GRIGORENKO A.M.

 SOME PROBLEMS of the THEORY
         of a TECHNICAL INFORMATION

(Site "Through thorns to stars". Anatol Grigorenco)
                  
Copyright©2001 Feedback

INTRODUCTION

There is no necessity to prove, that the measuring information is one from main component during creation and use of modern engineering, and also in a set of other areas of human activity. Really, any more - less complicated engineering, has systems of automatic control and monitoring, and such branches of human activity, as a meteorology, metrology and others generally only in that also occupied, that collect and treat a measuring information. Probably, without the special exaggeration, it is possible to tell, that the measuring information is one from bases of a modern civilization.

However hardly it is possible to tell, that the theoretical basis in the field of a measuring information are in the full order and arrange practical specialists. The outcomes of researches, offered to the favourable reader,  can appear quite useful, especially for account of streams of a measuring information with allowance for it metrological exactitude and reliability as for creation and use of measuring - information systems (MIS), and systems of automatic control and management with use MIS. A number of positions developed by the author and stated in the given book approbated in practice  of tests of air engineering also have received a positive evaluation at the authoritative experts.

The problems of measurements frequencies definition of  changed in time and space parameters, systems analysis of errors MIS with allowance for errors of digitization, struggle with anomalous measurements and some other problems are included in the book. Besides taking into account, that frequently MIS are an element of systems of automatic control, there was a problem of clarification  some problems of the theory of feedback as a fundamentals of such systems.

As in itself theory of a measuring information, as well as theory of feedback, exist for a long time, there was a necessity of the certain criticism of being available theoretical base. Some examples of practical realization of offered theoretical positions are briefly circumscribed.

The author hopes, that the outcomes of this work will appear not useless, and will be grateful for design criticism.

THE CHAPTER 1. MAIN PERFORMANCES MIS

The measuring - information systems is complicated object of engineering that have a number of characteristics, main of which are informational and metrological. 

There are following main information characteristics:

Information stream from object on a system as a whole and on it separate elements;
                - Productivity of elements MIS;
                - Required volumes of storage devices of elements MIS.
                Main metrological characteristics MIS are an exactitude and reliability of measurements.
The information stream from object depends from dynamic characteristics and requests to an exactitude of measurements. In this plan it does not depend from MIS and is specific magnitude. However, as a rule, this stream is largely redundant. The decrease of redundancy can be reached:
               By adaptation of a structure MIS to a solution of concrete problems and, accordingly, modification of the list of connected gauges and other sources of an information, their number and frequencies of their inquiry;
                - By the substantiation of really necesssary norms of an exactitude;
                - By the substantiation of really necessary number of samplings in unit of time (frequency of inquiry).
                As to the substantiation of the list and number of interrogated gauges, and also required norms of an exactitude, they are determined by the developers MIS, proceeding from concrete problems with reference to concrete object, and the discussion them within the framework of the present work is not obviously necessary.
                Here we shall consider problems of definition (determination) of frequencies of inquiry. A problem this not new. By the beginning of shaping of theoretical basisof definition (determination) of frequencies of inquiry (or the theories of digitization) usually consider in a works Kotelnikov V.A. [1] and Shannon [2]. Are present also number of other works on this problem. However their theoretical analysis and the analysis of their practical application shows, that effect from existing theoretical basis practically is not present any. In more detail criticism it is stated in APPENDIX 1. In this connection, till now frequency of inquiries was determined rather approximately and, as a rule, intuitively, because of of experience of the experts and, at the best, experimentally. For want of it the frequency of inquiries becomes essential redundant and can reduce in such requests to an equipment, that the creation MIS becomes problematic or, at least, it becomes much more bulky and expensive. The underestimation of frequencies of inquiry results to  metrological  unauthenticity of outcomes of measurements. Further will are considered problems of definition (determination) of sampling rate.

THE CHAPTER 2. DETERMINATION of SAMPLING RATE

1. The methods of digitization and restoring of signals after digitization can be divided on some groups depending on accepted indications of a classification. The following indications of a classification are selected [3]: 
                - regular of readout; 
                - criterion of an evaluation of an exacctitude of digitization and restoring; 
                - basis functions; 
                - principle of an approximation.
               In the correspondence with indications of a regular of readout it is possible to select two main groups of methods: uniform and irregular. Methods uniform the digitizations have found the most broad application. It is explained to that the algorithms of digitization and restoring are simple enough. However, because of a discordance of a priori performances  of a measured parameter to performances of a model of processing, the significant redundancy of readout is possible. From methods of irregular digitization two groups - adaptive and programmed are known. 

