq.(Sa) may be considered as a selection of the
?
units of measurement of V such that V2 = 1. Ole shall likewise term the nonmed value
of h the pulse height.
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5. The probability P(h0) that the pulse height exceeds the value h0, i. e., the
probability of the inequality h > h0, will be, on the basis of eq.(5a):
- hs _ r
-- h"
P (!r~) = e t h e s I? (/zs lr) dh =
h,,
h;
-ha ~ h2
= 1- ~ 2 f It e z Io t lrs h) dfr,
since (Bibl. G.) _ hs
(7)
~~h c '` Jo (h~ h) dh = e `
For h0 = 0, the value of P (h0) will be equal, as indeed it should, to P (0) =1.
(It should be noted that eq. (7) is in agreement, as indeed it should be, with the
definition of opposite events).
Let us now denote by PIl(h0) the mean number of pulses in unit time whose height
exceeds an assigned value h0. Then the mean number of pulses during the period
T = ~ will be equal to P1T(h0) = P11(h0) T. During the course of any time inter-
0
? val of duration T, one pulse occurs (since the number of pulses in unit time, equal
to the number of naxima of the "sinusoid', is equal to the frequency ,i = ~ ). For
this reason the number of pulses, during a period, whose height exceeds a certain
value h0 > 0, may be equal either to 1 or to 0. The probability of the former
event (h > h0) equals P(h0), the probability of the latter event (h ~
On the basis of the well known integral expression for a Bessel function
x ~?s
I? (x) = r ~ e dx
h~ h cos (4o t + gyp)
~=s h cos 4?!
r.
1
r
-
e
_ e
?
r. 1
0
ehs/t cns xd ~ _ j s
,(h ft~.
(26 )
By virtue of eq. (26 ), after averaging over time, the integrand in eq.(25) takes
N~ (ho) _ ~,,, ~ 2 (~ h e ' to (hs /1) dh. (27 )
~ ,~ I
? The expression so obtained for the mean number of pulses agrees with that found
earlier by a different method (cf.Section /~, eq. (9) ).
9. r,quation(5a) for the law of pulse-height distribution follows from eq. (9 ).
Indeed, let n pulses occur in unit time, where n is a random quantity. If t~1h is
the pulse-height probability density, then the mean number of pulses per unit time
exceeding h in height, is equal to
N, (h) = is .~ ~h (X) dx, (28 )
h
where n =mean number of pulses, in unit time, which is obviously equal to n ~ Nl (0).
and on the basis of eq.(27) (or of eq.(9)), the expression for the wanted pulse-
It follows from eq.(28) that
height probability density will be
h'+hs
~~h (fl) = fl L ~' ~0 (fTsTt)~
(30)
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?
i~rhich agrees Frith eq. (Sa).
10. Equations (3 0) and (27 ), or egs.(5a) and (9), so obtained, represent the
solution of the problem posed (cf. also eqs. (11 ), (12) and (11~)).
Thus the law of pulse height distribution is defined by the expression of eq.(30)
h~ + h~
Wh(h)=1tc 7 1~(Jr h) (I)
and the mean number of pulses related to unit time and exceeding the value h0 in
height, is equal to
N~ (ll..) = 2 ~ Nr (fro
h~
c
l1' It ~ -
h,
ht h~ (II)
-- - h - __.
= 1 -~ r = j h ~ to (hsh) dh
(cf.eq. (7)). The quantity P~T(h0) is equal to the mean number of pulses for the
period T = ~ = f of a height exceeding the value h0.
? o
At great values of h, or of hs, at which hs h ? 1 (cf.eq.(I)), or (cf.(II)),
the well known asymptotic expression for a Bessel function may be used:
CA.
'L>t .r
the form
Then the expression for the distribution law of eq. (I) may be represented in
~n-hs~=
e - ; hs h~~ 1),
while the expression for the mean number of pulses for a period may be expressed by
a" (!t -h
_ 1 - --t
NT(hu) ~/Yr J Y ~l e 2 dlt. (IIa)
S
By performing the exchange of the integration variable x = h - hs in the latter
equation, we get?
lV r (hog Yen J y 1 t ~, e 2 dx.
S
h.-hs ~ (IIf a )
?
