42nd Annual Frequency Control Symposium - 1988 STANDARD TERXINOLJXY FOR * FUNDAMENTAL FREQUEXY AND TINE NEnloucY

David Allan, National Bureau of Standards, Boulder, CO80303 Helmut Hellwig. National Bureau of Standards, Caithersburg. KD 20899 Peter Kartaschoff, Swiss PTT, RAD, CH 3000 Barn 29, Switzerland Jacques Vanier. National Research Council, Ottave. Canada KIA OR6 John Vig. U.S. Army Electronics Technology and Devices Laboratory. Fort Konmouth. NJ 07703 Cernot U.R. Uinkler, U.S. Naval Obsewatoty, Washington. DC 20390 Nicholas F. Yannoni, Rome Air Development Center, Hanrcom AFB. Bedford, HA 01731

1. InCroducCLon S,(f) of y(t) S,(f) of 4(t) Tachniques to characterize and to measure the frequency and phase fnstabilities in frequency and tine devices s;(f) of I(t) and in recelvrd radio signals are of fundamental impor- S,(f) of x(t). tance to a11 manufacturers and users of frequency and technology. These spectral densicias are related by the In 1964, a subcommittee on frequency stability was form- equations: ed vithin the Institute of Electrical and Elcccronics 2 Engineers (IEEE) Standards Committee IA and. later (in SyW = + S,(f) 1966). in the Technical Committee on Frequency and Time vithin the Society of Instrumentation and Measurement "0 (SIX). to prepare an IEEE standard on frequency scabili- ty. In 1969. this subcommittee completed a document S)(f) = (21f)2 Sb(f) proposing dcfinltions for measures on frequency and phase stabilities (Barnes, et al.. 1971). These rccom- mended measures of instabilttics in frequency generarors have gained general acceptance among frequency and time Sx(f) = --+ S&(f) . ) users throughout the vorld. (2*v 0

In this paper, measures in the time and in the frequency A device or signal shall be characterized by a plot domains are reviewed. The particular choice as to which of spectral density vs. Fourier frequency or by domain is used depends on the application. Hovever, the tabulating discrete values or by equivalent means users are remindad that conversions using mathematical such as a statement of pover law(s) (Appendix I). formulations (see Appendix I) from one domain to the other can prasent problems. According to the conventional definition (Kartarchoff. 1978) of Y(f) (pronounced ‘script Yost of the major manufacturers now specify instability all”), Y(f) is the ratio of the power in one sida- characteristics of their standards in terms of these band due to phase modulation by noise (for a 1 Hz recommended measures. This paper thus defines and bandvidth) to the total signal power (carrier plus formalizes the general practfca of more than a . sidebands) , that is, s 2. Kcasurcs of Freauencv and Phase Ins.abil&y Rmr dwicf, am thw-raise sc&l.atLcn sftkkri& Y(fl = Total sigrml per Frequency and phase instabilities shall be measured in ~crms of the instantaneous, normalized frequency depar- The conventional definition of Y(f) is ralaced co t.Jre y(t) from the nominal frequency ws and/or by phase S,(f) by de?artura j(c), in radians, from the nominal phase 2wjt as follow: f(f) = q(f)

only if the mean squared phase deviation, <$* (f)> = the fntegral.of S,(f) from f co a., is much smaller than one radian. In other words, this relationship is valid only for Fourier frequencies f far enough BLfl x(t) = *av from the carrier frequency and is always violated 0 near the carrier.

uhere x(t) is the phase departure expressed in units of Sfnca S,(f) is the quantity that is generally mca- =ime. sure4 in frequency standards metrolo~. and Y(f) has become the prevailing measure of phase noise 3. ~haracccri-arton of F:eaucncv and Pm among manufacturers an4 us*rs of frequency stan- ns:abi!i:ies dards, s!(f) is redefined as

Y(f) = M,(f) .

In the frequency domain, frequency and phase in- This redefinition is intended to avoid erroneous a;abflity is defined by any of the follovlng one- use of Y(f) in situa:ions where the small angle sided s?ec:ral denal:ies (rho Fourier frequency approximation is not valid. In other vorbr. S,(f) rmgrr from 0 co 0): l See Appendix Note + 17 419 TN-139 ir ch. prJfJrrod maasura, sfnca, unmbfguou~ly. It to + c alway, can be mbAJur.d. TIE(t) = x(to+t) - x(tO) = y(t')dt' . I '0 b. w: For fairly ~fmple models. r.gr.JJiOn analysfr can In the time domain, frequency inrtAbflity rhaLL bo provide efffcienc estimates of the TIE (Draper And d4fin.d by the NO-JlOpl4 deviation O,(V) which iJ Smith. 1966; CCIR, 1986). In general, there Are the JqrurJ root Of the t’v~-J~ph VArianCe U,‘(r). many .JtimAcorJ pOJJibL4 for Any JtatiJtfCAl quan- ThfJ VarfJnC., 4, ‘(r), hAJ no dAAd-tina between the tity. Ideally, ve vould like an efficient and frequency J~PL~J and in 81~0 called the ALLan unbiAJ.d estimator. Using the time domain measure varfanca. For the rampling timA r. if. writ.: urz(r) deffned in (b). the following 4Jtimat. of the Jtandard deviation (MS) of TIE and irr AJJOC- fated Jysr.macic depareur. duo to a Linear frequency drift (or its uncertainty) can be ured to where predict J probable time interval error of a synchronized at c=tO=O and left frae running there- after:

