BSTJ 50: 3. March 1971: Short-Term Frequency Stability of Precision Oscillators and Frequency Generators. (Barber, Raymond E.)

16.4 MB·English

Short-Term Frequency Stability of
Precision 0
Frequeney Generators

illators and

By Rasmond K. Barber

(Succi reseed Ana 3,180)

‘We eatin this par to deftones rm fagusuey sabi
(a tine domain, the espectod solue of the vavionce of the fractal
Freruency Bacvetions fom veined fequeney, (eHN, Py =), fn
Wea the pumber vf wanes, Ts te avraping time pla the deed fie
lets sample, ad 1 the oraging rime and (i) frauacy domain,
tha power epectal density of the frortat jroguency departure rom
onal frapueney, 8,i) We diss the opie of srverins Lrom the from
uency domain to he tina domain aid eonoreon mong Hane domain

‘Al avesene sera made ithe me doa, wing prod awning
tesinique dn ocatr tne ofl tu fraeacy Oy cing 0 specitiy
brett quota ery! nl pled Jor 10 RES bao the fegueney of te olher
foures, THs mclaor sas wel to obtain the beat frpuonch repurl By

2 rin az grouch,

Bince the we of (212, Ty o)) 28 a measure of shorter stability has
sonifieaneavanioge ster GAN, Ty }), the telationahsp betosey the
fae quant ne trestigoted Pur adapta Sines of ane za oF Ie
GIG, 2, Vb avd GEN, 3 wee alt eu

The ser ey of several quart ertl oxelalrs o preision
Srey georatre eos masre, Phe aby eer te abot erage
fines ous wooly aul fe wat teeing. At lenge ses, the
ality of enh sacar won senigue. The fegueney gonevotore demon
trees cave aveating Himes of £0 ae 200 milscond
ut were unique slelare, The comurey of al measurements wes Stl
ti systematic ata rom the environment aa he meaearenen ouipaent
‘tes
S82 mmm svermy rocanercay soumwaL, ate Lat

‘When the phage “short-term frequeney stability” is mentioned, one
con imumedintely infer that frequency varies with ime and al & shor
fine intersel i involved, Nay cuenlions are, shi poit w
answered: Tow short isthe interye? Tow i tabi deft? What
‘ete mallunotions can reaull from rxtesive instability? What,
Aetnitions ane appropiate for he spite appiction and why?

Some encwsrs to tise questions are aevolosed bere; others mut be
determined hy the weer i terms nf the specie eystm involved. If
‘a menanreimet ef sharlatern hequeney ability i to be of signifier
ee cach ef theze quetioneruet be role.

‘Although cear-insreasing intrest in shox-term stability hee existed
for 25 oF 80 years, no untverslly accepted definition of short-arm
slabilyy exit, Prinatily, obey swear in dha anee was eaomely
fpplietion orien. Title general work was performed with the
Fesull that many defitions wets developed. Individuals and organiza
Uns defined wad ued sfabiily in tera of dbeis own sppiations
‘The -taullans sonusion shewo at a mote general eoiiten ap
cable to the miiorly of otees wens cesirale in Lhe Logie of shorter
frequency eablity

“dey, sunny waes ‘ule about etsble willatorswichout wlerstarde
ing why or even if euch provision in neteeary, Manufactncere wld 10
the present can‘usion through lak of rier in tcir performance epesi-
foations, Tht i, they oem merely quote rummber wichost dling
‘hat, how, or why they ae mcasorng te sablity of heir equipmon:

Sophisticated eysteane exist taduy which rire more provision thaw
frequoney standards o: 10 to 16 yeare ago, Only recently hus the pre-
icin onblla‘or deen iherabad from 3 poscion in seluo’! portions el
fateully eonkmlled iehoratorce- Spacceraf: epplications sufico es
fool extmper of the strides rude in thin ans wide temperature
‘aviation, severe merhaninal ination, and varying esilaer vltage
fre wlan enustere, Ty the inunedigte fubne, hese requienents
‘ill Geiten even farther. 4 ppd eosin avoidance ayer for
Uvvrafe wil myuire stable onilatore. With higher and igher deta
Fate in comm stieation sjavens, even the Heperdahtetelepove afer
vill contain precision ecilators, Ae pression owilstors vom into
fener ise, accepted messirement theory and teehnigios nee tbeo=
Tce ceceeary.

