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

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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

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. 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)

OmSU Publishing

Omsk 2004

© Золотарев И.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

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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 µ = ∆ω s ω c and ε = 2 ∆ω c ω r , where ∆ω s – the width of a radiosignal spectrum, ωc – the filling high-frequency (HF), 2 ∆ω c – the bandwidth of oscillating system, ωr – resonant system frequency are usually considered. For narrow-band signals and systems we have the small parameters µ and ε ( µ 0 is satisfied for the abscissa of convergence. Then the first term in (3.11) will look like f (t )e − pt

0

∞

df (t ) − pt f1 ( p ) = ∫ e dt 0 dt f1( p ) = f (t )e

∞

0

∞

− ∫ f (t )de

− pt

= f (t )e

− pt

∞

0

0

0

− pt

∫ f (t ) e

dt = f ( p) .

0 ∞

+ p ∫ f (t )e

Thus − pt

dt .

(3.11)

0

The complex variable p = c + jω . The real part of it is Re{ p} = c is called the abscissa of convergence. The magnitude of c is selected so that the straight line p = c + jω , where c = const , parallel to the imaginary axis on a complex plane lays on the right of the imaging function f ( p ) poles (fig. 3.2).

17

∞

= 0 − f ( 0 ) and therefore f1( p) = p ∫ f (t )e − pt dt − f (0) , but ∞

and integrating in parts we shall obtain: − pt

∞

(3.12) f1( p ) = pf ( p ) − f ( 0 ) . If consider the first derivative of the function of a variable t as the initial function and the second derivative as the derivative of the first derivative f 2 (t ) =

df1(t ) d 2 f (t ) , → dt dt 2

than the second derivative image is written over the first derivative image according to the formula (3.12) as f 2 ( p ) = pf1( p ) − f ′( 0 ) , (3.13) df (t ) where f ′(t ) = . dt 18

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The recursion formula for the n th derivative image over the n − 1 derivative image is obtained similarly (3.14) f n ( p ) = p f n − 1 ( p ) − f ( n − 1) ( 0 ) , where f

( n −1)

=

d n −1 f (t ) dt n −1

.

Using the formula (3.13) or more common recursion formula (3.14) it is possible to write easily and sequentially the expanded formulas for the n th derivative images on the base of (3.12): f1 ( p ) = pf ( p ) − f ( 0 ),

Differentiating the primitive function (3.17) we have dF(t ) = f (t ) . dt

Hence according to the theorem of derivative image we can write: (3.19) f ( p ) = pF ( p ) − F ( 0 ) . But from (3.17) we have F(0) = 0 because of the value of a definite integral with equal limits is equal to zero. So f ( p ) = pF ( p ) . Then F ( p) =

2

f 2 ( p ) = p f ( p ) − pf ( 0 ) − f ′( 0 ) = f1( p ) − f ′( 0 ),

3.5. The Theorem Of The Real Variable Function Integral Image Let f (t ) → f ( p ) . Let's find the image of the integral: t

(3.17)

0

It is necessary to define: F ( p ) = L {F (t )} .

4. THE STANDARD FORM SIGNALS IMAGES Let's take advantages of foregoing theorems for determination of the images of some important functions defining signals which are frequently met at researches of radioelectronic circuits. At the same time we shall emanate from the unit step saltus image obtained before: 1(t ) → f ск ( p ) = 1 p . 4.1. The Exponential Impulse Image Let's define exponential impulse as f e (t ) = Ae β t 1(t ) ,

(4.1) where β – a real constant, which can accept both positive and negative values. At β = 0 it reduces to the saltus function considered earlier in the item 2.4.2, where f A (t ) → A ⋅1 p . Having taken advantage of the theorem of displacement in the images space we shall write: f e ( p) = A

*

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)).

19

(3.20)

(3.15)

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 initial conditions. Hence one of the essential operational calculus advantages is the fact that the initial conditions are taken into account at once when solving the differential equations. Simplification of writings is obtained if to put the initial conditions equal to zero (it is also important for practice but is a special case of system operation)*. Then from (3.15) we have a simple relation for zero initial conditions (3.16) f n ( p ) = pn f ( p ) .

F (t ) = ∫ f (ξ )dξ .

f ( p) . p

Thus the image of initial function integral is equal to its image divided by p.

f 3 ( p ) = pf 2 ( p ) − f ′′( 0 ) = = p3 f ( p ) − p2 f ( 0 ) − pf ′( 0 ) − f ′′( 0 ),

(3.18)

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1 . p− β

(4.2)

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At the same manner let's find the image for the function

4.2. The Image Of A Sine Wave Switching Function Let the signal defined with the function is given: f (t ) = A cos(ω сt + ψ )1(t ) , (4.3) i.e. the signal is given with a sinusoid truncated by the factor 1(t ) for the t < 0 area. We shall call the function (4.3) the radiosaltus function by analogy with the unit step saltus [5]. Having taken advantage of the Euler formula we shall rewrite (4.3) as ⎡ e j (ω сt +ψ ) e − j (ω сt +ψ ) ⎤ f (t ) = A ⎢ + ⎥ ⋅1(t ) . 2 2 ⎣⎢ ⎦⎥

(4.4)

Having applied the theorem of displacement in the frequency domain, we shall obtain

{

}

L A ⋅1(t )e ± jω сt = A

{

1 , p ∓ jω с

}

L A ⋅1(t )e ± j (ω сt +ψ ) = A

(4.5)

e jψ . p ∓ jω с

(4.6)

Having taken advantage of the sum image theorem we shall obtain the expression from (4.4) A ⎡ e jψ e − jψ ⎤ + f ( p ) = L { f (t )} = ⎢ ⎥. 2 ⎣⎢ p − jω с p + jω с ⎦⎥

(4.7)

It is sometimes convenient to represent the image of the truncated sinusoid in other more compact form. Let's execute trivial conversions with the expression (4.7) for this purpose: A ⎡( p + jωс )(cosψ + j sinψ ) + ( p − jωс )(cosψ − j sinψ ) ⎤ ⎥= f ( p) = ⎢ 2⎢ p2 +ωс2 ⎦⎥ ⎣ ⎡ pcosψ −ωс sinψ + j(ωс cosψ + psinψ ) + pcos ψ −ωс sinψ ⎤ −⎥ ⎢ p2 +ωс2 A⎢ ⎥. = ⎢ ⎥ 2 j(ωс cos ψ + psinψ ) ⎥ ⎢− ⎥⎦ ⎢⎣ p2 +ωс2

Finally we shall obtain f ( p) = A

p cosψ − ω с sinψ p2 + ω с2

21

.

(4.8)

f (t ) = A sin(ω сt + ψ ) ⋅1(t ) ,

f (t ) = A

f ( p ) = L { f (t )} = A

e j (ω сt +ψ ) − e − j (ω сt +ψ ) 1(t ) , (4.9) 2j

p sin ψ + ω с cosψ . p2 + ω с2

(4.10)

The formula (4.10) can be obviously obtained from (4.8) (and on the contrary the formula (4.8) from (4.10) if to take into account that the cosine and sine signals differ by a phase lag by π / 2 (compare the formulas (4.3) and (4.9)). Then for example A sin( ω сt + ψ )1(t ) = A cos( ω сt + ψ −

π 2

) ⋅1(t ) .

(4.11)

Hence substituting the ψ - π / 2 instead of ψ in the formula (4.8) for the image of a cosine function we shall obtain the image of a sinusoidal function as (4.10). 4.3. The Image Of Oscillatory Processes With Exponential Envelope Let's represent the signal for this case as f ( t ) = Ae β t sin( ω сt + ψ )1( t ) . (4.12) Here the formula (4.12) determines a damping or increasing (depending on that whether β 0) sinusoidal function with the envelope Ae β t 1( t ) . The formula (4.12) circumscribing the given signal differs from the expression (4.9) by the factor e β t . Then having taken advantage of the theorem of displacement in the frequency domain we obtain the image of the function (4.12) from the expression (4.10) ( p − β ) sin ψ + ω с cosψ . (4.13) f ( p ) = L { f (t )} = A ( p − β )2 + ω с2 For the function like (4.14) f (t ) = Ae β t cos( ω сt + ψ ) ⋅1(t ) , comparing (4.14) with (4.3) of the image (4.8) let's obtain in the same manner the expression applying the theorem of displacement

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f ( p) = A

( p − β ) cosψ − ω с sin ψ . ( p − β )2 + ω с2

(4.15)

4.4. The Image Of Secular Function Defined Signal Let's write a signal in the most common form: (4.16) f (t ) = At k e β t cos( ω сt + ψ )1( t ) , where the time goes outside a cosine (secular function [19]), k = 0, 1, 2..., i.e. k is integer, positive including zero. Let's obtain the auxiliary functions images before finding the images of the signal (4.16). For this purpose we shall remind that the image of δ –function (the unit impulse) f δ ( p ) = 1 (the formula (2.8)). Let's represent the unit step saltus image as in the formula (2.9): f ( p ) = L {1(t )} =

1 . p

Let's notice that it is possible to turn from (2.8) to (2.9) on the basis of the theorem of time function integral image taking into account the integral link between originals as (2.5): 1(t ) =

t

∫

−∞

f −3 ( t ) =

t

∫

−∞

f −2 (ξ )dξ = ∫

t2 1( t ), ..., 2

t

−∞

0

t2 1 1(t ) → L{ f −2 (t )} = f −2 ( p ) = 3 , 2 p t 1 . 1( t ) → L { f − n ( t )} = f − n ( p ) = n +1 n! p

t

∫ 1(ξ )dξ = ∫ ξdξ =

−∞

0

t

t

t

−∞

0

0

∫ ∫1(ξ )dξ = ∫ ∫ ξdξ = ∫

f − n (t ) = ∫ ∫ ...

23

t

∫ 1(ξ )dξ =

−∞

tn 1( t ) . n!

…

∫ 1(ξ )dξ = ∫ dξ = t ⋅1(t ) , t

f − n (t ) =

1 , t ⋅1( t ) → L { f −2 (t )} = f −1( p ) = p2

n

t

f −1(ξ )dξ = ∫

f −2 (t ) =

The inferior limit of definite integrals in (4.17) can be taken equal to zero accordingly. Outgoing from obviousness of the factor 1(t ) presence it is sometimes omitted in the links (4.17). In further we shall not stipulate the function truncation on the negative time half-axis. It is so as the integral in direct Laplace transformation is one-sided. Let's write the image for the function in (4.17) according to the theorem of integral image

∫ δ (ξ )dξ .

Let's find the functions

f −2 ( t ) =

f −1 ( t ) = t ⋅1( t ) ,

t

−∞

f −1(t ) =

It is necessary to be implied that the solution of integrals in the expression (4.17) is fair for t>0. It is the consequence of the fact that 1( t ) = 0 at t < 0 in the integrand. Thus is necessary to include the factor 1(t ) everywhere in the solutions (4.17) to show that they are fair for t >0. For example

ξ

From (4.18) we obtain the image for the function of the integer degrees of t truncated in time

t2 1(t ) , 2

2

1(t ) → 3

t 1(t ) dξ = , 2 2 ⋅3

tn ⋅1(t ) . n!

(4.18)

(4.17)

1 , p

t ⋅1(t ) →

1 p2

t 2 ⋅1(t ) → t 3 ⋅1(t ) →

… 24

,

2

, p3 2 ⋅3 p4

,

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t n ⋅1(t ) →

n! p т +1

(4.19)

Now we shall take advantage of the appropriate theorems to turn from (4.19) to the required image of the original (4.16). For this purpose we shall represent (4.16) as f (t ) =

[

]

At k ( β + jω с )t + jψ e + e ( β − jω с )t − jψ ⋅1(t ) . 2

(4.21)

Let's reduce the last link to a common denominator. At the same time we shall take into account that (4.22) {[( p − β ) − jω с ][( p − β ) + jω с ]}k +1 = [( p − β )2 + ω с2 ]k +1 . Then the imaging function f ( p ) from (4.21) is converted to f ( p) =

=

Ak ! ⎧⎪[( p − β ) + jωс ]k +1e jψ + [( p − β ) − jωс ]k +1e− jψ ⎫⎪ ⎨ ⎬= 2 ⎪⎩ ⎪⎭ [( p − β )2 + ωс2 ]k +1

Ak ! ⎧⎪[( p − β ) + jωс ]k +1(cosψ + j sinψ ) + [( p − β ) − jωс ]k +1(cos ψ − j sinψ ) ⎫⎪ ⎬= ⎨ 2 ⎪⎩ ⎪⎭ [( p − β )2 + ωс2 ]k +1

=

Ak ! ⎡{[( p − β ) + jωс ]k +1 + [( p − β ) − jωс ]k +1}cosψ + ⎢ 2 ⎢⎣ [( p − β )2 + ωс2 ]k +1

+

j{[( p − β ) + jωс ]k +1 − [( p − β ) − jωс ]k +1}sinψ ⎤ ⎥. ⎥⎦ [( p − β )2 + ωс2 ]k +1

At k = 0 the expression (4.23) takes the form ( p − β ) cosψ − ω с sin ψ , f ( p) = A ( p − β )2 + ω с2

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5. THE IMAGING EQUATION

(4.20)

According to the theorems of displacement in the frequency domain and the signals sum image we obtain the required image of the secular signal (4.16) from (4.19) and (4.20) ⎤ Ak ! ⎡ e jψ e − jψ f ( p) = + ⎥. ⎢ k + 1 k + 1 2 ⎢⎣ [ p − ( β + jω с )] [ p − ( β − jω с )] ⎥⎦

i.e. at k = 0 we turn from (4.23) to the formula (4.15) of the function (4.14) image which is obtained from the original (4.16) supposing in it k =0.

Let's apply direct Laplace transformation to the right and left part of the differential equation (1.7). ⎧⎪ n ⎧⎪ m d λ x ⎫⎪ d µ y ⎫⎪ L ⎨ ∑ aµ = L ⎨ ∑ bλ ⎬. ⎬ ⎪⎩λ = 0 dt λ ⎭⎪ dt µ ⎪⎭ ⎪⎩ µ = 0

(5.1)

To simplify reasonings we shall consider that the investigated system has zero initial conditions. At zero initial conditions showing the absence of initial energy stores in the investigated system the expressions for the function derivatives images (3.15) are turned to more simple appearance (3.16) since the initial values of the function and its derivatives up to the (n–1)th order are equal to zero i.e. n −1 f ( 0 ) = 0, f ′( 0 ) = 0, ..., f ( ) ( 0 ) = 0 . Applying the theorem of derivative image as well as auxiliary theorems of sum image and function multiplication by a constant we shall obtain at once ⎛ n ⎞ ⎛ m ⎞ ⎜ (5.2) aµ p µ ⎟ ⋅ y ( p ) = ⎜ ∑ bλ p λ ⎟ ⋅ x ( p ), ∑ ⎜ ⎟ ⎜ ⎟ ⎝ λ =0 ⎠ ⎝ µ =0 ⎠ where x ( p ) and y ( p ) – the input signal and system response images

accordingly. Let's find the system response image from the expression (5.2) as (5.3) y ( p ) = K ( p )x ( p ) , m

∑ bλ pλ

K ( p ) = λ =0 n

∑ aµ p

µ =0

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, µ

(5.4)

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where the fractional rational function K ( p ) is called the transfer function of a system (or the system function). From comparison (5.4) with the initial differential equation (the formula (1.7)) it follows that the transfer function is completely determined by the coefficients of the initial differential equation. In other words, there is a sole link between K ( p ) and differential equation. So the investigated physical system can be determined both over the differential equation and the transfer function K ( p ) . The link between the input signal x (t ) and the system response y (t ) over the turning to the images space and back is schematically shown at fig. 5.1.

L ↓

K ( p)

x ( p)

↑ L−1 y ( p)

Thus the Laplace transformation application to the right and left part of the initial differential equation turns it into the images space. The algebraic equation (5.2) is called the imaging equation. Putting that the input signal x (t ) is known the right term of the equation (1.7) can be represented as the function F (t ) which is sometimes called the forcing function [23, p. 61]. ⎧⎪ n d µ y ⎫⎪ L⎨ ∑ = F ( p) , µ ⎬ ⎪⎩µ = 0 dt ⎪⎭

(5.5)

where F ( p ) – the forcing function image. It is possible to rewrite the imaging equation as

∑ (aµ pµ )y ( p ) = F ( p ). n

µ =0

27

1

F ( p) .

n

∑ aµ p

(5.7)

µ

µ =0

Let's designate the factor at F ( p ) over K1( p) K1( p ) =

1 n

∑ aµ p

.

(5.8)

µ

µ =0

Then (5.7) will accept the appearance: (5.9) In the important case if only one signal x (t ) acts in the system input instead of the sum of a signal and its derivatives with weight coefficients bλ then as follows from (1.7) and (5.2) we suppose bλ = 0

Fig. 5.1

Then

y ( p) =

y ( p) = K1( p ) ⋅ F ( p) .

y (t )

x(t )

Solving the equation (5.1) with respect to y ( p ) we have

(5.6)

for all λ = 1, m . Retaining a generality of considering we can accept b0 = 1 and then F ( t ) = x ( t ) . Thus the transfer function (the system characteristic) K ( p ) = K 1( p ) is also determined only by the left part of the differential equation (1.7). The formula (5.8) corresponds to the specified left part. We consider only this case for practice. Such an approach simplifies a little result determination, does not require a modification of solution obtaining method at taking more composite perturbation F (t ) containing derivatives of x (t ) . Actually the presence of derivatives x (t ) influences the structure of transfer performance K1( p) numerator and does not affect its poles. The difficulty of y (t ) solution obtaining and its behavior are determined by the poles images and therefore by the K ( p ) poles (and also x ( p ) poles, formula (5.3)). In spite of presence or absence of initial storage of energy the application of direct Laplace transformation to the right and left part of the equation (1.7) allows to algebraize the ordinary differential equation. At the same time the right and left parts of the equation (1.7) are represented by polynomials of integer degrees of a variable p in the images space. The highest degree of a polynomial in the left part is equal to a degree of a differential equation. The presence of initial conditions will only lead to occurrence of additional terms in the right 28

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polynomial which degree is lower than the degree of the differential equation. After reducing similar terms we obtain polynomials of the integer degrees of p in the right and left part of the imaging equation for the initial differential equation like (1.7). Thus the presence of initial conditions will only change magnitudes of the coefficients at the integer degrees of p. For the majority of the practical tasks of radioelectronics the transfer performance K(p) is a fractional rational function (FRF), i.e. it is a ratio of polynomials of the integer degrees of p (formula (5.4)). Let's consider the class of signals for which the x ( p ) images are also FRF. Then the image y ( p ) as a product of fractional rational functions is also FRF.