At h0 < hs (i. e., for h0 - hs < 0), and at sufficiently great values for hs,
the following approximation will hold:
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ou _ x'
NT(h?~~ ~2n J e s dr __ i _f t
~ y~Yn
-lhs -hol
11. Equations (I) and (II) are graphically represented in Figs.2-6.
(31)
Figure 2 gives curves for the pulse-height probability density h H/a y at var-
ious values of hs; i.e., Hs representing the pulse height in the absence of noise
(or in other words, ac various values of the signal amplitude Es at the input, or,
more accurately, at various values of the signal-noise ratio at the input (cf.eq.ll).
The curves are constructed by eq.(1). The curves for hs < 5 are constructed by the
approximate (asymptotic) eq.(Ia). As hs increases, i.e., as the signal amplitude
increases, the most probable values of the pulse height (i.e., the values of H, cor-
responding to the maxima of probability density) approach, as indeed was to be expec-
?
ted, the corresponding values Hs = Evhs, equal to the pulse height in the absence of
-1
2
3
5
r
-
-
~.
] t '2 3
4
5 6 7 8
9 10 11 t
~
Fig.2
On Fig.3 the curves are plotted (solid lines) for the mean number of pulses ex-
6vh0 is plotted against
the ratio,h? , i.e., of the ratio of HO to the height Hs, which the pulses would
c
? have in the absence of noise ( 1? = H? ). The mean number of pulses during the
s s
ceedng the value h0 in a period, or in other words, HO
period represents, in other words, the ratio between the mean number of pulses for
i
any desired time interval in the presence of noise to the mean number of pulses for
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?
1
the same time interval in the absence of noise. The curves are constructed by eq.(II)
by mean's of numerical integration. The curves for hs > 5 are constructed 'by the aid
of the approximate formula (IIa). At hs -~ ~ (~ -- m or oU -. G), i. e., in the limit,
in the absence of noise, the form of the curve for the mean number of pulses NT
plotted against h0 tends, as was to be expected, to rectangular. Namely
N,.(h~)=~ I art (32)
~t,
if hs -- ~. Equation(II) in the limit as hs ~ ~ passes over into eq.(32). Indeed,
n~hol. N~(h?I ~Nr~hu~l/~r.ll
Fig.3
h
at ho
s
< 1, i. e., at hs - h0
> 0, and at h ~ ~ , we have
s
{'..~ .1
~,
> 1, i. e., at h0 - hs > 0 and hs
On the other hand, at ho
s
?~ , the value of
NT (h0) approaches 0, as follovrs directly from the approximate eq. (IIa).
On Fig.3 are also constructed (with dashed lines) curves for the quantity
_ , NT(hp) - [NT (h0)]hs = 0, equal to the difference between the mean value (for the
period) of the number of pulses in the presence of the signal on the corresponding
? quantity in the presence only of noise. This difference is obviousl e
ratio N(h?) -[N(h?)]hc = 0 ~ Y ,qual to the
_ which. in a cPrt.a;n co?QO ,,.,,,.,,.,...,...,._ iL_
U
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. noise ratio at the output. In this connection the fact that these curves have maxima
must be taken into account. At large values of the ratio h-? = H? ~ the dashed cur-
- hs Hs
ves for the difference NT(h0) - [I~IT(h0)]hs
? the quantity t1T(h0).
Nr~hi)"~.lr(~t)~h -0 Ns~hu~-~Nr~hi)~hs-o
0 tends to coincide with the ct~rvc~ for
z~
~1--
!,6 1,6 ..iRs
Fig. /~
? On Fig. S, curves for the same quantities are constructed as in Fig.3, but here
h0 is the independent variable. The curves given in Fig.l~ (solid lines) for the
NT (!~) - [Nr (l~u)1 h,-o - 1 -' [N (ho)1~5 _o
1Vr (!i,) n inn/
likewise in a certain sense characterize the signal-noise ratio (more exactly, the
Nr~he)? ^'r(n.)-I~'rrlho~~h?0
s
-
~_
i
-
~~
--
.7
-
-
- ?---.._. .. _ _._ --
J ~~
~
~ ~ ~
~
~
lho~
N
r
r
r 2
-:
-
---- Nr(hr)-~Nr~ho)Jh
s-0
~ -
3
5
-0
~
~
?
