APPENDIX 2. STROBOSCOPIC EFFECT

The stroboscopic effect in theory is unsufficiently known to a measuring information. In this connection we shall consider it more in detail. The application here of frequent methods here is rather inconvenient. In work [60] the attempt of a solution of this problem is indicated. Here assumed function represented as a sequence of readout multiplied on a number equidistant d Functions.

For want of it the resulting function acquires a kind: , Where  - Period of digitization,  - Number of readout. In an outcome of the rather bulky calculations is shown, that “ the spectrum of quantized magnitude has appeared distinct from zero and on low frequencies, though these frequencies in continuous function did not contain ”. In conventional understanding of the theorem Котельников (Shannon) of emerging new spectral component be does not owe, as this distortion of a signal.

Fig. 60  

In this connection, we shall stay on consideration of stroboscopic effect. The initial sinusoid is given. From not ё the samples with frequency of the a little bit greater double frequency of a sinusoid are made. Further function is restored by a filter. In an outcome the modulation with some frequency is received W .
             This process was  modeled  on installation, which scheme on Fig. 60. Into structure of installation enter: the generator of sinusoidal oscillations (ГСК); the shaper of clock pulses (ФСИ); the modulator (М); the generator of impulses (ГИ); a filter (Ф) and   The oscilloscope. A filter  - Low-frequency with a rectangular passband. The generator of impulses - with an adjustable porosity. In an outcome of a research of stroboscopic effect the oscillograms shown on indicated below photos are obtained.

Photo 3  

Photo 4

Photo 5

Photo 6

Photo 7

Photo 8

Photo 9

 Photo 10

Photo 11

On a Photo 3 the initial sinusoid is shown. On a Photo 4 obtained samples, which frequency is a little bit more than the double frequency of a sinusoid. On a Photo 5 - signal obtained after a regenerating filter. The origin of beatings on frequency approximately on 8 % more double frequency of a sinusoid is obviously visible. For want of increase of frequency samples within the limits of one multiplicity the beat frequency is increased (Photo 6 and Photo 10). And for want of magnification of a multiplicity of frequency samples in relation to frequency of a sinusoid the amplitude of beatings decreases (Photo 7 and Photo 11). For want of there is enough  large multiplicities (Photo 8,9) the restored sinusoid visually becomes indistinguishable from an initial sinusoid (Photo 3).

Obviously, that if the frequency samples is equal to the double frequency of a sinusoid, that, depending on a phase shift, the samples can hit on zero significances of a sinusoid, extremums or intermediate significances. In this connection, as is a priori     phase   samples  rather  sinusoids to us not  is known, after restoring a signal by a filter a sinusoid generally it is possible to not see. Therefore it is possible to tell, that the error is equal transfer of a sinusoid for want of to frequency samples to the equal double frequency of a sinusoid to hundred percents. Already it is enough of it for validation of conclusions stated in the second chapter.

Nevertheless, we shall make some theoretical researches of this effect. So, the initial sinusoid with frequency is given  . From it the samples with frequency are made    . For want of it the bandpass filter of low frequencies with frequency of a shear is used  a little bit greater  . In an outcome, if the frequency of samples is a little bit more to the double frequency of a sinusoid, after a filter the sinusoid appears modulated by other low-frequency sinusoid with frequency W
            The distance between sites of beatings is determined by attack of a phase between samples and sinusoid. For want of it each site corresponds measuring of sample with zero of a sinusoid.In this case, shift on p/2 is received for  , where - number samples, and   - residual of phases between samples and halfcycles of a sinusoid. But the residual of phases can be shown to a difference of frequencies. From here beat frequency W Is equal - .

It is possible to select such parities  and , that component of a spectrum of beatings will appear in the field of a passband of a filter so, that mutual influence of a spectrum of multiple frequencies samples is possible to neglect. In these conditions observance of requests of the Котельников theorem  available in full volume. For want of it the duality, in spectral submission of process is exhibited. On the one hand, the outcome of restoring of a sinusoid on a filter is possible to present as beating two close on frequency and sinusoids, equal on amplitude, what not absolutely correctly as the carrier frequency of a restored signal follows frequency samples and not clearly concurrence of amplitudes for want of nonuniformity AFC of a filter, or modulation rather complicated and rich in the spectral relation by function. It more corresponds reals, taking into account nonlinearity of transformation. Really, through accounting number of points it is possible to conduct infinite number both sinusoids, and functions generally. The filter “does not know” THAT FROM IT WAIT. It simply selects such function, which would require a minimum of energy on a principle of least action [61].