6. THE TIME SYSTEM CHARACTERISTICS The time system (radioelectronic circuit) characteristics are responses of a system to standard signal forms. They are definitively linked with its transfer performance and with the system differential equation as well. The impulse and transient responses of a system are the most widely used. 6.1. The Impulse Response Of A System The impulse response of a system is a response of it to an energizing signal like δ -function. The impulse response has a series of synonyms: the impulse response, the impulse characteristic, the system memory. The Green function is an analog of the impulse response at physics. Let's designate impulse response by the character g (t). Then for the differential equation like (1.7) n

dµg

µ =0

dt µ

∑ bµ

= δ (t ) .

n

( ∑ bµ p µ ) g ( p ) = 1 ,

(6.2)

µ =0

where g ( p ) – the system impulse response image. Hence g( p) =

1 n

∑ bµ p

= K ( p) .

(6.3)

µ

µ =0

It follows from (6.3) that system impulse response image is equal to system transfer performance to within dimension. Let's apply inverse Laplace transformation to (6.3) to determine the impulse response. Then (6.4) g( t ) = L −1{ g ( p )} = L −1{K ( p )} . It follows from this equation that there is a simple link between the impulse response and the transfer function of a system namely follows: the impulse response of a system is determined by ILT of transfer performance to within dimension3. As the system is given with the transfer performance K ( p ) (or differential equation, uniquely linked with K ( p ) ) then the impulse response uniquely determines the system. In other words impulse response is the function of time defining a system in the time domain as well as transfer performance in the field of a complex variable p, or differential equation connecting the input and output of the system. 6.2. The Transient Response Of A System The transient response of a system (transient function) is a response of a system to the unit step saltus. Let's designate transient re-

(6.1) 3

Applying direct Laplace transformation to (6.1) and taking into account that L {δ(t )} = 1 we obtain

It follows from (6.1) and (6.2) taking into account the appearance of DFT (the formula (2.1)) that the dimension of the one in the right term of (6.2) as the δ -function image is equal to [δ (t )]⋅ c . The impulse response dimension [g ( p )] = [δ (t )]⋅ c ⋅ [K ( p )] from (6.3) as well. Here square brackets mean dimension of the appropriate variable (function).

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sponse by the character h(t). Appealing to the differential equation (1.7) we have n

d µh

µ =0

dt µ

∑ bµ

= 1(t ) ,

⎧⎪ n ⎞ 1 d µ h ⎫⎪ ⎛⎜ n ⇒ L ⎨ ∑ bµ = ∑ bµ p µ ⎟ h ( p ) = , ⎬ µ ⎜ ⎟ p dt ⎪⎭ ⎝ µ = 0 ⎪⎩ µ = 0 ⎠

System Differential equation K ( p ), h(t ), g (t )

x(t )

(6.5) (6.6)

y (t )

Fig. 6.1

where L{1(t )} = 1 p ; h ( p ) – the transient response image. Hereinafter the symbol ⇒ means " whence follows". It is obtained from the formula (6.6): h ( p) =

1 K ( p) . = p µ p ∑ bµ p 1

n

7. THE DIFFERENTIAL EQUATION INTEGRATION

µ =0

Thus there is a simple link between the transient response image h ( p ) and the system transfer performance K ( p ) defined by the ratio (6.7). Let's rewrite (6.7) as c + j∞

1 K ( p ) pt h(t ) = L {h ( p )} = e dp 2πj c −∫j∞ p −1

(6.8)

(6.9)

As the transient response of a system is uniquely linked with its transfer performance it determines as well as the system impulse response a physical system in the time domain. So the system is determined: by the differential equation, or transfer performance, or time response, or transient response (fig. 6.1). All these definitions are uniquely linked among themselves.

31

Applying inverse Laplace transformation to the equation (5.4) we obtain the required differential equation integral of the investigated system: y (t ) = L −1{ y ( p )} = L −1{K ( p ) x ( p )} ,

(7.1)

or y (t ) =

and therefore ⎧ K ( p) ⎫ h(t ) = L −1 ⎨ ⎬. ⎩ p ⎭

7.1. The Operational Calculus Application For Linear Differential Equations Integration

(6.7)

1 2πj

c + j∞

∫

y ( p )e pt dp =

c − j∞

1 2πj

c + j∞

∫

K ( p ) x ( p )e pt dp .

(7.2)

c − j∞

The integral (7.2) is equal to the sum of residues in integrand poles: y ( t ) = ∑ res ( pν ) , (7.3) ν

where res ( pν ) – the residue in the н -th pole of the integrand; ν = 1, r ; r – the number of all poles of the integrand; the dash means the integer values into the interval 1… r . It follows from (5.3) that K ( p ) – is a fractional rational function of a variable p, i.e. a ratio of two polynomials of integer degrees of p. In general case of a signal representation by a secular functions sum (see the formula (4.16)) the signal image is also represented by a sum of FRF like (4.21) that also gives a resulting FRF of the signal image. The secular signal is generality understood in the sense that all other signals considered above (except for the function δ (t)) can be obtained from 32

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(4.16) supposing the parameters k , β , ω н and ψ equal to zero separately or jointly. We can concern the same to their IF which can also be obtained from (4.21). Then the image y ( p ) of the system response as a product of K(p) and x ( p ) is also FRF according to (5.3). Residues of a fractional rational function can be found applying the inversion formula according to which the turning from the images to the originals space is carried out. Let's obtain this formula. 7.2. The Inversion Formula For The Image, Defined By FRF We consider the cases of FRF with simple poles. Let's write y ( p ) such as: y ( p) =

F ( p) . Q( p )

F(p )

∑ Q ′( p ) ν

p = pν

ν

⋅

1 , p − pν

fν ( p ) =

1 ← e pν t 1( t ) . p − pν

(7.6)

The expansion formula for y ( p ) contains a sum of fractions like 1 ( p − pν ) with constant coefficients F ( pν ) / Q′( p ) p = pν . Then according to the theorem of multiplication of function by a constant we obtain y (t ) =

r

F( p )

∑ Q′( p ) ν

e pν t 1( t ) .

(7.7)

p = pν

ν =1

This formula which is called the inversion formula provides turning from the image to original for the important case of imaging function simple poles K(p). For FRF y ( p ) with multiple poles the approach remains the same but the formulas are more composite and then less obvious [2, 5].

(7.4)

F ( p ) and Q( p ) are polynomials of the integer degrees of a variables p . The radicals of the denominator polynomial Q(p) defined over the relation Q (p) = 0 are the poles of FRF y ( p ) . According to the requirement stipulated above for the sake of considering simplification we put that F ( p ) has first order poles, i.e. simple (not multiple) poles. Then the fraction f (t) can be represented by common fractions sum: y ( p) =

we have

(7.5)

where Q ′(p) – the derivative of the denominator by p ( Q ′( p ) = dQ / dp ) , pν – the radicals of the denominator Q( p ) . We put that there are r radicals, i.e. ν = 1, r . Let's turn from (7.4) to the original y (t ) . For this purpose we shall take into account that 1 p is the unit step saltus image. Then taking into consideration the theorem of displacement (the formula (3.7)): fν ( p ) = f ( p ∓ pν ) ← f ( t ) e ± pν t ,

8. THE EXAMPLES OF TRANSIENT PROCESSES CALCULATION OVER THE INVERSION FORMULA FOR FRF WITH SIMPLE POLES

i (t )

The Example 1. Obtain the voltage in parallel oscillatory circuit u k ( t ) at energizing it by the current source i(t ) = A01(t ) sin(ωн +ψ ) (fig. 8.1). Let's accept zero initial conditions. The Solution. Having taken advantage of the Ohm law in the operator form we shall obtain

L

C uk (t ) R

Рис. 8.1

u k ( p ) = K ui ( p) i ( p ) = z ( p) i ( p ) ,

(8.1)

where K ui ( p ) – the transfer performance of the circuit. The current operates in the input, the voltage is obtained in the output that corresponds to the transfer performance dimension [K ui ( p )] = [V] / [A] = 33

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Ohm, i.e. the transfer performance has a dimension of resistance: hence we have introduced the index ui for K ( p ) . Thus in this case K ui ( p ) = z ( p ) that is obvious from the circuit (fig. 8.1). Let's find z ( p ) . We have two parallel branches of the oscillatory circuit: the inductor one with the operator resistance z L ( p ) = pL + r and the capacitor one zc ( p ) = 1 / pC . Hence 1 ( pL + r ) z z 1 p + 2α pC , (8.2) z( p) = L c = = z L + zc pL + r + 1 C p 2 + 2α p + ω 2p pC where α – the oscillatory circuit damping factor, α = r / 2 L ; ωr – the

resonant frequency of oscillations, ω r = 1 LC . It is necessary to determine uk (t ) . Let's find the imaging function for a current radiosaltus switching to the oscillatory circuit: i ( p ) = L{A01(t ) sin(ω ct + ψ )} . According to (4.9) and (4.10) p sin ψ + ω c cosψ . (8.3) i ( p ) = A0 p2 + ω c2 Substituting (8.2) and (8.3) in (8.1) we shall obtain 1 p sin ψ + ω c cos ψ p + 2α . u k ( p ) = A0 ⋅ 2 2 2 C p + ωc p + 2αp + ω r2

(8.4)

We find the function uk ( p ) poles as radicals of the denominator Q( p ) supposing Q(p) = 0 . Q(p) = (p 2 + щc2 )(p 2 + 2бp + щr2 ) = 0 .

(8.5)

Hence the p1,2 are obtained from the condition ( p2 + ω c2 ) = 0 ; we find the p3,4 supposing (p

2

+ 2αp + ω r2 ) = 0 .

Then the complex conjugate pairs

p1,2 = ± jω c ,

35

p3,4 = −α ± jω 0 ,

where ω0 – the eigen oscillations frequency ω 0 = ω r2 − α 2 ,

(8.7) are the poles of the imaging function uk ( p ) . The denominator derivative Q( p) of the expression (8.4) by a variable p yields Q′( p ) = 2 p( p2 + 2αp + ω r2 ) + ( 2 p + 2α )( p2 + ω c2 ) .

(8.8)

Substituting denominator radicals in (8.8) we shall obtain: Q′( p ) p = p1 = 2 jω c [( jω c )2 + 2αjω c + ω r2 ] = 2 jω c (ω r2 − ω c2 + 2αjω c ), Q′( p ) p = p2 = −2 jω c [(− jω c )2 − 2αjω c + ω r2 ] = −2 jω c (ω r2 − ω c2 − 2αjω c ), Q′( p ) p = p3 = [2( −α + jω 0 ) + 2α ][(−α + jω 0 )2 + ω c2 ] = = 2 jω 0 (α 2 − ω 02 − 2αjω 0 + ω c2 ),

or, taking into account (8.7), Q′( p ) p = p3 = 2 jω 0 [α 2 + α 2 − ω r 2 − 2αjω 0 + ω c2 ] = = 2 jω 0 [2α 2 − 2αjω 0 + (ω c2 − ω r2 )]. Similarly for the pole p4

Q′( p) p= p4 = −2 jω 0 [2α 2 + 2αjω 0 + (ω н2 − ω p2 )].

Let's determine the numerator F ( p ) from (8.4) as F ( p) =

A0 ( p sinψ + ω н cosψ )( p + 2α ). C

According to the inversion formula (7.7) the solution for uk (t ) looks like uk (t ) =

4

F( p )

∑ Q′( p ) ν

e pν t 1(t )

p = pν

ν =1

or substituting the value Q′( p) p = pν we obtain

(8.6)

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⎡ F( p2 ) F( p1) e− jωнt + e jωнt + uk (t) = ⎢ 2 2 2 2 −2 jωн (ωp −ωн −2α jωн ) ⎢⎣2 jωн (ωp −ωн +2α jωн ) F( p3 ) + e(−α+ jω 0)t + 2 2 2 jω0 (ωн −ω0 −2α jω0 +α 2 ) +

R uin

(8.9)

⎤ e(−α− jω 0)t ⎥1(t). −2 jω0 (ω −ω0 +2α jω0 +α ) ⎦⎥ F( p4 )

2 н

2

Fig. 8.2

2

In this solution the first two terms determine the forced component of transient process (FCTP) (the residues in the "forced" poles p1,2 = ± jω c ); the last two terms determine the free component of transient process (FrCTP) (the residues in the "free" poles p3,4 = −α ± jω0 ). The result found as (8.9) basically yields the required solution. However, for the purpose of writing the real signal uk (t ) in the obvious form it is required to execute a series of intricate and difficult operations with complex functions in (8.9). Besides it is difficult to pick out evidently the phase of the oscillation uk (t ) in radiosignal writing uk (t ) obtained after such transformations. Therefore such an approach especially hampers the analysis of phase systems operation at researching of signals passing through selective circuits. On the other hand, the radiosignal phase microstructure allows in many cases to obtain much more information than it can be picked up in RED operating on radiosignal envelope. It has led to wide phase methods and tools using at creating modern RED. The explained in this tutorial method simplifying ILT allows to decrease greatly the difficulty of result obtaining at researching of oscillatory processes and to pick out relatively easily the phase of the studied radiosignal in the obvious form. The Example 2. Switching of a constant voltage to integrating circuit (Fig. 8.2).

37

uout

С

The Solution. uout ( p ) = A ⋅1( t ) , K ( p ) = K ( p) =

1 pC

zc ( p ) , zc ( p ) + R

1 1 R+ pC

=

1 , 1 + pτ

(8.10)

where the time constant τ = RC . uin ( p ) = L {uin (t )} = L {A ⋅1(t )} =

A , p

uout ( p ) = uin ( p ) ⋅ K ( p ) .

Thus uout ( p ) =

A 1 . ⋅ p 1 + pτ

(8.11)

The poles of the imaging function uout ( p) lay in a complex plane in the points p1 = 0 , p2 = −1 τ , Q( p ) = p(1 + pτ ) , Q′( p ) = (1 + pτ ) + pτ = 1 + 2 pτ , Q′( p ) p = p1 = 1 , Q′( p ) p = p2 = 1 − 2 τ τ = −1 . Having taken advantage of the inversion formula we have 2

F ( pν ) pν t A ⎡A ⎤ e 1(t ) = ⎢ e0 t − e−t τ ⎥ ⋅1(t ) = A(1 − e−t τ ) ⋅1(t ) (8.12) ′ Q ( p ) 1 1 ⎣ ⎦ p= pν ν =1

uout (t ) = ∑

The signal found in (8.12) as a response of integrating circuit to switching of the function proportional to the unit step saltus is proportional to transient function of integrating circuit. A is a proportionality constant. Thus uout (t ) = A ⋅ h(t ) , where h(t ) – the transient function of integrating circuit. Let's notice that the first term in (8.12) – the residue in 38

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the pole p1 = 0 , i.e. in the pole defined from IF of energizing signal ("the forced pole"). This term determines FCTP. The second term in (8.12) – the residue in the pole p2 = −1 τ , it characterizes FrCTP. Here p2 – the "free" pole defined from the transfer performance (system function) K ( p ) . Let's notice that the circuit shown at fig. 8.2 is not the ideal integrator. At enough large time constant τ when the duration of the process is much less than τ the effect of integration (accumulation) owing to "storage" of the capacity C takes place. According to the theorem of time function integral image the ideal integrator transfer performance looks like k int .id . ( p ) = 1/ p . The Example 3. Obtain the impulse response (impulse characteristic) of integrating circuit. The Solution. The impulse response is a response of a circuit to δ -impulse. The δ -impulse image fδ ( p) = 1 (the formula (2.8.)). The transfer performance of integrating circuit K ( p ) = (1 + pτ ) −1 is determined in (8.10). Then for the integrating circuit impulse response we have IF g ( p ) = K ( p ) fδ ( p ) = (1 + pτ ) −1 , (8.13) with the pole at p1 = −1 τ . As follows from (8.13) we have only "free" poles when determine the impulse response. The IF circuit response i.e. g(t ) represents a free process determined only by circuit (system). To turn (8.13) to the originals space we shall take advantage of the inversion formula (7.7). Then from (8.13) Q( p ) = 1 + pτ , Q′( p ) = τ , F ( p ) = 1 and hence g( t ) =

1 −t τ e 1( t ) .

