IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. IM-27, NO. 3, SEPTEMBER 1978

Improvements of the Procedures Used to Studythe Fluctuations of Oscillators

ETIENNE BOILEAU

Abstract-Considering that procedures currently used to study thefluctuations in oscillators do not give directly useful results, we havetried to define a procedure more suitable for experimentalists. Wehave tempted to obtain a function a(T) which would be a kind ofvariance of the,frequency fluctuations observed in a duration T.Taking a frequency domain point of view, we find a procedure whichcan yield also an estimation of the spectral density of the frequency.

I. INTRODUCTIONIN EXPERIMENTS with an oscillator, it is often impor-

tant to have some statistical knowledge of the frequencyfluctuations occurring in a time interval of duration T.Because of slow frequency drifts, the amplitude of thesefluctuations increases with T. The variance (defined forT -+ co) even seems to be infinite, however, this last point isnot important here, since measurements are always per-formed in a finite time T. A single parameter, independent ofT is currently given, supposed to describe the "short-termstability" of an oscillator, but a "finite time variance" v(T)could have a more precise meaning, and be very useful.Several attempts have been made to give a good definition ofsuch a function.Two procedures associated with two expressions of the

true variance and yielding such functions have beenproposed at first, and in a previous paper [1], a comparisonof these two functions a 1(T) and A2(T) has been given. Thiscomparison was done from several points of view, but themost important one seems to be in frequency domain. Inboth cases, some kind of low-frequency cutoff is obtained,wrongly defined around 1/T. This observation led us to lookfor a procedure which would give, at best, a cutoff at v = 1/T.In other words, assuming that the frequency F of theoscillator under study has a spectral density YF(V), we arelooking for the procedure which would give the best ap-proximation of

00

a2(T) = 2 YF(v) dv. (1)1IT

Besides, such a procedure could be used to get an estimationof YF(V) by simple derivation of the function obtained bytaking the new variable f= 1/T (for v > fm = i/TM if TM isthe maximum value of T used).

In a first approach, we seeked the optimal procedure using

Manuscript received October 27, 1976.The author is with the Laboratoire des Signaux et Systemes, Ecole

Supeieure d'Electricite, Plateau du Moulon, 91190 GIF/Yvette, France.

a linear filter of the frequency F [2], and later, we obtainedanother improvement with a quadratic procedure [3]. Theaim of this paper is to present these results to the experimen-talists studying oscillators.We note that the Allan variance [4] yields another descrip-

tion of the fluctuations of an oscillator; YF(v) being wellapproximated by a sum Ei (Ai/ v j'x), if one ofthese terms ispredominant in some v interval, the Allan variance gives oti.However, the knowledge of these ai is clearly not as useful forthe practician as that of the curve of a finite time variance.

II. REVIEW OF THE FIRST PROCEDURESIt is assumed that sampled measurements ofthe frequency

Fn = F(to + nT1) are available, and a procedure yielding agood "finite time variance" is wanted. We have T = pT, and,except for the first case, p will be odd and we put p = 2N + 1.The result a priori will depend on both p and T1, and also onthe time constant - of the frequency measurements; indeedthe instantaneous frequency is not directly measurable, butonly its average on a certain time interval T. The dependenceupon T will disappear only if r is choosen short enough sothat fast fluctuations be not filtered off [1].

If the spectral density YF(V) of F(t) exists, the sampledprocess Fn has a periodic (period 1/T1) spectral density y(v)which is

1 I=- 1 (2)

If T1 is small enough, YF(V) is negligible outside the periodv < 1/2T1, and (2) becomes

y(v) T YF6V) (3)

Then (1) becomes

l/Tia2(T) 2T1 ( y(v)dv.