2 2 x(t 1 R!!TLEeJt= t r = t) + ( O)* 4, Y( t The JymbOL < > d4nOt.J An infinitb time Average. 1 In practice, the requirement of infinite time average 1~ navar fulfilled; ura the of the forego- vherc “a” in the normalized Linear frequency drift ing terms shall be permitted for finite time per (aging) or the uncertainty in the drff: eJcfmat4, o, the tvo-~.ampie deviation of the initial frequehcy Adjustment. o,(r) the tvo- sample deviation describing the random frequency givJr Jero dead time between frequency measurb- instability of the clock at t--r. and x(t,) ir the m.ntJ. “Residual” implies the known JyJCcmJtic fnftiaL Jynchronfzation uncertainty. The third effects have been removed. in the brackets provider an optimum and un- biased estimate (under the condition of an optimum If dead time exists betveen the frequency departure (RXS) prediction method) in the cases of vhic. measurements and this LJ ignored in the computation noise M and/or random valk Fn. The third term LJ of u,(r). resulting instability values vi11 be too opcfmf~cic, by about A factor of 1.4, for biased (except for vhite frequency noise). Some of flicker noise M. and too p4ssfmiJ:ic. by about a the biases have been studied and some correction factor of 3, for vhfte noise PX. tables published [Bam.~. 1969; LOJA~, 1983; Barnes and Allan, 19861. Therefore, the term O,(T) Thfr l stfmaca is a useful and fairly Jfmple approx- shall not be UJJd to describe ruch biased mcJJure- imation. In general, a more complete error mbncs. Rather. if bfaJed inJcabfLicy m4AJur.J are analysis becomes difficult; if carried out, such an made, the information in the references should be analysir n44dJ to include cha methods of time pre- used to report an unbiared 41cim~t.. diction, the uncertainties of the clock parameters. using the confidence Lfmics of mcasuremencs defined If the fnicfA1 sampling rate LJ specified as l/r., belov. the decalled clock noise modeLJ. Jystemacfc then 1: has been shown that, in gbnaral. YC may effects, etc. obtain a more efficient estimate of u,(r) urfng vhat ir called “overlapping 4JtfmateJ." ThiJ 4. ConfiOanca - estimate LJ obtained by computing An l srimacc for O,(I) can be made from a finite data Jet N-2m vi:h h meaJu:amentr of y2 as foLLovJ: c $+*a- *xi+m+v* n-1 i=l 0 (r) 6 C (Jj+L - ?j)* I ’ Y = I where X fJ the number of original time reJfduJ1 j-1 meaJur.m.ntJ spaced by r,(N=X+l, where h ir the number of orlglnal frequency measurements of ramp18 or, if the data ara time readings x2 : ttme rg ) and I = mr, . X-1 uyw = - 1 (xj+* - 2x,+l + xQ2 4 . From the above equation. v. ~44 that or2(r) acts I I Like a -difference operator on the time ZrI(Y-1) j=l deviation raJiduALJ--providing a s:a:fonary maaJure of the Jcochastic behavior even for nonstationary The 68 percent confidancc fnterval (or error bar). I,. pr0c.ss.s. Additional variances. vhich may be ured for Gaussian noLs4 of a particular value a,(r) obtained to describe frequency inJ:abfLicf4J, arc defined in from a finite number of samples can be cscima:cd as Appendix II. foLLovs:

c. ELock-‘Clmr PredfccLep I, = o,(r)ire.y-i

The variation of the tfse difference between A real vhers : clock And an ideal uniform tin. rcale, also known AI time intervaL error, TIE, obrerved over a tbJ8 n = total number of data points used in the inter?a? r:ar:ing a: time to and ending at t,,+t 4stima:4, JhaLL be defined AJ: 0 = an integer as defined in Appendix I. "2 *=I - 0.99, TN-140 420 =a = 0.87, 0 the environment during oeasurement; II- 1 - 0.7;. 1-1 = 0.75. 0 if A p&JJfVe element, such as a ClytAl filter, IJ being measured in contraat to AJ .n l xampla of the C~wsian nod.1 with H=lOO, o = -1 a frequency and/or time generator. (flicker frequency nolre) and o,(r = 1 second) = lo-“, ve may vrico: 6. - 1. - o,(r)*(0.77)*(100)‘~ - o,(r)*(0.077) Barnes, J.A., Tables of bias functions, 1, and Da, for variances based on finite samples of processes vith vhich givas : power lev spoetral densities, NBS, Washingcon. DC, Tech. Note 37S, (Jan. 1969). u,(r = 1 second) = (1 f 0.01)) x 10-12. BarneJ. J.A. end Allan. D.U., ‘Variances Based on Data vith Dead Time Between the Keasuremancs: Theory If K in ~~11, then the plus and minus confidence intar- and Tables,’ NBS Tech Note 1318 (1988). vals become asymmetric and the ae coefficients are not valid; hovaver, these confidence intervals can be calcu- Banes, J.A.. Chi. A.R.. Cutler, L.S., Healay, D.J., laced (Leraga and Audoin. 1973). Learon, D.B., HcCunLgal, T.E., gullan, J.A., Saich,U.L., Sydnor, R., Vessot, R.F. and Wtnkler, If ‘overlapping” ertLoaCer are used, AJ outlined above, C.H.R., Characterization of frequency Jtability, then the COnfidJnce Lncerval of the escirate can be IEEE Trans. Inrtr. and He&r., Vol. u. 105-120, shown to be less than or equal to I, AJ given above (Hay 1971). (Hove, Allan. Barnes, 1981). CCIR Report 898, Dubrovnik. (1986). 5. 7 for~orReoQgg&g Draper, N.R. and Smith. X., Applied Regression Arulysia. John Wiley and Sona,(l966). a. Nonrandom phenomena should be recognized, for example : Hove, D. A., Allen. D. W., and Barnes, J.A., Propertier of rigr~l sources and maasurament methods. Proc. 0 any observad time dependency of the JCA- 3Sth Annual Symporfum on Frequency Control. 669- tLstLca1 measuras should be atatad; 717, (hay 1981).

0 the method of modeling systematic Kartaschoff, P. , Frequency and TLne, Academic Press, New behavior should be specified (for ox- York (1978). ampla. an l JtiMte Of the lfnear fre- quency drift vas obtained from the LeJage, P.. end Audoin, C. Charactariration of frequency coefficients of J linear lJJJC-JqUJraJ stability: uncartaincy due to the fLni:e number of ragraJsLon to H frequency measureoanCs, ae*suramenCs. IPXE Trans. Instr. and Hear., Vol. each vith a specified averaging or sample m, 157-161, (June 1973). time t and measuramenc bandvidth f,) ; Lesage. P., Characterization of Frequency Stability: 0 the l nvLronmenca1 sensLtLvLtLes should be Bfar due KO the juxtapostcfon of time fncerval JtJtJd (for example, the dependanca of measurementr. IEEE Trans. on Instr. and near, E frequency and/or phase on temperature, x2. 204-207. (1983). magnetic field, barometric pressure, vibration, etc.); Stain. S. R. Frequency and Time: Their measurament and characterizatfon. Pracirfon Frequency Control, b. Relevant mearuremenc or JpeCLffCatLOn Vol. 2, Academic Press, New York, 191-232. (1985). parametars should be given:

0 the method of measurements;

0 the characteristics of the reference 1. rover-Lv SoecfI+Lgensi-ha e Ji&‘d ; Povar-lav Jpeccral dewities ara often employed as rea- 0 the nominal rignal frequency us: sonable and accurate models of the random fLuc:uatiolu fn precision oscillators. In practice, these random 0 the measuramant .systam bandvidth f, and fluctuationr can often be raprasanced by the s’um of five the corrarponding low P~JJ filtar independent nofse procaJJeJ. and hence: reJponJe; +2 0 the tocal measurement time and the number 1 hobo for 0 c f < fh of measuremeno H; Sy(f) = o=-2 the calculatLon techniques (for example. 0 0 for f 2 fh decaflr of the window f*mction vhen t esCiaaCing power Jpectral dens?ties from time domain daea. or the asrumpcionr where he’s are conrtantr, Q'J are LntegcrJ. ar.d f, is about effects of dead-time when artimat- the high frequency cut-off of a lov paJS filter. HLgh ing the two-r~mplJ deviation u,(r)); frequency divergence LJ elLmtnaced by the res:riCtionJ on f in ChiJ eqUACiOn. The idencfficatton and cb.srac- 0 the confidanca of the ertlmare (or error tarira:lon of the fLve noise processes are given in bar) and ICJ statistical probability Table 1, and shown in Fig. 1. (e.6. ‘thrae-sigma’);

421 TN-141 where S,(f) fs the spectral density of fraqwncy fluc- tutions. l/r is the measurmmt rate (T-r is the doad The operation of the eountar, ovaraglng the frequency time betwean measurements). In the case of the ~lo- for l the t, may bo thought of as a filtering opora- ramp10 varlanca IH(f)lz Is 2(rin 'rrf)/(rrf)l. 'Iha tvo- tion. Thr transfer funetfon. H(f). of this equivalent sample variance can thus be computed from flltar is then the Fourier transform of tha impulse rarponso of the filter. Tha tina domain frequency instability Is than given by fh oy2(r) = 2 S (f) + df . Y (*rf) I0 0’ (H.T*r) - J; s,(f) I H(f) I’ df, Specifically, for the power law model given, the time domain measure also follows a power lav.