`
-
- -
-
-
--
~~
~
-~-
--
-
---
-
,'
_
3,
i
~
~~.
a
n r
n r
~ rs
Fig.S
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signal noise ratio at the output). The ratio
N (hor
~~~ (/ro~~h ?o , representing the
necessary signal-noise ratio, tends to increase without limit when the level of hO
increases without limit, i. e., at a signal as weak as may be assigned; the ratio of
? the mean number of pulses of height ecceeding a certain value h0, in the presence of
a signal, to the mean number of such pulses, in the absence of a signal, tends to
increase without limit as h0 increases. Of course, however, the actual number of
pulses in both cases does approach zero.
Nrrho~-~'Nr2~ho1~~.c . N?(h~)-INr (hu)~hs 9
. n Nr 1!rp~ __
~`, --~` Lr~,,,, Pr.~r49~y-p r0~
'. ~ I ~ ~ ~ ~
.. - 1 I I~
Fig.b
7 6 9 rn rr >Z h~
Figure 1~ also gives curves (dashed lines) for the difference NT(h0) - [NT(h0)]hs=
= 0 which were also represented in Fig.3. These curves at small values of the ratio
, coincide with the curves for the
uantit
h
q
y s
s N (h
Figure 6 represents the same quantities shown in'Fi Tl~ ? ra hicallT
- g~ g P 3, as a func-
tion of h0. As large values oh2hs, the quantity NT(h?) ~NT(h?)]}'s = ~
NT(li? tends to
coincide with tY,e curve 1 - e ~ (beginnin in
g practice with values of hs equal,
let us say, to 5),
?Paper Received by the Fditors 30 September 1953.
BIBLIOGRAPHY
? ~l. Granovskiy,V.L. -Electrical Fluctuations. Moscow-Leningrad, 1936.
2. Bunimovich,V.I:, and Leontovi.ch,T4I.A. '- Doc. AN SSSR 53, 21-24, (1916 ).
3.` Bunimovich,V.I.'- Fluctuational Processes in Radio Receiving Installations.
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Moscow, 1951 -
1+. Ryzhik,I.M. and Gradshteyn, I.S. -Tables of Integrals, Sums, Series, and Pro-
ducts. Moscow-Leningrad, 1951.
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RECEPTION OF PULSE SIG?7ALS~BY THE t~THOD OF MUTUAL CORRELATION
by
V.I.Chaykovskiy
Full t4ember of the Society
The ratio of signal to fluctuation noise on the reception of pulse signals by
the method of mutual correlation is determined; the ratio so obtained is compared
with the corresponding ratio at the output of an ideal band filter.
The purpose of this paper is to analyze the noiseproof features on the recep-
tion of pulse signals by the method of mutual correlation (Bibl.l, 2~ 3)? From the
block diagram of the receiving installation shown in Fig.l, it follows that the mix-
ture of the useful signal UF(t) and the fluctuation noise xF(t) passing through the
input filter, is multiplied by the signal of the local heterodyne e(tl. The product
of these quantities is averaged by the means of an integrating device. Thus there
exists at the output of the correlation receiver a discrete series of consecutive
values of yT(ti)~ which may be written in the form:
T
T
VT(t) - ~ (~ rtF (f~ ,' a:F (l)~ c' (~f ;- ~, cll = T .J t~' (f ~ t' (l -;- ~l clf
` ~~
_ `' (1)
+ T .J ~f ~f) r' (f -}- T) [If ='?u,~T (T~ ~- t"~xrT ~t~'
n
This quantity is the sum of the short-time functions of mutual correlation of
the noise xF(t) and of the useful signal uF(t) with the signal of the local hetero-
cyne e(t). The noise at the output of the correlation receiver is represented by a
function of mutual correlation between the noise and the local signal cp~XeT(~)~ and
the useful signal is represented by a function of mutual correlation between the
useful and local' signals ~ ueT(~ )? Since the interval of averaging T i ~ a finite
quantity, ~xeT(T~ `nd TurT(T) are functions of the time, the function ~xeT(~) as the
mean of a random quantity over the dine in a finite interval is a stochastic func-
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tion of time and is characterized, as usual, by the mean square of the fluctuation.