1/T

If T1 is not small enough, YF(V) has components for v >1/2T1 which should be estimated previously with analogicapparatus, in order to calculate their contribution to (1); theknowledge of (1) is then equivalent to that of ( f = 1/T):

1/2TiI(T) = J(f) = 2T1 f 7(v) dv. (4)

0018-9456/78/0900-0210$00.75 ©) 1978 IEEE

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BOILEAU: PROCEDURES USED TO STUDY FLUCTUATIONS OF OSCILLATORS

A. Two Procedures Built on the Two Expressionsof the Variance [1]For each set of p samples, the following quantities are

calculated:1 P

P1 =- E Fn2P n=l1

(P 21)I

(5)

(6)P2 F=(- 2N + I _E N )

With a certain number ofsuch sets, the following averagesare estimated

52= E(P1)

52= E(P2).

If yF(V) exists, we have. +0 G )

(Ti | / F(v) Gi(v) 12dv*- ,0

. 1!2T1

= T,!T

(7)

(8)

(9) 1

y(v)IGi(v)12 dv

sin p7rvT, 21G ,(v')12 = -_~p sin rvT1

IG2(V)12 = (I

(10)

(11)sin pivT1 2

p sin nvT1 /

These two functions are even and periodic (period l/T1)and are plotted on Fig. 1 for p = 31 (i.e., N = 15). We see

that they modulate > (v) in (9) quite differently, and are notgood approximations of the periodic function H(v) used in(4) and defined by

H(v)= lo, for lvi <fforf< IvI < 1/2T1.

(12)

These observations led us to look for an optimal approxi-mation of H(v).

B. First Optimization

We first tried to use a procedure using a filter whichgeneralizes (6); let us put

.'I 2

P3 = R,F-nn = -

732 E(P3)

(13)

(14)

- G11 2

--- IG210 -

1/(2 N +1) T1 1'/ T1Fig. 1. Frequency ponderation obtained with the procedures P1 and P2.

! v

I 'I._-i

i

| f =12 f.1.

f. 4f. 12 F.2T,

Fig. 2. Case of the procedure P3 obtained with the discontinuousoriginal.

As it is important to get rid of the singularity of ;(v) forv = 0, we impose the supplementary condition

G3(0)= 0. (17)

It is easily shown [2], [5], that the optimal R, are thenobtained from the coefficients cn of the complex Fourierseries development of H(v):

1 N

Rn= cn cnnn2N + In=n- Nco= 1 - 2fT1

c = - sin 2rnfTI, n 0.nx

(18)

(19)

(20)

With the same assumption as before, we have (9) withi = 3 and

N

G3(v)= E R,, exp (in2ivT1). (15)n =-N

We want that G3(v) 12 be the best approximation (for a

given N) of H(v), but this condition yields nonlinear equa-

tions; to simplify, we search the Rn such that G3(v) be thebest approximation of /(v) = H(v), in the mean-squaresense. In other words, we minimize

. 1/TI

D =

I

H(v) - G3(v) 12 dv. (16)' l-/Ti

One will use f= l/T = 1/(2N + 1)T1 to get the curve

a3(T). On the contrary, in order to obtain an estimation ofy(f) we must use different values off, but we can use thesame value of T (i.e., same sets of measurements). Somecurves of G3(v) 12 obtained with N = 15 (i.e., T = Cte) anddifferent values offare given on Fig. 2. We have put

1 1

T (2N +)T1(21)

It can be observed that the overshoot increases slightly asfis increased abovefo, but increases quickly asfis decreased

t forf<fo.

with

211

v

1

IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. IM-27, NO. 3, SEPTEMBER 1978

G30 2

F 4 F. j f=12F.I,-'

V

F. Af. 12 f. /2T,Fig. 3. Case of the procedure P3 obtained with a continuous original,

a =f0.

00

2

v

f.

Fig. 4. Case of

4 F. 1/2 T

procedures P3 obtained with different continuousoriginals.

Besides, for f- 0, Cn (n $ 0) -- 0, so that

P3 (oF- 1 ZF)f-0O 2Nn+n-N

Thus in the limit, P3 is the process P2, i.e., U2 is hereoptimal, if we want to get rid only of the component v = 0.On the other hand, P3 can be used for a given N with fvarying in (fo, 'T1). In this interval, 53(f) gives an estima-tion of (4), whence, by derivation, an estimation of 7(f ) canbe obtained.