Tw view othe above, rwo definitions of fraqueney stability are pre-
sented in this paper. One defsitin iia the time demain: the ober
SORT-oeNN FORAY BENIN 885

ss in Ue frequency domain, Conversion been the demins, an @x-
trewely icoportant subject, is glen vested. Every stiempr has been
tmnde to be convent with the TEED fulseommittes on Frequency
Slailty-” Le it anteipated that the svbcomssitce will publith ite
{ermal definition eoretime i 1971

a4 Genet Dito

‘The general <efinition of irataatyrens angolar frequency i he
cimerato of hangs of phase, That

a

a = a8 w

An oeilltor otpot egal may he Aeseribed ae
of) = [+All cco ot + HO, ea)
if) = [A +A oe ek + WOT, ey

here

«o = nominel angular frequcney ef the osslator,
BL gree = roti frequency in heres of ube usllator,
‘A = nominal emplitude of the evils,

(9) ~ smal, slowly time-varying nemylitude Auctions, and

4H) = Slowly vime-varging ral foneton (shes).

rare, o() may be conidered as vither a voltuge or surment. If the
oilatr isto ually a & preven olisto, i is required that

a

”

sus Statist Precenses

For the peetnt, assume Esa ube variable of equation (2) ar rane
om procestes. Te is gouty tested in the ty of fequeney saT-
Sty that anpltide deviation, 0), done: dicety alert frequency ot

coe St ava a tm es
sea Ba eo orate
88 nacemmuysvenne mavens sou, MARCHE NTL

phase? 1c must be strated that gi) presumably contains all fe
‘quency and phase fusbastions and the inain quand eonedetet

‘Tha rudn price ix salionnry in he are sea, He unalerled
Ly transitions of the orgie for timo This implies tha rhe pbui-
ity distribution of vie ho gscule wil be Ue sa mn 80
insanis, Henes, examition of the envemble yields no dutta 0
‘which istant of vise the examination. cecurred

‘Tents on rarlom aoiee und stachaeie processes esmmue that mart
olan fonecona teay be resesental or apacoxinttad ue sasineary
tcwssinn radon ponossse with aera averegce* The juscdeson Jot
reaming 2 gansien dicebubion lice i the centre lt theoecn. ie
‘Fell-known theorem sttce shat eo distbutionw! the sutn of large
‘nirober of independent raadom vatinbls will eporonth mien dite
Iribution. $. 0, Rien showed that noe dee, im general, confor 40 8
fsussian distribution, poovided that x safeeny lange sample 5
Taken Zero aversger ofeach randoms variable arc assumed for com

‘A process car be dotned as srgedie when the station of ene eye
‘em over an ir foie pavid of Lime are equivtert to che states of
an ininite essemble of systems at eny given instant. (Note tba: sta
Eematity ‘3 @ necenery but not subcient condition for egodioty]
Since the prosesos are saticrary and gaussian, the noke furetiuns
fcoraiered in tha paper are cesumed egodie.

1 tbe eal mori o proces aun lee naty ie the sre wee, To
be stationary, the peohshility distributions cess the enscibie rt
Tae the seine for any sec isian of Line. This tiation inludes
time a i approaches infty TT the proves i terminated se some
fature ine, the eonee of slaionyrty inthe snes seme is vlad
‘The raziom progesy, X;, elle siaoracy inthe wid see, ithe

fist und eoeand onter moments of ite random yoriblet eset and
ale
FAR) = content
GR} = vonuat,
Bl eoy = Bl, — BOLI = 260,
where

RG) = autocorcation function, and
1 = some acbitrr nite Ge interval

‘The well verifntion of slatinurity is net feusible, The mein 26-
Som-semse HULENCY eTABaLTTY 8

‘quzement ie Une the ploysieal proves: he consistent with the concept
Of seationarity. That i, if moamieraln gov to he of any use, they
tut zensonably deseribe the gress ae el ces is the future, prior
to termienion of he proms

2a Syatonatie

‘When a study is made ofa precision esilator or frequency genres
to, ert pine are often taken to iolste the cireuits from eletriesl
Aistzvances, ‘Tht bone to mininive the systomarie eee so thst
foniy tae truly rardom apie car be obverse}. Tor exelation af the
fnraron fel, fia ine sound practice, owevce, in the eal world,
bleetieal drturbunces do evs. For neample, ina data processing
fyotom there ero card readers, mogpelir Lape tranapats, and other
deve, ach ofthese devices has numero cletzo-merhanical relay
‘phich are eotinally eetering. us, i 4 deere ta Snwute tha
Xt cooevey qenereion has the neeesicy stabil, Thie obviously
meane it mute dezarsltote adequate ability in te apereting ex
‘noniuent with the contrition of She stems fect