It follotrs.from this that the signal-noise ratio .at the output of the correlation
_
---
_
_
~
~'
"L
~ ~
__
F
ig.l
receiver may be determined as the ratio of
the square of the short-time function of
the mutual correlation of the useful and
local signals QS, calcutated at the in-
stant of existence of the signal, to the
mean square of fluctuation of the short-
time function of mutual correlation of the
a) Input filter;~~c~c; b) Zhzltiplier; 2
noise and the local signal 6 x. It will be
c) Integrating (averaging) unit; d) Gen-
shown below that the quantity Q x decreases
erator of local signal e(t)
with increasing T; however, an increase of
T leads to a reduction of accuracy in determining the position of the useful signal
pulse on the time axis (the duration of the interval of averaging equals the maximum
? error in determining the instant of occurrence of the useful signal pulse). For this
reason, by comparing the signal-noise ratio at the output of the correlation receiver
with the corresponding ratio at the output of an ideal band filter, it is necessary
to adopt the condition that the error in determining the position of the pulse on the
time axis at the--output of an ideal band filter shall equal the corresponding error
.at the output of -the correlation receiver.
Let a use.ful~~ signal u(t ), in the form of a segment of a sinusoid of duration
and frequency silo, act on the input of a receiver with band ~c~c:
(2)
? and let the fluctuation noise x(t), determined as a certained stationary stochastic
- process, likewise .act on that input.
The transfer ratio of .the filter at the receiver input will be:
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11 .___ , J - t) , J
The local signal, which frequency is always taken equal to the frequency of the
local signal, ~ ,
(~r? )
Let us determine the noise at?the output of the correlation receiver, i. e., the
mean value of-the"square of the fluctuation of the short-time function of mutual cor-
relation of the noise and the local signal 6 x.
It is t,rell known (Bibl.4) that the mean value of the square of the fluctuations
on repeated calculation of the mean value of some stochastic quantity Z(t) with time,
over a finite interval T, may be expressed in the form:
n == V at a, ~ u,~- ----dRO cu ] W~1 i
'J ?J
(5)
? In our case ~ zZ(T) is a function of the autocorrelation of the product
xF(t) e (t + T); Z(t) is a function of the mutual correlation of noise and local sig-
nal, which, by virtue of the statistical independence of noise and local signal, is
equal to zero. .-
The functionYof autocorrelation of ,the products xF(t) e (t + T) may be repre-
sented (Bib1.5)~a? the product"of the function of autocorrelation of the noise at the
filter output eq. (3 ), and a function of autocorrelation of the local signal e(t):
where bz is the noise power on unit bandx.
COS' wa,, (61 -
Here and below we shall consider the value of the power given off across a resis-
tance of 1 ohm. ,
18 STAT,
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~~o~ T
y' _ ~In -
oX = LAG=' l - l l COS= wot d T.
. T ~( TI ;
Integration of this expression gives;_
a2 = giG~ sl A~o~ 1' _ 2 1 -COs ~m~ T /~ -}- 2T, { si [(~A 2 - Zwol Tl -
x T{ 2 OwA T C l 1
2
?
(9)
(10)
ow~'r
Figure 2 shows the values of ~I' ( 2 ) ?
The value of the short-time function of the mutual correlation of the useful and
local signals c~ueT(T), calculated at the
0,5
instant of existence of the useful signal,
Q4
1~ ~ ~ ! .~
depends on the duration of the interval of
I 'II'I
~ I I i. ~ I ~ ~~;
~
I
.
~ ` ~ ~ ~ ' ~ i
I ~
-
i
~
I'
~~
~
- -
}-
~I
~-~~,~
?Fig:2
(7)
L1 -COs I?(~ 2 + `~wol T~l } . (8~
At ~r`~c ? 2~~0, the sum of the last two terms of eq. (8) may be neglected, and
~~
then
_~~
z ~w< ~ ~
_~_ are based on the same method, which allows, to the maximum extent, satisfaction cf ~
~ the basic requirements`for modern iJSW fre~uency converters`:`--
.`3:~
!! The selection of one of these circuitflts or the other depends mainly on design
^_, considerations, in particular, on?the tuning element in the USW band and on the quesi
~ ' - -i t i
?
tion of its mechanical connection with they tuning elements on the other bands of the,
I
receiver. Vlhen a block of variable capacitors is used as the tuning element for the?
_ ? USj~T range, the converter circuit using the inductive bridge should be employed, 1
while when a unit of_.variable inductances,is used for tuning, the capacitative
bridge circuit should be used instead.
It must be noted that when modern triodes, having high transconductance charact-
eristics and small interelectrode capacitances (S 5 - 6 ma/v, Cpc 0.2-0.25??)
are used in the high frequency amplification stage, it is possible to obtain stable
amplification at high frequency, of the order of 10-12.
Thanks to so considerable a high-frequency amplification, the converter noise
will have practically' no effect on the noise factor of the receiver, and this factor
will be determined by the noise of the high-frequency amplification stage.
