7. 8. 9. i0. ii. 12. 13. 14. 15. 16.

Yu. V. Kurtikov et al., Metrologiya, No. 5 (1979). All-Union State Standard (GOST) 16493-70: Product Quality: Statistical Acceptance Control on an Alternative Feature [in Russian]. All-Union State Standard (GOST) 8.051-73: Errors Tolerable in Measuring Linear Dimensions up to 500 ram [in Russian]~ N. N. Vostroknutov et al., Izmer. Tekh., N-. 8 (1975). N. N. Vostroknutov et al., Izmer. Tekh., No. 7 (1977). V. D. Frumkin et al., Reliability in Controlling Means of Radio Measurement and Control Tolerances [in Russian], Standartov (1975). All-Union State Standard (GOST) 16263-70: The State System of Measurements: Metrology, Terms and Definitions [in Russian]. All-Union State Standard (GOST) 8.280-78: The State System of Measurements: Automatic Potentiometers and Balanced Bridges: Methods and Means of Checking [in Russian]. M. A. Zemel'man et al., Izmer. Tekh., No. 2 (1969). D. J. Cowden, Statistical Methods in Quality Control [Russian translation], Fizmatgiz, Moscow (1961).

STATISTICAL CHARACTERISTICS OF TEST SIGNALS FOR MEASUREMENT INFORMATION SYSTEMS G. D.~Sverdlichenko

UDC 621.391.37.6.019.3:621.3.088

The theory and practice of modeling~signals with a given type of correlation function Rxx(T) or spectral power density G(~) are well understood. A signal with a given function Rxx(~ ) has a distribution function of instantaneous values W(x) closely approximating a normal distribution. This result is attributable to the fact that the form of Rxx(T ) is synthesized in linear transient systems, whose passband is considerably narrower than the spectrum of the primary signal. Signals with a given function W(x) are modeled by nonlinear transformations of a certain primary signal. In this case the distribution of the frequency components of that signal is changed. The mutual influence of the generated functions W(x) and Rxx(T) or G(m) poses the principal difficulty of modeling the required signals. To obtain prescribed characteristics W(x) and Rxx(T) corrections are introduced into one of them to compensate for the distortions induced by the other. To simplify hardware implementation a practical technique is to generate a signal with the required function W(x) from a mixture of signals, each having a normal distribution [I, 2]. Using a linear transient transformation to form a signal with a prescribed function Rxx(T ) and then mixing signals with the known function Rxx(T ) and with normal distributions W(x) having different parameters, we can obtain a signal with a prescribed form of W(x). The distortions of the correlation and spectral characteristics of the signal in this case can be calculated and taken into account for particular hardware solutions [3]. On the basis of an analysis of the distortions the prescribed function Rxx(T) we have investigated istics of a modeled signal.

induced by the modeled function W(x) in the correlation and spectral character-

A block diagram of the system used to generate signals with prescribed forms of W(x) and Rxx(T ) or G(~) is shown in Fig. i; it includes: the unit 1 for generation of the primary signal, e.g., a pseudorandom sequence of maximum length; the unit 2 for synthesis of a prescribed form of power spectrum~ e.g., a digital filter;~the multiplication unit 3; the addition unit 4; the matching unit 5; and the switch 6. The design principles, eonfiguration~ and operation of the first two units are described, for example, in [4]. The output signal from unit 2, in the form of a step function~ is sent through the switch to the multiplication unit 3, where each "step" is multiplied by a switch-selected coefficient. Then the resulting signal is sent to the addition unit 4. where it is summed with a certain shift selected bv the switch synchronously with the selection of the multiplication factor. With the arrival of a new "step"~the switch selects a new multiplication factor and a new shift. The selection of the multiplication factor and the shift is limited to n. Therefore, the (n + l)-th ordinate of Translated from Izmeritel'naya

438

Tekhnika, No. 6, pp. 12-15, June, 1981.

