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 time 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 decade. 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 clock 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 unit of time (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. term 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 second-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.