The metod simplifing inverse Laplace transformatijn at ossillatory processes researchet: Учебное пособие

66 Pages·0.642 MB·Russian

by  Zolotarev I. D.

Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»
Ministry of Education and Science of Russian Federation UDC 621.396.6+517.442(075)
Z 82
Omsk State University

Reviewers:

Dr.Sc., professor (Head of Applied Mathematics Chair
of Omsk State University of Transport) V.K. Okishev

Translated by D.A. Timochenko

Zolotarev I.D.
I.D. Zolotarev Z 82 The Method Simplifying Inverse Laplace Transformation At
Oscillatory Processes Researches. The "Amplitude, Phase, Fre-
quency" Problem In Radioelectronics And Its Solution / Transl.
THE METHOD SIMPLIFYING INVERSE
by DA Timochenko: Tutorial. – Omsk: OmSU Publishing,
LAPLACE TRANSFORMATION AT
2004. – 132 p.
OSCILLATORY PROCESSES RESEARCHES.
ISBN 5-7779-0472-6
THE "AMPLITUDE, PHASE, FREQUENCY" PROBLEM
IN RADIOELECTRONICS AND ITS SOLUTION The research method of transient processes in oscillatory systems
is stated. It permits essential simplification of the most difficult op-

eration in finding solution of a system differential equation – inverse
Tutorial
Laplace transformation. It is shown that complex signal provides cor-

rect definition of an envelope and phase of a real signal using this
This book is recommended by Siberian regional department
method. The obviousness of obtained solutions is achieved by engag-
of teaching methological association education in the field ing the spectral method. The examples of transient processes calcula-
of energetics and electrotechniques as educational textbook tion in developing radioelectronic devices are given.
for interuniversity practice This tutorial is intended for students, post-graduate students, en-
gineers, scientific employees of both radio and electrotechnical speci-
alities and also specialists in the field of measuring and automation
technology while researching dynamics of oscillating systems.

UDC 621.396.6+517.442(075)