0543-1972/81/2406-0438507.50

9 1981 Plenum Publishing Corporation

........ 7 Fig. i the signal is multiplied and added with the same multiplication factor and shift as the first one. Oonsequentlv, the system output siKnal x(t) is a composite siKnal, whose components have a normal distribution of instantaneous values W(zi) with known variances and expectations specified by the multiplication factors k i and the shifts a i, respectively (with, of course, a sufficient sample size for each of the components). The total distribution function of the composite signal is ,

w (x) = ~ j ~v (z~).

(I)

i=I

Thus, by proper selection of the parameters of the functions W(zi) governing their positions on the abscissa axis and the scale on the ordinate axis it is possible to obtain any predetermined configuration W(x). The correlation function of each of the components of the signals zi(t ) differs from the prescribed function by a factor k l2." Thus the correlation function of the ordinates of the synthesized signal at successive intervals [(i + n) AT + T] differs from the prescribed function only by the ordinate axis scale. To determine Rxx(T ) for the composite signal it is practical to represent the sequence of ordinates of that signal in the form

(2)

x ( l ) ~ zn; z,1;...;zi~;...z,.~l; z,~_...zi2 ... zti...zi.~, '

where zij is~the j-th ordinate of the i-th signal component, zij = kixi+jn + ai, xi+jn is the instantaneous ordinate of the signal at the output of unit 2; j = 0, i, 2, ..., N -- i, and N is the sample size for each signal component z i. If a multiplicative algorithm is used for the quantized signal and allowance is made for the fact that: the deterministic periodic structure of the signal at the output of unit 2 can be used to ensure a sample size N for any value of the argument T; the signal at the output of unit 2 is centered; N is large enough to permit neglect of the variances of the estimates Rxixi(T ) ; the order of "inspection" of the coefficients k i and a i is invariant, we can write

kiki+P "~-

R'xx (pAT) = P x x (pA'r) I n "=

kiki-~n-P ~- ~ i=1

aiai+n_ p -- m~ for P < n.

aiai+P -~i=1

(3)

i=1

For p = 0 the expression is considerably simplified:

RL(o) =R~(O)--~

ki ~=1

(4)

~ ~ ~--,.~. i=1

The mean value of the resulting composite signal has the form I

n

(5)

mx------n , ~ ai" i=1

For a shift greater than n with respect to the argument T the nature of the distortions of the ordinates Rxx(T ) repeats after every n values thereof. Consequently, any value of Rxx(T ) is determined from the expression

R'~.~ [(Qn + p) ATI = R~x [(Qn + p) A~I x --n--

k'ki+P +

h~ki+~-~ + t'=l

in which Q = 0, i, 2, ..., 0 ~

p ~

n.

Thus, the statistical characteristics of the signal generated by the system (Fig. 71) are determined as follows: W(x) from (i); the variance D x from (4); m x from (5); and Rxx(T ) from (6).

439

!

The distortions o f the synthesized function Rxx(T ) can be investigated in two directions with a view toward diminishing them. First, we can aim for complete or partial elimination of the synthesis error by introducing "predistortions" into the unit for synthesis of this statistical characteristic (unit 2); second, we can investigate the dependence of the error of synthesis of R ~ ( T ) on the parameters of W(x), namely D x and mx, in terms of the coefficients k i and a i and then minimize the unknown error with the imposition of constraints on the range of variations and the relations between k i and ai. We discuss each of these approaches below. If unit 2 (see Fig. i) is made in the form of a linear digital filter, it will realize the integral of the convolution of the input signal as a pseudorandom sequence of bipolar pulses of equal amplitude and different durations withthe pulse response of the modeled linear system h(t) represented by a set of '~eighting" factors h(pAt). Inasmuch as the passband of the modeled linear system is much narrower than the frequency band of the pseudorandom signal arriving at its input, the correlation and spectral statistical characteristics of the output signal will almost exactly coincide in form with the lamplitude and frequency responses of the system, respectively. We can therefore write R~x l(@n + p) A~l R~ (o)

h l(qn + p) At] h (o)

and require that h [(Qn@p) All

{Rx~ l(On -+- p) Azl--A(p)} K(O) K (p) [n,x (C) --A(O)]

