A METHOD

FOR C O R R E C T I N G

IN MEASUREMENT

SIGNALS

DISTORTED

CIRCUITS

(UDC 62-301.2.681.142.353.2) K. Y a .

Shvetsov

Translated from I z m e r i t e l ' n a y a Tekhnika, No. 3, pp. 29-34, March, 1966

It is often necessary in e x p e r i m e n t a l practice to measure signals* with very small amplitudes by means of amplifying stages which have a combined gain expressed in four and five digit numbers. Each amplifying channel as w e l l as the receiving and recording elements have their own frequency characteristics which distort the original signal in some part of the spectrum. Therefore, the representation of the signal thus obtained does not reflect the true nature of the phenomenon at the input of the receiving element. The reproduction of the true spectral characteristics of stationary random processes or regular signals, which can be defined by corresponding spectral densities or spectra, can be m a d e in the course of their recording by a w e l l known method. This method consists in multiplying the spectral density of the signal by the squared modulus of the inverse transfer function of the appropriate amplifying stage. However, the experimenter often does not require the signal probability characteristics only, or even as much as a true representation of the receiving e l e m e n t ' s input signal. The reproduction of the signal, and not of the prob a b i l i t y characteristics, becomes, in the case of a transient input process, a most important procedure, and even the only possible one. The necessity of obtaining a true input signal is indispensible, for instance in certain types of analyses when the presence or absence of the useful signal is determined precisely by the shape or extent of its transient characteristic. Below we discuss one of the methods for reproducing the receiving e l e m e n t ' s input signal. This method was applied by the author of this article in investigating low-frequency e l e c t r o m a g n e t i c signals at a very low power level. It should be noted that we are not dealing with the reproduction of information which has been lost in the internal noise of the transducer and the f i a t amplifier stage. The level of the components in the basic information part of the spectrum should be high enough as compared with the measuring-channel nois'e in the corresponding parts of the spectrum to allow us to ignore this noise without a substantial loss of information, in a general case this noise will be reproduced in the above method at the output of the measuring channel side by side with the input signal. The essence of the method consists of an inversion of the recorder signal by means of a continuous system whose transfer function is inversed as compared with that of the measuring channel, or by means of a discrete system whose weighted function is inversed with respect to that of the measuring channel. An inversed system for a continuous conversion of a low-frequency signal can be obtained by means of an electronic model, and for a discrete conversion by means of a digital computer. Certain additior~al specific features involved in continuous signal conversion by means of a m o d e l should be mentioned. The m o d e l should be fed with a physical signal in the form of a voltage and, therefore, the signal should be fed in this form or reproduced from recordings; e x t r e m e l y l o w - l e v e l signals can be recorded at the output of the m o d e l only after considerable undistorted amplification which, it would appear, should be made at the end of each conversion stage, so that the signal should exceed considerably the instability level of the model; this a m p l i fication should be eventually accounted for in evaluating the actual signal level. *In this a r t i c l e the term "signal" is understood for the sake of convenience to include everything fed to the input of the measuring channel.

334

, -[__[_d

, -L_J._J

, -I

Reproductis I o f t h e proc~?s

~

Fig. 1

2OO0

12

!600

8

1200

4

a00

0

qO~

0 fHz

zo Fig. 3

~0

All the a b o v e - m e n t i o n e d limitations and difficulties of continuous conversion obliged us to adopt the second method for producing an inversion system, namely, the method entailing a digital computer. A digital computer's c a p a c i t y for recording any signal level depends only on its digital network, whose range in modern computers extends from I0-19-I019, which c o m p l e t e l y covers any possible cases of its application. Moreover, a computer used as a digital converter has a constant a m p l i f i c a t i o n in any frequency range and, therefore, its digital network can be used most effectively for raising within possible limits the level of the original signal.

/gOf Hz Fig. 2

ii c

Fig. 4

The loss of a certain amount of information which occurs at higher frequencies and is due to discrete conversion is discussed at the end of this article.

Let us now deal directly with the composition of the measuring channel which was used in our experiment, with the recording of the signal at tile output of this channel and with the algorithm for transforming the input signal into an information converter. It should be noted that the signal values are given in terms of voltage m a g n i tudes without reconverting them into physical quantities of an e l e c t r o m a g n e t i c field. Side by side with the discussion of a specific system we shall also provide more generalized formulas which can serve to apply this method in any other linear measuring system. The block s c h e m a t i c of the e x p e r i m e n t a l installation is shown in Fig. 1. The units of this installation consist of linear devices whose transfer functions can be represented in a general case by the formula fn

, ~ bi P~ P (p)

(m < n)

-

n

(1)

~ a]p] ]=0 However, in a very large number of cases as, for instance, in the case of our system, the transfer functions of linear links can be represented by a very simple expression:

avp F (p) - -

T2p 2 + (2--~)Tp+ 1"

(2)

335

This expression corresponds to second-order linear links which have a resonant amplitude-frequency characteristic, i.e., to various types of passband amplifiers. By adjusting parameters A. T and B in expression (2) it becomes possible to select from experimental amplitude-frequency characteristics an amptifier m e a n frequency equal to r = 1/T, and then to set the slope of the ascending characteristic from the value of ~ and, finally, to determine the m a x i m u m gain for the m e a n fiiter frequency according to the formuIa

s,? 0.8

40

A = (2 - - ~) r.ma x(3)

OI

20

40

Go

|,

80 [, .7.