© Золотарев И.Д., 2004

© Омский госуниверситет, 2004

© Zolotarev I.D., 2004

ISBN 5-7779-0472-6 © Omsk State University, 2004
OmSU Omsk
© Timoshenko D.A., transl. from
Publishing 2004
Rus. into Engl., 2004
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INTRODUCTION transient processes in electronic devices always attracts a serious atten-
tion of specialists [l–6].
At development of radioelectronic devices for different purposes The most prevalent at researches of transient processes in radio-
an engineer frequently has to solve a researching problem of impulse systems is the method of slowly varying envelopes (SVE) designed by
radiosignals passing through linear circuits. For solving the task in the S. I. Evtyanov. In this method essential decreasing of difficulty in solv-
time domain the operational calculus based on integral Laplace trans- ing linear differential equations (DE) while researching TP in oscillat-
formations is widely used. When considering the task in the frequency ing systems is achieved applying some particular simplifying assump-
domain the spectral method based on integral Fourier transformations tions (asymptotic method of the small parameter). In this case the ini-
is applied. Both these research approaches are tightly interlinked tial DE communicating the response of the linear system and the ener-
among themselves and sometimes are considered as a uniform method gizing radiosignal are converted into truncated symbolical equations
(the method of Fourier transformation). regarding to SVE [2]. The more narrow-band signals and systems are
When finding a response of a radioelectronic device (RED) to an studied the more precise solutions are obtained using the SVE method.
impulse energization applying the operational calculus the most diffi- As a measure of band narrowity of radiosignals and systems the ratios
cult operation is the inverse Laplace transformation (ILT) execution µ=∆ω ω and ε=2∆ω ω , where ∆ω – the width of a radiosignal
s c c r s
[1]. The difficulty of ILT especially increases for important radioelec- spectrum, ω – the filling high-frequency (HF), 2∆ω – the bandwidth
c c
tronic applications when a radioimpulse signal affects RED and if se-
of oscillating system, ω – resonant system frequency are usually con-
lective filters are included in a signal tract of RED (oscillating system). r
sidered. For narrow-band signals and systems we have the small pa-
It is stipulated by the fact that for radioimpulse signals and such reali-
rameters µ and ε (µ<< 1, ε<< 1). For wide-band and ultra wide-
zations of RED the imaging function (IF) of the researching system
response for the input disturbance has complex conjugate poles (CCP) band systems these parameters are comparable to the unit.
pairs. In these cases even for rather simple IF the difficulty and awk- Although the method of S. I. Evtyanov allows to simplify essen-
wardness of conversions when turning from the images space to the tially the difficulties in finding an enough precise solution for an enve-
originals one essentially raise in comparison with finding solutions for lope of a signal in the output of a radiosystem it does not provide the
real poles IF [2]. In the meantime the existing tendency of extreme in- authentic description of thin (phase) structure of an output radiosignal.
creasing of information processing speed in radiosystems requires to Considering the greatest possibilities and advantages of phase informa-
develop RED working in the dynamic mode when the conversions of a tion radiosystems operating in the dynamic mode [7] we notice that the
signal, taking off and processing its informative parameter are executed specified disadvantage of the SVE method is rather essential.
not after the termination of transient processes (TP) in the output of the Designed in [5–10] method simplifying inverse Laplace trans-
informative channel but during these processes. Generally because of formation ensures the same reducing of difficulties in solution obtain-
inevitable TP presence at energization of a radioelectronic system by ing as the SVE method. However when using the method [5–10] the
an impulse signal the form of it is distorted. The specified distortions precise (accurate to phase) solution for the radiosignal in the output of
corrupt the informative parameter of a signal (originate definite dy- an investigated radiosystem is obtained. Thus it is not necessary to in-
namic errors in system functioning). The researching TP in a system troduce simplifying assumptions which are peculiar to asymptotic
for the purpose of minimization of an error imported to the signal in- methods including the SVE one.
formative parameter by transient processes is one of necessary devel- Apart from great simplification of a solution determination by
opment stages for modern RED operating in the dynamic mode. There- the method [5–10], application of it for the important cases of oscilla-
fore the problem of development of methods simplifying researches of tory processes researches allows to obtain a system response definition
as a complex signal (CS). It facilitates dynamic modes in radiosystems
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researches and ensures more obvious interpretation of the obtained re- 1. INITIAL STATEMENTS
sults.
Complex signal is also used for radiosignal envelope and phase 1.1. The Function Of Time (Signal)
determination. So the complex function absolute value circumscribing
a radiosignal determines the envelope and its argument – the phase of a Let's represent a signal by the sum of monoharmonic compo-
radiosignal. The initial physical signal is accepted as a real part of CS. nents
But in such a setting CS is indefinite because of the possibility of find- n
f(t)= ∑A cos(ωt+ψ ), (1.1)
ing uncountable number of CS meeting to the same real signal. For ν ν ν
ν=1
elimination of the indeterminacy the operator uniquely linking the
where ω – the angular frequency of the νth component, ψ – the
imaginary and real parts of CS should be introduced. The envelope and ν ν
phase obtained from such CS should correspond to physical matter of initial phase, Aν – the amplitude of the νth signal component. If the
these parameters in the initial radiosignal. frequency ω is multiple to some frequency ω, where ω – the signal
ν 1 1
The task of radiosignal envelope and phase single-valued defini- first harmonic component frequency, the signal f(t) is the periodic
tion is one of the fundamental problems at modern radioelectronics (the
function of time with the period
problem "Amplitude, phase, frequency" – the APF problem). At pre-
2π 1
sent a popular solution of this problem at radioelectronics is the com- T = = .
ω f
plex representation of a radiosignal as analytical signal (AS). AS is 1 1
In this case the sum (1.1) is a Fourier series (FS).
determined from the initial real radiosignal over integral Hilbert con-
In more general case each component of a signal is powered up
versions [11–13]. An essential shortage of AS is some lack of fit of
obtained envelope and phase values in comparison with their physical in any moment tν. Then we shall rewrite (1.1) in the form
matter. Therefore alongside with AS the other forms of radiosignal n
f(t)= ∑A cos(ωt+ψ )1(t−t ), (1.2)
complex representation were also studied [14–16]. ν ν ν ν
ν=1
The method simplifying ILT considered in the tutorial allows to
where 1(t−t ) – the unit step saltus which is powered up at t =t , i.e.
obtain the radiosignal envelope and phase adequate to their physical ν ν
representation also for ultra bandwidth signals [9, 10, 17, 18]. It is im- 1(t−tν)=⎨⎧0, t<tν. (1.3)
portant because of the striving for more and more wide-band signals in ⎩1, t >tν
modern radioelectronics by virtue of their high self-descriptiveness [19,
It is the first kind rupture. The function is often redefined at the rupture
20].
point t as
ν
The given examples of transient processes calculation in radio-
f(t −0)+ f(t +0)
systems are intended for mastering operational calculus application for f(t )= ν ν ,
ν
2
researches of impulse signals passing through a signal tract of RED and
where ±0 means as much close as necessary on the right and on the left
in particular to acquisition of skill on application of the method simpli-
of the rupture point.
fying ILT. The generality of the spectral method and operational calcu-
Therefore in the case of a saltus in an arbitrary point t =t
lus has stipulated the expediency of their combined considering in the ν
tutorial. 1(t−t ) =1(t−tν−0)t=tν+1(t−tν+0)t=tν =1 .
ν t=tν 2 2
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Here the factor 1(t−t ) takes into account that the νth compo- convenient to consider the setting of these tasks on the basis of the
ν
nent is powered up at the moment (t−t ), i.e. it cuts off a part of the fig. 1.1
ν