The adaptive methods allow to reduce redundancy of an information, however realization them is connected to a number of difficulties, namely: 
   - complexity of algorithms; 
   - necessity of a prediction; 
   - as a whole channel should be designed for a maximum stream of an information, that reduces advantages of this method.

Owing to these and other defects the method of adaptive digitization has not found broad application and will not be considered hereinafter. For want of programm methods of digitization the modification of frequency of inquiry is made in the correspondence with the beforehand composed program of measurements. Within the limits of the given stage of measurements it is reduced to uniform digitizations.
                2.From criterions of an evaluation of an exactitude are known maximum, root-mean-square, integrated and probability - zone [3].
                The probability - zone criterion, defined ratio, is most widely used:  
                               P{ e ( t ) < D} = Po       ( 2. 1 )

Where Ðî - The allowable probability that an error will not exceed significance D . Usually consider a Ðî specific and in the correspondence with [4] it can be accepted 0.05. In these conditions the magnitudes only D are set which in practical cases express, as a rule, indicated to maximum magnitude of a range of a measurement in percentage.


                3. The problem of digitization is indissolubly connected to a problem of restoring of function. For want of it it is necessary to deliver lattice function represented by samples, in the correspondence continuous function which would differ from assumed function on magnitude a not exceeding specific error that is:

            ï x (t) – V (t) ê £ D                                                ( 2.2. )

Function      V(t)  -  reproducing function. As reproducing functions the orthogonal numbers (Fourier serieses, Kotelnikovs etc.) can be applied, the degree polynomials etc. For want of to processing on the  computer the greatest distribution have received ascending power serieses, due to simple enough algorithms of their realization. The regenerating functions are called basis, as from  choice of these functions a depends method of the substantiation of characteristic of digitization and restoring of an information.
                4. On a principle of an approximation three groups of methods are chosen:
                - interpolational;
   p;              - extrapolational;
                 - combined.
    ;            The advantage of interpolational methods is them higher exactitude, but they introduce delay on period of digitization, therefore occurs by an additional dynamic error. Extrapolational methods are applied in the event that MIS enters into a closed loop of contro of object, when the temporary delay are invalid. But they require large frequency of readout and, therefore, greater stream of an information. In conditions of concrete problems the combined methods can be applied.
                Thus, further digitization will be understood, first of all, as program - adaptive (uniform) digitization. The criterion of an exactitude accepts a probability - zone criterion. The choice of basis function will be determined and the problems both interpolations, and extrapolation are considered.
                5. Measured  parameters time-dependent can be refered to analytical functions. They are limited on magnitude, have, as a rule, extremums, have no ruptures neither first, nor second kind neither at the function, nor at it of derivatives. It is necessary to mark, that some parameters can have so fast modifications, that can be accepted for ruptures of the first kind. However it is an elemination from a general rule and can on occassion in appropriate way be taken into account.
                6.As the speech goes about sampling rate, that, obviously, that first of all should be raised the question about dynamics of function. 
                Dynamics of function can be expressed with the help of of following characteristics:
                - autocorrelated functions;
                 - frequent spectrum;
                 - magnitudes of maximum derivatives of function.
               More often MIS is designed for objects with parameters, which dynamic characteristics are precisely unknown. For example, MIS for tests of mobile engineering (in particular of flight vehicles) can be applied to various types of engineering and various stages of tests. In this connection to define stochastic or spectral reflectance  of parameters  measured it is not obviously possible, as they can be determined during tests, that is then, when MIS is already made.  Besides this, actual measured parameters can be refered to not stationary processes, which a priori indeterminacy does to impossible deriving of representative sample. In this connection, the application of stochastic methods generally and autocorrelated analysis in particularly, is not obviously possible. As the autocorrelated function conjugates with a spectral denseness under the formula of the Wiener - Hinchin, same it is possible to tell and about the spectral analysis. Proceeding from this, preferable  is the using of maximum derivatives magnitudes for  dynamic characteristics. The analysis of practice of actual measurements shows, that measured parameters have as a minimum up to a second  derivatives, which maximum magnitudes simply to receive by theoretical, experimental or other way. And even the intuitive suppositions about their magnitude made by the expert in the given area are authentic enough. 
                Such approach has that advantage, that dynamics of process expreses through visual enough characteristics. These characteristics have the certain technical sense. For example, for want of measurement of height the first derivative it not that other, as vertical speed, and flexon -  transhipment. Thus, knowing as much as possible allowable vertical transhipment it is possible to define frequency of inquiry of the gauge of height.
                7. As the application of stochastic methods is not obviously possible, the mathematical means constructed on application of determined functions should be applied. However measured parameters are is a priori uncertain, and, therefore, casual. An exit here one to reduce is a priori not known, stochastic function to is a priori of known function adequate to a that request, that their dynamic characteristics were identical. Obviously, that for want of it and the frequency of inquiry at them will be identical. 
                Requests to such function the following: 
                - at first, it should as much as possible correspond to characteristics of actual parameters; 
                - secondly to have simple enough expression. As is known, any analytical function can be decomposed in a Fourier series. Therefore sinusoid is enough convenient function for comparison to analytical function mapping a measured parameter. In the practical plan it is important at that, that some parameters are close enough to a sinusoid. For example, oscillation of configuration items of a flight vehicle.

