Sc., professor (Head of Applied Mathematics Chair
of Omsk State University of Transport) V.K. Okishev
Translated by D.A. Timochenko
Z 82
I.D. Zolotarev
THE METHOD SIMPLIFYING INVERSE
LAPLACE TRANSFORMATION AT
OSCILLATORY PROCESSES RESEARCHES. <...> THE "AMPLITUDE, PHASE, FREQUENCY" PROBLEM
IN RADIOELECTRONICS AND ITS SOLUTION
Tutorial
This book is recommended by Siberian regional department
of teaching methological association education in the field
of energetics and electrotechniques as educational textbook
for interuniversity practice
Zolotarev I.D.
The Method Simplifying Inverse Laplace Transformation At
Oscillatory Processes Researches. <...> The "Amplitude, Phase, Frequency" Problem In Radioelectronics And Its Solution / Transl.
by DA Timochenko: Tutorial. – Omsk: OmSU Publishing,
2004. – 132 p. <...> ISBN 5-7779-0472-6
The research method of transient processes in oscillatory systems
is stated. <...> It permits essential simplification of the most difficult operation in finding solution of a system differential equation – inverse
Laplace transformation. <...> It is shown that complex signal provides correct definition of an envelope and phase of a real signal using this
method. <...> The obviousness of obtained solutions is achieved by engaging the spectral method. <...> The examples of transient processes calculation in developing radioelectronic devices are given. <...> Д., 2004
© Омский госуниверситет, 2004
© Zolotarev I.D., 2004
© Omsk State University, 2004
© Timoshenko D.A., transl. from
Rus. into Engl., 2004
ISBN 5-7779-0472-6
2
INTRODUCTION
At development of radioelectronic devices for different purposes
an engineer frequently has to solve a researching problem of impulse
radiosignals passing through linear circuits. <...> For solving the task in the
time domain the operational calculus based on integral Laplace transformations is widely used. <...> When considering the task in the frequency
domain the spectral method based on integral Fourier transformations
is applied. <...> Both these research approaches are tightly interlinked
among themselves and sometimes are considered as a uniform method
(the method of Fourier transformation). <...> When finding a response of a radioelectronic device (RED) to an
impulse energization applying the operational calculus the most difficult operation is the inverse Laplace transformation (ILT) execution
[1]. <...> The difficulty of ILT especially increases for important radioelectronic applications when a radioimpulse <...>
The_metod_simplifing_inverse_Laplace_transformatijn_at__ossillatory_processes_researchet_Учебное_пособие.pdf
Ministry of Education and Science of Russian Federation
Omsk State University
UDC 621.396.6+517.442(075)
Z 82
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
THE METHOD SIMPLIFYING INVERSE
LAPLACE TRANSFORMATION AT
OSCILLATORY PROCESSES RESEARCHES.
THE "AMPLITUDE, PHASE, FREQUENCY" PROBLEM
IN RADIOELECTRONICS AND ITS SOLUTION
Tutorial
This book is recommended by Siberian regional department
of teaching methological association education in the field
of energetics and electrotechniques as educational textbook
for interuniversity practice
Z 82
The Method Simplifying Inverse Laplace Transformation At
Oscillatory Processes Researches. The "Amplitude, Phase, Frequency"
Problem In Radioelectronics And Its Solution / Transl.
by DA Timochenko: Tutorial. – Omsk: OmSU Publishing,
2004. – 132 p.
ISBN 5-7779-0472-6
The research method of transient processes in oscillatory systems
is stated. It permits essential simplification of the most difficult operation
in finding solution of a system differential equation – inverse
Laplace transformation. It is shown that complex signal provides correct
definition of an envelope and phase of a real signal using this
method. The obviousness of obtained solutions is achieved by engaging
the spectral method. The examples of transient processes calculation
in developing radioelectronic devices are given.
This tutorial is intended for students, post-graduate students, engineers,
scientific employees of both radio and electrotechnical specialities
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
OmSU Omsk
Publishing
2004
2
ISBN 5-7779-0472-6
© Omsk State University, 2004
© Timoshenko D.A., transl. from
Rus. into Engl., 2004
Стр.1
INTRODUCTION
At development of radioelectronic devices for different purposes
an engineer frequently has to solve a researching problem of impulse
radiosignals passing through linear circuits. For solving the task in the
time domain the operational calculus based on integral Laplace transformations
is widely used. When considering the task in the frequency
domain the spectral method based on integral Fourier transformations
is applied. Both these research approaches are tightly interlinked
among themselves and sometimes are considered as a uniform method
(the method of Fourier transformation).