III. SOME IMPROVEMENTS

A. Use of Continuous OriginalsWith a finite development such as (15), discontinuous

functions are not well approximated; this is known as theGibbs phenomenon. Better results are obtained with contin-uous originals (functions used instead of H(v) in formercalculations). f

We have first used the function Ho(v), even and periodic(period 1/T1) defined by Ho(v) = 0 for v < (0, f- a),Ho(v) = 1 for v c (f + a, 1/2T1) and by the linear functionwhich ensures continuity in (f- a,f+ a).The optimal coefficients Rn are again given by (18), with c0

given by (19), but, for n #& 0, the coefficients cn ofthe Fourierseries of Ho(v) are

(co).= _sin 2rnfT1 sin 2nnaT (22)~Co)fl- nir 2rrnaT122Corresponding curves, obtained with a = fo, are plotted

on Fig. 3, showing a real improvement with respect to thoseof Fig. 2. The definition of Ho(v) assumes f> a; we havenevertheless plotted the curve obtained with (22) andf =fo/2 = a/2; the result (dotted curve) is better than withH(v) but clearly worse than for f> a. If one wants to usef <fo, one should use smaller a, but, for a givenf, the over-shoot increases with decreasing a (for a -4 0, Ho(v) -4 H(v)).The choice a =fo seems good for f c (fo, 1/2T).The improvement obtained with Ho is associated with a' In fact we worked above on /(v) = H(v); in the following, originals

differ from their root-squares and we shall use functions Hj(v) which inthis 1' play the role of root squares of the originals.

steeper asymptotic decrease ofthe coefficients c,,, and we cango on by taking originals having as many continuousderivatives as wanted [3]. We can obtain such functions byan integration (and translation) of the functions yjx), equalzero outside (-a, a), even and such that

(23)| yj(x) dx = 1.*a

We use the index j to indicate the number of continuousderivatives that the corresponding original Hj(v) possesses.Thus the discontinuous function y0(x):

1/2a,Yo = O

for lxI <afor |xI >a

(24)

gives the original Ho(v) introduced above, which has nocontinuous derivative.The Fourier coefficients of Hj(v) are (n * 0)

(c ) = -sin 2nnf yj(x) exp (-i27rnxT,) dx. (25)

We choose for yl(x) the continuous triangular function(Bartlett window), which after integration gives the originalH1(v) and

( = sin 27rnfT1 (sin n7raT) (26)

For y2(x), we use here the following function (Hanningwindow)

Y2 = (1/2a)(1 - cos lrx/a),0,

for lxI <afor IxI >a.

(27)

It has one continuous derivative, is associated with theoriginal H2(v) and gives

sin 2nrnfT1 sin 2irnaT1G2)n = - nnr 2irnaT1 1-4a2T2n2 (28)

We call G3j(v) the best approximation of Hj(v), obtainedby the preceeding method. Some curves of I G34v)2 aregiven on Fig. 4. Withf= 4f0 we have taken a = 2fo. whichgives improvements easy to see from j = 0 to i = 2. Witha= f0, we observe that the advantage of large j is less

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BOILEAU: PROCEDURES USED TO STUDY FLUCTUATIONS OF OSCILLATORS

obvious; indeed, the overshoot increases more rapidly (as adecreases) with larger j.We can then say that the original Ho(v) is sufficient with

f= fo and a = fo. Nevertheless H l(v) and H2(v) can be usefulforf > fo and large N. Besides, they will be used thereafter.

Larger values ofj are easily obtained [3] but seem to havelittle practical interest.

