IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 24, NO. 2, JUNE 1975

served in the case of pure gases, is only approximate.Due to the size of the effect in pure gases it would notbe possible to observe nonlinearities; but in the case ofmixtures where the temperature coefficient is verysmall, these nonlinearities are then observed and secondorder terms in the temperature dependence of the fre-quency could become apparent.

CONCLUSIONIn this paper, we have described a simple method for

making Rb87 cells containing mixture of buffer gasesand we have given data on the pressure and tempera-ture coefficients of three commonly used buffer gases. Itis believed that the numbers given are more accuratethan those already published. These numbers couldthus be used with some confidence in the constructionRb87 cells with predetermined characteristics.

ACKNOWLEDGMENTThe authors would like to thank Dr. M. Tetu for his

comments on the text.

REFERENCES

[11 R. J. Carpenter, E. G. Beaty, P. L. Bender, S. Saito and R. 0.Stone, "Prototype rubidium vapor frequency standard," IRETrans. Instrum., vol. I-9, pp. 132-135, Sept. 1960.

[2] J. M. Andres, D. J. Farmer, and G. T. Inouye, "Design studies fora rubidium gas cell frequency standard," IRE Trans. Mil. Elec-tron., vol. MIL-3, pp. 178-183, Oct. 1959.

3 R. H. Docke, Phys. Rev., vol. 89, p. 472, 1953.4 H. G. Dehmelt, Phys. Rev., vol. 105, p. 1487, 1957.5 Proc. IEEE, (Special Issue on Time and Frequency), vol. 60, May

1972.[61 P. L. Bender, E. C. Beaty and A. R. Chi, Physical Review Let-

ter vol., p. 311, 1958.7I M. Arditi and T. R. Carver, Phys. Rev., vol. 124, p. 800, 1961.8 P. Davidovits and R. Novick, "The Optically pumped rubidium

maser," Proc. IEEE (Special Issue of Frequency Stability), vol.54, pp. 155-170, Feb. 1966.

[91 J. P. Gourber, Ann. Radio-Elec., vol. 20, p. 191, 1965.[10] M. E. Frerking and D. E. Johnson, in Proc. 26th Annu. Symp.

Frequency Control, p. 216, 1972 (available from Electronic In-dustries Association, 2001 Eye St., North West Washington, D.C. 20006).

[11] P. Cerez, These de doctorat, Universite de Paris, Paris, France,no. CNRS AO 6534, (unpublished).

[121 J. P. Barrat and C. Cohen-Tannoudji, J. Phys. Radium, vol. 22,p. 329, 1961.

[13] J. Vanier, Can. J. Phys., vol. 47, p. 1461, 1969.14] G. Busca, M. Tetu, and J. Vanier, Can. J. Phys., vol. 51, p. 1379,

1973.

Statistical Error of Suboptimum Spectrum Analysis: Review

of- Estimates

BRANKO BAJIC

Abstract-Statistical error estimates of spectrum analysis aregiven for any number of degrees of freedom and any relativepassband width for continuous line and mixed spectra. The con-tinuous component is assumed to be normal but not necessarilywhite.

INTRODUCTION

A BASIC PROBLEM in spectrum analysis is thelachievement of sufficient resolution and satisfacto-

ry accuracy of the results. Resolution is determined bythe analyzer parameters or calculation algorithm andthe capacity of the digital processor memory; a highenough accuracy in specifying the frequency is easilyachievable. What remains is the accuracy of the esti-mate of the power spectrum density, this quantity beingthe proper spectrum description if random signals areconsidered.

Manuscript received October 9, 1974. This work was supported bythe Fund for Scientific Work of SRH, the Brodarski Institut, Zagreb,and the Electrotechnical Institute "Rade Koncar," Zagreb.The author is with the Brodarski Institut, Zagreb, Yugoslavia.