τ

(8.14)

The Example 4. Obtain the integrating circuit response to a radiosaltus uin (t ) = A sin(ω ct + ψ )1(t ) . The Solution. According to (4.10) the imaging function for input signal looks like uin ( p ) = A ( p sin ψ + ω c cos ψ )( p2 + ω c2 ) −1 . 39

Then taking into account the expression (8.10) for the integrating circuit transfer performance we have IF for output signal as A p sin ψ + ω c cosψ 1 , (8.15) uout ( p ) = τ p +1/ τ p2 + ω c2 where the "forced" poles p1,2 = ± jω c , the "free" pole p3 = −1 τ , F ( p ) = А / τ ( p sin ψ + ω c cosψ ), Q( p ) = ( p2 + ω c2 )( p + 1 / τ ), Q′( p ) = 2 p( p + 1 / τ ) + ( p2 + ω c2 ) = 3 p2 + 2 p / τ + ω c2 , Q′( p ) p = p = 2( jω c )2 + 2 jω c / τ = 2 jω c ( jω c + 1 / τ ), 1

Q′( p ) p = p = 2( − jω c )2 − 2 jω c / τ = −2 jω c ( − jω c + 1 / τ ), 2

Q′( p ) p = p = 3( −1 / τ )2 − 2 / τ 2 + ω c2 = ω c2 + 1 / τ 2 . 3

Then according to the inversion formula (7.7) we shall obtain the expression for the required integrating circuit response to a radiosaltus uout(t ) = +

− jωc sinψ + ωc cosψ A ⎡ jωc sinψ + ωc cosψ exp jωct + exp(− jωct ) + ⎢ 2 jωc ( jωc +1/τ ) − 2 jωc (− jωc +1/τ )

τ⎣

(−1/τ )sinψ + ωc cosψ

ωc2 +1/τ 2

⎤ exp(−t /τ )⎥1(t ) ⎥⎦

or A⎡ 1 exp j (ω ct + ψ ) + ⎢ τ ⎣ 2 j ( jω c + 1 / τ ) 1 + exp[− j (ω ct + ψ )] + − 2 j ( − jω c + 1 / τ )

uout (t ) =

+

(8.16)

⎤ (−1 / τ ) sinψ + ω c cosψ exp(−t / τ )⎥1(t ). 2 2 ωc + 1/ τ ⎥⎦

Let's notice that the first two terms in the formula (8.16) (the residues in the "forced" poles p1,2 = ± jω c ) determine FCTP, the third term (the residue in the "free" pole p3 = −1 τ ) determines FrCTP. We can rewrite the expression (8.16) as 40

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⎧1 uout ( t ) = A⎨ K ( jωc ) exp j( ωct +ψ ) − ⎩2 j 1 − K ( − jωc ) exp[ − j( ωct +ψ )] + 2j − sinψ + ωcτ cosψ ⎫ exp( −t / τ ) ⎬1( t ), + 1 + ωc2τ 2 ⎭

1

where is taken into account that

K( ± jωc ) = K( p )p=± jωc = (1+ pτ )p−=1± jωc follows from (8.10). The Example 5. Obtain the impulse and transient responses of the circuit (fig. 8.3). С +

–

uin (t )

uout (t )

R

Fig. 8.3 The circuit is widely used both for separating off the direct component (for example as a "not distorting" transitional circuit between cascades) and for quasidifferentiating (a circuit for signal shortening). At realization of equivalent inequalities 1

ω lowest C

R , >> τ . ωh ω hC The differentiation requirements for the circuit are the inequalities which are inverse to the inequalities (8.17). In this meaning we can say that the differentiating circuit is a circuit of a small time constant. The boundaries of spectrum are picked from the worst for realization of the inequalities (8.17) and (8.18). So for differentiation the inequalities (8.18) are more difficultly fulfilled for the upper frequencies of a signal spectrum. The frequency ω h is introduced there. For distortionless signal transmission the inequalities (8.17) are more difficultly fulfilled for low-frequency components of a signal spectrum. Therefore the frequency ω lowest is introduced there. Thus for the same circuit the differentiation area is concentrated in the vicinity of low frequencies; the area of distortionless transmission – beginning from some frequency ω lowest to the high-frequency area. The high-frequency spectrum components are accentuated at differentiation. The ideal differentiator has the transfer performance K d .id . ( p ) = Ap (the theorem of derivative image). As in both cases (differentiation and distortionless transmission of a signal) the circuit at fig. 8.3 remains the same, it is described by identical mathematical model (one differential equation). It means that the transfer performance K ( p ) determined by the circuit at fig. 8.3 does not depend on realization of the inequalities (8.24) and (8.25) and the transient process for both cases calculates under the same formulas. The Solution. Let's find the transfer performance K ( p ) for the circuit at fig. 8.3. This circuit is a potentiometric divider. So R R pτ p K ( p) = = = = . (8.19) R + zc ( p) R + 1/ pc 1 + pτ p + 1/τ Impulse response imaging function pτ p g ( p) = fδ ( p ) K ( p) = 1 = . (8.20) 1 + pτ p + 1/τ

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Transient response imaging function 1 1 . (8.21) h( p ) = fus( p )K( p ) = K( p ) = p p+1/ τ In both cases (the formulas (8.20) and (8.21)) fractions substituter is the same i.e. Q ( p ) = p + 1 τ , its radical (the pole of IF) p1 = −1 τ , Q ′( p ) = 1 . As the signal IF pole in the circuit output is found as a pole of transfer performance (i.e. the "free" pole) that the impulse response y(t) and the transient response h(t) determine in this case a free process (FrCTP). To determine the originals of IF ((8.20) and (8.21)) let's take advantage of the inversion formula (7.7) according to which we obtain at once: (8.22) g(t ) = −1 τ e −t τ 1(t ) , h(t ) = e −t τ 1(t ) .

(8.23) Let's notice that as the impulse response g(t) as the transient response h(t) of the circuit at fig. 8.3 are determined by the same exponential-damping function which damps as faster as less the circuit time constant. But the impulse response has the inverse sign concerning the sign of the action and its magnitude in the initial moment is as more as less the time constant. It is physically clear. As from δ -signal affection (for mathematical model its duration is infinitesimal, amplitude is indefinitely large and square is equal to the unit from the normalization requirement, i.e. its energy is finite) the capacity is charged up to the voltage value as greater as less the circuit time constant; for this case the signs of charge on the capacity are shown at fig. 8.3. After the δ impulse termination the capacity is discharged on the resistance R; thus we have a minus of voltage on the resistor lead connected to the capacity concerning the second one4. For transient response representing a response of the circuit to a positive unit step saltus we have a mode of the capacity C charging at forming the output signal taking off from the resistor R. Thus the out4 In the formula (8.22) circumscribing a circuit response to δ-impulse only the regular signal component is determined. Here it means that the circuit response to the impulse (perturbation) is considered at approaching on the right to the time axis zero,

put signal initial value is equal to "+1" (all input signal is applied to the resistor R at t=0 owing to the supposition that uc ( 0 ) = 0 and does not depend on the circuit time constant τ ). The polarity of output exponential impulse h(t) is positive and corresponds to the capacity C charging mode. The Example 6. Obtain the response of the transitional circuit at fig. 8.3 on switching of the radiosaltus uin ( t ) = A sin( ω ct + ψ )1( t ) . The Solution. The imaging function for input signal from (4.10) looks like p sinψ + ω c cosψ . uin ( p ) = A p2 + ω c2 Then taking into account (8.19) for the circuit transfer performance we have the output signal imaging function as: p sinψ + ω c cosψ p , (8.24) uout ( p ) = A 2 2 p + 1/ τ p + ωc where the forced poles are p1,2 = ± jω , the "free" pole is p3 = −1 / τ . The IF denominator is Q( p ) = ( p2 + ω c2 )( p + 1 / τ ) , and the numerator is F ( p ) = A ( p sin ψ + ω cсosψ ) p . The denominator derivative is 1 Q′( p ) = 2 p( p + ) + ( p 2 + ω с2 ) = 3 p 2 + 2 p / τ + ω с2 . τ As the denominator Q( p ) and the output signal IF poles are coincide with the example 4 than the values calculated for the corresponding poles are coincide too, namely: Q′( p ) p = p = 2 jω с ( jω с + 1 / τ ), Q′( p ) p = p = −2 jω с ( − jω с + 1 / τ ), 1

Q′( p ) p = p = ω с2 3

2

2

+ (1 / τ ) .

Then having taken advantage of the inversion formula (7.7) we obtain the required expressions for the output signal of the circuit at fig. 8.3 similarly to the solution of the example 4:

i.e. from the moment t = 0 + . In this case we consider that δ-impulse is not enveloped remaining on the left at the coordinates origin. At δ-impulse enveloping the impulse response would contain δ-signal as well [11, 30].

43

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⎡ jω c uout ( t ) = A⎢ exp j( ω ct +ψ ) + ⎣ 2 j( j ω c + 1 / τ ) − jω c exp[− j( ω ct +ψ )] + + − 2 j ( − jω c + 1 / τ ) +

where zk ( p ) = pL +

1 p , = e ( p) 2 2 1 L ( p + 2 α p + ω ) r pL + +r pC

(8.28)

the

damping factor α = r / 2 L , resonant frequency . The energizing signals images at impulse and transient ω r = ( LC ) responses determining are eδ ( p) = 1 , euss( p) =1/ p accordingly. Substituting them in (8.28) we shall obtain IF for the impulse response gi (t ) and the transient response hi (t ) of the series-tuned circuit: (8.29) gi ( p ) = p[L ( p 2 + 2α p + ω r2 )]−1, hi ( p ) = [L ( p 2 + 2α p + ω r2 )]− 1 The poles of both IF are p1,2 = −α ± jω 0 , where ω0 – the fre−1 / 2

⎧1 1 uout (t ) = A⎨ K ( jωc ) exp j(ωct + ψ ) − K ( − jωc ) exp[− j(ωct + ψ )] + 2j ⎩2 j ⎫ sinψ − ωcτ cosψ + ⋅ exp(−t / τ ) ⎬1(t ), 2 2 1 + ωc τ ⎭

where according to (8.18)

± jωc K ( ± jωc ) = K ( p ) p = ± jω c = . ± jω c + 1 / τ

The Example 7. Obtain the impulse and transient responses for the series-tuned circuit (fig. 8.4). C

i ( p) = e ( p)

where

or, similarly to the relation (8.16 а) obtaining we have:

L

y k ( p ) = 1/ z k ( p ) .

Then

(8.25)

⎤ sinψ − ω cτ exp( −t / τ ) ⎥1( t ). 2 2 1 + ωc τ ⎦

1 + r, pC

r

quency of eigen oscillations ω0 = (ωr2 − α2 )1/ 2 . Then the p1,2 are the system function yk ( p) poles, so they determine the system free oscillations. The denominators Q( p ) of imaging functions in (8.29) are identical: Q( p ) = L( p 2 + 2αp + ω 2р ) . The derivative Q′( p) = 2 L( p + α ) . Then Q′( p) p = p = 2L ( −α + jω0 + α ) = 2 jω0 L , 1

i(t)

Q ′( p ) p = p = − 2 j ω 0 L . e(t)

Fig. 8.4 The Solution. The impulse and transient responses for the series-tuned circuit are found as responses of the circuit on switching of the electromotive force (EF) source like δ -impulse or unit step saltus accordingly. Let's write the expression in the operator form starting from the Ohm's law: (8.27) i ( p) = e ( p) / zk ( p) = e ( p) yk ( p) , 45

2

Substituting found relations in the inversion formula (7.7) and taking into account (8.29) we shall obtain the required expressions for the impulse and transient responses of the series-tuned circuit: 2 F( p ) i e pit = −α + jω0 e( −α + jω0 )t + −α − jω0 e( −α − jω0 )t = g (t )= ∑ i=1 Q′( p ) 2 jω0 L − 2 jω0 L i = =

−α t

[ (

)]

) (

e jω t − jω t jω t − jω t −α e 0 − e 0 + jω0 e 0 + e 0 = 2 jω0 L −α t

e

ω0 L

(−α sinω t +ω cosω t )1( t), 0

0

0

46

(8.30)

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h (t )=

1 2 jω 0 L

( −α + jω 0 )t

e

+

−α t

1 e ( −α − jω 0 )t e = sin ω 0t 1(t ). (8.31) − 2 jω 0 L ω0 L

It follows from the obtained expressions (8.30) and (8.31) that g (t )=

dh ( t ) dt

, i.e. the link between impulse and transient responses is

the same as between energizing signals as δ (t ) = d1(t ) dt . It affects the images of these functions that they differ by the factor L{δ (t )} = 1, L{1(t )} = 1 p ⇒ pL{1(t )} = L{δ (t )}. The multiplication by p in the images space means the derivative determination in the originals space. The Example 8. Obtain the impulse and transient responses for the voltage on a parallel oscillatory circuit at its energization by a current source (fig. 8.1). The Solution. The impulse and transient responses for the voltage on a parallel oscillatory circuit are found as responses of the circuit on switching of the current source like δ -impulse or unit step saltus accordingly. Let's write the expression in the operator form according to the Ohm's law (8.32) uк ( p ) = i ( p )z к ( p ) , where the operator resistance of the circuit zc ( p ) is determined by the relation (8.2). The signals images are iδ ( p) = 1 , i ( p ) = 1 / p. us

Then we can write IF for the impulse and transient responses of a parallel oscillatory circuit according to (8.32): p + 2α p + 2α 1 1 gи ( p ) = = , (8.33) 2 2 C p + 2αp + ω r C ( p + α )2 + ω 02 p + 2α 1 1 p + 2α = . (8.34) hи ( p ) = 2 2 pC ( p + α )2 + ω 02 pC p + 2αp + ω r For IF g u ( p ) we have the poles p1,2 = −α ± jω 0 , for IF hu ( p ) the poles p1,2 = −α ± jω0 and p3 . For IF defined by the formulas (8.33) and (8.34) we accept the function F ( p ) = ( p + 2α ) C . In the function gu ( p ) 47

the denominator is Q δ ( p ) = p2 + 2αp + ω r2 , in hu ( p ) the denominator is Q uss ( p ) = p( p2 + 2αp + ω r2 ) , which coincide with the corresponding

functions of the previous example. Then as well as in the example 7: Q ′δ ( p ) = 2 ( p + α ), Q ′uss ( p ) = 3 p 2 + 4 α p + ω r2 , that gives Qδ′ ( p ) p = p = 2 jω 0 , Q′δ ( p ) p = p = −2 jω 0 . 1

2

Similarly

Q′uss ( p p = p1 ) = 2 jω0 ( −α + jω0 ), Q′uss ( p p = p2 ) = −2 jω0 ( −α − jω0 ), Q′uss ( p p = p3 ) = ω 2p .

Let's find the looked for impulse and transient response of a parallel oscillatory circuit having taken advantage of the inversion formula (7.7): ⎡ − α + jω 0 + 2α ( −α + jω0 )t − α − jω 0 + 2α ( −α − jω0 )t g и (t ) = ⎢ e + e 2 jω 0 C − 2 jω 0 C ⎣ e −α t = (α sin ω 0t + ω 0 cosω 0t )1(t ), ω 0C

−α − jω0 + 2α (−α + jω0 )t 2α 1 ⎡ −α + jω0 + 2α (−α + jω0 )t + 2 e + hи (t ) = ⎢ e − 2 jω0 (−α − jω0 ) с ⎣ 2 jω0 (−α + jω0 ) ωp

⎤ ⎥1(t ) = ⎦ (8.35)

⎤ ⎥1(t ) . ⎦

The last relation can be converted to more convenient appearance: ⎛ α 2 + ω 02 ⎞⎤ 1 ⎡ (8.36) hи (t ) = 2 ⎢2α − e −α t ⎜⎜ sin ω 0t + 2α cosω 0t ⎟⎟⎥1(t ) . ω p C ⎣⎢ ⎝ ω0 ⎠⎦⎥ We can draw the following conclusion from the examples considered above: the originals finding on a given imaging function becomes greatly complicated if it is necessary to convert a function containing complex conjugate poles. The presence of them indicates the oscillatory system response to an energizing signal.

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9. THE APPLICATION OF FOURIER TRANSFORMATION FOR THE ANALYSIS OF SIGNAL PASSING THROUGH THE INVESTIGATED TRACT The great importance of the spectral method at radioelectronics has stipulated wide elucidation of it in scientific, engineering, teaching and methodical literature. The work of Kharkevich A.A. was the establishing in this direction [26]. It contains the material on the spectral method and analysis of signal spectrums which is complete enough from the point of view of methodic.

Let's consider a periodic function f (t ) = f (t + nT ) ,

(9.1) where T – the least period, n – any integer number including zero. The Fourier series in the trigonometric form for such a function looks like ∞

∑(a k cos

k =1

2рk 2рk + bk sin ), T T

1 c0 = T

∫

−

f (t )dt .

(9.3)

T 2

ak =

2 T

∫

T − 2

f(t) cos

2рk 2 tdt, bk = T T

T 2

∞

∑Ck e

f(t) =

j

2р ⋅k t T

(9.8)

k = −∞

⎞ ⎛ 2рk ⎞⎤ ⎡ ⎛ 2рk ⎞ ck ⎢ j ⎜⎝ T −ϕ k ⎟⎠ − j ⎜⎝ T −ϕ k ⎟⎠ ⎥ ⎛ 2рk ck cos⎜ +e − ϕk ⎟ = ⎢e ⎥= ⎠ 2 ⎢ ⎝ T ⎥ ⎣ ⎦

= Ck e

j

2πk t T

+ C −k e

−j

2πk t T ,

(9.9)

where Ck and C− k – the oscillations complex amplitudes. c C k = k e − jϕ k , 2

c C − k = k e jϕ k . 2

(9.10)

Hence it follows that

∫

f(t) sin

T − 2

2рk tdt, T

∞ ⎞ ⎛ 2πk t − ϕk ⎟ , f (t ) = c0 + ∑ ck cos ⎜ ⎠ ⎝ T k =1

(9.11)

where the sign (*) designates the conjugate magnitude. From (9.7) and (9.10) (9.4)

Let's notice from (9.4): ak = a− k , bk = −b− k . The formula (9.2) for Fourier series by simple transformations can be reduced to

49

(9.6)

C k = C −* k ,

The coefficients ak and bk are determined by the formulas T 2

bk , ak

whence it follows that ck = c− k , ϕ k = −ϕ − k . The formulas (9.5) and (9.6) are obtained supposing in (9.2) that (9.7) ak = ck cos ϕ k , bk = ck sin ϕ k , The Fourier series (9.5) can be represented in more compact exponential form

(9.2)

where the average value for the period (constant component) is equal to T 2

ϕ k = arctg

which is obtained applying the representation of a cosine function over the Euler formula to the series (9.5). According to the formula each term of the sum can be filled as:

9.1. The Fourier Series

f(t) = c0 +

ck = ak2 + bk2 ,

(9.5)

c a b C k = k (cos ϕ k − j sin ϕ k ) = k − j k . 2 2 2

(9.12)

Substituting the formulas (9.4) in (9.12) we obtain the expression for the complex amplitude C k determined over the initial signal: T −j 1 2 Ck = ∫ f (t )e T −T

2π k ⋅t T

T

dt ,

2

1 2 C k = 0 = c0 = ∫ f (t )dt . T −T 2

50

(9.13)

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The complex amplitude C k is uniquely linked with the function f (t) (to the signal f (t) kind). In other words each signal has its own spectrum, i.e. each function f (t ) can be represented only by its own Fourier series (the theorem of uniqueness of Fourier series representation). Let's designate the frequency щ1 = 2р T , where ω1 – the first harmo-nic component frequency. The period of it coincides with the typematic period T of initial function. Then the frequency k ω1 = 2πk T = ω k , where ω k – the frequency of k-th signal harmonic component. A signal writing using one of the Fourier series forms is sometimes called spectral signal representation. The signal spectral representation can obviously be shown at the graphs of dependence of the amplitudes ak , bk and c0 (fig. 9.1) for the Fourier series in the form (9.2) or of dependence of the amplitudes ck and component initial phases (fig. 9.2) for the Fourier series in the form (9.5). ak

bk

с0

ω ω1

0

3ω1

2ω1 3ω1

ω1

0

ω

2ω1

Fig. 9.1 ϕk π

ck с0 0

ω1

ω1

ω1

ω1

ω

0 −π

Fig. 9.2 51

ω1

ω1

ω

Such a discrete spectrum which has the distance between contiguous spectrum components equal to the repetition rate (to the frequency of first harmonic) is called harmonic spectrum and its components – harmonic components. The amplitudes of some harmonic components can be equal to zero. The expansion into a series is possible if the amplitudes ak , bk and c0 (the formulas (9.3) and (9.4)) or the complex amplitudes C k (the formula (9.13)) can be found. For this purpose the initial function should meet the Dirichlet conditions: the function f (t ) is restricted, piecewise continuous and has a finite number of extremes on the period. The physical sense of Fourier series is easy to follow and has practical application. So it is physically possible to collect a periodic signal if to sum up separate spectrum components combining, for example, signals from sine oscillations generators of necessary frequencies (the frequencies of first and higher harmonic components). Thus "to collect" the required oscillation it is necessary to set the appropriate amplitude ck and initial phase ϕ k for any harmonic component of the Fourier series (9.5) according to the formulas (9.6), where ck and ϕ k are found over ak and bk determining the amplitudes of cosine and sine components of the Fourier series (9.2). In practice it is more convenient to use the values of complex amplitudes C k to determine ck and ϕ k defined by the integral (9.13). Then from (9.10) we have ck k ≠ 0 = 2 C k , ϕ k = − arg C k , where the enclosure of complex magnitude into dashes means that the absolute value is taken, i.e. the same as the lack of a point above complex magnitude. In the tutorial both absolute value designations are applied, i.e. Ck = Ck . The systems using the "harmonic" components sum for obtaining of the required form signal are created practically. Here it means that the appropriate amplitude ck and initial phase ϕ k of oscillation is set for any "harmonic" component. However these components are cut off in the time outside of the chosen time interval enclosing the time of 52