  Fig.1

           The idea of a method consists of the following [a Fig. 1]. In an instant t0,   appropriate to a local extremum of function f(t), in which |f(t)| is maximum, the function f(t) is approximated by a sinusoid F(t). For want of it, |F”(t)| max=|f(t)|max. As the instant t0, generally speaking, is not determined, it is supposed, that, in conditions of a measurement of the given parameter, such event is possible, for want of which in some instant t0 the flexon will reach the maxima. And from conditions of a measurement of a parameter it not   can to be more this maxim:  {| f "(t)|}  £ |f"(t)max|.
              The instant  t0 is convenient for accepting in quality   t = 0, for want of it F (t) = max. That is, F(t) there is that other as a cosinusoid. For want of rushing of a neighbourhood t0 to zero,   | F(t) - f(t) | ® 0. Therefore, in a neighbourhood t0 F (t) is approximating function. In this connection, the method is called as a method of an approximating cosinusoid.
                 The amplitude of a cosinusoid, for want of specific | F "(t) |max, is unequivocally connected to frequency and can be of any magnitude, including error, equal to a range, of a measurement of the given parameter. 
                In turn, having frequency of a cosinusoid, it is possible to determine frequency it of inquiries, from a condition 100 % of an error of possible distortion of this cosinusoid in connection with stroboscopic effect, which will be considered below.
                Hereinafter it will be possible to reduce the obtained formulas in a kind convenient for practical use.
               8. So F(t) =cos(wt + j) . As t0 is accepted for a beginning of readout and for want of it F (t) maximum,  that  j=0 .  Then  F(t) = cosw t. Taking into account that

| f" (t) max | = | F" (t)max |  and   F"(t)=w 2 cosw t    that |f"(t)max | = w 2cosw t.

In a point  t =0    | F"(t) |  is maximum, for want of it |F"(t)max| = w 2 .

From here:  w 2 = |f"(t)max| , therefore: w= Ö | f"(t)max |  .  Then the formula of an approximating sinusoid will accept a kind: F(t)=cos((Ö |f"(t)max|) ´ t) .
           As was specified above, the amplitude of a cosinusoid is indefinite. It is necessary, that it is equal to an error of digitization a parameter  -D ä, then:

   F(t) = D ä ´ cosw t ;        F"(t)max = D ä ´ w 2  = | f"(t)max | . From here: 
              _______________    
w
  =   Ö ( | f"(t)max | ) / D ä .

Then in a final kind the formula of an approximating cosinusoid will accept a kind:

                            _____________  
F(t)=D ä ´ cos{[Ö (|f"(t)|max)/D ä ] ´ t } .          (2.3)

To connect an obtained approximating cosinusoid with frequency of readout it is necessary to apply concept of stroboscopic effect. In more detail problems connected to stroboscopic effect, are considered in APPENDIX 2. Here we shall mark, that for want of fulfilment uniform  readout from a sinusoid, in case if the frequency of readout is equal to the double frequency of a sinusoid, at the expense of phase indeterminacy  the significances of readout  can accidentally hit on anyone it, including on both extremums, and zero. In this connection, for want of it there is 100 % an error in map of a sinusoid. Taking into account, that the amplitude of an approximating cosinusoid is equal to an allowable error of a measurement of a parameter, the frequency of readout equal to the double frequency of an approximating cosinusoid just and will be required frequency of inquiry  of a measured parameter, or   fä  =  2w ò.ê./2p    w à.ê/p . Or:

                  ___________ 
fä  =  1/p Ö| f"(t)max | /D ä .     (2.4)

Thus, the formula of determination of sampling rate is obtained. Now it is necessary to reduce it in a kind convenient in practical use. 
                9. It is usually accepted to express an error as the indicated error to maximum magnitude of an effective range of measuring in percentage. From here:

                 ________________                _______________   
fä=10/p Ö | f"(t)max | /(D ä´A)    »   3Ö | f"(t)max | /(D ä´A) ,                     (2.5)

where  fä  -  sampling rate, D ä  -  the indicated error of digitization in percentage, A - range of a measurement.
                This formula is already suitable to use. Is valid, if it is known, for example, that height of flight of a airplane in a range 0 - 10000 ì, allowable error of digitization - 0.5 % and for want of to as much as possible allowable transhipment on a vertical axes of a airplane - up to 5g, taking into account, that f"(t)max=(Ng-1)õ9.8, we shall receive:

              ________________
fä  =  3Ö [(5-1)õ9.8]/(0.5õ 10000) »   0.3gz.

10. In a number practically of important cases the graphic submission of a flexon is desirable. It is known [5], that:

                  ________     
k = f"(t)/(Ö 1+[f'(t)]2)3,

where k -  curvature of function f (t).
              In a bending point , f'(t) = 0 therefore k = f "(t). As k = 1/r , where  r - radius of a curvature, then:

               ___________
 fä = 3 /Ö r min ´D ä ´ À .                                    (2.6)

It is the second variant of the formula (2.4). 
                 11. On occassion to receive a flexon it is inconvenient, and in other cases the radius of a curvature is less than an allowable error. In these cases it is more convenient to use the first derivative of function for determination of frequency of inquiries.

  Fig.2

Really (Fig. 2) if to consider  r ® 0, in a point t = t0  and the time scale is selected in such a manner that an angle of tangents to an approximating cosinusoid in points it of maximum first derivatives (points of inflection) - a = p/4 , the neighbourhoods of a point of a maximum curvature will represent a break of function with an angle p/2 . The range from a point of a break up to a line (a, b) is selected equal to an error of digitization - D ä, in which the behaviour of function can be anyone. Including, it can be and cosinusoid with such period, that in points a and b it has by the tangents of a line of a maximum steepness. The radius of a curvature of a cosinusoid  in the point t0 is equal r = (2/p)D ä.

If to substitute in the formula (2.4) obtained magnitude r = |f"(t)max|, that

                             ___________
   fä  =  (1/p )´1/ Ö D ä´(2/p)´D ä ,   and also taking into account, that magnitude of an error D ä  is necessary also multiply on magnitude  2/p ,
                                    ________
  that: fä = (1/p )´ 1/Ö D 2 ( 4/p 2)      =   1/2D . (2.7)

This formula can be received and easier. Really, the interval t1- t2 is equal to two half-intervals (t1 – t0) and ( t0 - t2) each of which is equal D ä. From here t2 - t1 = 2D ä and, in turn, fä=1/2D . But it would be desirable to conclude from general items. The last conclusion once again shows a regularity of a circumscribed method.
             Further, taking into account concept of the indicated error, we shall receive:

                                     fä = 50/D äA.

Taking into account, that the time scale t is selected specially for a conclusion of the formula (2.7), at that, that the frequency of inquiries varies linearly with a rescaling of time, and also that D x/D t = tg x, it is possible to write :

fä = 50tg(a max) / D äA,  èëè fä = 50f'(t)max / DäA  (2.8)

The obtained formulas are effective for want of r £ D , in case r > D these formulas will give overstated significance fä. It is possible changing a time scale t, to supply fulfilment of a condition r £ D , for want of it the formula (2.8) will acquire a kind: fä = Ì50tg amax / D äA, where Ì - time scale.

Quite reasonably it is possible to raise the question about an exactitude of cosine approximation. This problem is considered in APPENDIX 3.

                12. In summary unit we shall reduce a population of the obtained formulas:
                                               fä = (1/p) fñ arc cos(1 - D ä).                 (2.9)

This formula is received by a solution of the formula (Ï2.1), where N = fä/fñ, and fñ frequency of oscillations of sinusoidal function f (t). For want of indicated error it looks like: fä = (f /p) arc cos (1-D ä/100). This formula can be applied to parameters have an obviously expressed oscillatory character. For example, oscillation of configuration items of a flight vehicle.