B. Use of a Quadratic Process

Noting that P1 defined by (5), does not belong to the set ofprocesses defined by (13), among which we have looked foran optimum, we have tried to improve our results usingquadratic processes of the form

N

P4= E AijFiFj.i,j= -N

(29)

Assuming again that F has a spectral density, we have1/2 T1

E(FiFj) = T1 f 7(v) exp [i2it(i -j)T1] dv (30)-1/2T1

and we obtain

E(P4) = T1 1 2Ti y(v)Q(v) dv (31)1/2T1

withN

Q(v) = E Aij exp [i2ir(i-j)vT1]i,j= -N

2N

- E Bk exp (i2rkvT1) (32)k -2N

Bk= E Ai. (33)ij=k

The sum (32) is ofthe same form as the one which gives G3(v)in (15), but it is made of (4N + 1) terms whereas (15) hasonly (2N + 1) terms. This observation makes us hope forbetter results with an adequate choice ofthe Bk. However, inthe interval where the original is zero, the fact that Gi(v)is squared in (9) will be an advantage for processes of theform (13).

Let us notice that it is not necessary to keep (2N + 1)2terms in (29); indeed, P4 is determined by the sums A ij + A i(coefficients of the Fi Fj) and the value ofP4 is not changed ifwe take for instance

Aij=0, for i>j (34)

after replacing the Aij for i < j by the former value of(Aij + Aji).We want to choose the Bk so that Q(v) be the best

approximation of H(v) defined by (12); however, as before,we get better results with smoothed originals. We use theHj(v) considered above; noting Q,(v) the correspondingoptimal Q(v), it minimizes

1/2 T1

-1/2T1Hj(v) - Qj(v) I2 dv

and satisfies

Q.

v

12 f. 1'2T,F. 4 f.

Fig. 5. Case of procedure P4 obtained with a continuous original (QD isobtained with the discontinuous original), a = f0.

The same calculations as above give the coefficients ofthedevelopment of Qj(v) with those from Hj(v):

I 2N

(Bj)k = (Cj)k - 4N +1 2N Cjk (37)

As the originals Hj(v) are even we have (cj) - k = (Cj)k andfrom (37), (Bj)_k = (Bj)k. This is clearly incompatible with(34) which gives Bk = 0 for k > 0. However, if we use (34),we must notice that in (31), the odd part of Q(v) does not

bring any contribution; it is the even part of Q(v) whichmust be the best approximation of the original and put in(35). From (32) we see that the even part of Q(v) is also itsreal part, and with (34) we obtain

(Bj)k = 2 (cj)k - 4 k-2N ] k <0

1 2N

j= (c)o - 4N + 1 k 2N(Cj)k(Bj)k = 0,

(38)

k > 0.

It is easily verified that (37) and (38) give functions Q]jv)which have the same even part.The curves2 of the Fig. 5, compared with those of Fig. 3

show the improvement obtained with P4; the number ofsamples is the same (p = 31) as well as the selected values off(f= fo/2, fo, 4fo, and 12fo). With a smaller value of a(a = fo /2), we have a smaller overshoot in a broaderfrequency interval.Forf= 4fo the curve QD(V) has been added, obtained with

the discontinuous original H(v). It can then be comparedwith those of Fig. 2; the overshoot is smaller, but we seeoscillations forf < fo which are much higher. The advantageof P4 would then be doubtful with the discontinuousoriginal.We have shown on Fig. 6 the improvement obtained with

originals having one and two continuous derivatives (j = 1and j= 2). The effect is best seen with a = 2fo (used withf= 4fo) but exists also with a = fo, with still a very small

2 As well as for Fig. 6, we have plotted the even parts of the functionsQj(4Qi(0)= 0.

213

(a...v

cc

iII

I1'iiiI

ii

I.

IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. IM-27, NO. 3, SEPTEMBER 1978

o = 2 f.f =4 F.

-J=O-- j = 1

..j =- 20 O AXfO

F.

Fig. 6. Case of procedure P4 obtained with different c(originals.