For want of frequencies samples to multiple frequency of a sinusoid and large  beatings also are observed. For want of it the principle of superposition is observed. For example, for want of  » process is possible to present as joint operation of two frequencies of inquiry with shift on DT/2 . Is obvious, that the greatest amplitude of beating will be in a moment equidistant from two maximums. As bending around of beatings - sinusoid, the amplitude of beatings is equal  . It is simple to show, that the number 4 is the relation .

It is similarly possible to receive the same ratio for any multiplicities of samples frequencies and sinusoid. The general formula will look like:

, Where  .                                               (П2.1)

Fig. 61   

Generally speaking, relation of samples frequencies and sinusoid not an integer. At the expense of it the dependence of an error on a ratio of samples frequencies  and sinusoid a little becomes complicated, but these delicacies already leave for frameworks of the present work. Graphically this dependence is shown in a Fig. 61.

 

APPENDIX 3. AN EXACTITUDE COSINE APPROXIMATION

It is known, that any analytical function can be presented as a degree polynomial [60]:  .

 Fig. 62

In that specific case it is possible to consider an ideal model, for want of with which , for want of  , is equal to zero, and for want of  equally to unit. For want of it,  . The population of such functions is shown in a Fig. 62.
           Curvature functions  express by the formula [5]                                                                 .
 Generally:  ;
 Then

 .                 (П3.)
                             
                           

Extremum  Is determined for want of  . Then, having equated this expression to zero and having simplified, it is possible to receive: .

Having substituted obtained  In (П3.1) it is possible to receive:

.

Graphically this function will look like, shown on a Fig. 63:

Fig. 63                                             Fig. 64

For determination of intervals of inquiries it is necessary to develop functions so that a variable  was normal to a vector  . In points  , appropriate  line, perpendicular vector  coincides with  . General expression in this case:  . Graphically this dependence is shown in a Fig. 64.

Further, for comparison of these functions on intervals of digitization is necessary them to normalize to r =1. It is achieved by a diminution of a curvature functions at the expense of magnification of scale . For want of normal position of an axes of time to a vector  . Then  .
             If to enter scale  on  , that                                                    (П3.2),

 then.

For want of  . With allowance for (П3.2),. Thus, for want of the transition to the equation is received:. That is, for normalization on it is necessary to increase in time. Using the above-stated reasons, the accounts on the COMPUTER were made, in which outcome the outcomes reflected on a Fig. 65 are obtained.

Fig. 65

On an axes n the degree is postponed   Component of a degree polynomial. On an ordinate axis half of interval of digitization. As the curve of the normalized degree function is nonsymmetric, we have two curves, defining maximum and minimum magnitude of a halfcycle of digitization.   On Figure the magnitude of a halfcycle of digitization defined on a method of cosine approximation - is reflected  . From Figure it is visible, required intervals of digitization, proceeding from the analysis measured of a parameter as ascending power series, as a whole more, than difiniendums on a method of cosine approximation. For want of it it is necessary to take into account, that the actual parameter practically never consists of one degree function. And, than above degree of the member of an ascending power series, the is less than it factor , That is it is less than it influence. In this connection, sum of a number actual measured of a parameter determines such intervals of digitization, which are in a zone (а) shown on Figure.

Thus, we can make a conclusion, that the determination of intervals of digitization on a method of cosine approximation, at first, is exact enough, secondly, practically unsuperfluous.

 In separate, practically rare cases, when the parameter is close to quadratic function, the a little bit large frequency of measurements can require. For this case the appropriate formula is entered into a population of the formulas of account of frequency of inquiry.

Besides it is necessary to take into account at that, that for want of metrological account of a measuring circuit, the error of digitization should be determined proceeding from a principle of an insignificant smallness, for want of it error in determination of sampling rate is magnitude of the second order of a smallness and it, in this case it is possible to neglect.

It is possible to state and more general reasons   for the benefit of cosine approximation. The survey of materials of actual measurements creates the certain impression, that many from them, especially in the most dynamical part, gravitate to a sinusoidal kind. And it not accidentally. Objects of measurements, as a rule, are gomeostatical as a systems supporting it specific status or changing it under the specific law. This status is determined by a population of feedbacks. The response of such systems on revolting effect is described by the equations giving or aperiodic, or oscillatory solutions, which  well approcsimated by sinusoids.