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the signal existence. The following approach is used as a fundamental of such circuits realization. An investigated signal is periodically prolonged by the chosen typematic period along the time axis. For such an iterated hypothetical signal ck and ϕ k are found, thus its harmonic components are determined. Then to determine the required signal its components are summed up on the interval equal to the chosen typematic period. It is achieved by cutting of continuous components representing Fourier series on the right and on the left with regards to the chosen summing time interval. At such a signal simulation the remarkable property of Fourier series – fast convergence to the initial signal is used. It allows to be restricted by minimal number of sum terms which are necessary for ensuring of a required precision when describing a given signal. 9.2. The Integral Fourier Transforms

⎛1 ⎜ ⎜ k = −∞ ⎝ T ∞

∑

T /2

∫ f (t )e

−T / 2

− jk ω1t

⎞ dt ⎟ e jk ω1t . ⎟ ⎠

(9.14)

Let's apply not absolutely strict but visual proof of conversion from a periodic to a non-periodic signal. Having fixed one of periodic 53

f(t)

f(t)

t

Т

Т

T →∞

t

T → −∞

a

b

Fig. 9.3 Then one impulse of a periodic sequence stays. Thus

Any real physical process is limited by time, i.e. once having been begun it will inevitably be finished. Obviously considering a real electrical signal as a some physical process it is possible to state that it always has a beginning and ending on the time axis. It means that the periodic signal considered above which is represented by Fourier series (i.e. by the discrete spectrum of the harmonic components sum) in the frequency domain was a useful abstraction as, strictly speaking, a periodic signal repeating by the period of T on the time axis up to t → ±∞ doesn't physically exist. The same concerns the harmonic spectrum components with the appropriate amplitude which should be prolonged in the time to t → ±∞ and consequently can not be physically implemented. Let's notice that the signals existing on some restricted time interval on which they are defined are called finite signals. Let's try to take advantage of Fourier series for the case of a nonperiodic one-time signal. For this purpose we shall rewrite the series (9.8) having substituted the formula (9.13) for C k in it. We obtain: f (t ) =

signals on the time axis we turn the period of T to the perpetuity ( T → ∞ ) (fig. 9.3.a, b.).

ω1 =

2π T

→ dω . T →∞

Hence the distance between spectrum components at T → ∞ aspires to dω , i.e. to infinitesimal. Besides k ω1 T →∞ → ω . Thus, k ω1 takes not only discrete first harmonic multiple frequencies, but all the values ω on the frequency axis. We obtain a continuous spectrum instead of a lined one. In further let's multiply and divide the factor 1 T in (9.14) by 2π . Then 1 2π 1 = = dω . T 2πT 2π Hence at T → ∞ we can rewrite (9.14) as f (t ) =

1 2π

∞

∞

∫ ( ∫ f (t )e

− jω t

dt )e jωt dω .

(9.15)

−∞ −∞

The expression (9.15) is called double Fourier integral. The interior integral depends on a current frequency value ω . Let's designate it by a complex frequency ω function S (ω ) =

∞

∫ f (t )e

−∞

54

− jω t

dt .

(9.16)

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Substituting (9.16) in (9.15) we can write f (t ) =

1 2π

∞

∫ S (ω )e

jω t

dω .

(9.17)

−∞

The function S ( ω) is called spectral density or spectral function. Sometimes owing to shortening it is simply called spectrum. The integral (9.16) is called direct Fourier transformation (DFT). Let's symbolically designate DFT over an operator F . S (ω ) = F { f (t )} =

∞

∫ f (t )e

− jω t

dt .

−∞ ∞

∫

f (t ) dt , i.e. if this integral has a finite value.

−∞

The integral (9.17) is called inverse Fourier transformation (IFT). It is designated over an operator F −1 . Then f ( t ) = F −1{S (ω )} =

1 2π

∞

∫ S (ω )e

jωt

dω .

(9.19)

−∞

The integrals (9.16) and (9.17) are called integral Fourier transformations. 9.3. The Complex Amplitude And Spectral Density Comparison As Well As The Fourier Series And Fourier Integral Let's compare the formulas (9.13) and (9.16) for a complex amplitude C k and spectral density S (ω ) . For this purpose we shall rewrite these formulas again as: Ck =

1 T

T /2

∫ f (t )e

− jk ω1t

∞

S (ω ) = ∫ f (t )e − jω t dt .

dt ,

C (ω k ) =

Let's notice that C k is calculated for the frequency ω k = k ω1 , i.e. we can write T /2

1 − jω k t f (t )e dt . ∫ T T /2

55

1 S (ω k ) , T

(9.20)

or if not to fix on the concrete frequency: C (ω ) =

1 S (ω ) . T

(9.21)

Spectral density S ( ω) depends only on function f (t ) and complex amplitude C (ω ) depends on function f (t ) and typematic period as well. If the period Т grows, the magnitude of C (ω ) diminishes. Spectral density S (ω ) is usually shown in tables. To determine C (ω ) for the required typematic period it is necessary to divide S (ω ) by the value of T. The convenience of the function S (ω ) against the function C (ω ) is that the first of them depends only on frequency but does not

depend on typematic period. To within dimension is possible to put that C (ω ) = S (ω ) at T = 1c. Now let's compare dimensions. If to appeal to (9.13) it is obvious at once that C ( ω) has the dimension of initial function f (t ) , i.e.

[C (ω )] = [ f (t )] . The dimension of S (ω ) (9.16) is [S (ω )] = [ f (t )]⋅ c = [C (ω )]⋅ c .

as follows from the formula

Now let's compare Fourier series and Fourier integral (inverse integral Fourier transformation). For this purpose we shall rewrite Fourier series in the form (9.8) and inverse Fourier transformation in the form (9.17)

−∞

−T / 2

C k = C (ω k ) =

with the limits ±T 2 for C (ω k ) will be same as the integrands are identical. Then from comparison (9.13) and (9.16) we can write

(9.18)

The spectral density S (ω ) exists if there is an absolute convergence of an integral

In further if C (ω k ) and S (ω k ) are evaluated for a signal of the same form than the values of the integral with the limits ±∞ for S (ω k ) and

f(t) =

∞

∑ C k e jщk t ,

f (t ) =

k = −∞

1 2π

∞

∫ S (ω ) ⋅ e

− jω k t

dω .

−∞

From comparison of these relations it is obvious that we sum components up in Fourier integral as well as in Fourier series. But in Fourier integral the infinitesimal amplitude components are summing up at their infinite number: dC (ω ) = (1 2π ) ⋅ S (ω )dω . Spectrum for non56

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periodic process is continuous, i.e. all components are present. The distance between contiguous components is equal to dω , i.e. infinitesimal. Thus Fourier integral (as well as Fourier series for periodic function) forms non-periodic process. But as distinct from Fourier series Fourier integral forms this process at the expense of summing of infinite number of components (continuous spectrum instead of discrete one). The amplitude of any component is infinitesimal. Thus if a nonperiodic function has the same form as a periodic one we have the relation (9.21) linking complex amplitude C (ω ) and spectral density S (ω ) .

where c – the abscissa of convergence, c ≥ 0 . It is obvious that when t is growing at t > 0 we have a diminution of the function f1(t ) with respect to f (t ) under the exponential law. It is possible to ensure the convergence of the integral (9.16) for the auxiliary function f1(t ) (fig. 10.1а and b) for physical tasks at enough large c . We can always refer the beginning of a time scale to the moment of signal arising. Then

∫

∞

S ( p) =

−∞

∫ f (t )e

− pt

0

∞

(10.2)

(10.3)

−∞

If f (t ) = 0 at t < 0 then f1(t ) = 0 at t < 0 . So we shall rewrite the integral (10.3) as one-sided DFT: ∞

S 1 (ω ) = ∫ f1 ( t )e − jωt dt

There is the closest link between integral Fourier and Laplace transformations. These transformations are so close that they are sometimes extended as a one transformation (for example naming both transformations Fourier transformation). At the same time the obviousness of the spectral method of signals and electronic circuits researches based on integral Fourier transformations allows to give by analogy the clear physical explanation to outwardly formal operational calculus methods beared on integral Laplace transformation. For comparison of the common approaches we shall write in the beginning the integral relations (2.2) and (9.16) for direct Laplace and Fourier transformations: f (t )e − jω t dt

−∞

S1(ω ) = ∫ f1( t )e − jωt dt ,

10.1. The Link Between Integral Fourier And Laplace Transformations

∞

∞

(Here the requirement of f (t ) = 0 is taken into account at t < 0 .) Now let's consider the integral

10. INTEGRAL FOURIER AND LAPLACE TRANSFORMATIONS

S (ω ) =

∞

S (ω ) = ∫ f (t )e − jωt dt = ∫ f (t )e − jωt dt .

0

and taking into account (10.1) ∞

S1 (ω ) = ∫ f ( t )e − ( c + jω )t dt .

(10.4)

0

Comparing (10.4) with one-sided direct Fourier transformation (10.2) with respect to the function f (t ) , we notice that as a result of the transformation we obtain the function S ( c + jω) , where S ( jω) – the spectral density of the function f (t ) *. Thus, it is possible to rewrite (10.4) as ∞

S1( jω ) = S ( c + jω ) = ∫ f (t )e −( c + jω )t dt .

(10.5)

0

Introducing a complex variable p = c + jω , we shall rewrite the last relation as (2.2), i.e. as direct Laplace transformation (DLT):

dt .

−∞

∞

S ( p) = ∫ f (t )e − pt dt = L{ f (t )} .

Let's introduce a new (auxiliary) function

(10.6)

0

f1(t ) = f (t )e −ct ,

57

(10.1)

Here the spectral density argument is introduced with the factor jω . It is out of principle because there is always a factor at ω in Fourier transformation. *

58

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So the difference between DFT and DLT is that the convergence of the integral at DLT is ensured at the expense of introduction of the factor e −ct ( c > 0 ) in the initial function f (t ) .

Taking into account (10.1) and (10.5), we shall rewrite (10.7) as f ( t )e − ct =

1 ∞ jωt ∫ S ( c + jω )e dω . 2π − ∞

(10.8)

Let's multiply the right and left-hand parts of (10.7) by e ct . It is possible as the factor e ct does not depend on the variable of integration ω . We obtain f (t ) =

1 ∞ ( c + jω )t dω . ∫ S ( c + jω )e 2π − ∞

(10.9)

Multiplying the numerator and denominator of (10.9) by j and taking into account that c = const , djω = d ( c + jω ) we shall rewrite (10.9) as a.) A video signal f (t ) and an auxiliary function f1(t ) which corresponds to it.

f (t ) =

1 ∞ ( c + jω )t d ( c + jω ) . ∫ S ( c + jω )e 2πj − ∞

Let's introduce a new variable of integration p = c + jω . Then for the integral limits at ω = ∞ p = c + j∞ ; ω = −∞ p = c − j∞ , and the expression (10.10) is rewritten as f (t ) =

Fig. 10.1 Let's reveal the link between inverse Fourier and Laplace transformations. For this purpose we shall write IFT (9.17) for the function f1(t ) . 1 ∞ jω t ∫ S 1 ( ω )e d ω . 2π − ∞

c + jω

pt ∫ S ( p )e dp .

(10.11)

c − jω

(10.7)

respect to the initial function f (t ) (at the expense of the factor e −ct ). Turning from (10.7) to (10.9) by multiplication of the right and left-hand parts by e ct we come to inverse Fourier transformation of the initial function f (t ) . Thus the separate components amplitudes under the integral sign are equal: dC (ω ) =

59

1 2πj

The formula (10.11) coincides with the expression (2.2), determining inverse Laplace transformation. Thus we turn from IFT of an auxiliary function f1(t ) to ILT of the function f (t ) . Let's come to a halt on the physical explanation of the link between inverse Fourier and Laplace transformations. For this purpose we apply to (10.7). This relation represents IFT of the auxiliary function f1(t ) which has a higher convergence with

б.) A signal f (t ) as a radiosaltus and an auxiliary function f1(t ) which corresponds to it.

f1 ( t ) =

(10.10)

1 1 S ( c + jω )e ct dω = S1(ω )e ct dω . 2π 2π

60

(10.12)

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So if at DFT the auxiliary function f1(t ) which has a higher convergence was introduced, now at IFT we introduce a divergence for each partial component (fig. 10.2). The turning from (10.9) to (10.11) reducing the initial relation to ILT is carried out at the expense of substitution of a real integration variable ω for a complex variable p = c + jω . But in this case the formula (10.11) (i.e. ILT) becomes more strong than the formula (9.19) determining IFT for physical tasks solving. The advantages of ILT (10.11) are ensured by a possibility of engaging of powerful methods of linear differential equations integration which are based on the theory of complex variable functions. It allowed to construct a special mathematical apparatus for researches of transient processes in linear systems – the operational calculus. 1 S (ω ) dω 2π

they ensure a build-up of a common method of the linear differential equations solving. The difference is that the integral Fourier transformations used at the spectral researches of electronic circuits and signals allow to achieve rather obvious representation of signals passing through circuits in the frequency domain. The use of integral Laplace transformations permitting, that is very important, to gain results immediately in the time domain is a bit formal. The unity of both these methods shown here allows in a particular degree to eliminate the formality of the approach at use of the operational calculus based on Laplace transformations. The achievable physical similarity of understanding of the operational calculus application allows to avoid probable errors caused by the exterior formality of its procedures. The generality of the spectral method and operational calculus allows to construct a uniform apparatus for researches of processes in linear electronic circuits as it, for example, is done in the work of M.I. Kontorovich [1]. 10.2. The Generality And Differences Of The Operational Calculus And Spectral Researches Of Linear Electronic Circuits 10.2.1. The Generality Of The Imaging Function And Signal Spectral Density

1 S (ω ) e ct dω 2π

Comparing DFT and DLT (the formulas (9.18) and (2.1)) we notice: if the beginning of a time scale envelops the time of a signal existence so, that at t < 0 we have f (t ) = 0 , then at formal substitution of a variable for jω we obtain the spectral density from the signal image. Obviously, the same rule is valid in the inverse direction. Actually comparing S ( p) =

Fig. 10.2 The closest link between integral Fourier and Laplace transformations follows from the consideration which was carried out. Generally speaking they can be considered from unified positions because 61

∞

∫

f (t ) ⋅ e − pt dt

and

S ( jω ) =

∞

∫ f (t ) ⋅ e

0

0

we can write at once S ( jω ) jω = p = S ( p) .

62

− jωt

dt ,

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Thus on the time function image the spectrum can be found at substitution p for jω and, on the contrary, at substitution jω for p the image of time function can be found. The relation (5.4) expresses the link between the images of input and output signals over the system transfer function y ( p) = K ( p) f ( p) . Substituting formally p for jω we obtain . S y ( jω ) = K ( jω ) S f ( jω ) .

The turning "image ↔ spectral density" at substitution p ↔ jω requires a particular accuracy at considering of signals circumscribed by discontinuous (singular) functions. At the same time we notice that it is physically impossible to form such signals. The spectrums of them spread up to infinitely high frequencies and to generate of such signals it would be required to have a system with infinite bandwidth that is physically impossible. However it is very convenient to determine mathematical models of real signals over standard discontinuous functions. It is widely used at researches of signals and circuits. To illustrate stated above we shall at first apply to the definition of spectral density over DFT (9.16) which can be rewritten as

∫ f ( t )e

−∞

− jω t

dt =

∞

∫ f (t )[cosωt − j sin ωt ]dt .

feven (t ) = feven (−t ) , f odd ( t ) = − f odd ( t )

(10.13)

(10.15)

– the requirements of component parity and oddness accordingly. Even and odd components of initial function are determined by the formulas f even (t ) =

f (t ) + f ( −t ) f (t ) − f ( −t ) , f odd (t ) = . 2 2

(10.16)

Substituting (10.14) in (10.13) we obtain S (ω ) =

∞

∞

−∞

−∞

∫ f even (t ) cos ω t dt − j ∫ f odd (t ) sin ω t dt = M (ω ) − jN (ω ) , ∞

∞

−∞

−∞

where M (ω) = ∫ feven(t )cos ω t dt , N (ω ) =

10.2.2. Some Limitations At Turning "Signal Image – Signal Spectral Density" At The Substitution p ↔ jω

∞

where

(10.12)

Thus at substitution p for jω in the imaging equation (5.4) we obtain at once the link between spectrums of input and output signals over the performance of a system.

S (ω ) =

Any real function can be uniquely represented by a sum of even and odd components*: (10.14) f (t ) = feven(t ) + fodd (t ) ,

∫ f odd (t ) sin ω t dt .

(10.17) (10.18)

It follows from the formula (10.18) that the real component M (ω ) is determined only by the even signal component feven (t ) ; the imaginary spectrum component N (ω ) is determined only by the odd signal component f odd (t ) . It also follows from (10.18) that M (ω ) = M ( −ω ) , −N (ω ) = N ( −ω ) , (10.19) i.e. for the real time function f (t ) the real component of the spectrum M (ω ) is the even function of frequency, and the imaginary component of the spectrum N (ω ) – the odd function of frequency. The most important relation follows from (10.17) and (10.19) (10.20) S (ω ) = S * ( −ω ) , i.e. for a real signal its spectral density is described by the complexconjugate function at frequency sign inversion.

−∞

Here and below the principal, as per Cauchy, values of integrals are taken, i.e. the integrals are calculated at equal (symmetric) trend to the upper and lower limits.

* The regular (smooth) functions up to the first order singularity (first order rupture) described by the saltus function are meant here. The higher order singularities

( δ (t ), dδ ( t ) dt d 2δ (t ) dt 2 etc.) demand for special consideration in the theory of generalized functions (the theory of distribution) [30].

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Let's write a spectral density of a signal f (t ) as S (ω ) = S (ω )e

jψ s (ω )

f sing ( t ) =

.