When finding a response of a radioelectronic device (RED) to an
impulse energization applying the operational calculus the most difficult
operation is the inverse Laplace transformation (ILT) execution
[1]. The difficulty of ILT especially increases for important radioelectronic
applications when a radioimpulse signal affects RED and if selective
filters are included in a signal tract of RED (oscillating system).
It is stipulated by the fact that for radioimpulse signals and such realizations
of RED the imaging function (IF) of the researching system
response for the input disturbance has complex conjugate poles (CCP)
pairs. In these cases even for rather simple IF the difficulty and awkwardness
of conversions when turning from the images space to the
originals one essentially raise in comparison with finding solutions for
real poles IF [2]. In the meantime the existing tendency of extreme increasing
of information processing speed in radiosystems requires to
develop RED working in the dynamic mode when the conversions of a
signal, taking off and processing its informative parameter are executed
not after the termination of transient processes (TP) in the output of the
informative channel but during these processes. Generally because of
inevitable TP presence at energization of a radioelectronic system by
an impulse signal the form of it is distorted. The specified distortions
corrupt the informative parameter of a signal (originate definite dynamic
errors in system functioning). The researching TP in a system
for the purpose of minimization of an error imported to the signal informative
parameter by transient processes is one of necessary development
stages for modern RED operating in the dynamic mode. Therefore
the problem of development of methods simplifying researches of
3
transient processes in electronic devices always attracts a serious attention
of specialists [l–6].
The most prevalent at researches of transient processes in radiosystems
is the method of slowly varying envelopes (SVE) designed by
S. I. Evtyanov. In this method essential decreasing of difficulty in solving
linear differential equations (DE) while researching TP in oscillating
systems is achieved applying some particular simplifying assumptions
(asymptotic method of the small parameter). In this case the initial
DE communicating the response of the linear system and the energizing
radiosignal are converted into truncated symbolical equations
regarding to SVE [2]. The more narrow-band signals and systems are
studied the more precise solutions are obtained using the SVE method.
As a measure of band narrowity of radiosignals and systems the ratios
= ∆2
= ∆ s
spectrum,
c and
of oscillating system,
and
(
c
c – the filling high-frequency (HF), 2∆ – the bandwidth
r – resonant system frequency are usually conr
, where ∆ – the width of a radiosignal
c
s
sidered. For narrow-band signals and systems we have the small parameters
<<
1,
<< 1). For wide-band and ultra wideband
systems these parameters are comparable to the unit.
Although the method of S. I. Evtyanov allows to simplify essentially
the difficulties in finding an enough precise solution for an envelope
of a signal in the output of a radiosystem it does not provide the
authentic description of thin (phase) structure of an output radiosignal.
Considering the greatest possibilities and advantages of phase information
radiosystems operating in the dynamic mode [7] we notice that the
specified disadvantage of the SVE method is rather essential.
Designed in [5–10] method simplifying inverse Laplace transformation
ensures the same reducing of difficulties in solution obtaining
as the SVE method. However when using the method [5–10] the
precise (accurate to phase) solution for the radiosignal in the output of
an investigated radiosystem is obtained. Thus it is not necessary to introduce
simplifying assumptions which are peculiar to asymptotic
methods including the SVE one.
Apart from great simplification of a solution determination by
the method [5–10], application of it for the important cases of oscillatory
processes researches allows to obtain a system response definition
as a complex signal (CS). It facilitates dynamic modes in radiosystems
4
ω
ω
ω
ε
ω
µ
ω ω
µ
ω
ω
µ
ε
ε
Стр.2
40. Zolotarev I.D. Simulation of a radio-frequency pulse with rectangular
envelope by the analytical signal // Omsk science bulletin.
1997. Issue 1. P. 52–55.
41. Zolotarev I.D. The problem "Amplitude, Phase, Frequency" and
its solution in radioengineering // Radiocommunications technique.