Fig. 8

If a wideband amplifier and a preamplifier are required in a given circuit, the values of the a b o v e - m e n t i o n e d factors c a l c u l a t e d from the amplirude-frequency characteristics (Figs. 2 and 3)should amomlt to Aa =- 6200; T l = 0 . 0 1 6 sec ;

~l-~-- 1.1;

A t -----42; T l = 0,00532:sec ;

I ~ l = - - 1.

The transfer-functfon parameters of an inductive transducer were determined theoretically. The computations were checked by comparing the generalized t h e o r e t i c a l and experimental amplitude-frequency characteristics of the converter and the preamplifier. The converter was represented by an equivalent circuit shown in Fig. 4, where L and C are its actual parameters, and R is the input resistance of the preamplifier. In such a circuit the complex voltage of the equivalent generator can be represented as

(4)

the output v o k a g e as

Uout = u,~ =

R/,

(s)

V out

so[~ D :

V''V v ,

/X .z,'a

~" ::A,,

V:' A^ ,d',- A , m. z!

Fig. 6

336

2., .

(lf 10j

IlI4U

V Fig. 7

s(f) ~l# 102

0.10.I0~

6

If

I#

2@

7#

##

#2

CY

f4

f0 f, Hz

Fig. 8 and the transfer voltage function as Y (p) == U ~

ui n

--

RCp

CCp: + Rcp + I

(6)

Expression (6) shows that the converter's transfer function fully meets (2) if it is assumed that r 3 = }'/LC;

(2--~)

T:~ = RC;

A3==2--,~3.

(7)

For the converter components values of L = 170 H, C= 3/IF and R= 40 k~ the transfer-function parameters become Ta=0,0227see;

A ~ = 1,7;

}a=0,8

(s) The amplitude-frequency characteristic of a linear circuit represented by (6) for parameters given in (8) is shown in Fig. 5.

33"/

Vrcst

g ~" ~"~

UV" v

lwV -V -u V~ V'"

V "~ w

*~oc

Fig. 9

;9/05

Fig. I0 It should be noted that a comparison of the converter's calculated and e x p e r i m e n t a l amplitude-frequency characteristic adequately confirms the computation results. A study of the frequency characteristic of an N-700 loop oscillograph used in the experimental installation has shown that the characteristic is constant over a sufficiently wide frequency range and, therefore, its effect can be neglected. Thus, in our case the inversed conversion of information is reduced to setting on an electronic model three transfer functions which are inversed with respect to (2), or providing for a digital computer algorithms which correspond to these functions. The reproduction of transfer functions of the type tt

aj pJ 1

H(p) = e(~)- --

]=o

~m b~ pz

(n>m),

(9)

does not present any difficulties, since it simply entails the simulation of a linear differential equation of the order m with its right-hand side of the order n. The only difficulty arises when n is of a high order and it is necessary to use a large number of differentiating links for reproducing the right-hand side of'the equation. This difficulty, however, can be overcome if expression H(p) is represented as a summation of elementary links of the firstand second order (aperiodic and oscillatory). 338

sO 1.040

~r 6

t?

tg

2~

70

YB

Of

ag

5o

80 f, Hz

Fig. II In our case the transfer function for inverse conversion is obtained in the form of three series-connected circuits of the form H(p) = ~

,

Tp @ (2 - - [~) @

(lO) which can obviously be easily simulated by a model. In the case of inversion by means of a digital computer, the discrete weighted functions are found from the continuous functions which in turn are evaluated from the expression

'I

~D

W (t - - z) = - t t ( p ) e p(t--z) dp. 2hi ,

(11)

--CO

Taking into consideration that, as a rule, the numerator in expression H(p) has a higher power than the d e n o m i nator, since in normal filters and amplifiers n > m, it is advisable in determining w ( t - r ) from (11) first to divide the numerator of H(p) by its denominator, and to separate from the weighted function its components with 6-functions and their derivatives. The weighted function in the remaining part of H(p) can be evaluated from. expression [1]: n

s 8

(12)

te~l

Another method for providing inversion on a digital computer consists of the solution of difference equations of the type of [2]: b m A 'n x (iT) ~- b,n_ ~ A rn-I x (iT) + , . .