APPENDIX 4. THE REVIEW and ANALYSIS of METHODS of an INTERPOLATION

The problem of an interpolation is indissolubly connected to a problem of digitization. The interpolation in an obvious or implicit kind is carried out always. In an obvious kind the interpolation is made in that case, when the parameters necessary for account of performances, are measured with different frequency, and the performances are calculated with frequency to equal maximum frequency of inquiry from frequencies of interrogated parameters. Besides the interpolation is made in a number of cases for want of graphic submission of parameters or performances. For want of it the interpolation will be realized by a programs or by hardware. In an implicit kind the interpolation is made in the event that the information is registered in a tabulared kind and an interpolation make consciously or unconsciously, making an evaluation of significances of function between samplings.

1. The problem of an interpolation for a long time attracted a mathematical idea [7]. Great mathematics CVII of century were well familiar with methods of an equidistant interpolation by polynomials and have advanced the theory of final differences up to a high level. They used ordinary, and also central differences. The basic work Grergori (1638-1679) was continued by a Newton. Stirling, and later Bessel have added  some formulas. The severe development of the theory of an interpolation begins with Ghana and Fihera. Dangers, connected with equidistant interpolating by polynomials were detected the Runge (1904) and Borel (1903) is independent from each other. The development of methods of an interpolation with the help of of orthogonal functions was developed the Fourier, Ostrogradskii, Whittaker. The methods of an interpolation of discrete sequences were developed Wolsh, Haar etc.

Recently large popularity have received a method of spline-functions developed in 1948г. [62,63 etc.]. Besides to problems of an interpolation it is possible to refer a method of the Gauss (method of least squares). The stochastic methods of an interpolation were developed [45].

As a whole, the known methods of an interpolation can be divided into three main groups:
-  Methods with application of ascending power serieses;
-  Methods with application of orthogonal functions;
-  Methods of the stochastic analysis.

             2. The essence of methods of ascending power serieses consists of the supposition, that initial (before digitization) the function can be represented of degree numbers of a kind:  for want of .
If to assume, that  , that is usually executed, 
The sum of the remaining members of a number.
              Then: there will be an interpolating series.
              For definition of a series  it is necessary to define, for what it is necessary to decide a set of equations for want of known significances . Methods of definition  , on known  , are be various, as determines distinction of methods of an interpolation by polynomials.

One from the most known interpolational algebraic polynomials is the polynomial of the Lagrange [3,48,45 etc.]. Known varieties of a polynomial of the Lagrange - interpolation formulas of a Newton, Gauss, Bessel, Stirling, Eferett, spline-function etc.

Advantage of methods of degree polynomials is their simplicity for want of realizations on the COMPUTER, therefore now they are main for want of to information processing on the COMPUTER. Defects of methods of degree polynomials are emerging effect of magnification of an error for want of magnification of a degree of a polynomial and unsufficient validity of application them to stochastic functions. It is connected that the methods of an interpolation originally were developed for needs of an astronomy, in particularly for account of parameters of orbits of celestial bodies (planets, comets, companions etc.) and for drawing up of astronomical calendars. In these cases the curve of movement of celestial bodies was described with an adequate accuracy by final degree polynomials, for example, the movement of a satellites round a planet  is described by a curve of the second order. Thus, the interpolating polynomial will be also final, final the necessary number of samplings is also.

Besides the methods of an interpolation by degree polynomials have found a use there, where the circumscribing polynomials beforehand are known olso them it is necessary to restore on known significances in isolated points. Such problem is decided, for example, in drawing automatons.

As if to stochastic functions, the interpolation by final polynomials is connected to errors, which essentially influence not only quality of restoring of function, but also on possibility to apply an interpolation to restoring assumed function.