                           ______________
Formula  ä = 3Ö |f"(t) max|/ D äA                    (2.10)

 Is applied to parameters, the deriving of magnitude of a maximum flexon concerning which does not represent large difficulties.

                             _____
Formula fä  = 3/Ö rD äA,                   (2.11)

 

is applied then, when probably geometric construction of dependence of a parameter from time. In particularly, then, when there is even one realization of a parameter in it the most dynamical kind, (for example recordings of test of a drive) .Beside that, this formula can be applied for want of account of an amount of data points on a space coordinate, for example amount of strain gauges on a range of a wing, For want of it the diagram of mechanical voltages is created and on its curvature the interval between gauges is determined.

                Ôîðìóëà fä = 50tg amax / DäA                                               (2.12)

is applied concerning parameters have r <D ä, and which graphic submission is simple enough. For example, the deviation of organs of management of a flight vehicle, which can be registered on ground. This formula can also be applied in case r > D ä. Factor Ì on a time scale is in this case applied and the formula acquires a kind: f=M50tg a max/DA .

                Formula  fä = 50f'(t)max / D äA                                       (2.13)

is applied concerning parameters, for which more preferable is the use of the first derivative. For example, velocity. However it is necessary to mean, that the given formula can give overstated significance of frequency of inquiries.
               In the event that beforehand it is known, that the dependence of a parameter on time is very close to square-law, the formula (2.9) will give an error in 1,5 times more. In most cases (see is lower) taking into account, that the error of digitization is determined on a criterion of an insignificant error in relation to an error of a measurement of a parameter, it can be neglected. The a little bit modified formula from [6] on occassion can be applied:

                 ______________
 fä  = 3,5Ö| f"(t)max| / D äA .

The circumscribed method has a number of properties from the point of view of restoring a signal (interpolation) and it of stochastic performances. In the following chapter these properties are considered.

 THE CHAPTER 3. AN INTERPOLATION And EXTRAPOLATION

The problem of an interpolation and extrapolation is indissolubly connected to a problem of digitization. The interpolation in an obvious or implicit kind is made always. In an obvious kind the interpolation is made when the parameters necessary for account with what or performance are measured with different frequency, and the performance is calculated with frequency greater, than frequency inquiry all or some part of 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 is made program or is 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.
                 The problem of an interpolation for a long time attracted a mathematical idea. A history of this problem, it the analysis and criticism are considered in APPENDIX 4. 
                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.

               Their analysis (see. THE APPENDIX 4) shows, that concerning digitization there are enough of convenient methods of digitization is not present, that forces us to try to decide this problem independently. A technique of an interpolation implying from a method of cosine approximation is circumscribed below.
              1. There are two next samplings of a parameter in instants t1 and t2. (Fig. 3). As the interval t2 - t1 is determined because of of cosine approximation, it is known, that the function of a parameter is limited to halfcycles of a cosinusoid. As the magnitude of a flexon in the formula (2.3) undertakes as the module, the halfcycle of a cosinusoid has either positive, or negative sign. Obviously, that the area between cosinusoids is area of a probablis determination of an actual option value.

Fig.3 

 The distribution of probability of an option value in this interval is symmetric under the relation of a line (a - b). Therefore line (a - b) is a geometric place of points of expectation of option values. It is quite natural, that the interpolating line should coincide a line (a - b), that is the interpolation should be piecewise linear and look like: f (t) = a + bt, where a = f (ti); b is determined as follows:
  f(t)  = f(ti )+bt ;  f (ti+1)  =  f(ti) + bti+1 ; b  =  [f(ti+1) - f(ti)]/ti+1 .

Thus, the settlement formula for definition of current significance f (t) between points of samplings  ti and ti+1 is determined by the equation:

f(t) = f(ti) + {[f(ti+1) - f(ti)] / ti+1}t.

Where ti+1   is determined from a beginning of readout, with which it is accepted  ti, that is ti+1   = Ò, where Ò an interval of digitization. From here  f(t) = f(ti) + {[f(ti+1) -  f(ti)] /T}t .

Advantage of a linear interpolation is the simplicity and convenience of its realization on the COMPUTER. Significantly, that, despite of repeated attempts of application of more complicated methods of an interpolation, in practice in main apply a linear interpolation.