1) If one wishes to simplify as much as possible thecalculation of P4, one takes a single Aij I 0 for eachi-j < 0, and, in order to have a factorization, we can take

P4 = F- N(BOF-N+ BF-N+1 + + B2NFN). (39)2) In order to get an estimation of E(P4) we must

calculate a certain number of samples and take their aver-age. If we can dispose of a number of Fn which allows astrong integration on P4, we thus obtain a good precision onE(P4). On the contrary, if, for instance, we have a given total

l/2T, number of Fn which does not allow strong integration, it is)ntinuous desirable to reduce the variance of P4 as much as possible;

we can then make another simple choice of the A ijby takingthem all equal for each value of i - j:

overshoot (unlike when the process P3 is used; see Fig. 4).Even with a =-o we have an overshoot very small anddecreasing with j (respectively, about 0.7 percent for Q0 and0.5 percent for Q1 and Q2). For a =f0 /2, the reverse is true(about 1 percent for Qo, 1.8 percent for Q1 and 2.8 percentfor Q2).A reasonable choice ofthe ratio a/fo consists in taking it as

small as possible compatible with an accepted value of theovershoot; we see then that this ratio can be taken smallerwhen j is smaller. This fact moderates the interest of large jvalues. With Qo and an accepted value of 1 percent for theovershoot, we then choose a =f0/2 and the procedure canbe used to get an estimation of y(v) for v c (1/2T, 1/2T1) asseen on Fig. 5.For v -+ 0, QJ{v) is of first order in v, whereas G3,V) 12 is of

second order and G2(v) 2 of fourth order; this could beconsidered as a drawback of our procedure [1]. This draw-back can be easily removed by adding supplementaryconditions: dQ3{O)/dv = 0, d2Q,{O)/d2v = 0, , to (36) (aswell as for G3(v)we can add dG3(0)/dv = 0, ). The solutionof the problem is well known [5], and gives similar curves asfar asf> a > fo [6]. However, these supplementary calcula-tions are necessary only if y(v) has a severe singularity forV =0.We have seen how to choose the coefficients Bk, but (33)

evidently does not determine the Aij from the Bk. One cantake advantage of this indetermination to minimize thevariance ofP4 [3], but it implies a preliminary determinationof the ]7k = E(F, F,-k) which is not very feasible here(because of the slow drifts). In practice the following choicesare sufficient:

A -j~: Bi-jj2N+ 1- li -jl-

(40)

The calculation of the samples of P4 will then cost moretime, but each P4 sample will already be a combination ofaverages on the terms Fn F, + i j. The variance ofP4 will thenbe smaller. (The choice (40) minimizes the variance of P4 ifF(t) has a correlation time short compared with T1 [3].)

IV. CONCLUSIONBy introducing a quadratic processing of the measure-

ments, we have defined a procedure which gives a moreaccurate description of the short-term fluctuations of anoscillor than the previous ones. Moreover our procedurecan be used to obtain a good estimation of the spectraldensity of the frequency fluctuations in the accessibleinterval.

REFERENCES[1] E. Boileau and B. Picinbono, "Statistical study of phase fluctuations

and oscillator stability," IEEE Trans. Instrum. Meas., vol. IM-25, pp.66-75, 1976.

[2] E. Boileau, "Elimination optimale des derives lentes dans les mesuresde fluctuations," Ann. TH&ommun., vol. 30, pp. 163-166, May-June1975.

[3] E. Boileau and H. Clergeot, "Optimisation de certains traitementsnumeriques avec une forme quadratique," Ann. Telecommun., vol. 31,pp. 179-189, May-June 1976.

[4] J. A. Barnes et al., "Characterisation of frequency stability," IEEETrans. Instrum. Meas., vol. IM-20, pp. 105 120, May 1971.

[5] J. C. Radix, Introduction au Filtrage Numerique. Paris, France,Eyrolles, 1970.

[6] E. Boileau and Y. Lecourtier, "Quelques problemes poses par l'analysespectrale basse frequence," in Proc. Cinquieme Colloque National sur leTraitement du Signal et ses Applications, Nice, France, pp. 141-149,June 1975.

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