ψ s = − arctg

N (ω ) . M (ω )

S (ω ) = Re{S (ω )} + j Im{S (ω )} = S (ω ) cosψ s (ω ) + jS (ω ) sinψ s (ω ) ,

⎧1, t > 0 ⎪ sgn t = ⎨0, t = 0 ⎪−1, t < 0 ⎩

(10.21)

In general case the spectral density can be represented as (10.22)

so comparing (10.22) with (10.17) we have Re{S (ω )} = M (ω )

Im{S (ω )} = −N (ω ) .

(10.23) Let's turn to the signal with a regular component and ruptures of the first type superimposed to the regular one. Then it is possible to pick out separately the regular and impulse (singular) components of a signal [11, P. 136]. Let's take a saltus of a signal at t = 0 . The analog of a signal with such a saltus is obtained applying DLT to a regular signal which begins at t < 0 . In this case at the expense of a zero value of the integral inferior limit in DLT the regular component f (t ) is cut off at t < 0 , i.e. only the function f (t ) value at t > 0 is taken into account. Then f (t ) = f sing (t ) + f reg(t ) = f (0) ⋅1(t ) + f reg(t ) . Here it is obvious that f reg (0) = 0 and the saltus of the function f (t ) at t = 0 is taken into account in the first singular term of the sum. The image of singular component f sing ( t ) = f ( 0 ) ⋅1( t ) is the funcf (0 ) . p

(10.27)

The expression (10.26) determines a singular signal in the form of a saltus function as the sum of the odd component (the sign function (1 2) ⋅ f (0) ⋅ sgn t ) and the "degenerated" even component ("degenerated" in the meaning that the even component here simply represents a stationary value component, equal to (1 2) ⋅ f (0) along all time axis) (fig. 10.3). The spectrum (10.25) can be formed only by the odd component of the signal (10.26) determined by the sign function.

fsing(t )

f (0) 0

t

=

tion f sing ( p ) =

0,5f

(10.24)

0

t

The turning p → jω gives a spectrum S (ω ) =

f (0 ) = − jN (ω ) . jω

(10.25)

Here Re{S (ω )} = M (ω ) = 0 . The spectral density S (ω ) in (10.25) contains only the imaginary component. This one according to (10.18) should correspond in the time domain to the signal circumscribed only by the odd time function. But, on the other hand, a singular signal in the form of a saltus function f sing (t ) = f ( 0 ) ⋅1(t ) contains both the even and odd components. Actually f sing (t ) can be represented as the sum [32]. 65

(10.26)

where sgn t – the sign function (the function of a sign):

Then from (10.17) S (ω ) = M 2 (ω ) + N 2 (ω ) ,

1 f ( 0 )[1 + sgn t ] = f ( 0 )1( t ) , 2

+ f (0) 0 Fig. 10.3 66

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Let's check it up having executed IFT with respect to the spectral density S (ω ) = f ( 0 ) jω .

δ (t ) =

The Example 1. We find y (t ) = F −1{ f (0 ) / jω } . 1 y (t ) = 2π

I1 =

Let's determine δ -function over IFT taking into account that its spectrum is S δ (ω ) = 1 :

∞ ∞ sin ωt ⎤ f (0) f (0) ⎡ cosωt e jω t ω ω f ( 0 ) d d j dω ⎥ = [I1 + I 2 ] . = ⎢ + ∫ ∫ ∫ jω jω 2π ⎢⎣−∞ jω −∞ −∞ ⎦⎥ 2π ∞

∞

cos ω t dω = 0 by virtue of the oddness of the integrand at symjω −∞

∫

1 2π

∞

∞

∞

sin ωt sin ξ sin ξ ∫ ωt dωt = ∫ ξ dξ =2 ∫ ξ dξ . −∞ −∞ 0

The right integral is obtained from the previous taking into account the parity of the integrand. The sine integral is determined as t

Si t = ∫

0

sin ξ

ξ

dξ ,

Si ∞ = π / 2, ⇒ I 2 = π at t > 0 .

The inversion of the sign of t inverts the sign of I 2 , i.e. I 2 = −π at t < 0 . So at t = 0 we have a saltus I 2 from −π to π , i.e. I 2 = π sgn t and, hence y (t ) =

f (0) f (0) π sgn t = sgn t . 2π 2

Thus the odd component of a singular signal determined by the function ( f ( 0 ) 2 ) ⋅ sgn t has the spectral density S (ω ) = ( f ( 0 ) jω ) as it was to be proved. It is required to find the spectral density of the resulting signal stationary value component (the first term in (10.26)) to obtain the spectrum of the signal (10.26). For this purpose let's at first obtain some auxiliary relations.

67

S δ (ω )e jωt dω =

∫

−∞

1 2π

∞

∫e

jωt

dω .

(10.28)

−∞

Let's formally substitute t for ω . Then 1 2π

δ (ω ) =

∞

jωt

∫e

(10.29)

dt

−∞

or inverting the sign of ω we obtain δ ( −ω ) =

metric integration limits. I2 =

∞

∞

1 2π

∫e

− jωt

dt .

(10.30)

−∞

Let's consider δ ( x ) as the even function of the argument. It means that δ ( x ) = δ ( − x ) . Let's multiply the right and left-hand parts in (10.30) by a constant A0 . Thus we obtain A0 2πδ (ω ) =

∞

∫ A0e

− jωt

dt = F { A0 } .

(10.31)

−∞

There is DFT with respect to a constant A0 in the right-hand term of the expression (10.31), i.e. the spectral density of a constant which is determined over δ -function. In other words (10.32) S A0 (ω ) = A0 2πδ (ω ) . The Example 2. Let's find the initial value A0 having applied IFT to (10.32) to check up the obtained relation (10.32): F −1{S A (ω )} = A0 2π 0

1 2π

∞

∫ δ (ω )e

jω t

dω = A0 ,

(10.33)

−∞

as it was to be expected. Thus the spectrum of a constant is determined over δ -function in the frequency domain. Hence the spectrum of a constant 0,5 ⋅ f ( 0 ) from (10.32) S f 0 (ω ) = πf ( 0 )δ (ω ) . (10.34)

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Then the required spectrum of a singular signal (the function of a saltus with the magnitude of f ( 0 ) ) (the formula 10.26)) is determined as the sum of the spectrums (10.25) and (10.34): Sf

sing

⎡ 1 ⎤ (ω ) = f ( 0 )⎢ + πδ (ω )⎥ . j ω ⎣ ⎦

(10.35)

Consequently at considering signals circumscribed by the functions with ruptures of the first type it is necessary to be careful at turning from the imaging function of a signal to its spectral density by substitution p for jω , as the δ -singular component of the spectrum can be lost. U. V. Tronin also indicates this [33]. The necessary requirement of lack of δ -singularity in the signal spectrum is the finite magnitude of the absolute value integral of the function circumscribing a signal, i.e. ∞

∫

f ( t ) dt ≠ ±∞

−∞

(sometimes the requirement of absolute integrability is put as the necessary one at DFT [26, P. 64]). The factor of convergence e −ct , c > 0 is introduced to go round this difficulty. Thus the signal is considered to be determined on the positive time semiaxis. The spectrum of such a function is found as S e (ω ) =

1 c + jω

.

Then directing c to zero we come to the spectrum as (10.25). The δ singularity of the spectrum (10.35) is just also lost at such an approach in definition of a signal spectrum circumscribed by a function of saltus. The error occurs owing to the exponential function which at t → ∞ aspires to zero regardless of the smallness of c. Consequently this function has a limited square, so the stationary value component is equal to zero and the δ -singular component of the spectrum is also equal to zero. The other matter when DLT is discussed. In this case the factor −ct e is specially introduced for ensuring of the convergence of the initial function f (t ) . Then we use the substitution of the variable

69

p = c + jω in expression for the spectrum S e (ω ) and obtain the strict

image of the saltus function S ( p ) = 1 / p . Obviously the inverse turning from a saltus function spectrum to its image by substitution jω for p in (10.35) can not be performed as the term with δ -singularity does not allow to make the corresponding substitution. The factor of convergence-assisted turning is often met in the literature [26, 32 etc.]. Then we obtain Sf = lim S e (ω ) = 1 / jω . sing

c→0

We obtain the same result as at turning from the saltus image S ( p ) = 1 / p to its spectrum substituting p for jω . In most practical cases such an approach is appropriate. A stationary value component doesn't carry information and appears owing to the spectrum δ singularity at a saltus function modeling. As a rule this component is not usually considered for signal lines because it is not passed by transitional RC-circuits between cascades. 10.2.3. The Turning From The System Imaging Equation To The Equation For Spectrums Of Input And Output Signals The system transfer performance K ( p ) is determined according to (5.4) and (5.8) from the system imaging equation (5.2). It is obtained by turning of the system differential equation to the images space (the formula (5.1)). Thus there is a sole link between transfer performance and differential equation of a system (1.7) and (1.8). The initial conditions can be taken into account in the right term of the differential equation. In this case they come into as equivalent sources [1, P. 8-9]. At such an approach the transfer performance K ( p ) is determined only by left-hand part of the differential equation (really by the system differential equation (SDE), the formula (5.8) for K1( p ) ). Considering the link between the spectral method and operational calculus it allows without reducing a generality of the reasonings to put that an "empty" quadripole is investigated, i.e. the circuit has zero initial stores of energy in accumulative devices (the capacities and inductances) of a system. The acceptance of the hypothesis of the "empty" quadripole allows to simplify the reasonings. Such an ap70

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proach has also a great practical value because many tasks of radioelectronics assume zero initial states of a system. Besides the spectral method in usual statement does not take into account the initial conditions (a modification of the spectral method where the initial conditions are taken into account by analogy with DFT is undertaken in [1, P. 175]). Let's apply to the writing of the link (5.3) between the images of input and output signals over the system transfer function y ( p) = k ( p) f ( p) . Substituting formally p for jω we obtain

S y ( jω ) = k ( jω ) S f ( j ω ) ,

(10.36)

where S y ( jω ) and S F ( jω ) – the spectral densities of output and input signals accordingly. It follows from stated above that S y ( jω ) p = jω = y ( p) , S f ( jω ) p = jω = f ( p) .

(10.37)

The frequency complex function K ( j ω ) p = jω = K ( p )

(10.38)

is called the system complex frequency performance (or simply the system frequency performance). It follows from (10.36) that S y ( jω ) = k ( jω ) S1( jω ) .

(10.39)

The formula (10.39) can also be written in the equivalent form* (10.39а) S y (ω ) = k (ω )S F (ω ) .

* To emphasize that the dependence of the complex functions (CF) absolute values on frequency is determined in (10.39 a) the factor j at ω is omitted. It is possible owing to the fact that at finding of a complex function absolute value (which is a real function of frequency) the real and imaginary CF parts are added in quadratures. In this case in final relations for absolute value the real and imaginary CF parts are not already divided (the factor j at ω is absent). The same approach is applied at finding of the complex function argument. It is obtained as anti-tangent of the ratio of imaginary and real parts.

71

Here as it mentioned before the absence of points above the functions means that the absolute values are taken. It follows from (10.15) (or from (10.39а)), that the absolute value of the complex frequency performance indicates the link between the corresponding spectrum component amplitudes of input and output signals. This performance (i.e. K ( jω) ) is called the system frequency response (FR). Let's represent the functions which are included in (10.36) as S y ( jω ) = S y (ω )e

jΦ y (ω )

, S F ( jω ) = S F (ω )e

K ( jω ) = K (ω )e

jΦ k (ω )

jΦF ( jω )

,

, (10.40)

whence we can write Φ y (ω ) = arg S y ( jω ) , Φ F (ω ) = arg S F ( jω ) , Φ k (ω ) = arg k ( jω ) . (10.41)

It is obvious that the equation (10.36) determines the link not only between the amplitudes of input signals but also between their arguments. Really it follows from (10.36) that (10.42) arg S ( jω ) = arg k ( jω ) + arg S F ( jω ) or in other writing (10.42а) Φ y (ω ) = Φ F (ω ) + Φ k (ω ) . As from (9.20) the complex amplitudes of a periodic signal spectrum C y (ω ) and C F (ω ) are determined over the spectral densities as 1 1 S y ( jω k ) , C F ( jω k ) = S F ( jω k ) , T T 1 1 C y (ω ) = S y (ω ) , C F ( ω k ) = S F ( ω k ) , T T

C y ( jω k ) =

(10.43) (10.44)

The complex amplitudes arguments determining the initial phases of harmonic components for the Fourier-series expansion of input and output periodic signals are linked with the same dependence as the link between arguments of the corresponding spectral densities (the formulas (10.42) or (10.42а)): arg C y ( jω ) = arg S y ( jω ) = Φ y (ω ) , arg C F ( jω ) = arg S F ( jω ) = Φ F (ω ) .

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(10.45)

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So Фy (ω ) and ФF (ω ) can be treated as the frequency dependence of initial phases of relevant spectral component which amplitudes are finite for a periodic signals (they are the functions C y (ω ) and C F (ω ) ) and are infinitesimal for non-periodic signals (they are the functions 1 1 S F (ω )dω . S y (ω )dω and 2π 2π The function Φ k (ω ) = arg k ( jω )

(10.46) determining according to (10.42) or (10.42а) the dependence between the initial phases of input and output signals spectral components is called the system phase response (PR of a system). Therefore the system complex frequency response K ( jω) describes at once two frequency characteristics: the absolute value K (ω ) – FR of a system, the argument arg K ( jω) – PR of a system. So the complex equation (10.36) determines at once two links between the spectrums of input and output signals: by absolute values over FR and initial phases – over PR. It corresponds to the fact that the spectrum of any signal is determined by two performances of spectral density – the frequency dependencies of the absolute value and argument, relevant to the amplitude and initial phase of each spectral component. Let's notice that here we speak about the spectrum mathematical representation. In this case (using the Euler formula for a sinusoid representation) alongside with the positive frequencies the negative ones are formally introduced (take a look through the turning from (9.5) to (9.8)). Negative frequencies doesn't carry any physical sense because the frequency of oscillations – it is a number of periods per second which can be only positive. But it is a very convenient mathematical abstraction allowing to represent many relations in the spectrums theory in the complex form (compare, for example, the formulas (9.2), (9.3) of the Fourier series with (9.8)). Thus at physical spectrum representation only positive (physical) frequencies of the spectrum are present (the formulas (9.2) and (9.3)). At equivalent mathematical representation both positive and negative frequencies are present. A one of advantages of mathematical representation except for compactness of writings improving the clearness of obtained results is the possibility of oscillations representation as complex functions and also their visual 73

representation as vectors (where the amplitudes and phases of corresponding sine components are indicated simultaneously) on a complex plane. The physical spectrum component amplitudes are two times more than mathematical spectrum amplitudes (except for C 0 ) ( the formula (9.14)), i.e. ck = 2 C k ≡ 2 C (ω k ) , where the complex amplitudes C (ω k ) of the series are determined by the formulas (9.13). Let's indicate similar relations for the amplitudes of physical and mathematical spectrums of non-periodic real signal. Let's rewrite the expression (9.18) for the spectral density as S ( jω ) =

∞

∫

f (t )e − jωt dt =

−∞

∞

∫ f (t )(cos ωt − j sin ω t )dt .

(10.46а)

−∞

The important relation follows at once from (10.46) (10.47) S ( jω ) = S * ( − jω ) , It gives the relations for the spectral density absolute values and arguments (10.48) S (ω ) = S ( −ω ) , arg S ( jω ) = − arg S ( − jω ) . Then taking into account (10.47) let's write IFT (the formula (9.17)) as 1 ∞ 1 ∞⎡ − jω t ⎤ jω t jω t + S (−ω )e dω = ∫ S (ω )e ∫ S ( jω )e ⎥⎦ dω = 2π − ∞ 2π 0 ⎢⎣ (10.49) 1 ∞⎡ jω t − jω t ⎤ * = + S ( jω )e ∫ S ( jω )e ⎥⎦ dω . 2π 0 ⎢⎣ f (t ) =

Having written the spectral density as S (ω ) = S (ω )e jψ S (ω ) and substituting this expression in (10.49) we obtain at once f (t ) =

1

π

∞

∫ S ( ω ) cos( ω t + ψ S

)d ω .

(10.50)

0

Here the integration is performed only along a positive semiaxis of frequencies. That is we have only a physical spectrum.

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10.2.4. The Tables Of The Spectral Method And Operational Calculus Comparison At Linear Differential Equations Integration Let's consider some tables illustrating the link between the spectral method and operational calculus application at researches signals passing through physical systems. In the table 10.1 the link between the spectral densities and signals images in the system input and output is illustrated. Also the link between the system differential equation and both transfer and frequency performances of a system is indicated. In the third column the turning from spectrums to the images of signals and the inverse turning are shown. Just here is revealed that at the zero initial conditions the same turning is valid between transfer and frequency performances. The Table 10.1

The initial function 1

The operator 2

The function in the frequency domain 3

L

f ( p)

f (t )

p = jω F

S f ( jω )

L

y ( p)

y (t )

S y ( jω )

At zero initial conditions

Differential equation ∑ aµ µ

d y dt µ

Spectral density Image

p = jω F

µ

The function name 4 Image

= f (t )

L

Spectral density Transfer performance

k ( p) p = jω k ( jω )

F

Complex frequency response

The table 10.2 shows the sequence of operations at system analysis with both the spectral method and operational calculus. Comparing these two methods of analysis we can see the deep cognation of them. 75

In the table 10.3 for comparison of the spectral method and operational calculus the sequence of operations at solving of the system synthesis task is given. The unity of both these methods is also visible from this table. As the synthesis task is ambiguous the choice of concrete build-up of a synthesized radioelectronic device is determined by the existing element basis and technical-technological back-log. Sometimes for analytical researches it is sufficient if the result of a system synthesis is the system differential equation obtaining. The solution of it describes a given response of a system y (t ) to particular energization f (t ) . Both the transfer or frequency performances can be found from the differential equation of a system (tab. 10.1) and the differential equation can be obtained from these performances [1]. Thus as it has already been stated there is a sole link between SDE and its performances K ( p ) and K ( jω ) . The Table 10.2

Analysis It is given: f (t ) ; differential equation or the system structure. It is required to find: y (t ) The spectral method The operational calculus We find the spectral density We find the image S f ( jω ) = F{ f (t )} f ( p) = L{ f (t )} We obtain from the differenWe obtain from the differential equation or from the system tial equation k ( p)

k ( jω )

We determine the image

We get the spectrum S y ( jω ) = k ( jω )S f ( jω )

y ( p) = k ( p) f ( p)

4. We search for the system reWe search for the system sponse y (t ) response y (t )

{

} {

{

}

}

y ( t ) = L − 1{y ( p )} = L − 1 k ( p ) f ( p )

y(t) = f −1 S y ( jω) = f −1 K( jω)S f ( jω)

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The Table 10.3

Synthesis It is given: f (t ) , y (t ) . It is required to find the system differential equation or K ( p ) or K ( jω) . After all it is required to find a circuit which realizes the turning f (t ) → y (t ) The spectral method The operational calculus 1. We find the images 1. We find the spectrums S f ( jω ),

S y ( jω )

f ( p ), y ( p ) f ( p) = L{ f (t )}, y ( p) = L{ y (t )}

S f ( jω ) = F{ f (t )}, S y ( jω ) = F{ y (t )}

2. We determine the frequency S y ( jω ) response k ( jω ) = S f ( jω ) 3. We synthesized the system on the k ( jω) found

2. We determine the frequency response k ( p) =

y ( p) f ( p)

3. We synthesized the system on the k ( jω) found

Unlike the analysis task the synthesis one is ambiguous. The same operator can be realized by different system build-up.