Let Gx(fo) be the ideal value of the power spectrumdensity at frequency fo of the ergodic stationary randomsignal x (t), and let t.x (fo) be the estimate of this quanti-ty obtained by analysis of finite duration T and finitepassband width B (fo is the passband centrum frequen-cy). The results &x(fo) are distributed around (kx(fo)),this value in general being different from Gx(fo). (Here( ) denotes ensemble averaging). The root-mean-square error of the results of estimation of I:x(fo) isgiven [1] by

= (Es2 + (b2)1/where

(b = [(Gx(f0)) - Gx(fo)]/Gx(fo)

is the bias term caused by the finiteness of B and thespectrum nonlinearity, and

(S = ([GO((fo) -(GX(f"))]2)1121Gx(fo)

184

BAJIC: STATISTICAL ERROR ESTIMATES

is the so-called statistical error measuring random dis-persion of the results caused by the finite sample dura-tion. This latter quantity remains well defined also inthe case of deterministic signals.

In general, the problem of the statistical error estima-tion was solved by Davenport et al. [2]. However, withfinite B and T their result is effective only in the case ofa white spectrum (white within the passband), when

BT > 5

if a true integrator is used as the averager, and

185

To and T are the RC constant and the duration of mea-surement, respectively.)

In this paper, only results are given. The derivationscan be found in [5].

NORMAL NOISE CASELet c-, be the statistical error that would correspond

to given analyzer parameters in the white-noise case.Then, for normally distributed noise of any (continu-ous) spectrum, the following bound for the statisticalerror is valid:

2BT,, > 5 (2)

if an RC :ntegrator is used (ro is its RC constant).There is no simple result in the case of colored noise. Itis easy to see that the error estimate of the real mea-surement according to [2] should require data thatcould be gained only by an ideal measurement!The estimate valid for the case of white noise that

broadens validity of (2) to the case of any 2BTr, was de-veloped by Lovyagin [3]. However, his result is validonly for the long duration, narrow-band analysis, T >>TO, B << 2fo. Similarly, another result [4] could be usedto broaden the validity of (1), but again within theselimits.

In practice, cases quite often occur when the whitenoise assumption that transforms the results of [2] intoan effective form is not justified. This happens, e.g.,with analog analyses when the bandwidths are fixed.Further, in some applications with fixed bandwidthsthere are cases when these widths and central frequen-cies are comparable, so that it is uncertain whether highfrequency terms of fluctuations could be neglected as inthe case when B << fo. Similarly, there are many cases inpractice when in addition to the continuous spectrumcomponents there are also the discrete ones, one or sev-eral of them. Finally, apart from the case of high BT,those with BT < 1 are of interest as well.

All of the aforementioned cases, where difficultiesarise when statistical error is to be estimated, might becharacterized as suboptimum analyses. In this paper,statistical error estimates are given for the followingpractical and important cases of suboptimum analyses.

1) Effective estimate for colored noise which assumesminimum knowledge about the signal.

2) Estimate valid for any and specially for the highrelative bandwidth.

3) Estimate valid for small values of BT or 2BTO.4) Estimate valid in the case of mixed and line spec-

tra.These error sources are universal-they are present

in all methods of analysis. Therefore, the estimates areapplicable in the case of digital analysis too. The esti-mates are derived by use of the almost always justifiedassumption that the noise is normally distributed. Also,an ideal bandpass filter is assumed. In the case of an RCintegrator the stationary case is assumed To << T. (Here

(3)

(4)

e '< kes 0

k = max XG(f)IG/ ( passbanld

When noise is white, k = 1 holds, so that relation (3) be-comes an equality; therefore, it is clear that this is aneconomical bound. There are no signal parameters nec-essary for the white-noise error Es, estimation; it is onlythe factor k that describes a signal. This estimate is thesolution of problem 1).