νth sine component at t <t .
ν
More common form of a signal writing which takes into account
fin(t) L i n e a r fout(t)
the exponential form of sine components envelopes is often introduced
System
as:

n
f(t)= ∑A eβνtcos(ωt+ψ )1(t−t ). (1.4) Fig. 1.1
ν ν ν ν

ν=1
Much more common form of a signal writing When solving the analysis task on a given linear system and in-
put action it is required to define a signal in the system output. Mathe-
n
f(t)= ∑A tkνeβνtcos(ωt+ψ )1(t−t ). (1.5) matically the system can be given by some operator L. It can be the
ν ν ν ν
ν=1 differential equation of a system, the transfer response – К(p) or the
In these relations ν=1,n, where the line overhead means the integer frequency response – К(cid:5)(jω)or system time responses. Time responses
values of area, from 1 up to n. Thus the writing ν=1,n means that ν of the system are:
accepts integer values from 1 up to n. 1) the impulse response g(t) defined as a response of the system
In the formula (1.5) we consider that k accepts the integer val- to a signal of the δ-function kind;
ν
ues, and at the same time the magnitude of k depends on the form of 2) the transient response h(t) considered as a response of the sys-
ν
tem to a perturbation of the unit step saltus 1(t) kind.
the signal νth component envelope. We suppose that the νth compo-
There is a sole link between DE of the given system and all these
nent envelope is featured in (1.5) by the factor
responses [26]. The system response can be found over time character-
Aνeβνttkν1(t−tν). (1.6) istics applying the integral of superposition (Duhamel integral) repre-
Obviously the formula (1.4) is obtained from (1.5) supposing that k = senting a convolution of input signal and one of system time responses
ν
0 (regardless of theνnumber). The formula (1.2) is gained from (1.4) if [3,11]. If the concrete physical circuit for solving of the analysis task is
to put β = 0 regardless of the νnumber. taken then the operator of any kind is founded by usual way starting
ν
from the given circuit realization (for example, determining the fre-
Thus the formula (1.5) of a signal writing is the most common.
quency response or DE of the system on the given circuit).
The formulas (1.2) and (1.4) can be obtained from (1.5) as special
There are different definitions (synonyms) for signals in the in-
cases.
put and in the output of a system. For the input signal: energization,
The terms of the sum (1.5) sometimes are called secular func-
perturbation, input, input action. For the output signal: system (circuit)
tions of time [9]. Notice that the formula (1.5) generally represents a
response, response, output. Any of these terms are used here.
solution of the ordinary linear differential equation.
When solving the synthesis task of a system on given input per-

turbation and system response it is required to find its structure. For the
1.2. Setting The Task At Linear Systems Researching
case of mathematical system simulation the synthesis requires a system