(7)

h (o)

'

where P

n--p

)

aiai+pq -

A (p) = -'-~-

a,ai-l.n-p

;

i= 1

K (p) = The

values

of

h[Qn

+ p)At]

determined

n

"=

in

i=l

this

way

result

in

the

synthesis

of

a

signal

with

a "predistorted" function Rxx(Z) at the output of the linear system. The distortions subsequently introduced by units 3 and 4 (see Fig. i) cause complete or partial restoration of the form of the modeled function Rxx(T ), depending on the relations between the parameters a i and ki of W(x). The dependence of the error of synthesis of Rxx(Z) on the parameters W(x) can be determined from the expression for the relative error of synthesis of the ordinates of this statistical characteristic:

5n~~ (p) _

n--p

p

~.

t=l

i=I

i=t

-k

n--p

p

~-z

"=~

i=1

i=1

2q

=6~+5"(p).

(8)

Rxx [(Qn+ p) A~I ~ k~

i=1

i=1

For subsequent transformations we arrange k i and a i in ascending (or descending) order and write them in the form te~=k--(i--1)hk;

ai=a--(i--l)ha,

where k and a are the largest values of the corresponding coefficients in the constructed variational series, and Ak and Aa are the intervals between successive values of the corresponding coefficients : Ak =

kmax - - ~ m i n -- ; n--I

Aa-

amax - - arnin n--1

We now write the errors 6'(p) and 6"(p) in the form Drt

Ak2---~ (p-- n) ~' (p) = nk ~ . - kAkn ( n--t ) @ A k ~ - (2nZ--3n-{- 1)

440

(9)

pn

6"

Aa ~ --2(P -- n) Z (p) = n

(i0)

Rxx'[(Qn + p)'ATI

2

1)+Ak2--c(2n - - 3 n + l ) ] X

|nk ~ - - k A k n ( n -

The multiplicative component 5'(p) and the additive component 6"(p) attain their maximum values when p = E(n/2), where E(') is the integer closest to n/2. The dependence of the error of synthesis of Rxx(T ) on the values of the set o i is determined from (9):

k =

(lzma'-----~x;

Ak

a pri

k

__

Affz

Aa =

ffzm~x--

azmax

Crzrl'in

n -- 1

apr i is the standard deviation of the signal at the output of unit 2. Figure 2 gives a nomogram.reflecting the function Ak/k = f(8'; n), which can be used for a prescribed value of 6' and known value o f n to d e t e r m i n e the ratio of the mean increment Ao z to the maximum value a z max, ensuring an admissible 5'. Relying on the ratio Ak/k so determined, we determine the relatiDnship between 5" and Aa from the expression Aa

Ao

O'max

ffma•

/ - 6"

where ~xx[(Qn + p)AT] is the value of Rxx[(Qn + p)AT] normalmzed " 2 i. A n o m o g r a m reto Opr flecting the dependence found for several values of n, 6', A a / a m a x with P.xx(T) = 1 and Pxx(T) = 0.i is given in Fig. 3. The component 8" § = for PXX(T) = 0. It is sensible, therefore, to speak of 6" for a certain Pxx min(T) # 0. The parameters of W(x) are given in relative values in the foregoing expressions. The absolute values are determined by t h e scale of this characteristic a l o n g the abscissa axis, which is selected in accordance with the admissible ratios between the level of the output signal and the levels of the intermediate transformation signals. A practical criterion is ~ max = apri (i.e., k = 1). If we assume that 6' = 6", then Aa

_

~m ax

a n d Aa = O.1Aa f o r

Pxx(T)

= 0.01,

Aa

Voxxi(Qn+p)

A~]'

(~m a X

providing

a rather

stringent

restriction

for

selection

of

the parameters of the components of the Gaussian functions. In practice, such a requirement can only be met for a limited class of modeled functions W(x), for example those with a positive kurtosis. Usually Aa and Ao d i f f e r much less. The closer these numbers, the stronger will be the inequality 8" > 8', which can sometimes produce an appreciable variation of the form of the modeled function Pxx(T). In the general case the order indices of the variational series of coefficients k i and a i may not necessarily coincide. Consequently, the upper bound of 8Rxx(P) obtained by the ahove-described procedure may turn out to be far too high. The stronger the inequality 5" > 6', the smaller will be the indicated excess. But if 5" and 8' are of the same order, then the upper bound of 6Rxx(P) is better sought in the form [ORxx (P)lmax = [8' (p) + 6" (P)]max,

where

6'(p)

and

5"(p)

are

evaluated

from

(8)

for

0 < p < n and Rxx[(Qn + p)AT]min.