In the following, the dependence of the normal white-noise statistical error on the analyzer parameters is ex-amined. If the notation

K = 2f,,Band

{BT,17 =

for true integratorfor RC integrator

are used, K being equal to the reciprocal value of the rel-ative pass bandwidth, the white-noise statistical error

(I ( =__ (PI + p/)112is given by the following:

PI= I + 2Fj17) + F-(2rO

PI,=I [(K + 1)Fj((K + 1n) + k K - 1)F1((K - 11)

- 2KF1(K?7)f + 22 22[F(27r(K + 1)71)

+ F,(2r(K - 1)17) - 2F,(X2WrK7)if the true integrator is used as an everager, and

9 1P, = - arctan 7n - 2, In (1 + 2q72)

P = (K + 1) arctan1 + 1771 + r2~~K(K + 1q)17

- (K -1) arctan 1 + 7K7 -1)17]1 ( 1 + T K- 1)-71In7r q [I1 + T2(k -1)27-2][ + 2( + 1--

if the RC integrator is used. Here

F,(x) = si(27rix) - sin2 TX7FX

FJx) -= dz = ci(x) - Inx - Cz

(SO I'll, ll(BT )1/2,

CIO I-, 11(2BTo)'I',

IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, JUNE 1975

eso

as

BT

0Q01 a1 10

Fig. 1. Statistical error for normal noise with a white spectrurin the passband, with a true integrator at the analyzer outpiusual approximation is drawn too. In the colored spectrum cerror is higher at most k times, this factor being given by (4).

1.5 -.--r A,\ s \\1.0

eo

IE.

0.5

2 BT,

aD1 01 1 l 1008

Fig. 2. Statistical error for normal noise with a white spectrum with-in the passband, with an RC integrator at the analyzer output. Theusual approximation is drawn too. In the colored spectrum case theerror is higher at most k times, this factor being given by (4).

C = 0.57721 ... is Euler's constant, and

si(x) Xin z dz

andA- COS Zci(x) = f cozdz

are integral-sine and integral-cosine, respectively.Pl and Ph are low- and high-frequency components of

the fluctuations. As can be observed, the statisticalerror depends on two parameters, the number of de-grees of freedom BT or 2Bro and the relative passbandwidth. If the relative width is small (K >> 1), Ph << Pholds, so that the error is completely determined by thelow-frequency components and it is only number of de-grees of freedom that remains. In this approximation,the RC integrator result becomes that obtained by Lov-yagin [3].The obtained results are represented graphically in

Figs. 1 and 2. The approximations given by (1) and (2)are also drawn.The given formulas are not convenient for the direct

use, so that simpler estimates and approximations areof interest. In addition to known estimates (1) and (2),

100

n with- Fig. 3. Statistical error bounds for normal noise with a white spec-ut. The trum within the passband: 1 equations (10) and (11), 2 and 3-*ase the equation (12), 4-equations (8) and (9). In the colored spectrum

case the values should be increased by the factor k given by (4).With true or RC integrator the variables BT or 2Bro should beused, respectively. The bounds are valid as follows: 1 with a trueor RC integrator, without restrictions, 2-with a true integrator,without restrictions, 3 with an RC integrator, without restrictions,4 with a true or RC integrator, with T > 1/(2fo) or ro > 1/(4fo). Inthe case with an RC integrator, T >> ro is assumed.

that are valid only in the small error case (see Figs. 1and 2), the following estimates hold:

0 < i/2, (true and RC integrator, withoutrestrictions) (5)

'o <`~ [min 1, 1/(BT)I + min 1, [r(2f0/B - 1)BTY -2f/2,(true integrator, without restrictions) (6)

E o < [min 1l, 1/(2B To)t + 11 + [r(2fo/B - 1)(2BTo)]2r11/2,(RC integrator, without restrictions) (7)

$O< min {1,1/(BT)1,

$.O< min 11,1/(2BTo)1/2t,

f,o < min W2,1/(BT)1/2,restriction)

(so) < min [F2, 1/(2BTo)112t,restriction)

(s$)< (s(I2fo/B = 1,without restrictior

(true integrator, T > 1/(2fo))(8)

(RC integrator, To > 1/(4fo))(9)

(true integrator, without(10)

(RC integrator, without(11)

(true and RC integrator,(12)

The last estimate could be used by means of Figs. 1 and2. Some bounds are shown in Fig. 3. The rigorous esti-mates and given approximations and bounds solve theproblems 2) and 3).