operator determination. At physical system synthesis it is required to
There are two fundamental tasks at electronic systems research-
define concrete structure (for example, electronic circuit) which meets
ing and developing: the task of synthesis and the task of analysis. It is
the given operator. As distinct from the analysis task the synthesis one
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is ambiguous because different system (electronic circuit) realizations or compactly:
can meet the same operator. The synthesis task is usually considered in n dµy m dλx
a context of system optimization by selected criterion. In the whole the ∑ aµ dtµ = ∑ bλ dtλ . (1.8)
synthesis task of a system is much more complicated than the analysis µ=0 λ=0
one and assumes application of analysis methods in searching the op- The analysis task is reduced to a solution of the differential equation
timum realization [27, 28]. (1.7)
Let's concentrate the attention on the system analysis task, i.e.

the task when the system output signal is searched on the given input x(t) n dµy m dλx y(t)
action and conversion operator. Generally a system can be acted by ∑ aµ dtµ = ∑ bλ dtλ
several signals applied to one or several points of it. Similarly output µ=0 λ=0
signal can be taken from several (or only one) points of the system.
Then it is possible to suppose that the vector x =[x ,x ,...x ], where m Fig. 1.3
1 2 m

– the number of input actions (in other writing
Schematically the "relation" input→system→output at definition
x ≡ f =[f ,f ,...f ]), operates in the input. We also have the vec-
in in1 in2 inm of a system by the differential equation is shown at fig. 1.3.
tor in the system output: y =[y ,y ,...y ], where l – the number of sig-
1 2 l
nals taking off from the system output (outputs). At multivariate ac-
tions and output signal we have a multivariate system operator L, i.e. 2. THE OPERATIONAL CALCULUS APPLICATION
y(t)=L{x(t)} (fig.1.2). FOR SOLVING LINEAR DIFFERENTIAL EQUATIONS

2.1. Initial Statements
х y
L
The operational calculus represents a powerful apparatus for
solving linear differential equations. So the image of the differential
Fig. 1.2
equation in the images space is searched. The simplification of a solu-

tion obtaining is reached at the expense of algebraization of the equa-
In this case the conversion operator, for example, can be represented by
tion when turning to the images space. It is achieved by application of
a set of equations. For simplification of further considering we shall
direct Laplace transformation (DFT) to the right and left part of the
take a system with one input and one output. System with several in-
differential equation (1.7). Besides the operational calculus allows to
puts and outputs can be researched in a similar way.
decrease essentially the difficulties of DE solution terms obtaining. The
Let the L operator is given in the form of the ordinary differential
values of the specified terms are determined by initial conditions (ini-
equation. Then in general appearance we shall write the link between
tial quantities of energy in energy-storage circuit elements) [1, 3, 29,
the input x(t) and output y(t) as follows:
30].
dny dn−1y dy
a +a +...+a +a y =
n dtn n−1 dtn−1 1 dt 0
(1.7)
dmx dm−1x
=b +b +...+b x
m dtm m−1 dtm−1 0
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2.2. Direct Laplace Transformation 2.3. Inverse Laplace Transformation

Direct Laplace transformation looks like The inverse turning from the image to original is carried out un-
∞ der the formula
f(p)= ∫ f(t)e−ptdt , (2.1)
c+j∞
1
0 f(t)= ∫ f(p)eptdp, (2.2)
where p=c+iω – the complex variable, c – the abscissa of conver- 2πjc−j∞
gence. As a result of integral conversion (2.1) realization we obtain the which is symbolically written as f(t)=L−1{f(p)}, where L−1– the op-
function of a variable p which depends on the appearance of the inte-
erator of inverse integral Laplace transformation. The formulas (2.1)
grand f (t):
and (2.2) sometimes are called Laplace inversion formulas and opera-
f(p)÷ f(t).
tions L1 and L−1– reversions of the original or image accordingly.
The function f(p)is called the image of the function f(t), and f(t) – Sometimes (especially in old books on the operational calculus) Kar-
the original; symbol ÷ expresses the correspondence of the image and son transformation is used instead of direct Laplace transformation.
original. Sometimes a symbolical writing f(p)=L{f(t)}, where the Karson transformation differs from Laplace transformation by a factor
operator L symbolizes direct Laplace transformation, is applied instead p, i.e. we shall define the image by Karson fk(p) as
of (2.1). There is a sole correspondence between the original and its ∞
image (the theorem of uniqueness of the image), i.e. each original has fk(p)= p∫ f(t)e−ptdt = pL{f(t)}.
the image uniquely corresponded to it. This statement is formulated 0
Then the turning from the image by Karson to original is carried
more strict to within a measure of zero original, when f(t) can have
out applying the reconversion based on the Bromvich integral.
isolated points outside of "smooth" definition of the function (for ex-
c+j∞
ample, M1 and M2 at fig. 2.1). f(t)= 1 ∫ fk(p)eptdp.
2πj p
c−j∞
f(t) ti+0 At present time the operational calculus based on Laplace transforma-
Than ∫ f(ξ)dξ=0,
M1 tions is mainly used.
ti−0