+ box (iT)

== a n A n y (iT) -~ a n _ l A n - l y (iT) + . . . - t - ao Y ( i T ) ,

(13)

where AJx(iT) are the j - o r d e r amplitude differences of the reproduced signal at the input of the measuring channel or the amplifier stage; Aiy(iT) are the i-order amplitude differences of the recorded signal; aj and b i are the coefficients of the transfer-function numerator and denominator terms. In order to use directly the amplitudes instead of their differences, expression (13) can be written in the form B m x (iT--roT)

q- B i n _ t x ( i T - - m T @ T ) + . . . q -

B o x (iT) : An y ( i T - - nT)

+A~_~y (ir--nT + T ) + . . . + A oy(iT). (14)

339

The coefficients in (14) are evaluated from those in (13) according to the relationships

&% 18

n

14 10

fn

= (-

(15) 2 10

30

50

70

Fig. 12

The algorithms for the solution of difference eqi~ations can be compiled according to the well-known methods of Runge-Kutt or Adams. In our case the general output to input conversion algorithm comprises three particular conversion algorithms which correspond to transfer function (10) and differ only by their parameters.

Each of these algorithms can be written in the form of the following integral-differential equation t

x (0 = 7

' I.r'dY'"dt

'i"tov ( 0 dt +

+. -~-

]

(2 - 8) v (t) ,

(16)

which corresponds to the discrete equation of the form n-I x (n Tn) = ~

[y (n T n + Tn) - - Y (n Tn ~ Tn)l

(17) i=0

Algorithm (17) was used for programming a "Ural-2" digital computer. The graph of the signal subject to reconstitution is shown in Fig. 6; the correlation function and the spectral density computed from this graph are shown in Figs. 7 and 8, respectively. A threefold application of algorithm (17) with parameters corresponding to the appropriate amplifying stages produced a new graph shown in Fig. 9. The correlation function and the spectral density calculated from the reconstituted signal graph are shown in Figs. 10 and 11, respectively. A comparison of the correlation functions and spectral densities of the initial and reconstituted signals confirm the advisability of using the above technique for reconstituting the input signal. The absolute value of the signal thus obtained corresponds to the actual quantitative values of the input signal. The main spectral-density peak is substantially reduced as compared with the general background noise, which corresponds to the theoretical premises. The hardly noticeable high-frequency spectral density peaks (in the range of 35 and 50 Hz) are increased appreciably, thus indicating the relatively large part played by them in the formation of the input signal, which should obviously-be taken into account in designing equipment for operation in the above range. The relatively smatl rise in the low-frequency range of the reconstituted spectral density is probably due to the insufficiently prolonged recording of the original signal and, therefore, the impossibility of analyzing it in the low-frequency range. In conclusion it is advisable to mention the loss of information produced by the discrete reconstitution of the input signal and by its inaccurate recording. In order to determine the quantization error in a signal passing through a line filter, we used the simplest case of half a sinusoidal wave transmitted through a discrete filter of an aperiodic link type with a transfer function of k/(Tco + 1) with k = 1, T = 1 sec and a sinusoidal frequency of ws = 1/sec.

340

The discrete weighted function of this filter is

q (it.) = ~

k

e

Tn i r

(lS)

In our case for T = 1 sec and k = 1 it assumes the form q (iTn) = e - rni

(19)

The m i n i m u m quantization period was taken as T n = 0.0785 sec, which corresponds to 80 signal counts over a sinusoidal period, with a l l the remaining periods assumed to be multiples of the m i n i m u m period. The error was evaluated by changes in the signal ordinate which correspond to the t i m e of half a sinusoidal period transmitted through a filter with different values of T n as compared with an ordinate obtained by continuous filtrations. Thus, the r e l a t i v e input signal error due to discrete measurements was evaluated instead of the absolute filtration error produced by the transient process. The value of the relative error as a function of the number of measured input-signal ordinates over a sinusoidal period is shown in Fig. 12. The shape of the curve indicates that for a number of readings n = 8-10. The error does not exceed 0.1 of the absolute signal value. By assuming this number of discrete measurements to be sufficient, we can determine the quantization interval of the signal subject to reeonstitution according to the above technique. This interval is equal to 0.1-0.12 periods of the highest-frequency spectral component which can still be distinguished in the recorded signal. In addition to the above considerations about the selection of a measuring interval, it is also necessary to take into account the steepness of the frequency characteristic slopes in the measuring-channel units at higher frequencies. Particular attention should be paid to the selection of measuring intervals in the case of steep slopes in the c h a r a c teristics and large filter or amplifying stage gains at their resonant frequencies, since in such a case high-frequency components which have s m a l l amplitudes on the oscillogram can grow in the course of reconstitution into very substantial signal components. It is characteristic that the above technique is more effective with respect to the Iowfrequency signal components whose reconstitution accuracy increases with a reduction of the frequency. The information loss in recording the output signal cannot be recovered. In order to reduce these losses it is necessary to reduce the relative width of the loop-oscillograph b e a m and increase the sharpness of signal recording, The above technique is not suitable for the reeonstitution of a signal which is transmitted through nonlinear elements and, therefore, the nonlinear amplitude distortions which arise owing to an insufficiently wide dynamic range of the measuring channel must also be considered as irretrievable losses of information. LITERATURE

1. 2.

CITED

V. S. Pugachev, Theory of Random Functions [in Russian], Fizmatgiz (State Press for Physical and M a t h e m a t i c a l Literature), Moscow (1962). L. T. Kuzin, Computation and Design of Discrete Control Systems [in Russian], Mashgiz, Moscow (1962).

341