MIXED AND LINE SPECTRATo solve problem 4), the following model of the signal

is assumed-in the passband there is an additive mix-ture of white or colored normal noise with a power spec-trum density Gn(f) and m lines with amplitudes aj (j =1,2, . . , m) and frequencies fj (j = 1,2, . . ., m). There isno simple result similar to that for a pure continuousspectrum, where k given by (4) was the only data on thesignal. To estimate the error more detailed knowledge

186

I

:5e

\, fIE

BAJI(: STATISTICAL ERROR ESTIMATES

on the spectrum is needed. In addition to the factor

kn = max JGn(fVG,(f0)jf- p ssb nl

analogous to k, quite a lot has to be known about thediscrete component. In general, frequencies and relativeamplitudes must be known. Only if high-frequencycomponents can be neglected due to the action of theintegrator, are less data needed-relative amplitudesand differences of line frequencies should be known.A general result for the mixed spectrum will not be

given here. Only some estimates suitable for practicaluse will be noted. With

m

A = L(a2/2)/[Gn (fo)B]j=l

equal to the ratio of powers of discrete and continuousspectrum components within the passband, and bjl = 1or bji = 0 if j = 1 or j # 1, respectively, the following isvalid for statistical error-the relative rms value of thefluctuations of the result:

I + A L2kn2 1 + 6m- mm2 l, 26f 2T2J

+1I+X) min{1,BT

+ I + + 2k 2) min {1, r2(9f -2B)2T2}](true integrator, mixed spectrum, withoutrestriction) (13)

< kl, rXA2I -n1 11 + XL2kn, 1 +am 1 + 4w2bf2Tro+ (I + )m {1, 2BT }

+ (1 + k + 2k 21 + 4w72(2f(, - B)2T2o]r(RC integrator, mixed spectrum, without

restriction). (14)

The important quantity

bf = minji= 1 ..... m

is the minimum distance between neighboring lines.Mainly it determines the strength of a part of the fluc-tuations caused by the discrete spectrum components.Nearby lines interfere with long beating periods, so thata very long integration time is necessary for fluctuationsto be supressed.From (13) and (14) the universal bound follows:

t < 1 + [2k1 + 1 ia) + k + 2

(true and RC integrators, mixed spectrum withoutrestriction). (15)

In the limit 0 (k =- k), equations (13) and (14)

yield pure continuous spectrum estimates (6) and (7)with (3), and (15) transforms into (5) with (3). In the

opposite case of a pure line spectrum X c, equations(13)-(15) yield the following:

L, 1+ < mir {l w1, f2T2I

+ min {1, w2(2f -B)2T2 }jl2(true integrator, line spectrum, without restriction)

' L21 + a 2 1 + 47r2bf2To21 1 ]1/22 1 + 4wr2(2f0 - B)2T02J

(RC integrator, line spectrum, without restriction)

11r2, for one line< }F3/2, for two lines

L1, for more than two lines, (true andRC integrators, line spectrum, without restriction).

In essence, this last estimate is analogous to (5).For the important case of small fluctuations, it fol-

lows that:

<I

xI/f{1,S J276fT U,for two linesfor more than two lines,

(true integrator, line spectrum, with two or morelines, 2f»>> B, r2lbf2T » 1) (16)

e, < 1 X 1/1/2, for two linesv1/97r(2b ) 1i for more than two lines.(RC integrator, line spectrum with two or morelines, 2f )>> B, 4w T0f2T09>»1) (17)

s < 1/(2/-27rf0T), (true integrator, line spectrumwith one line, 2fo >> B)

S < 1/(4ir27rf To), (RC integrator, line spectrumwith one line, 2/,, >> B).

In this last case, the error is small; pure sinusoidaldeterministic component could be accurately measuredalready with low time constants. If in addition to thediscrete components there is also the continuous oneand the passband is narrow (2fo >> B), then the latterone will nearly always be dominant.

In the case of a spectrum with two or more close lines,the discrete component contribution might be dominanteven if the continuous one is strong! It is best to com-pare the errors due to small distances between the linesand those due to the continuous component by means of(1) and (2) with (3):

< k/(BT)"'",(true integrator, continuous spectrum, BT > 5)

<S< k/(2BTo)l'2,(RC integrator, continuous spectrum, BT0 > 5)

and the analogous estimates (16) and (17) for the linespectrum. All these estimates are valid in the case of ac-

187

IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 24, NO. 2, JUNE 1975

curate analysis, i.e., when the errors are small. In orderthat the errors may be equal, the following should hold:

bf/B = 1/[rk(2BT)1'2], for true integrator

bf/B = 1/[Lrk(4BT,)121, for RC integrator.