where t – the argu-
i 2.4. The Images Of Basic Singular Functions
ment of the function

f(t) in which it un-
2.4.1. The δ-Function Image
dergoes a rupture of
M2
a zero measure (i.e. Sometimes it is required to represent formally a physical signal
t
the rupture with zero by discontinuous (singular) functions of the order higher than 1(t),
square) [29]. d1(t) d21(t)
Fig. 2.1 δ(t)= , δ (t)= etc.
1 dt 2 dt2

The delta function δ(t) is widely applied (fig. 2.2).

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2.4.2. The Unit Step Saltus Image
∞
1(t) δ(t)
Let's obtain the image of a unit step saltus having applied direct
Laplace transformation to the function 1(t)
∞ ∞
f (p)=∫1(t)e−ptdt=∫e−ptdt.
us
0 0
Substituting an integration variable
0 t ⎧t =∞ x =−∞,
0 t −pt =x, at ⎨
⎩t =0 x =0.
Fig. 2.2
Then

1−∞ 1 −∞ 1 .
The function f (p)=− ∫exdx=− ex =0+
us p p 0 p
0
⎧0 at t <0,
δ(t)=⎪⎨∞ at t =0, (2.4) Thus fus(p)= 1p.
⎪
⎩0 at t >0.
+∞ +ε 3. SOME OPERATIONAL CALCULUS THEOREMS
∫δ(ξ)dξ= ∫δ(ξ)dξ=1. (2.5)
−∞ −ε The operational calculus theorems (for the images and originals
The condition (2.5) – the condition of δ-function normalization.
of functions) apart from their immediate application corresponding to
t the title of the theorem allow to obtain images of signals circumscribed
1(t)= ∫δ(ξ)dξ. (2.6)
by more composite functions over the images of simple signals. So, for
−∞
example, having taken the image of unit step saltus it is possible to ob-
Taking into consideration (2.4) and (2.5) we can write for any τ, in-
tain the image of exponential impulse or sine signal.
cluding zero:

∞ τ+ε τ+ε
∫ f(t)δ(t−τ)dt = ∫ f(t)δ(t−τ)dt =f(τ) ∫δ(t−τ)dt = f(τ). (2.7) 3.1. The Theorem Of Delay

−∞ τ−ε τ−ε
The expression (2.7) determines the so-called "filtering" property of Let's find the image
f(t) f (t)
δfunction. The integral like (2.7) gives a value of the integrand which τ f (p) of the lagging function
τ
goes into by a factor to δ-function for the argument value at which δ– f (t)=f(t−τ), τ>0 (fig. 3.1).
τ
function is equal to ∞. The δ-function image taking into account (2.7) ∞
is determined as f (p)= ∫ f(t−τ)e−ptdt . (3.1)
τ
∞ ε 0
fδ(p)= ∫δ(t)e−ptdt =e−p0∫δ(t)dt =1. 0 τ t But the integrand of the inte-
0 0 gral (3.1) is equal to zero at
Let's consider here that the inferior limit envelopes δ-function. Thus, Fig. 3.1 t <τ, so for such t we have
f (p)=1. (2.8) f (t)= f(t−τ)=0. By virtue of
δ τ
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it the equality is reduced to f (t)= f(t)e±αt
α
∞
f (p)= ∫ f(t−τ)e−ptdt. (3.2) we have
τ
f (p)= f(p∓α). (3.6)
τ α
Let's substitute the integration variable t =t−τ, at t=ε we have
1
t =0, at t =∞,t =∞,dt =dt, So in this case (at multiplication of the function in the originals space
1 1 1
±αt
∞ ∞ by e ) we have a displacement of coordinates along the real axis on
f (p)= ∫ f(t )e−p(t1+τ)dt =e−pt∫ f(t )e−pt1dt =e−pτf(p).
τ 1 1 1 1 the magnitude ±α. At last if to multiply the initial function by e±pνt,
0 0
Thus fτ(p)=e−pτf(p). (3.3) i.e. fpν(t)= f(t)e±pνt, where pν– a complex variable, we have the link
between the images
3.2. The Theorem Of Displacement In The Frequency f (p)= f(p∓ p ) (3.7)
pν ν
Domain (The Theorem Of Transposition)