3. The methods of application of orthogonal expansions have begun to develop from time of development of expansion of functions on sinusoidal component, which development connect to the name of the french mathematician the Fourier. The mathematical significance of expansion of periodic functions in a number on sinusoidal functions was realized in XVII century by the Euler and Lagrang. By a merit the Fourier was distribution of possibilities of such expansion on nonanalytic functions. The theory of orthogonal functions was widely considered by Russian mathematician by the Ostrogradskii a m.t. [64]. He marked, that “ a row

is one from the most simple and useful rows enveloped by general expansion. ” The sinusoidal expansions have found broad application in connection with development of a radio communication. Their efficiency has appeared is so high, that many have begun to consider spectral component of a signal real-life [65.66]. In connection with discrete transfer, as interpolating function  the number was offered by Kotelnikov
  orthogonal functions of a kind: .
                 Orthogonal functions are the functions of the Chebyshev etc. The application of these methods in practice of measurements has not found broad application for want of to processing of a measuring information in connection with complexity of accounts, connected to them.

5. The methods of the stochastic analysis in the essence consist in determination of expectation of an error of digitization. Now for want of  the metrological analysis of process of digitization go by this way (for example [45] etc.) more often. These methods yet have not found broad application owing to difficulties of determination of stochastic performances of an initial information. Besides these methods are aplicable for stationary casual processes, while is real measured parameters by those, as a rule, are not. Though methodically these methods are more acceptable, as they reflect a casual character of an measured information.

Thus, despite of an abundance of methods of an interpolation, till now problem on choice of an optimum method of an interpolation in application to processing a measuring information up to an last time is not solved.

THE CONCLUSION

So, dear colleagues, all  that I wanted necessary to state concerning the theory of a technical information, I have stated. Certainly, that that remained on different reasons not live in this book. In particularly, some problems of a structurally - technological character, and also materials of approbation of the developed techniques and design solutions. If someone will be interested with these problems, can to me address.

Beforehand I want to answer to those critics, which will tell, that the theoretical means of the techniques, stated in the book, is too simple. My  judgement is those, that attempts to apply without special on that necessities any last squeak of a mathematical style seldom result in something acceptable. Sometimes think, that it is necessary to find any magic formula, and all problems will be allowed. So it does not happen. More often it happens on the contrary.

Really, the actual processes in a nature are complicated.  More over, they up to an extremity to human reason are not conceivable. At the best we form in the consciousness some model, which in any measure can correspond to physical process. For the mathematical description such model frequently should so be simplified, that, even if the mathematical means is developed irreproachably, the received conclusions can not correspond at all to real. An example such, on my sight, invalid, simplification is restrictions superimposed on a system with feedback by Nikewist.

Nevertheless I am ready with thanks to accept design criticism, offer and wish, which will be certainly reflected in the following issuings of the book, if those will follow.  Are present a number of problems, which are not considered in the book. For example, problems connected to a statistical analysis  of parameters and structure of gauges, problems of standardization and unification, perspective design-technology solutions on the basis of modern circuitry and systems engineering etc. I think, that would be useful within the framework of creative group to prepare more volumetric book.

I think, that now it is possible to raise the question about creation of the base concept of construction MIS with development of standardized and unified requests to interfaces (in particularly to interface of connection with gauges), programm shells etc. In this case, the firms which have executed this work, have chance to leave in the leaders not only for area of MIS, but also in the field of gauges. That is, to create to themselves conditions similar of themes, which have supplied success to Microsoft . As the market for such equipment can be very broad.

The experts in area MIS hardly need to list filds of application of the theory and means of informational - measuring engineering. Nevertheless, for those who is more connected to organization and production management, briefly I shall prompt some from them:
- The space engineering (regular and test facility);
- Ground mobile engineering (automobiles, iron roads etc., regular and test facility);
-  Surface and underwater ships;
- Meteorological systems and systems of monitoring, including space;
- Plants (chemical  and biological reactors);
- Power (nuclear reactors, boiler etc.)
- Biotelemtry.

In all these areas of application, methodical and means developed with materials, stated in this book, will supply a high degree of reliability of measurements, them metrological validity, decrease of the costs on development of an equipment and  reduction of terms of tests etc. And consequently also considerable material profits.

For example, for want of realizations of space monitoring, proceeding from a necessary exactitude of measurements, the amount and orbits of the satellites of the Earth is justified. It is natural, if this substantiation is correct enough, the amount of the satellites can be optimum. If it is not correct, the number them can be or less necessary (for want of it the reliability of an information, and, therefore, and it value and, certainly, cost) is reduced. Or it is more, and then will be the large costs of creation of a satellite system. And that the manufacturing and start even of one satellite costs very dearly, it is known to all. And to save these means it is necessary simply to count their necessary number with the help of techniques based on a material, stated in this book.

ON MAIN PAGE

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