On the basis of the foregoing discussion we propose the following scheme for taking into account and compensating the distortions of the modeled function Rxx(T): i. From the existing parameters a i and a i of the Gaussian f u n c t i o n s approximating the prescribed function W(x) construct variational series and determine k, Ak, Ao, and Aa. 2. If Ao and Aa are of the same order, use the nomograms of Figs. 2 and 3 to estimate the multiplicative and additive components. Determine the error of synthesis 6Rxx(r ) as the sum of the values obtained for the components. 441

r,~z; ~'--o,o5 ] Z;0,~ l

a,.., 65

g~

#; O,#Cl # #; fl,f . / f; O,OL}~!

#,J

z; o,z

'

u,z ~, o,o5 ]

6

gz

~; #,2

o,I

,g,l I I

#,1

]

~z

I

!

J,7

0,~

o,1

I

{_-O.f

8; 0,05 []'-' ..8;3':#.Z ] l

~z

I

]

t

g,J

0,@

0,5

I

d"~/z10~

0,~ 5]f00%

Fig; 2

Fig. 3

3. If Ao and Aa differ appreciably (say, by an order of magnitude), determine the error of synthesis of Rxx(T) as the largest value of ~Rxx(T) from (8) for 0 < p < n, Rxx[(Qn + P)AT]min. 4. Calculate the corrections A(p) and K(p). 5. If the resulting ~R~.(T)max falls within the interval of admissible values, the modeled function Rxx(T ) c a n ~ e regarded as consistent with the prescribed function Rxx(T ) within certain error limits. If necessary, the latter can ! be eliminated by introducing the corrections A(p) and K(p) into each measured ordinate or Rxx(T ) according to the expression Rx~ [(Qn%p) At] --A (p) K (p) (corrected) 6. I f t h e r e s u l t i n g ~Rxx(~)max d o e s n o t f a l l w i t h i n t h e i n t e r v a l of admissible values, then the corrections A(p) and K(p) must be introduced into the calculated coefficients h(t) [see (7)]. The form of the modeled function Rxx(T) in this case can correspond exactly to the prescribed function or be close to it.

Despite the fact that the application of step 6 o f this scheme provides a significant reduction in the distortion of the modeled function Rxx(T), it is inadvisable to limit the procedure to this step, because then a rigid relationship will exist between W(x) and Rxx(T ). If it is required to model another form of W(x) for the same Rxx(T), then h(t) must be altered. The introduction of the corrections A(p) and K(p) requires a known and fixed sequence of "inspection" of the coefficients k i and a i by the switch. Changing the order of this "inspection" will alter the sequence of introduction of the corrections A(p) and K)p) (the set of values of the corrections is left unchanged in this case). If it is acceptable merely to take into account the distortions o f Rxx(T ) (steps 1,5), then the only requirement on t h e sequence of "inspection" of the coefficients k i and a i is invariance. The modeled functions W(x) and Rxx(T) are completely independent in ~ this case. The manner of imposing constraints on the parameters of themodeled statistical characteriatics varies with the stringency of the requirements on the error of synthesis of Rxx(T ) and W(x). It is assumed here that the error of synthesis of W(x) must be smaller than the error of synthesis of Rxx(T). Consequently, the error 6Rxx(T ) and the corrections A(p) and K(p) are determined from the prescribed parameters of the modeled function W(x). But if it is more important to keep the errors of synthesis of Rxx(T ) within specified limits then the nomograms of Figs. 2 and 3 must be used to determine the admissible ratios AO/Oma x and Aa/Oma x for given values of B' and B". The subsequent approximation of W(x) must be carried out with regard for the constraints on the interval between successive k i and a i. The methods of determination of the set of these coefficients and the methods of approximation of W(x) will not be described here, because they are to be the subject of an independent investigation.