11. THE METHOD SIMPLIFYING INVERSE LAPLACE TRANSFORMATION 11.1. The Task Statement Using the operational calculus for linear differential equations integration usually the most difficult operation is inverse Laplace transformation. At researches of radiosignals passing through electronic circuits containing oscillatory units the difficulties especially increase. In this case the imaging function (IF) describing the system response to the energizing radiosignal contains complex-conjugate poles (CCP) pairs. It is clear visible from the examples of the inversion formula application in the usual form (7.7) considered in chapter 8. Even in the 77

elementary case when IF has one pair of simple (i.e. the first order of multiplicity) CCP the finishing of the researches up to the final result appears rather intricate and difficult. The indicated circumstance has reduced to numerous searches for the methods and paths facilitating the task of transient processes analysis in radiosystems. The method of complex slowly varying envelopes is the most widely used one. It allows to simplify researches of oscillatory transient processes in radiosystems regarding to the tasks of radioelectronics. This method was designed by S. I. Evtyanov [2]. The main point of this method is the substitution of a radiosystem for the relevant lowfrequency analog. In this case we obtain truncated symbolical equations. It corresponds to the depression of the system differential equation in two times. A.D. Artym [8] offered the method of simplification of the solution obtaining which is close enough to the above-mentioned one. However the methods of determination of transient processes in oscillatory systems considered in [2, 8] give approximated solutions. These solutions asymptotically aspire to the precise ones as faster, as more narrow-band signals and filters are investigated. The method simplifying ILT in the case of oscillations in the investigated transient process worked out and offered in [5–7, 11, 12] is free from this disadvantage. The main point of this method is the imaging function residues calculation only for one of them in each CCP pair, i.e. the residues are determined in one (upper or lower) half-plane of a complex variable p. The obtained complex signal (CS) allows to determine the envelope and phase of a signal in the investigated circuit output relevant to their physical adequate regardless of the signal bandwidth [20-23]. It is rather important at modern radioelectronic systems development because of the tendency towards wideband and ultra-wideband signals for the purpose of information processing speed increasing [24, 25]. The offered approach [5-7] allowed to obtain the formulas simplifying ILT very much at researches of transient processes in radiosystems.

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11.2. The Formula Simplifying ILT (The Case Of Simple CCP Of The Imaging Function) Let's consider the fractional rational imaging function (FRF) to obtain the formula simplifying the turning from the images space to the originals one y ( p) =

f ( p) . Q( p )

(11.1)

Let's represent the polynomial Q( p ) in the denominator of the expression (11.1) as Q( p ) =

m 2

r

ν =1

ν = m +1

∏ ( p − pν )( p − pν* ) ⋅ ∏ ( p − pν ) ,

(11.2)

where m / 2 – the number of pairs of complex-conjugate radicals of the polynomial Q( p ) ; r – the number of all radicals; ( r − m ) – the number of real radicals. The symbol (*) means* the complex-conjugate radical. The complex-conjugate radicals of the function Q( p ) look like Pν ⎫⎪ ⎬ = βν ± jων , Pν * ⎪⎭

y (t ) =

r

f (p )

∑ Q′( pν ) e pν t ⋅1(t ) .

ν =1

ν

Let's find the first derivative of Q( p ) then for the real poles of IF from (11.3) we obtain Q′( p ) = W ( p ) + ( p − pν ) ⋅W ′( p ) . (11.5) For the complex-conjugate poles of IF (11.6) Q′( p ) = ( p − pν* ) ⋅V ( p ) + ( p − pν ) ⋅V ( p ) + ( p − pν )( p − pν* ) ⋅V ′( p ) , Substituting in (11.5) and (11.6) the values of the pole pν we obtain for real and CCP accordingly Q′( pν ) = W ( pν ) , (11.7) Q′( pν ) = 2 jωνV ( pν ) . (11.8) If to take into account (11.4) then (11.7) turns into (11.8). We shall use the representation Q′( pν ) as (11.8) for CCP and as (11.7) – for real poles. Then the inversion formula is represented as y (t ) =

) m / 2 F ( p* ) p * t r ν e pν t + ν e ν + ∑ ∑ *

m/ 2F( p

∑

ν =1 Q( pν )

ν =1 Q( pν )

F ( pν )

ν = m +1 Q′( pν )

e

pν t

. (11.9)

Obviously the second sum is complex-conjugate to the first sum. It allows to write

the real radicals pν = βν . Let's write the formula (11.2) as Q( p ) = ( p − pν ) ⋅W ( p ) = ( p − pν )( p − pν* ) ⋅V ( p ) ,

[

(11.3)

]

−1

where W ( p ) = Q( p )( p − pν ) −1 , V ( p ) = Q( p ) ( p − pν )( p − pν* ) , i.e. the function W ( p ) is a residual of division of the polynomial Q( p ) by the term ( p− pν ) , V ( p ) – the residual of division Q( p ) by the square trinomial ( p − pν )( p − pν* ) . The following expression is obtained from (11.3) W ( p ) = ( p − pν* ) ⋅V ( p ) .

Let's apply to the inversion formula (7.7)*

m 2

r f ( pν ) pν t f (p ) + e (11.10) ∑ Q′( p ) ∑ Q′( pν ) e pν t . ν ν ν =1 ν = m +1 Let's find the real signal y (t ) performing symbolical operations of tak-

y (t ) = 2 ⋅

ing real or imaginary parts of the complex signal y (t ) , i.e. y (t ) = Re {y (t )} = Im { jy (t )} .

(11.11)

(11.4) *

As it mentioned above the factor 1(t ) in the inversion formula (7.7) indicates the

fact that if the operational calculus is used in the usual form, at DLT the values f (t ) *

We shall consider ILT simplification for the case of simple poles of the imaging function. If multiple poles are present the approach remains the same but the formulas become more intricate [5, 7].

79

are cut off at t < 0 . Accordingly by virtue of the causality principle the response y (t ) should be also equal to zero at t < 0 . Owing to the obviousness of this statement the factor 1(t ) will be omitted in the inversion formula in further.

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Now let's substitute in (11.10) the values of derivatives Q′(Pν ) from (11.7) and (11.8). We obtain y (t ) =

m 2

∑

ν =1

r F ( pν ) F ( pν ) pν t e pν t + ∑ e . jων ⋅V ( pν ) W ν = m +1 ( pν )

(11.12)

We have just obtained the formula for calculations simplifying ILT very much for oscillatory processes. According to the formula the residues for the complex-conjugate poles are taken for one of each CCP pair (only one sum instead of two in the formula (11.9)). The residues in real poles are taken conventionally. Since at the formula (11.12) development we used the representation of the imaging FRF denominator Q (p) as Q( p ) =

m 2

r

ν =1

ν = m +1 r

∏ ( p − pν )( p − pν* ) ⋅ ∏ ( p − pν ) ,

which has a coefficient at the highest term of p , i.e. the term of the polynomial Q (p) with the highest degree equal to the unit, then in order to prevent an error at IF denominator Q (p) determination it is necessary to satisfy this requirement. It always can be done having taken out the coefficient at the highest term of the r degree as a multiplicand of the polynomial Q (p). So if in the initial IF representation we have r ap in the denominator, where a – the coefficient at the highest term then having rewritten IF as a −1 ⋅ F ( p) / Q( p) we ensure the coefficient at the highest term of the denominator Q (p) equal to the unit. It is the best to illustrate by examples the efficiency of the method simplifying ILT for oscillatory transient processes which reduces to the inversion formula (11.12). 11.3. The Examples Of Application Of The Inversion Formula Simplifying ILT The examples showing how to solve the tasks of oscillatory transient processes determination in linear electronic circuits for the same requirements as in chapter 8 are given below. But the inversion formula for the turning from the imaging function to the original is used in chapter 8 in a usual appearance (7.7). Here the inversion formula is applied for determination of the given circuit response to impulse energization in the appearance (11.12) that allows to simplify very much 81

the most difficult part of work at TP researches – inverse Laplace transformation. The comparison of the ways of solution obtaining indicates that even for considered relatively simple IF the difficulty and awkwardness of transformations are essentially reduced when using the formula (11.12) in comparison with the usual inversion formula (7.7). For simple IF considered in the examples analytical solutions requiring intricate transformations at using of the inversion formula (7.7) can be obtained mentally at a small skill of using the inversion formula (11.12). The Example 1. Find the voltage on the parallel oscillatory circuit at its energization by a current source as a radiosaltus. Put the initial conditions equal to zero (zero initial stores of energy in the circuit). The Solution. This task has been solved (the example 1, chapter 8) using the inversion formula (7.7). The imaging function found in chapter 8 for the required circuit response looks like (the formula (7.6)): 1 p sinψ + ω c cosψ p + 2α , u ( p )k = A0 = 2 C p 2 + ω c2 p + 2αp + ω р2 with the poles p1,2 = ± jω c p3,4 = −α ± jω 0 . Now to turn to the original we shall take advantage of the formula (11.12) simplifying ILT: uout (t ) =

m 2

∑

ν =1

r F ( pν ) F ( pν ) pν t e pν t + ∑ e , jων ⋅V ( pν ) W ν = m +1 ( pν )

where the first sum is found for one of each CCP pair (we take the poles in the upper half-plane of a complex variable p). The number of them is equal to m/2. The second sum is taken for each real pole. The number of them is equal to (r-m), r – the total number of imaging function poles. The functions V(p) and W(p) are determined by the formulas (11.4):

[

]

−1 Vν ( p ) = Q ( p ) ( p − pν )( p − pν* ) , Wν ( p ) = Q( p )( p − pν )−1 .

The required real signal is found taking the real or imaginary part of the relation (11.12) according to the symbolical writing of this operation (11.11): u k (t ) = Re {u k (t )}= Im {ju k (t )}. 82

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In this example the IF has two CCP pairs. Therefore in the formula (11.12) of the turning from the images space to the originals one we have two terms of the first sum for the poles p1 and p3. Let's determine the function F(p) as before A f ( p ) = 0 ( p sinψ + ω c cosψ )( p + 2α ) . C The function V(p) depends on which of CCP the term of the first sum of the formula (11.12) is found. According to the formula (11.4) the function V(p) is obtained by rejection of one of multiplicands in the appearance of square trinomial ( p − pν ) p − pν* in the denominator of

(

)

fractional rational IF. At the same time p and pν* just the CCP pair the residue (the term of the sum in (11.12)) for one of which is obtained. As from (8.4) the denominator Q(p) of the imaging function uk ( p ) is determined by the relation Q( p ) = ( p2 + ω c2 )( p2 + 2αp + ω r2 ) , then at finding of the function V(p) for the poles p1 and p2 = p1* we re-

ject the multiplicand p2 + ω с2 = ( p − jω с )( p + jω с ) in Q(p), i.e. in this case (11.13) V1 ( pi ) = ( jω c )2 + 2αjω c + ω r2 . At finding of the function V(p) for the poles p3 and p4 = p3* we

reject the multiplicand p2 + 2αp + ω r2 = ( p + α − jω 0 )( p + jω 0 ) in Q(p),

where ω 0 = ω r2 − α 2 – the free-running frequency of the parallel oscillatory circuit. Then V 3 ( p3 ) = (− α + jω 0 )

2

+ ω с2

. (11.14) From this for IF (8.4) we obtain the solution as a complex signal for the response of the parallel oscillatory circuit to a radiosaltus according to the inversion formula (11.12) u k (t ) =

p t F ( p3 ) F ( p1 ) e 3 . e p1t + jω V 3 ( p3 ) jω сV1 ( p1 )

(11.15)

We obtain the real signal on this solution according to (11.11) as u k (t ) = Im {u k (t )} . In the enhanced form we write the expression for the 83

voltage on the parallel oscillatory circuit at the energization by a current source as a radiosaltus as uk = +

A0 C

⎧⎪ ( jω sin Ψ + ω cos Ψ )( jω + 2α ) − jω t c c c e c + ⎨ ⎪⎩ ω c ( jω c )2 + 2αjω c + ω r2

[

]

[( −α + jω0 ) sin Ψ + ω c cos Ψ ]⋅ ( −α + jω0 + 2α ) e( −α + jω 0 )t ⎫⎪ =

[

ω 0 ( −α + jω 0 )2 + ω c2

]

⎬ ⎪⎭

= U k forced (t ) + U k free (t ) = U m forced e jω c t + U m free e ( −α + jω 0 )t ,

where U k

and U k

forced

free

(11.16)

– the transient process forced and free

components in a complex form accordingly which are determined by the corresponding term of the curly bracket of the expression (11.16); U m forced and U k free – the complex amplitudes (CA) of the forced and free components of the response of a parallel oscillatory circuit to a current radiosaltus. Let's execute trivial transformations in the formula (11.16). For this purpose applying to the expression (8.3) we notice that the multiplicand of the first fraction of the curly bracket of the formula (11.16) 1 jω c + 2α , (11.17) z ( jω c ) = 2 C ω r − ω c + 2αjω c i.e. is equal to the complex resistance of a parallel oscillatory circuit on the frequency of HF filling of the energizing signal i(t) and the multiplicand A0 ( jω c sin Ψ + ω c cos Ψ ) / ω c = 2 ⋅ A0e jΨ . Then taking into account that the current energizing the circuit can be written in the complex form as i (t ) = A e j (ω с t + Ψ ) ⋅1(t ) = A ⋅1(t )e jω 0 t , where CA of a current 0

0

jΨ

radiosaltus is A0 = A0 e , and the energizing real signal is determined by the operation i (t ) = Im {i (t )}, we can write for the TP forced component complex amplitude (11.18) U k forced ( t ) = A0 z ( jω c ) = U m forced exp jβ , whence the FCTP amplitude is U m forced = A0 ⋅ z (ω c ) , and the FCTP initial phase is β = arg U k

forced

= arg A + arg z ( jω c ) = Ψ + arg z ( jω c ) .

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The expression (11.18) represents the Ohm law in the symbolical form for the voltage CA on a parallel oscillatory circuit energized by a monoharmonical current source of the frequency ω с and the complex amplitude A0 = A0 ⋅ e jΨ , where A0 – the sine-wave current amplitude, Ψ – the initial phase of it. Thus for determination of the forced component voltage on a parallel oscillatory circuit at its energization by a current radiosaltus it is enough to take advantage of the Ohm law in the symbolical form (11.18). Let's notice that the generalization of the Ohm law application (11.18) at FCTP finding is the determination of the forced component of the circuit response at equal dimensions of signals in the input and output (voltage – voltage, or current – current). In this case the link between input complex amplitudes and FCTP is determined over the dimensionless complex frequency performance of the circuit K ( jω с ) . It

implies that K ( jω с ) can be found, for example, using a double application of the Ohm law in the symbolical form. So if the voltage in the circuit input is given and we search for the voltage in the output, then to find CA of the forced component of the circuit response we at first obtain CA of the current which flows through the circuit element the signal is taken from using the Ohm law. Then having taken advantage of the Ohm law (11.17) CA of FCTP is found (such a task arises in frequently used circuit build-up as a potentiometric divider). The examples of tasks solving of TP determination when the frequency performance K ( jω с ) is used are given below. The TP free component complex amplitude is determined according to (11.16) by the relation Uk

free

=

A0 [(− α + jω 0 )sin Ψ + ω c cos Ψ ](α + jω 0 ) ⋅ ω 0C (− α + jω 0 )2 + ω c2

(11.19)

Comparing the expression (11.16) obtained here for the response of a parallel oscillatory circuit to a radiosaltus with the formula (8.9) developed before we notice that the first term in (11.16) coincides with a doubled value of the first term in (8.10) (it is FCTP). The second term in (11.16) coincides with a doubled value of the third term in (8.12) (it is FrCTP). Obviously the same concerns the concurrence of CA of the free component of transient process (the formula (11.19)) with the mul85

tiplied by two value of the fraction in the third term of the formula (8.4). The multiplication of the corresponding terms of the formula (8.9) by two is explained by the fact that in (8.9) the real signal is found by summing up of the first and second, third and fourth complex-conjugate terms. In the case of use the formula (11.15) the real signal is obtained by the operation of taking the real part of uk (t ) . Besides in both cases the same result is achieved by doubling the magnitudes of the terms in the inversion formula (11.15) and as a result in (11.16) with respect to the magnitudes of the corresponding (the first and third) terms in (8.9). The indicated concurrence of the corresponding terms of the formulas (8.9) and (11.16) shows that the real signals obtained by both paths coincide as well. But in the second case (when using the formula (11.12) simplifying the turning from IF to the original) the amount of transformations is much less and the result is more obvious. The formula (11.12) gives generally complex terms which can be represented by rotating vectors on a complex plane. The projections of these vectors to the relevant axes give the corresponding real signals – the free and forced TP components. Here we have an analog of vector representation of a sine wave on a complex plane which is widely used for obvious and convenient calculation of alternating current circuits by the symbolical method (the method of complex amplitudes). The representation of TP for the voltage on a parallel oscillatory circuit at energizing of the circuit by a current source as a radiosaltus can be illustrated as the sum of two vectors on a complex plane of the forced and free components of TP (the formula (11.16)). If the axes of projections are rotating with the angular speed ω с then the vector of FCTP stands motionlessly on a complex plane and the damping vector of FrCTP spins around the FrCTP vector head with the angular speed Ω = ω с − ω 0 outlining an equiangular spiral. This spiral is a hodograph of the resulting signal vector taking off a parallel oscillatory circuit [5, 11]. Obviously at time growing the signal taking off a parallel circuit aspires to FCTP.