3.3. The Theorems Implying From Linear Properties
It is given:* f(t)→ f(p).
Of Laplace Transformation
Let's find the image f (p) of the function as f(cid:5) (t)= f(t)e±jΩt.
Ω Ω
In other words it is necessary to find a link between the real vari- There are several consequences concerning Laplace transforma-
able initial function image and a function multiplied by e±jΩt. Let's tion implying from linear properties of definite integrals which remain
execute direct Laplace transformation: fair even for improper integrals in particular for the Laplace integral.
∞ ∞ They are:
fΩ(p)= ∫ fΩ(t)e−ptdt = ∫ f(t)e−(p∓jΩ)tdt. (3.4) А. The theorem of image of real variable functions sum.
0 0 The image of real variable functions sum is equal to the sum of the
Comparing (3.4) with the formula (2.1) of direct Laplace trans-
components images. Let
formation we notice that after the integration by t in (3.4) we obtain
f(t)=∑ f (t), (3.8)
the image f(p∓ jΩ) with the same functional dependence on p∓ jω k
k
that took place concerning a variable p for the image f(p) obtained as where the symbol ∑ means the sum of all k signal components.
a result of initial function f(t) DLT: k
Then applying direct Laplace transformation to (3.8) we shall obtain:
f (p)= f(p∓ jΩ). (3.5)
Ω
∞ ∞ ∞
Thus a time function multiplication by the factor e±jΩ⋅t displaces f(p)= ∫ f(t)e−ptdt= ∫[∑ f (t)]e−ptdt =∑[∫ f (t)e−ptdt]=∑ f (p),
k k k
the origin of coordinates of an independent variable p on a complex 0 0 k k 0 k
plane on the magnitude along the imaginary axis. i.e. if f(t)=∑ f (t), then f(p)=∑ f (p). (3.9)
±jΩ k k
Completely similarly for k k
The statement the functions sum integral is equal to the sum of inte-

grals of these functions is used here by virtue of linearity of Laplace
* Hereinafter the sign → means the correspondence, i.e. the writing f(t)→ f(p)
means, that the image f(p) corresponds to the function f(t).
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transformation (the function f(t) goes into the integral sign at the first
The poles of the imaging
degree).
function are marked with as-
B. The theorem of original and image multiplication by a con-
jω p=c+jω
stant. с terisks; 0 σ – the complex
variable p real axis. It is pos-
If f(t)→ f(p), then af(t)→af(p). (3.10)
sible to reveal, that the condi-
Here a– a constant. This theorem does not require any special proofs.
* tion mentioned above ensures
At direct Laplace transformation the constant factor is simply brought *
* direct Laplace transformation
out the integral. * 0 σ
convergence.
*

The poles of the imaging
3.4. The Theorem Of The Time Function Derivative Image
function f(p) usually (for
Fig. 3.2

steady systems) lay in the left
Let f(t)→ f(p). Let's find the image of original derivative half-plane (it is an indication
f (p), so of damping of oscillations occurring in a system).
1