TABLE I- Tbo functional characteristics of the independent noise processes used in modeling froquoncy instability of oscillators

7 3 Slope charactarlstics of log log plot

Frequency domain lima-domain

D*ser.iptfon of s,(f) or S,(f) or Noise Process SJ(f) S,(f) 3 (1) 0, (r) Mod.o, (1)

a I 8 P r/2 co Random Walk Fraq Kodulrtion -2 -4 1 l/2 1

Flicker Frequency Modulation -1 -3 0 0 0

White Frequrncy Modulation 0 -2 -1 -l/2 -1

Flicker Phase Modulation 1 -1 -2 -1 -2

bhito Phase Hoduhtion 2 0 -2 -1 -3

S,(f) - ,+$+--S,(f) - h,P $

s, (0 - & h=f”-’ = & h,f’ Xod.o,(r) - 1~1”

itiLE 2 - Translation of frequency instability measures from spectral donsitfes in fraquency domain to varfances in tlma domain end vicm varsa (For Sxf,,r D 1)

$eseription of nofso process of(r) - I S,(f) * ‘Random Walk Frrquency AIf S,(f)lrl 1 Kodulatlon

jFLfckor Frrquency Modulation Blf S,Wr” $[9 *:(r)]f-'

$iiite Frequency Modulation Cl!? S,(f)lr” $1 c; (r)]fO I IFlfckor Phase Hodulacion Dlf-'S,(f)lr" ;[9 #; w] fl t / ,b%i:r Phase Hodulacfon EIf-2S,(f)Ir” I

c = l/2 1.038 + 3 log,(2wfbr) D= 4ti TN-142 422 o,~O) = hm2+ r+h a1 =3&,2 + ho k

1.038 4 3 log (2sf T) +h h +h2--lfh’ I. s or w ‘Itro-sAd&duiui 1 (242 v2 (2*)2,2 nuLz,‘Lzl Instead of the use of o s(r), a Wodifiod Variance. Uod This implicitly assumes that the random driving mech- =I *(r) may be wed to characterize frequency instabili- l nism for oath term is independent of the others. In ties (Stein, 1985; Allan. 1987). It haa the property of addition. there is the implicit assumption that the yielding different dependencr on r for white phase nois* mechanism is valid over all Fourier frequencirr, vhich and flicker phase noise. Tha dependence for hod u,(r) may not slvays be true. is f-ala and rml rorpoctively. Hod u,t(r) is defined 4,: The values of he are characteristic models of oscillator 2 frequbncy noise. For integer values (as oftan l eeaa to be the eere for reasonable models), p = -a - 1, Hod +, = for -3 I o < 1, and p = -2 for Q > 1, vhere ~,~(r)-?. J=l 1 i=J Table 2 gives the coefficients of the translation smong J the freq6ency stability measures from time domafn to - frequency domein and from frequency domain to time vhere N is the original number of time measurements spaced by rO and T = IT. the sample time of choice domain. (N=h+l). A device or signal shall be characterized by a plot of u,(r) or u,~(v) or Hod u,(r) or Xod u,‘(r) vs. The slope eharacceriscics of the five indepandant noise sampling time r. or by tabulating dfscretr values or by processor are plotted in the frequency and time domains equivalent means l uch as l statement of power law in Fig. 1 (log-log scale). (Appendix I). 2. Oehct

Several other variances have been introduced by vorkers in this field. In particular, before the introduction of the two-ramp10 variance, it vas standard practice to usa the ssmplr varimca, sz, defined as

Log Fourier Frequency, f s2 = ,2 S,(f) (+$+ df.

In practice it may be obtained from a set of mesruro- ihll II I I I I I I mencs of the frequency of the oscillator as t i’\’f4 ’ i 1 I I I I

The sample variance diverges for some types of noise and, therefore, is not generally useful.

Other variances based on the structure function approach can also be defined (Lindsay and Chi, 1976). For example, theta are the Hadeaard variance. the three- sample variance and the high pass variance (Rutoan 19iS). They are occasionally used in research and scientific vorks for specific purposes. such as differentiating betwssn different types of nofse and for dealing vith rys:eoaticr and sidabands in tha spectrum.

Log Fourier Frequexy, f

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