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Applying to the expression (11.16) we shall determine the real signal taking off a parallel oscillatory circuit as u k (t ) = Im {u k (t )} = U m forced sin (ω ct + β ) + U m free e −αt sin (ω 0 + γ ) , (11.20) where the oscillations amplitudes U m forced and U m free as well as their initial phases β and γ are found from the relations (11.18), (11.19). It is the required result. Let's notice that to obtain the expression (11.20) for uk (t ) from the formula (8.15) developed before the additional rather intricate transformations would be required. So it is necessary to take into account that the solution developing as (8.15) is much more difficult than the complex signal (11.16). The simplification of determination of the solution (11.16) is achieved using the formula (11.12) of modified ILT offered in [5-7]. A great reduction of ILT difficulty at oscillatory processes researches is obtained at the expense of a rejection of the square trinomial regarding one of which radicals in the imaging function denominator the residue is obtained. Also a double diminution of residues number which are found in the complex-conjugate poles of the imaging function (the residue is found regarding only one of each pair of complex-conjugate poles) has the same effect. The solution obtaining in the form of complex signal provides essential advantages at researches of dynamic modes of functioning of the radioelectronic circuits [5, 11, 12]. These advantages are in a way similar to that which are applied for calculations by the method of complex amplitudes (the symbolical method) in the theory of alternating currents. However the method of complex amplitudes is used for the restricted class of steady modes of circuit operation. At radioelectronics at dynamic modes researches the representation of actual signal as a complex analytical signal is widely used. That also gives advantages corresponding to application of a complex signal at researches of transient processes. However the analytical signal gives a particular error at determining of the amplitude, phase and frequency of a radiosignal. This error leads to paradoxical results (for example, to violation of a fundamental principle of causality). The researches carried out in [12, 20-23, 34] show that the solutions obtained as a result of modified ILT application [5-7] apart from the essential 87

reducing of difficulty of the solution determination ensure the obtaining of amplitude, phase and frequency of a radiosignal corresponding to their physical adequate. It is very important because as a rule these parameters of a signal carry a basic informative load [5]. In the given example the procedures of the inversion formula (11.12) application simplifying the difficult ILT for the important case at radioelectronics – the oscillation of the investigated TP – are developed in details. The solution obtaining has not required difficult operations and at a small skill could be carried out without intermediate writings (at this path of the result determination the task could be solved mentally). Besides at considering of this example a great attention was paid to physical interpretation of obtained results. The clearness of physical understanding allow to avoid errors at TP researches. As at considering of other cases of TP researches we have the similar interpretation of solutions then it is not necessary to discuss so detailed the examples given below. They illustrate the application of the inversion formula (11.12) for determination of transient processes in electronic circuits that allows to fix in memory the skill of modified ILT application. The Example 2. Find the impulse response and transient performance of a parallel oscillatory circuit. The Solution. We find the impulse response gu (t ) and transient performance hu (t ) for the voltage on a parallel oscillatory circuit as a response to energization by a current source in the form of δ -impulse or unit step saltus accordingly. This task has already been solved in chapter 8 (the example 8) applying of the known inversion formula (7.7). Let's solve it here having taken advantage of the formula (11.12). The Ohm law in the operator form for the voltage on the circuit (8.32) gives uk ( p ) = i ( p ) ⋅ z ( p ) . The signals images for determination of the impulse response and transient performance are iδ ( p ) = 1 and i salt ( p ) = 1 p .

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Then we have the formulas (the example 8, chapter 8) for IF of the impulse response and transient performance (8.33): 1 p + 2α . gu ( p ) = iδ ( p ) ⋅ z ( p ) = z ( p ) = 2 C p + 2αp + ω r2 The poles of IF (8.39) are p1,2 = −α ± jω 0 , the denominator is

(

)

Q( p ) = ( p − p1 ) p − p1* ⋅ν ( p ) = p2 + 2αp + ω r2 and from here ν1( p ) = 1 , F ( p ) = (1 C )( p + 2α ) . Then according to the formula (11.12)

gu (t ) =

−α + jω0 + 2α ( −α + jω 0 )t e 1( t ) , jω 0C

(11.21)

whence it is obtained at once gu (t ) = Re{ gu (t )} =

e −αt (ω0 cos ω0t + α sin ω0t )⋅1(t ) . ω0C

(11.22)

According to the determination of the impulse response the imaging function for it is equal to the system function K ( p ) (for this example K ( p ) = z ( p ) ). Hence the impulse response is equal to the free poles residues of the integrand K ( p ) exp( pt ) in ILT, i.e. it is FrCTP. IF for the transient performance accroding to (8.34) has three poles p1,2 = −α ± jω 0 and p3 = 0 . Let's take advantage of the formula (11.12). For this purpose we shall take into account that F(p) =

1 (p + 2б), C

Q (p)=p(p-p н )(p-p н* )=p(p 2+2бp + щr2 ) .

Then from (8.34) for the complex transient performance we obtain the expression: hu(t) =

1 C

⎛ − б + jщ + 2б ( б + jщ )t 2б ⎞ u 0 ⎜ ⎟ ⋅1(t ) . + e 2⎟ ⎜ jщ0 ( −б + jщ0 ) щ 0 ⎠ ⎝

(11.23)

1 C

⎛ 2α j (α + jω 0 )2 ( −α + jω 0 )t ⎞⎟ ⎜ e ⋅1(t ) . + ⎜ ω 2 ω (ω 2 + α 2 ) ⎟ 0 0 ⎝ r ⎠

1 C

⎡ ⎤ α 2 − ω02 sin ω0t )⎥ ⋅1(t ) . (11.25) ⎢2α − e −α t (2α cos ω0t + ω0 ⎣⎢ ⎦⎥

It coincides with the expression for the transient performance (8.42) found in chapter 8 (the example 8). But here the formula (11.25) is obtained much easier. Taking into account that 2α 2r 2 rLC = = =r, 2 2 ω r C ω r c2 L 2 LC we shall rewrite the relation (11.25) for the transient performance of a parallel oscillatory circuit once again as ⎡ ⎤ α 2 − ω02 1 hu (t ) = ⎢r − e −αt ( r cos ω 0t + sin ω0t )⎥ ⋅1(t ) , 2 ω r ω0C ⎢⎣ ⎥⎦

(11.26)

The first term of square brackets here determines FCTP, the second term – FrCTP. The Example 3. Determine the impulse and transient performances of the current in a series-tuned circuit (fig. 8.4). The Solution. This task has already been solved in the example 7, chapter 8 using the inversion formula (7.7). Let's solve the same task having applied the inversion formula (11.12) simplifying ILT. The images of impulse and transient performances of the current in a seriestuned circuit are determined by the formulas (8.29) developed before: gi ( p ) = p[L ( p2 + 2αp + ω r2 )]−1, hi ( p ) = [L ( p2 + 2αp + ω r2 )]−1 .

Hence the functions 1 p , Fh ( p ) = , L L the denominator Q g ( p ) = Q h(p) = Q(p) = p 2 + 2αp + ω r2 that gives the Fg ( p ) =

Let's convert the expression (11.23) to the form hu (t ) =

hu (t ) = Re{ hu (t )} =

(11.24)

ω 02 + α 2 = ω r2 (α + jω 0 )2 = α 2 + 2αjω 0 − ω 02

poles for both IF p1,2 = −α ± jω 0 , ω 0 = ω r2 − α 2 . To turn from IF to the original under the formula (11.12) we have for both IF V( p ) = Q( p )[( p − p1 )(p − p2 )]-1 = 1 .

89

90

Let's find the real transient performance taking into consideration that

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Then we obtain at once the expressions for the complex impulse and transient performances −α + jω0 ( −α + jω 0 )t (11.27) gi (t ) = e 1( t ) , Lj ω0 hi ( t ) =

1 e ( −α + jω 0 )t 1(t ) , jω 0 L

(11.28)

Real impulse and transient performances are determined from (11.27) and (11.28) by the functions g i (t ) = Re{gi (t )} =

1

e −αt (ω 0 cos ω 0t − α sin ω 0t )1(t ) ,

ω0L

hi ( t ) =

1

ω0L

e −αt 1( t ) sin ω 0t .

(11.29) (11.30)

The expressions (11.29) and (11.30) coincide with the formulas (8.30) and (8.31) for the impulse and transient performances obtained before applying the inversion formula (7.7). But here the solutions were obtained at once. There was no need in intermediate transformations. The Example 4. Determine the response of the integrating circuit to a radiosaltus uin (t ) = A sin(ω сt + ψ ) . The Solution. This task has already been solved in the example 4, chapter 8 using the inversion formula (7.7). Let's find the solution applying the inversion formula (11.12) simplifying ILT. The imaging function for the response of the circuit to a radiosaltus is given by the relation (8.20): A p sinψ + ωc cosψ 1 uout ( p ) = , τ p +1 τ p2 + ωc2 The "forced" poles are p1,2 = ± jω c , the "free" pole is p3 = −1 τ . The IF numerator is F ( p) =

A

τ

( p sin ψ + ω c cosψ ) ,

the denominator is Q ( p )=( p2+щc2 )( p+1/τ ) . Then according to the formula (11.12) we have V1( p ) = p + 1 / τ , W 3 ( p) = p2 + ω c2 . 91

Substituting the pointed functions in the formula (11.12) we shall obtain at once uout (t ) =

− 1 / τ sin ψ + ω c cosψ −1/ τ ⎤ A ⎡ jω c sin ψ + ω c cosψ jω ct +j e e ⎢ ⎥ ⋅1( t ) . τ ⎢⎣ ω c ( jω c + 1 / τ ) ( −1 / τ )2 + ω c2 ⎥⎦

After trivial transformations ⎡ 1 − sin ψ + ω cτ cos ψ −1 / τ ⎤ uout (t ) = A ⎢ e (ω ct +ψ ) + j e ⎥ ⋅1(t ) = 1 + (ω cτ )2 ⎢⎣1 + jω cτ ⎥⎦

⎡ − sin ψ + ω cτ cosψ −1/ τ ⎤ = ⎢ A k ( jω c )e(ω ct +ψ ) + jA e ⎥ ⋅1(t ) = 1 + (ω cτ )2 ⎢⎣ ⎥⎦ = uout (t ) + u free (t )

(11.31)

where FCTP uout (t ) is determined by the first term and FrCTP u free (t ) – by the second term in square brackets of the obtained formula. The real response of the integrating circuit to a radiosaltus is determined from (11.31) according to the formula (11.12) as ⎡ − sin ψ + ω cτ cosψ −1/ τ ⎤ uout (t ) = Im {uout (t )} = A ⎢k (ω c ) sin( ω c + β ) + e ⎥ ⋅1(t ) . 1 + ω c2τ 2 ⎣⎢ ⎦⎥

11.4. The Substantiation Of The Method Simplifying Inverse Laplace Transformation At Electronic Circuits Dynamic Oscillatory Modes Researches The operational calculus is the basic instrument at researches of radioelectronic circuits dynamic modes. However as it has already been mentioned the essential hindrance at practical usage of the operational method for transient processes researches is the difficulty of ILT. It especially increases at considering of the important class of oscillatory processes and systems at radioelectronics. The method of slowlyvarying amplitudes developed by S. I. Evtyanov [2] is one of the most important methods simplifying the solution determination in this case. More over it is widely used at researches of radioelectronic circuits. The serious disadvantage of this method is the proximity of obtained solutions. This fact restricts the possibility of application of this 92

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method for researches of modern RED which use the radiosignal microstructure. A graphical method [35] permitting to simplify inverse Laplace transformation for the imaging function with simple poles is worth mentioning. At this the coefficients F ( pi ) / Q′( pi ) of the expansion formula (7.7) are determined as the ratio of product of difference vectors between the i-th pole and the imaging function g( p ) zeros to product of difference vectors between the i-th pole and other poles of this function. At first the locations of zeros and poles of g( p ) on the images complex plane is found. Then the absolute values and angles of difference vectors between the i-th pole and each of zeros and poles are measured. In the case of complex-conjugate poles the difference vectors are found on a deviation to one of them introducing the extra factor 1/ jω . The graphical build-up of zeros and poles of the function g( p ) on a complex plane allows to value visually the influence of their mutual disposition to the circuit response. However these build-up and measuring require for working hours of actions and do not ensure enough accuracy. In some cases it is possible to reduce a little mathematical manipulations having decreased the number of required residues by one. It is done when at any rate one pair of poles lays on the imaginary axis of the images complex plane. The energization of the investigated circuit by a harmonic signal concerns, in particular, this case. Sometimes it is possible to simplify inverse Laplace transformation by simultaneous displacement of all poles of the imaging function g( p ) . However both these cases allow to obtain perceptible effect only for rather restricted number of g( p ) representations [3]. The first case corresponds to the complex form of representation of a signal such as a radiosaltus. Let's consider more general case of switching-on of sine waves with exponential modulating functions.

(

)

α µt e

f (t ) = ∑ eα i t cos ω µ t + ψ µ = ∑ e µ

. = f1 (t ) +

(

j ω µ t +ψ µ

) + e − j (ω µ t +ψ µ ) 2

µ

. ⎤ ⎡. f1* (t ) = 2 Re ⎢ f 1 (t )⎥ . ⎥⎦ ⎢⎣

=

(11.32)

The expression (11.12) simplifying the turning from the images space to the originals one at researches of oscillatory processes and systems very much was obtained before on the basis of formal transformations. In this paragraph a visual substantiation and interpretation of the offered method of researches of transient processes in oscillatory systems are given*. The image for f (t ) : L { f (t )} =

jψ − jψ µ 1 ⎛⎜ e µ e + ∑ 2 µ ⎜ p − pµ p − pµ* ⎝

⎞ ⎟. ⎟ ⎠

(11.33)

Let's write the energizing function in the complex form as: . . α t j (ω t +ψ ) α t f (t ) = 2 f 1(t ) = ∑ e µ e µ µ = ∑ e µ cos ω µ t + ψ µ + j sin ω µ t + ψ µ w µ

[ (

µ

)

)]

(

hence: f (t ) = f (t ) + jf€(t ).

(11.34)

where f€(t ) – the function conjugate to the original signal f (t ) . According to this the image of the complex energizing function is € f ( p ) = f ( p ) + jf ( p ) =

jψ

e µ . p − pµ µ

∑

(11.35)

The image of the system response to a real signal is determined by the relation g ( p) = f ( p) K ( p) , (11.36) where K ( p ) – the system function (transfer performance). *

Here for the purpose of simplification the imaging functions with simple poles are considered. However there is no limitations connected with a multiplicity of poles. Therefore the same approach can be applied for imaging functions with multiple poles. A more common case of multiple IF poles is considered in [5 – 7, 12].

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Let's find the image of the response g f ( p ) at energization of a system by the complex signal f (t )

L{k (t )} =

€ g f ( p ) = k ( p) f ( p) = k ( p ) f ( p) + jk ( p ) f ( p).

Then taking into account (11.36) we have g ( p ) = Re{g f ( p )} ,

The image of the impulse response of physically realizable systems in the case of simple poles looks like:

(11.37)

1 ⎛⎜ e jψ λ e − jψ λ + ∑ 2 λ ⎜⎝ p − p λ p − p λ*

⎞ ⎟. ⎟ ⎠

(11.40)

Let's introduce a complex impulse response K (t ) : ⎧⎪ e jψ λ ⎫⎪ −1 K (t ) = L −1 ⎨∑ ⎬ = L {K ( p ) + jK I ( p )} . ⎪⎩ λ p − pλ ⎪⎭

or taking into consideration the property of commutativity of Laplace transformation and symbolical operations Re or Im [3] we shall obtain (11.38) g (t ) = L−1{Re g f ( p ) }= Re[g f (t ) ] .

Then the image of the spurious circuit response f ( p) to complex ener-

Thus for g f (t ) = L−1{k ( p) f ( p)} we have

gizing function K (t ) is:

[

]

{

}

g(t ) = Re g f (t ) .

(11.39)

As it follows from comparing (11.33) and (11.35) every pole in the image of the complex signal f (t ) corresponds to the pair of conjugate poles in the image of the real energizing function f (t ) . It allows to simplify inverse Laplace transformation at the forced component g forced (t ) determination. However it is easy to be convinced that the definition of residues for the free component of the response g free (t ) here will not be simpler. Moreover it can be even essentially complicated, in particular, because of the violation of conjugation of residues in the "free" poles of the function g f (t ) . So at such an approach the effective reduction of difficulty of turning from the images space to the originals one is not generally ensured. At the same time there is a possibility of considerable simplification of mathematical manipulations to obtain the original on the imaging function with conjugate pairs of poles. The analysis of transient processes in radiosystems often reduces to this just the most difficult case. Let's return to the formula (11.36) to consider this possibility. Taking into account the equality of f ( p) and k ( p ) we conditionally assume that the impulse response k (t ) is an energizing function of a spurious circuit f ( p) . Let's carry out the reasonings similar to performed above at a complex representation of the energizing function f (t ) . 95

(11.41)

(11.42) Taking into consideration (11.36) we shall write g( p ) = Re{ gK ( p )} . Therefore the system response g(t ) to the energizing function f (t ) is determined by the relation (11.43) g ( t ) = Re L − 1 [g K ( p ) ] . . g K ( p ) = f ( p ) K ( p ) = f ( p ) K ( p ) + jf ( p ) K I ( p ).

{

}

Comparing the turning from the image to the original under the formulas (7.7) and (11.43) we notice that in the last case we find the residues in one of poles of these pairs instead of determining residues in each of "free" poles of a conjugate pair. Therefore for the complex representation of K (t ) at determination of the free component of the circuit response inverse Laplace transformation usually becomes simpler in contrast to energizing function f (t ) complex representation considered before. At that the difficulty of mathematical operations at the forced component determining remains the same. From the explained above and taking into account that the response of a circuit is the sum of the forced and free components, i.e. g ( t ) = g forced ( t ) + g free ( t ) we determine the forced component at energization of the system with the transfer performance K ( p ) by a complex signal f (t ) . The free component can be found at energization of a spurious circuit f ( p) by complex representation of the system impulse response K (t ) . Then from (11.40) and (11.43) we shall obtain

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[

]

{ [

]}

g(t ) = Re g forced f (t ) + g free K (t ) = Re L −1 g forced f ( p ) + g free K ( p ) . (11.44)

The definition of the circuit response at conjugate poles pairs in the response image ensures simplification of mathematical manipulations as at the forced as at the free components determination. This path allows to develop an engineering procedure of calculation of transient processes in radiosystems providing essential reduction of mathematical operations at inverse Laplace transformation. The main idea of this method is the fact that the residue is found with respect to one of the imaging function poles instead of determination of residues in each pole of conjugate pair. It is possible to factorize the fraction g ( p) represented by the expression (11.1) into simple fractions. Each of these parts is the image of some complex function. Then for determination of the impulse response in the complex form g(t ) as a sum of residues determined in one of each pair of conjugate poles it is possible to apply the theorem of transposition in a complex field. It can be done performing a displacement of the pole to the coordinates origin. Here we mean the pole the residue regarding which is determined. It ensures further simplification of the response g(t ) determination. As an illustration of all described above we shall consider the example of definition of the series-tuned circuit response to energization by a sine harmonic signal (radiosaltus) f (t ) = 1(t ) sin ω c t.