df(t)
→ f1(p). In this case it is enough that the condition Re{p}=c>0 is satisfied for
dt
the abscissa of convergence. Then the first term in (3.11) will look like
In other words we find a link between the images of original and de-
rivative of a real variable t function. Let's apply direct Laplace trans- ∞ ∞
f(t)e−pt =0− f(0) and therefore f (p)=p∫f(t)e−ptdt− f(0), but
1
formation (2.1)
0 0
f1(p)=∞∫dfd(tt)e−ptdt ∞∫f(t)e−ptdt=f(p).
0
0
and integrating in parts we shall obtain:
Thus
∞ ∞ ∞ ∞
f1(p)= f(t)e−pt −∫ f(t)de−pt = f(t)e−pt + p∫ f(t)e−ptdt. (3.11) f1(p)= pf(p)− f(0). (3.12)
0 0 0 0 If consider the first derivative of the function of a variable t as
The complex variable p=c+ jω . The real part of it is Re{p}=cis the initial function and the second derivative as the derivative of the
called the abscissa of convergence. The magnitude of c is selected so first derivative
that the straight line p=c+ jω, where c=const , parallel to the imagi- df (t) d2f(t)
f (t)= 1 → ,
nary axis on a complex plane lays on the right of the imaging function 2 dt dt2
f(p) poles (fig. 3.2). than the second derivative image is written over the first derivative im-
age according to the formula (3.12) as
f (p)= pf (p)− f′(0), (3.13)
2 1
df(t)
where f′(t)= .
dt
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Copyright ОАО «ЦКБ «БИБКОМ» & ООО «Aгентство Kнига-Cервис»
The recursion formula for the nth derivative image over the n−1 de- Differentiating the primitive function (3.17) we have
rivative image is obtained similarly dF(t)
=f(t). (3.18)
f (p)= pf (p)− f (n−1)(0), (3.14) dt
n n−1
Hence according to the theorem of derivative image we can write:
dn−1f(t)
where f(n−1) = . f(p)= pF(p)−F(0). (3.19)
dtn−1
But from (3.17) we have F(0) = 0 because of the value of a definite
Using the formula (3.13) or more common recursion formula
integral with equal limits is equal to zero. So f(p)= pF(p).
(3.14) it is possible to write easily and sequentially the expanded for-
Then
mulas for the nth derivative images on the base of (3.12):
f(p)
f1(p)= pf(p)− f(0), F(p)= . (3.20)

p
f (p)= p2f(p)−pf(0)− f′(0)= f (p)− f′(0),
2 1 Thus the image of initial function integral is equal to its image divided
f3(p)= pf2(p)− f′′(0)= by p.
= p3f(p)−p2f(0)−pf′(0)− f′′(0), (3.15)

(cid:34)
4. THE STANDARD FORM SIGNALS IMAGES
Here f(0), f ′(0), f ′′(0)...– the magnitudes of the function f(t)

and its derivatives at the value of an independent variable t =0, i.e. the
Let's take advantages of foregoing theorems for determination of
initial conditions. Hence one of the essential operational calculus ad-
the images of some important functions defining signals which are fre-
vantages is the fact that the initial conditions are taken into account at
quently met at researches of radioelectronic circuits. At the same time
once when solving the differential equations. Simplification of writings
we shall emanate from the unit step saltus image obtained before:
is obtained if to put the initial conditions equal to zero (it is also impor-
tant for practice but is a special case of system operation)*. 1(t)→ fск(p)=1 p.

Then from (3.15) we have a simple relation for zero initial conditions
4.1. The Exponential Impulse Image
f (p)= pnf(p). (3.16)
n

Let's define exponential impulse as
3.5. The Theorem Of The Real Variable Function Integral Image
f (t)=Aeβt1(t), (4.1)
e
where β– a real constant, which can accept both positive and negative
Let f(t)→ f(p). Let's find the image of the integral:
values. At β= 0 it reduces to the saltus function considered earlier in
t
F(t)=∫ f(ξ)dξ. (3.17) the item 2.4.2, where f (t)→A⋅1 p. Having taken advantage of the
A
0 theorem of displacement in the images space we shall write:
It is necessary to define: F(p)=L{F(t)}.
1
f (p)=A . (4.2)
e
p−β
* Zero initial conditions, generally speaking, is not a key feature for consequent
taking up of the method simplifying ILT. They influence only values of coefficients at
the integer degrees of a variable p, obtained at their coercion for p identical degrees. So
the coefficients depend on the initial conditions (see the formulas (3.15)).
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