“in euation (2), i was wernt thatthe min mstrbation of pth
ans foun the oselstr, Now ite nevessiry ta Tagen te eeu
lo iucte hs sontainrione co pi) from he enevating environment
White thee nie sawenacales ane uot aud yy the elec
acts to ebverve 0} mill ielade Reon efets, ‘Pherdor, (01 30
{ation (2) cum be ronsdere t neha

ot = of +H + 000, ©
shane
i) ~ pose due to aging (it

10) ~ phase de tn enuronanataleystematio effet, ana
1) — phase de te wseillator random noise

‘The sonsihations of») ave sively long 2mm, afcoting time inter
vals af « thousind remade o*lonaer. Hezeo, fo tse peviode of ess

a4 thou seven, 2 Ineores most conecant sa thus is 02
‘Sghifans, The exact compotion ef s{0) ie unknown, although ic has
Toll Tonge and short-torm eFete, Again, We Togsera eouponeat of
the egeteratic aeets teomes hiqifeant for short time. periods
ur Faese shor: ivervu, (1 becomes

0) = 060) + mi, @
8850 Ramu,

49(0) = alue-tern components ofthe eystemato ets

‘For the pasa it will be awed thas the sber-term eomporent
but the leratie eee fx andor provoas whieh fe statoneay at
Teast in the wide sense: goursan, with zero averagns, and egadie
‘Tis etipalarion may later fall tore ia ane about the vale of
systnatieeflets, Certainly, ever if not srulyrnndor, the ects wil
fapene as short-term inti,

Any envirumnt ll entribave «tars a (8). Obsiouey, the
rystenaiee, (0, en ln ininiaed Lo le pl where her stb
‘lon i tveral ordre of maaritoe les than the oeillatar noise, nt)
Iieconee he only signifeant meseurvle. On the oiber bac the
envieorment is extremely noive contkcinald, the setae) owslstor
natciliticn may ho "smtraed” Hy the eysteieefeete, Conitions
uo exiee that azo eo eontarsintod cho: coherent operation of many
‘loetronieaysteme eno! possible, Lhe aysteratic eects wil eoncibte
‘a tho chort-term intabilities im an aditive manner on power base,
A on ertinate of the short-term ability of an oecillater or fe
queney generator een be obtained ia encefally extrelle, low-noise
Crsitonmon® and if subeaquent estimate ean bo mace in an oper ag
tmoephore, che ree.tng instablities may be cempared. ‘The rantie
hution of she noise due to tas eystemate effects of the env zannent
say thea be deseribed

a onerat

‘Many apotieations exist st noueesiate the as af highly stuble o=
ailntrs or fragzoney paeralors: Du to the aeraeney requirments of
Ise vation applieions, ome mensire of ovllstar ineabily
perstive. Alshongh many persons view sul a quantity money as @
Yelle for centr selene and eetparisn between. eacilaots,
Ue uses are erly aondemie.Stietly epeekng, a meseure of imal.
ity enables prediction of ceils per‘ormanoe or, i nob alent
tlnble, nonperfornuine ia a given eystom. More precicely, any devis-
‘on from the intended frequency will degrade the performance of any
syeiem. Specific case of the fete of Peeuenay slebility ot ah
operation are domes by J. A. Baenga! *

rsented brew are defitivons of shorliewn frequency stability
wonT-TRIDE FREaDESeY suas 887

hoth in be time and frequency stoning. ‘Trans
domains salen cheered

ta Tine Dunoon
1 this par, shos-term fesquenay stability ix the cme domain ie
ofr a the ensembe average of che varisnee of Tracional equoney
fuctustione from nominal fequeney. This deletion is awe on the
verte of DW Allan
‘eis convenient make the foTlawing desinition

Bah

where

$6) = eit = id (fom equations {1} une (2)}, wad
= Tong-tenn average Fequenes, hertz, of the esilator (trom
equation 2)

‘Taking the dhorl-tnrm average value of equation (7)

pol [nou Mae, «
nef ya a, o

were m= 1,2, °° im,
Tsing enunlione (8) and 19), a expression in tert of the eumple

variance in abtaines,
oT) = an

Eli

where

X= numberof expe,

TT imme between the begining of suceesivesemple intervals and

7 = simple Cine

‘Short-term feequeney atebity in Us ine domain is defined as the
experied value of craenble overnzr of ration (10) the vaxiaaee of
fruolionalfrqueney duetaations ‘vom nominal feqacey. Thats

voto sf kl-kSa)].
888 aN MeLL svar rieA WOME JOURNAL, MARAR TTL

where RIXT denotos the expeeted value or ensemble wrerwae of X.
[Noto dhe thie definition of frequeneyeanblty is dimensonles

‘The justilicai of the use of te term 1/(V — 1), ovate airectiy
tthe lth of the fa, scromaton in equations (105 sal 11, flow
run he Kieory of eetmation ove Re. 9). Thu in ue vo eb a
unbiased estimate of the popslition variance, the sample variance
ig muiplied by AViN ~ 1, raling in equstions (10) end (11)
CCleacly, for large sample, the popalstion and sample variances ee
come wary nea equel.