2. And now we shall consider the problem in the a little bit greater generality. If to put, that in some moment ti the readout is made and the point option value f (ti) is obtained, before the following readout it is possible to enter one condition, that measuring the parameter will not be changed more, than to magnitude of a specific error D . Really, any physical magnitude can not change the significance instantly. The modification of a parameter on D conjugates with some  D t. And, the interval between readout Ò should be equal D t, as under condition of Ò>D t the error above allowable will be increased, and for want of T<D t samplings ti+1 and ti become correlated and, therefore, is informational are redundant. For want of Ò= D t readout become not correlated. Indeterminacy of magnitude f (t) to a point  ti+1 limited by limits ±D . That is it is possible to tell, that for want of Ò = D t, the readout in limits ±D become a Markov chain, or, in more general understanding, martingale. It is natural, that the knowledge of significances f (t) in the past nothing can give in determination f (t) on an interval Ò. From here becomes clear, that the application of any kinds of an interpolation by degree polynomials is higher than the first degree (that is, using magnitude  f ( ti - n), where n = 1,2...), is senseless. In it and essence of appearing errors for want of polynomial interpolations with magnification of a degree of a polynomial [7].

In this case was considered, that the significance f(ti+1) is not known, that is the problem of an extrapolation was decided. If it is known  f(ti+1), it is clear, that f (t) owes from known f(ti)  proceed to known significance f(ti+1). It can make this transition is equiprobable deviating from a direct line connecting this point, that determines symmetry of distribution of probability to a line f(ti) - f(ti+1), and, therefore and choice of this line as interpolating.  For want of it the maximum deviation of this line under condition of restriction of function on a flexon and analyticity of function will be limited to halfcycles of a cosinusoid. Therefore, considering  f(ti) as the Markov chain, for want of Ò > ti+1  - ti  is possible to make a conclusion about an optimality of a linear interpolation.

3. It is possible to reduce and following reasonings. Recognizing that the approximating cosinusoid can be one from realizations of a researched parameter for want of it maximum, for want of specific conditions, dynamicallity, it is possible to receive autocorrelated function of this parameter R(t) , which will look like, as is known, sinusoid. Obviously, that the significance R(t) = 0 limits area, in which option values stochastically, and consequently is informational are connected. As the period R(t) is equal to period F (t), it is possible to make a conclusion, that outside of an interval Ò any readout can not give an additional information about a parameter, and consequently their use with the purposes of an interpolation on an interval Ò is senseless.

4. The extrapolation frequently is applied in MIS, especially then, when the information processing is made in real time, and in particularally, when MIS is included in an outline of management of object. The problem of an extrapolation represents in essence problem of a prediction [8]. As for determination of sampling rate the criterion of a maximum curvature is selected, that is maximum flexon, it is possible to suppose, that on an interval from t up to ti+1  (see. the Fig. 4) flexon will be a constant and maximum, that is| f"(t)max |= const. In this case f (t) will represent a parabola f (t) = (at)2 . For want of f (t) = 1 (D   is accepted for 0), t=p /2 From here: (ap /2)2 =1. Then:  a2p 2=4 ;  a=2/p ;  therefore f (t) = ( 2t/p )2. For want of f (t) =2, t=p /Ö 2, p /Ö 2-p/2  » p ´ 0,208. As the halfcycle of a cosinusoid (it(he) an interval of readout) is equal p, the attitude (relation) p /p ´ 0,208 » 4,8 gives the attitude (relation) of required sampling rate for want of to extrapolation in relation to sampling rate for want of interpolations.

Fig.4

Thus, the frequency of an extrapolation should be higher than frequency of an interpolation almost five times. On the other hand, the function f(t) can be after a point p /2 and sine wave(sinusoidal). In essence it is two extreme possibilities. In conditions of indeterminacy it is meaningful to select something average, namely line a tangent to a sinusoid in a point p/2. Thus, we can receive a point b, which divides an interval of digitization in p of time. That is the frequency of an extrapolation will be approximately three times more frequency of an interpolation.

Fig.5


            In that case, when the frequency of inquiry is determined on a maximum first derivative (Fig. 5),   f (t) can from a point d move or to a point b, or to a point and. Obviously, that that the parameter has not left for limits of the admission D it is necessary and enough to reduce magnitude of an interval twice, that is to increase sampling rate twice. Thus, for case of an extrapolation it is necessary the formulas (2.9,2.10,2.11) multiply on 3, and formula (2.12,2.13) on 2.

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