442

LITERATURE CITED io 2.

3. 4.

M. M. Baboglo et al., Inventor's Certificate No. 641564; Byull. Izobr., No. 1 (1979). A. N. Zhovinskii and V. N. Zhovinskii, in: Proc. Sixth All-Union Symp. Methods of Representation and Instrumental Analysis of Stochastic Processes and Fields [in Russian], Erevan (1973). V. A. Demkin et al., Metrologiya, No. 3 (1975). G. D. Sverdlichenko, Tr. Metrolog. Inst. SSSR, No. 154, 214 (1976).

APPROXIMATION OF THE ERROR DISTRIBUTIONS OF INSTRUMENTS E. I. Toneva

UDC 53.088:519.2

The introduction of probability and statistical characteristics into standards [i, 2] and the methodological support of these control documents continue to be a topic of debate [3]. Still-unsolved problems arise in connection with estimating the errors of measurements. Some of these problems are related to the determination of theoretical laws approximating the error distributions of measuring instruments. The solution of this problem is characterized by considerable difficulty, uncertainty, and ambiguity. Present-day experimentalists, despite their superior training over their counterparts of the 1960s, still feel intimidated by mathematical statistics, as has been remarked earlier [4]. One reason for this attitude lies in the fact that different theoretical methods often yield ambiguous solutions. This situation is attributable to the tendency to seek new mathematical methods divorced from the problems of their validity in comparison with other known methods. An example may be cited in [5], where distribution functions are obtained, approximating the specific histograms of errors of primary electrochemical transducers, with the application of four known methods. We note that nonuniformity of instrument errors has recently been observed [6]. Consequently, the approximation discussed below for histograms, unless further analyzed, cannot be extended to the errors of all instruments of the given class. The fundamental conclusion of [5], that the form of the approximating function depends on the method used, perplexes experimentalists. So seldom do approximating functions differ from one another (for the same set of data, bimodal and normal), that it is only natural to view the methods with suspicion or to doubt the correctness of their application. A method based on estimates of the antikurtosis • and the entropy coefficient k [7] is inconsistent with the method of moments, which is accepted as classical. Our present objective is to call attention to the ambiguity of statistical decision and to indicate some of its causes. In connection with the method based on estimates of • and k (cal2ed the information method in [8]), one must not overlook the inevitable scatter of these estimates. The scatter of the estimates of • and k depends on the number of results and on the experimental dataprocessing technique. Also, the number of clustering intervals of the results, the method of reading, etc., are of considerable significance [9]. In Figs. 1 and 2 confidence intervals are plotted for ~ and k for initial data having a normal distribution and for a confidence coefficient Pc =- 0.95. As in [5], Figs. 1 and 2 show the points of estimation obtained for and k from different histograms. The volume of experimental data is taken into account here. The results indicate that for a number of histograms the approximating function changes. The hypothesis of a normal distribution function is not refuted for the distributions of histograms 2, 6, 7, i0, and ii. For the distributions of histograms 7, i0, and ii there is agreement with the results of the method of moments, for the distribution of histogram 2 with the meahod of modeling, and for the distribution of histogram 6 with the method of generalized histograms. Variations in the other distributions can be expected if estimates of the entropy coefficient are once again made with the selection of m = 5 log n-- 5 intervals for clustering of the results (i.e., the nearest odd number of such intervals [8]). The resulting three or five intervals are much fewer in number than the nine intervals used in the given article, so that the bias of the esiimates of k is diminished, while the estimates themselves increase [i0]. People's Republic of Bulgaria. 16, June, 1981.

Translated from Izmeritel'naya Tekhnika 9 No, 6, pp. 15-

0543-1972/81.2406- 0443507.50

9 1981 Plenum Publishing Corporation

443