"A povalar expression of shor-tenm staility in the the domain is
the axpeecd veluo ofthe stendad deviation [nme value} of se Irae
tional Invgueney Fastations frm omine fecverey xprsesd fue
tionally, thie besomes

{1 shouldbe noted that the square reo of equation (1,
Ss qunl to eyurtion (12), (@ (47, x) onky when the saeeomive samples
fre tray sislonery in the wide wonee, Tf the amples appece only
Slightiy nonstationary, approximate equality may be nesumed ss
et, 2, 8 1
443 Promuney Donia
1D. 8. Culler and C. L, Searie showed tht w peteina definition of
shorten Frequency sability een be sive in terms of wutocoreltion

fnnetions and power spetoal densitien™
"The nuloworelationTanelion of pluwe is defined ns

aa)

ga = ny | ale + nota a as,

Ryfel ear uso bo detorained using numerical methode on a digital
mputer i eng amples cae ae eH

Ra
whore icone lange pei ntoge.

“Let Rn} be the aacoreationSvostion of puoces wie in ee
tionary fn he wd nea end sont non fhe WinnoeeKaiehine
theorem, crown chat aocrareaton faneions and power socal
denis nee Fourier nafons pits. Hone, the (ror soetal

2 Eau eae, a
SRORE-TuRAC yHNULIAY STARTLETY sv

donety of phate hecomestoe

sine [Ree (16)
20m this the owes ata denny of pst, $0), Devoe
say 22 [Rone nian «

Rie

2 [° 840 coe ate a as)

tn somal Fourier unl difezentiaton i the tne domain eon
responds t0 milinvation by jf inthe fnqueney down. In terme
this brooms [2ef, Thos

840 = aps. a9
fie the power epecteal density of froqueney.Auetuations

ee 8
{fo eblain the joer speteel density of instytatinoasTrctinal fre
Qquosey separa fm nominal fequeney, eaustion (19) must be

‘normalised

eo

“Thos, 8(D, equation (20), isthe dedniton of short-term frequeney
toby in the frsqueney domain, Note Unt S30) has the units Sper
Irerta”

4 Travaations etecon Domaine

sea Propuncy Dimi te Tone aan
1S. Cutler bse shown Ut the olling equation allows eleultion
of the dimedomsin taility fre the Trequaney dass"

wie, 7.0

wg aoliett [; gee], on

tase LD - sae

tr

hn ren sh communi ea ths top ia eaten (2) gud
ee ti ea nt esha Bo fn ae fe
890 symtoms. sven enna ona, SCE 10T

842 Trawabaions Amy PineDein ease

Jn certain apslioutions, translation among time-domain moveanee
is of interst, Uhe folloming method, presented by ares," alow
‘aleulaton of (e(0/ Ts, ra), given en estimate of (oN, 2, x), for
fanetions which have = power law spectral density

‘ince mnt pression osclistors and. frequenty generators have
poster Ine edral deniie, ia poible 1 defn tw bia Felon,
GO, 7 4) ab Bal gl a else
GN. a)
TO

1 ae

BAN, 2,01 = )

here

= apoctel type (elated to te shape of the spectrum. Tn Raf
‘Harmes shone how j nay be found, given 2).

Now an etiante of (f(Ws Ty, x} can be made

(ayaa
Ler Bi

8) aa, Ty wh

Les Te Te

an
Soe Ref, 1 fr a listing of zh his funotions for various epeetral types,
‘umber of cemple, end

41 Pesiad Covatng Tectaigue

"The basis junctional deeription oF this mello is hewn is Fig.
1. This orzongement i similar to that presented in Ref. 2, Two sie
secillators sow ofec in Sroqueney. This offs ean be produced by
fajustment of the Gining contra! ox the oti or we of ery
which produces a siskly diferent uvorage freq eney. Te $s wim
that these procedszce ehunge ea the long-orm average irequeey
ofthe cevilistons ed Ua h auor-Wnm jntaliltes ve not fest

As in equation (2), the eutpatsigcals from these sinilaroseiturs
‘re few mver or produc deletoe. ‘The reuttingoupal i fel ta 8
Tow paze Alter. This ede an cxpresion

Ail tooo aeth. — FOE LD — AOL. 8)