1997. Issue. 3. P. 3–10.
INTRODUCTION...........................................................................................3
1. INITIAL STATEMENTS............................................................................6
1.1. The Function Of Time (Signal) ............................................................6
1.2. Setting The Task At Linear Systems Researching................................7
2. THE OPERATIONAL CALCULUS APPLICATION FOR SOLVING
LINEAR DIFFERENTIAL EQUATIONS....................................................10
2.1. Initial Statements ................................................................................10
2.2. Direct Laplace Transformation...........................................................11
2.3. Inverse Laplace Transformation .........................................................12
2.4. The Images Of Basic Singular Functions ...........................................12
2.4.1. The
-Function Image..................................................................12
2.4.2. The Unit Step Saltus Image .........................................................14
3. SOME OPERATIONAL CALCULUS THEOREMS...............................14
3.1. The Theorem Of Delay.......................................................................14
3.2 The Theorem Of Displacement In The Frequency Domain
(The Theorem Of Transposition)...............................................................15
3.3. The Theorems Implying From Linear Properties Of Laplace
Transformation ..........................................................................................16
3.4. The Theorem Of The Time Function Derivative Image.....................17
3.5 The Theorem Of The Real Variable Function Integral Image.............19
4. THE STANDARD FORM SIGNALS IMAGES ......................................20
4.1 The Exponential Impulse Image..........................................................20
4.2 The Image Of A Sine Wave Switching Function ................................21
4.3. The Image Of Oscillatory Processes With Exponential Envelope......22
4.4. The Image Of Secular Function Defined Signal.................................23
5. THE IMAGING EQUATION...................................................................26
6. THE TIME SYSTEM CHARACTERISTICS...........................................29
6.1. The Impulse Response Of A System..................................................29
6.2. The Transient Response Of A System................................................30
7. THE DIFFERENTIAL EQUATION INTEGRATION.............................32
7.1. The Operational Calculus Application For Linear Differential
Equations Integration.................................................................................32
7.2. The Inversion Formula For The Image, Defined By FRF ..................33
8. THE EXAMPLES OF TRANSIENT PROCESSES CALCULATION
OVER THE INVERSION FORMULA FOR FRF WITH SIMPLE
POLES...........................................................................................................34
127
128
δ
Стр.64
9. THE APPLICATION OF FOURIER TRANSFORMATION FOR
THE ANALYSIS OF SIGNAL PASSING THROUGH THE
INVESTIGATED TRACT............................................................................49
9.1. The Fourier Series ..............................................................................49
9.2. The Integral Fourier Transforms ........................................................53
9.3. The Complex Amplitude And Spectral Density Comparison
As Well As The Fourier Series And Fourier Integral................................55
10. INTEGRAL FOURIER AND LAPLACE TRANSFORMATIONS.......57
10.1. The Link Between Integral Fourier And Laplace
Transformations ........................................................................................57
10.2. The Generality And Differences Of The Operational Calculus
And Spectral Researches Of Linear Electronic Circuits............................62
10.2.1. The Generality Of The Imaging Function And Signal
Spectral Density ....................................................................................62
10.2.2. Some Limitations At Turning "Signal Image ↔ Signal
Spectral Density" At The Substitution p j↔ ...................................63
10.2.3. The Turning From The System Imaging Equation To The
Equation For Spectrums Of Input And Output Signals .........................70
10.2.4. The Tables Of The Spectral Method And Operational
Calculus Comparison At Linear Differential Equations Integration .....75
11. THE METHOD SIMPLIFYING INVERSE LAPLACE
TRANSFORMATION..................................................................................77
11.1. The Task Statement ..........................................................................77
11.2. The Formula Simplifying ILT (The Case Of Simple CCP Of The
Imaging Function) .....................................................................................79
11.3. The Examples Of Application Of The Inversion Formula
Simplifying ILT.........................................................................................81
11.4. The Substantiation Of The Method Simplifying Inverse Laplace
Transformation At Electronic Circuits Dynamic Oscillatory Modes
Researches.................................................................................................92
12. THE COMPLEX SIGNAL AND THE "AMPLITUDE, PHASE,
FREQUENCY" PROBLEM FOR OSCILLATORY PROCESSES .............99
12.1. The Statement Of The "Amplitude, Phase, Frequency" Problem
In Radioelectronics....................................................................................99
12.2. The Analytical Signal.....................................................................103
12.3. The Spectrum Of The Analytical Signal. The Comparison
Of Spectrums Of The Analytical And Complex Signals.........................109
12.4. The New Approach At Solving The Problem "Amplitude,
Phase, Frequency" ...................................................................................114
129
130
12.5. The Representation Of AS Over CS For The Case Of Arbitrary
Amplitude-Phase Modulation..................................................................119
INFERENCE...............................................................................................122
THE LIST OF ACCEPTED ABBREVIATIONS.......................................123
THE BIBLIOGRAPHY...............................................................................124
ω
Стр.65
I.D. Zolotarev
THE METHOD SIMPLIFYING INVERSE
LAPLACE TRANSFORMATION AT
OSCILLATORY PROCESSES RESEARCHES.
THE "AMPLITUDE, PHASE, FREQUENCY" PROBLEM IN
RADIOELECTRONICS AND ITS SOLUTION
Tutorial
Translated by D.A. Timoshenko
Editor – L.M. Kitsina
Technical editor – N.B. Moskvichjeva
Editor of translation – L.K. Kondratjukova
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131
132
Стр.66