J
Paper Received by the Fditor 16 July 1951L.
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.~.
A 2~~THOD OF STUDY.ING_.TRANS T_ PftQCESSES_IN_LINEAR_SYSTk~ _ --T _ _ ~
~~
by
R.D.Leytes and L.N.Gutman
Full Members of the Society
~~~
' In this paper an approximate
-~ i
oped, based on the application of
t? _, -
_._ equation under consideration.
It is shown that after the introduction of special coefficients a simple ex-
pression is obtained, which relates the input and output voltages of the system.
This expression allows the solution of a number of problems relating to transient
processes in amplifiers. As an example, a stage with plate correction is investi-
gated.
? A method of approximate determination of the transient characteristics directly;
from the differential equation describing this circuit is presented.
__ The investigation of transient processes in linear systems reduces in most ~
.- cases either to the direct application.. of the inversion Svertyvaniya integral, or i~i
- i
_ to the solution of an integral equation of the Volterre type. ~ I I
I
The former problem is always solved,,but the functions which in ,practice enter
,,
into the equation may prove to be so complex that there are serious difficulties in~:-.
~, ;
when certain of the functions are assigned graphically and cannot be expressed with
finding the solution of the integral equation. In addition, cases are possible.
---- "Read before the All Union Scientific Ses'siori of VPJORiE [All Union Scientific Wand
.,..May 1953.?
method of studying transient processes is devel-
the theory of finite differences to the integral
3
?a.rc~yuv,__
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~" ;-;~----------- l ~~ - - ~~~---- - ical form. It is therefore
^, a sufficient+degree of accuracy in a convenient analyt'
~~ ~ ?,
-1 very desirable to use approx5lnate methods of investigating transient processes. A
_f^
large riumber~of studies conducted in various directions, have been devoted to this
question.
~' ..-
In a considerable number of studies, the f+method of time sequencest' (Bibl.l)
"is used. This method is based on the fact that a continuous function of time with
a bounded spectrum_i.s determined by the assignment of a finite series of numbers,
connected with the values of this function taken at discrete instants of time. A
proof of this theorem was introduced into communications theory by V.A.Kotelfnikov '
in 1933, and was subsequently (191L9) treated by Shannon.
In essence the method of time sequences is a method of the theory of finite
differences, applied to the study of transient processes in linear systems; this
connection has also been pointed out in the literature (Bibl.2). For this reason,
this group may also include works based on the approximate representation of differ-
. _ ential equations describing diagrams by equations in finite differences, and also
studies in which the methods of numerical solutions of integral equations are em-
ployed. (Bibl.3-6 ).
The approximate method of study of transient processes developed in the present
paper may be included in this group of worksy. A feature of the method is the con-
venient computational formulas, which allow the solution of a number of problems
connected with transient processes in complicated diagrams.
1. APPLICATION-OF THE I~"rHOD OF FINITE DIFFERENCES TO THE DUHAMII, INTEuRAL
Derivation of the Fundamental Relation between Output and Input Voltages
The voltage~=a.t,the output of the system U(t) is connected with the voltage at
__'the input V(t) by the well known relation.(Duhamel integral): '
??.,; "' I
.~
~_ ~'~he present paper was written under the influence of a study. _(Bibl.7)--belongingw to_
rr.: '1. L... P, n1~ n4' mA~'.PA't'AZ~LT~.
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U(1)= h(0)V (t)-}-.~k'(E)V (t - E)dE,
u
_~ _
_;
. ~ where t =time, h(t )
? _ tem (reaction of the
For application
=_transient characteristic or transient__admittance?of the_sys-
system to a single voltage shock).
of the method of finite differences it is convenient to intro-
duce ? the notation U (m T) Um, V(m T) Vm, h(m T) = hm, where T = step of interpola-
tion. Then eq. (1) takes the form ~ ?
Linear interpolation is most convenient, for it makes it easy
error of this approximate method. Be Newtonts formula (Bibl.8)
consequently,
V nt : E ,
( - E) (n -}- 1)V,~-? - nV~,-,~-~ ~' (V~-,~-i -Vm-~+~ - ~' r~-~(E)
(3)
(n 0, 1, 2,..., m), m-n is the residual term equal to
R{ S
.U,~ = hoV,~ -{- f h' (E)V (n:.T - E) d E. (2)
(nt - tt - 1)TC c~-~