(We omitted the factor 1/V/2. More than two lines areassumed.) High-quality analysis is characterized by BT>> 1 od 2Bro >> 1, so that it can be seen that the discretecomponents cause an error equal to that due to randomfluctuations in the case of the continuous spectrumanalysis, only if the distances between the lines are verysmall. The problem of the accuracy of the line spectrumanalysis will almost never be as difficult as that with thecontinuous spectrum.

ACKNOWLEDGMENTThe author is indebted to Dr. M. Konrad of Institute

"Rudjer Boskovic," Zagreb, Yugoslavia, for critical re-marks and suggestions.

REFERENCES[1] J. S. Bendat and A. G. Piersol, Random Data: Analysis and Mea-

surement Procedures. New York: Wiley-Interscience, 1971, pp.76, 171, 184, 188, 271, 278.

[2] W. B. Davenport, Jr., R. A. Johnson, and D. Middleton, J. Appl.Phys., vol. 23, pp. 377-388, 1952.

[3] A. S. Lovyagin, "K voprosu o tochnosti spektral'nogo analiza slu-chaynykh prosessov (On the problem of accuracy of the spectrumanalysis of random processes)," Akustika i ul'trazvukovaya tekh-nika (Acoustics and Ultrasonic Engineering, Kiev, USSR: Tekh-nika, 1970, pp. 61-63).

[4] W. A. Johnson, IEEE Trans. Inform. Theory, vol. IT-19, pp.115-117, (1973).

[5] B. Bajic, Elektrotehnika, vol. 17, pp. 94-102, 1974.

A New Method for the Measurement and Integration of Arc

Durations in Electrical Contacts

M. ABDUL KADER PRAMANIK, TASUKU TAKAGI, AND KUNIO MANO

Abstract-A unique method for the continuous measurementof electric-arc duration is presented. By using the method, it ispossible to know the distribution of arc durations at any in-stance of time and, at the same time, the integration of arc dura-tions has become much more convenient and less time consumingthan any other existing methods. It has also been found that thetotal integrated-arc durations bear a linear relationship withthe number of operations on a log-log scale.

I. INTRODUCTION

T IS WELL KNOWN that the electric arc

is one of the main agents to cause wear, transfer, andwelding which may lead a contact to failure. When an

arc occurs, the arc power dissipates on the contact sur-

face resulting in oxidation and evaporation of the con-

tact material at the surface. The total energy Et, themost important factor which cause damage, can be de-fined mathematically as

r3

Et = E ValIata J

where Va is the arc voltage, Ia is the arc current, t0 isthe arc duration, and n is an integer.

Manuscript received August 6, 1974.M. A. K. Pramanik and T. Takagi are with the Faculty of Engineer-

ing, Tohoku University, Sendai, Japan.K. Mano is with the Faculty of Science and Technology, Meijo Uni-

versity, Nagoya, Japan.

Considering Va and Ia as constants, it can be seenfrom (1) that in order to know the total energy over nnumber of operations, at first, it is necessary to knowthe total arc durations.

There are various methods for measuring arc dura-tions. They can be broadly divided into analog and digi-tal methods.

A. Analog MethodThe measurement of showering arc or high-frequency

arc has become possible with the development of oscil-loscope having wider bandwidth [2], [3].The measurement of arc duration by using a CRO

have been performed for a long time [1] there remains,however, several major problems in this measuring pro-cedure.1) For wide range of measurements, the setup of therange of the CRO is a major problem, because, long andshort arcs occur at random during the operations of anelectric contact.2) In order to take an accurate reading, it is necessaryto photograph and then read the results which consumemuch time.3) For accumulation or integration of arc durationsover certain number of operations, this method is tootedious and laborious.Another form of analog method is the condenser-

188