The response image for this case is proportional to the function g ( p) =

ωc p

2

+ ω c2

p p

2

+ 2αp + ω r2

=

⎤ 1 ⎡ 1 1 − ⎢ ⎥× 2 j ⎣ p − jω c p + jω c ⎦

⎡ − α + jω 0 ⎤ − α − jω 0 1 1 ×⎢ + ⎥, + − j ω p α j ω j ω p α j ω − + + 2 2 0 0 0 0⎦ ⎣

The poles of the function

are

p1,2 = ± jω c

and

p3,4 = −α ± jω 0 . Then the forced component of the circuit response is

found after displacement of the pole and image

{

}

L g forced f (t )e − jω ct =

1 p + jω c . p j ( p + jω c )2 + 2α ( p + jω c ) + ω r2

[

]

Let's put p=0 in the second multiplicand. So we obtain the forced component taking into account that the residue in the coordinates origin is equal to that multiplicand. jω c g forced f (t ) = e jω ct . 2 2 j ( jω c ) + 2αjω c + ω r Similarly we obtain the free component from the image ωc −α + jω 0 1 , L g free f (t )e (α − jω 0 )t = 2 2 p ( p − α + jω0 ) + ω c jω0

[

{

]

}

whence g free K (t ) =

−α + jω 0 ωc e (−α + jω 0 ) t . jω 0 (− α + jω 0 )2 + ω c2

Finally the series-tuned circuit response is determined according to (11.44) as

{

}

g(t ) = Re g forced f (t ) + g free K (t ) .

The solution of this task on the expansion formula would require much more intricate mathematical manipulations. The path shown here allows to obtain precise expressions for transient processes at passing of radiosignals through resonant systems in the analytical form. It is of interest of researches of modern radioelectronic systems where the fine phase structure of a signal is used as the information carrier.

where α – the damping factor of a circuit, ω r – the resonant frequency, ω 0 = ω r2 − α 2 – the free-running frequency.

97

g( p )

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12. THE COMPLEX SIGNAL AND THE "AMPLITUDE, PHASE, FREQUENCY" PROBLEM FOR OSCILLATORY PROCESSES 12.1. The Statement Of The "Amplitude, Phase, Frequency" Problem In Radioelectronics Oscillatory processes are widely used in radioengineering practice for information transmission. Considering the form of oscillatory process (for example, on the oscillograph screen) it is usually possible to draw mentally the lines "enframing" the oscillatory process by both sides. These lines are called the envelopes of oscillatory process. It is enough to recollect the envelopes at the amplitude modulation (AM) of a signal. In this meaning the envelope determines the law of radiosignal amplitude alteration. To obtain the envelope as a physical signal different types of amplitude detectors are used. The oscillatory process with altering in time amplitude is fed to the input of these detectors. A signal filling the interval between "enframing" upper and lower envelopes, i.e. the initial oscillatory process is called high-frequency filling of a radiosignal. At the angle modulation (AnM) the frequency (frequency modulation (FM)) or phase (phase modulation (PM)) of a radiosignal are altered under the law of the modulating signal. For only angle modulation the signal amplitude remains constant. The demodulation of signals with the angle modulation is carried out by frequency or phase detectors depending on the type of AnM. Principally both types of modulation (amplitude and angular) can be used independently for information transmission. At this information is supplied over AM and AnM channels either independently, or the information of one channel is used to refine the information of another channel (for example, the information about the envelope is used for improving the information about the phase of a signal in the navigational system "Loran-C") [5]. Besides different systems of line multiplexing are used in practice for maximum possible loading of the information channel. These systems demand for special circuit build-up of modulators and demodulators. 99

Generally an oscillatory process can be represented by the function f (t ) = A(t ) cos Φ (t ) ,

(12.1) where the first multiplicand of the right term A (t ) determines the envelope of the signal f (t ) , the second – cos Φ (t ) , characterizes the variability of the process (12.1). The argument Φ (t ) of the sinusoidal function cos Ф( t ) determines the phase of oscillations. Let's write Φ (t ) as Φ (t ) = ω 0t + Ψ(t ) + Ψ0 , (12.2) The oscillations frequency ω(t ) is found as a derivative of the phase, i.e. dΦ ( t ) dΨ ( t ) = ω0 + = ω 0 + Ω( t ) , dt dt

ω (t ) =

where Ω( t ) =

(12.3)

dΨ ( t ) . dt

There is an inverse (integral) law of the link between the phase and oscillations frequency accordingly t

t

0 t

0

Ф(t ) = ∫ ω (ξ )dξ + Ψ0 = ∫ [ω0 + Ω(ξ )]dξ + Ψ0 =

(12.4)

= ω0t + ∫ Ω(ξ )dξ + Ψ0 = ω0t + Ψ (t ) + Ψ0 , 0

t

Ψ(t ) = ∫ Ω(ξ )dξ ⇒ Ψ(0) = 0 .

(12.5)

0

In the expressions (12.2) – (12.4) ω 0 – some average oscillations frequency. As a rule the carrier frequency ωс of initial oscillations unmodulated on frequency or phase is understood under ω 0 . As it follows from (12.4), (12.5) the initial oscillations phase Φ (0) = Ψ0 . It implies from (12.3) and (12.5) that the presence of the phase modulation Ψ(t ) , i.e. the deviation of the phase alteration Φ (t ) from the linear law (look up the formula (12.2)), signifies the corresponding frequency modulation. Similarly the presence of FM causes simultaneous PM. Therefore 100

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both these types of modulation are called angle modulation. It is impossible to determine which type of modulation, the phase or the frequency, takes place if a signal with angle modulation is given. It is possible to determine with type of modulation, the frequency or the phase, is applied at generating of a signal with AnM. For this purpose we should find which of the informative signal parameters, the frequency or the phase, is altering under the law of modulating function. Let's notice that the envelope of oscillatory process can be determined as a videosignal used as a modulating one at AM (the term videosignal is taken from TV engineering). Obviously that the radiosignal f (t) offers much more advantages for messages transmission as in a radiosignal both the envelope and phase (frequency) of a signal can carry the informative load. It is obvious that the special case of correspondence between radio- and videosignals are radio- and videopulses. The difference between radio- and videopulses is clearly visible at considering the spectrums of these signals. Spectrums of videopulses are adjoined to the frequency ω = 0 , the spectrum of radio-frequency pulses is formed at a point of HF filling of radio-frequency pulses. At a diminution of the radio-frequency pulse duration the width of the signal spectrum increases. At duration of a radio-frequency pulse τ II about 2 − 5T0 , where T 0 = 2π ω 0 – the period of radio-frequency pulse filling, the width of its spectrum appears comparable with the carrier frequency ω 0 . In this case we have broadband and ultra-broadband signals (UBBS). At further shortening radio-frequency pulse duration up to parts of the period is transforms to the category of videopulses both under the form of spectrum and under the form in the time domain. Thus, at turning to UBBS the noticeable border between radio- and videopulses is cleared [5, 24]. Let's also notice that the envelope itself in some cases is described by rather complicated function. So, its spectrum is complex as well. For the radiosignal with such an envelope the differences between the spectrum of radiosignal itself and the spectrum of the envelope (for example, if the envelope of a radio-frequency pulse is a part of sinusoid) can appear weaker. Great possibilities afforded by usage of modern communications and navigational systems have led to fast increasing a number of the 101

users. It has stipulated the requirement of optimal load of selected frequency bands at providing high quality and reliability of messages transmission, i.e. to the necessity of solving the problem of electromagnetic compatibility at high saturation of radio-frequency band. In this way the problems of optimal signal coding and processing accept special significance. It is impossible to solve these tasks properly with a lack of authentic description of informative parameters of a signal – the envelope and phase (the problem "Amplitude, Phase, Frequency" (APF)). The essence of the problem is that it is formally possible to find an uncountable set of multiplicands pairs combinations A(t) and cos Φ ( t ) for the real oscillatory process, introduced in the form of the composition f (t ) = A(t ) cos Φ (t ) , meeting the same signal f (t). The existing indeterminacy of analytical representation of radiosignal informative parameters – its APF, is prohibitive principally, as it hinders with correct considering processes of signals transformation in radioelectronic systems. The indeterminacy of APF is usually eliminated by introduction of a complex signal (12.6) f (t ) = f (t ) + jf€(t ) , The imaginary part of the complex signal f€(t ) is linked to the initial real signal with some transformation [13-20]. The absolute value A(t ) =

f 2 (t ) + f€2 (t ) ,

(12.7)

of such CS determines the envelope, and the argument ∧

f (t ) , f (t )

Φ(t ) = arctg

(12.8)

– the phase of the initial physical signal. The instantaneous frequency of a complex signal is determined from (12.8) as ∧

dФ(t ) f ′(t ) f (t ) − = ω (t ) = dt f 2 (t ) +

∧

f ′(t ) f (t ) ∧ 2

f (t )

102

∧

=

∧

f ′(t ) f (t ) − f ′(t ) f (t ) A 2 (t )

. (12.9)

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The problem is that it is necessary to introduce such a link between the .

original signal f (t ) and the signal f€(t ) = Im{ f (t )} conjugate to the first one, that the informative parameters amplitude, phase, frequency, defined according to (12.7)–(12.9) meet the physical adequate of these parameters in the initial radiosignal f (t ) . 12.2. The Analytical Signal The complex analytical signal introduced by D. Gabor in 1944 [13] has taken a dominant place in modern radioelectronics. For the .

∧

.

f y (t ) = A01(t ) exp j (ω 0t + Ψ0 ) = f y (t ) + j f y ( t ) .

(12.16)

According to (12.7) and (12.8) for the envelope and phase of this signal we have the relations as Ф(t ) = ω 0 t + Ψ0 . (12.17) A(t ) = f y (t ) = A0 1(t ) , However the analytical signal found for f y (t ) reduces to other results. Actually, having applied HT to f y (t ) we shall obtain [36]

analytical signal f a (t )

∧

∧

.

f a (t ) = f (t ) + j f c (t ) ,

∧

f ( t ) = H { f (t )} = −

1

π

∞

v. p.

∫

−∞

f (τ ) dτ , τ −t

(12.11)

where H – the Hilbert operator, v.p. – the symbol of the principal on Cauchy value of the integral. According to (12.7) and (12.8) the envelope and phase is determined from AS as f 2 (t ) +

⎤ ⎡ ∧ f (t ) Φ a ( t ) = arctg ⎢⎢ c ⎥⎥ . f (t ) ⎥⎦ ⎢⎣

∧ f c2 ( t ) ,

f Ну (t ) = A0{sin(ω 0t + Ψ0 ) +

(12.10)

the function f€c ( t ) conjugate to the original signal is determined by Hilbert transformation (HT):

Aa (t ) =

The amplitude modulation should not disturb the phase of oscillatory multiplicand (the principle of phase invariance). Then according to the invariance principle the amplitudes and phases of CS should be determined as

(12.12)

Then we can rewrite (12.10) in the form

−

.

(12.14) f (t ) = Re{ f a (t )} . Let's take a truncated radiosignal (radiosaltus) which is frequently met in applications (12.15) f y ( t ) = A0 1( t ) cos(ω 0 t + Ψ0 ) ,

103

π

π

si (ω 0t ) sin(ω 0t + Ψ0 ) −

(12.18)

ci(ω 0t ) cos( ω0t + Ψ0 )},

where si(ω 0t ) and ci (ω 0 t ) – the integral sine and integral cosine. Let's introduce

a

.

complex-conjugate

signal

.

f Ну (t )

for

which

.

f Ну (t ) = Im{ f Ну (t )} . From (12.18) we have . . ⎡ 1. ⎤ . f Ну (t ) = A0 exp j (ω 0t + Ψ0 ) ⎢1 + r sici (ω 0t )⎥ = f (t ) N Ну (ω 0t ) , (12.19) ⎢⎣ π ⎥⎦ where rsici (ω0t ) – the radius-vector circumscribing a sici-spiral on a

complex plane .

r sici (ω 0 t ) = si (ω 0 t ) + jci (ω 0 t ) ,

.

(12.13) f a ( t ) = Aa ( t ) exp jΦ a ( t ) . Thus the original signal is found performing the same operation as in (12.10)

1

1

.

N Ну (ω0t ) = 1 +

1

π

.

r sici (ω 0t ) = N Ну (ω 0t ) exp jξ (ω0t ) .

The multiplicative function N Ну (ω0t ) determines the error of conjugate signal f€Hy (t ) at representation of a radiosaltus by the analytical signal. The diagram of the function N Ну (ω0t ) behaviour is given at fig. 12.1.

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These frequencies are cut off at the case of AS. It leads in turn to higher beat-frequencies of AS vector and to lower transition interval of the envelope and phase establishing in a real time scale as well. The energy of the spectrum part penetrates into the negative frequencies area becomes lower (if the narrowity of a signal band increases), but the transition interval of f€Hy (t ) decreases accordingly. We can see the

Fig. 12.1

event like Gibbs phenomenon of the spectrums theory [36]. However the behaviour of AS is described by more complicated relations than Gibbs phenomenon. Here the spectrum penetrates from positive frequency area is cut off on the left semiaxis and substituted for complexconjugate part of spectrum penetrates from negative frequency area. Let's consider AS for a truncated sinusoid (12.16) on the negative semiaxis of time. If to take into account that at t < 0 f y (t ) ≡ 0 and

A damping character of the conjugate function f€Ну (t ) oscillations ob-

more over ci ( − | ω 0 t |) = ci | ω 0 t | , si ( − | ω 0 t |) = −π − si | ω 0 t | , so we have (12.20) from (12.12) and (12.18)

viously follows from this figure. It reduces to the relevant damping oscillatory character of the representation of the envelope and phase of a radiosaltus over AS. Obviously, we shall obtain the same representation (track) after the moment of switching the signal off for a radiofrequency pulse with rectangular envelope as well. We have N Hу (ω 0 t ) → 1 , ζ (ω 0 t ) → 0 at t → ∞ for a radiosaltus. At the point ω 0 t = 0 the absolute value of the function N Ну (ω 0 t ) , consequently the

absolute value of the signal f Ну (t ) , and the envelope А0 (t ) of AS as well turn in perpetuity. Let's notice that the complex error function N Hy (ω 0 t ) depends on dimensionless time ω 0 t , so it is not a function of frequency. In other words the current defect of the analytical signal is determined by the relative time t T0 , T 0 = 2π ω0 which counts off from the moment of switching the radiosaltus. Hence it follows that the character of deviation of f a = Re {f a (t )} from the initial physical signal does not depend on the proximity of ω 0 to zero frequency, i. e. does not depend on the measure of signal broadbandness. This unobvious result can be interpreted easier if to take into account that the more narrowband signal is used the more difference between the frequencies of the spectrum part penetrates into the negative frequencies area and ω 0 . 105

∧

(12.20)

f а ( t ) = 0 + j f Ну ( t ), ∧

A0

[cos(ω0t + ψ 0t )ci(| ω0t |) − sin(ω0t + ψ 0t )si(| ω0t |)]⋅1( −t ) , (12.21) π Hence according to the definitions (12.12) for the envelope and phase of AS at t < 0 we shall obtain f Ну (t ) =

∧

Aа ( t ) = f

Ну ( t ) 1( −t ),

Фа ( t ) =

π 2

∧

sgn f

Н

(t ) ,

⎧1, at t < 0, ⎪ 1 where 1( −t ) = ⎪⎨ , at t = 0, 2 ⎪ ⎪⎩0, at t > 0,

⎧1, at x > 0, ⎪ a signum function (a function of a sign) sgn x = ⎨0, at x = 0, ⎪− 1, at x < 0. ⎩

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(12.22)

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If as before to emanate from the representation of conjugate on Hilbert function in the complex form f Н (t ) for t < 0 as well, for which the

model of radiosaltus described over AS (in correspondence with * (12.12)) is determined by a radius vector rsici (ω 0t ) , which has a rupture at t = 0 and damping oscillatory character at moving off from t = 0 in both legs. Thus, we have a paradoxical and absurd result from a physical point of view. The envelope and phase obtained from AS do not correspond to the initial physical signal. The initial physical signal (radiosaltus) generally has no both envelope and phase oscillation and envelope rupture to perpetuity at t = 0 . Moreover, at APF definition over AS one of the fundamental laws of physics – a principle of causality – is broken, as AS gives an oscillatory forerunner preceding a radiosaltus switching on [37, 40, 41]. The similar result is obtained after the radiosaltus switching off, i. e. when we consider a radio-frequency pulse with rectangular envelope in the form A0 [1(t ) −1(t − τ )] as an original signal. In this case AS apart from considered before oscillatory forerunner and damped envelope and phase oscillations after a moment of radio-frequency pulse switching on gives an increasing envelope and phase oscillation up to a moment of radio-frequency pulse switching off, and also an oscillatory

track. Thus, AS does not allow to obtain adequate description of the envelope and phase of radio-frequency pulse with rectangular envelope. The researches of AS for such a radio-frequency pulse apparently for the first time were carried out by A. K. Smolinski [37]. Both forerunner and track were obtained in his work. At the same time the violation of causality principle was pointed out. D. E. Vakman and L. A. Vainshtein, being apologists of AS, could not approach critically to the concept of this signal. The insufficient pretentions to the results obtained by A. K. Smolinski have reduced that fallacies have migrated from [37] to [15,16] together with borrowed results. The violation of causality, envelope and phase invariance principles peculiar to AS appeared the basis for a wide discursion and contest of applicability of AS [18-23, 34, 36, 38, 39 etc.]. The vogue on AS has reduced that it is considered as a universal and uniquely valid in a number of serious works in attempts to substantiate AS fundamentally. The other descriptions of a signal, distinct from AS, are declared physically inconsistent "a priori" ("naive" representations, "old" radioengineering). It was offered to consider correct, objective in mathematical, physical and engineering sense only results implying from AS applications. It was admissible to apply the other definitions of signal parameters only in so far as they meet AS [14-16]. The authors of [15, 16] guess even a power treatment of forerunner and track of radiofrequency pulse (a detection of them with the help of a special circuits), though physically both forerunner and track do not exist. The existence of forerunner should be eliminated at all from a causality principle. To avoid an error it is required to accept that AS gives one of ways of mathematical description of oscillatory process in the complex form with a particular asymptotics as it is actually [12, 20–23, 34, 36]. Achilles' heel of AS are two properties of it: 1) it is not local while real physical signals are finite in the time domain; 2) the spectrum of AS is truncated; considering this, it is easier to detect the nature of the APF defect. But the main difficulty is "bad" integral HT which are taken directly for restricted type of functions describing signals. The similar difficulty remains at finding AS by one-sided IFT. It is necessary to notice as well that AS does not coordinate directly with

107

108

∧

real function is f Ну (t ) = Im {f Ну ( t )}, we can write t