Allan variance - INDICO-FNAL (Indico)

Allan variance

R. Ansari - 16 July 2020

Updated July 23rd, to correct my mistakes , thanks to references provided by Peter & Anh

onds. Afterwards it starts increasing again in response to

rFotoartmionuloaf itnheseskcyt.ioTnhe6A.6lloanf tvhaeriadnicsehfopratpheerr(eFailgpa1r9t)of

the visibility is calculated as follows:

12 transits (2017 kHz bandwidth) declination aroun degrees in right a

2

(m?0)

=

2(m?0

)2

1 (N

2m)

arcmin resolution The visibilit

array, where 720

N? 2m

(Re [Vn+2m 2Vn+m + Vn])2 (3)

of observation, ea interval, and 121

n=1

cross-correlations

where

2

(m?0)

is

the

overlapping

Allan

deviation

at

an

av-

eraging time of ? = m?0, Vn is the time-series of the visi-

bility, spaced by measurement interval ?0 = 1 s with length N ? 438 103 seconds (9 nights), depending on the base-

lines. Roughly, the Allan variance can be understood as the

variance between chunks of data of equal size ? .

and 1. A comple comparing the o for the Tianlai a with only one po

Using the s construct a clean size, the four hou

parts, each cover

7 MAPS AROUND SOURCES

The 16 dishes of the array provide 16 auto-correxlai tions and

map tiles, each co guard area have to obtain the fu

120 cross correlations visibilities for each of the two lin-

in figure 20 (bot

y ear polarisations (HH or VV), as well as 256i crossThpios-corresmpoanpdqsutaolitthye is th

larisation (HV) visibilities. To illustrate the array pervfoar-iable wvhicsihblIec.aOllnZithe o

mance, we have reconstructed sky maps around few bright

duces low amplit

point sources by combining single linear polarisyatiioisn sHimHpolry the sGucacuesssisainvebeam i

VV signals. The sky maps through several algorithms

shown here have been obtained which characteristics aredibffreiereflnyces

We also ma

xFi+i1g-uxrei 21 and 2

outlined in Appendix C.

using 1 hour of d

frequency chann

From wikipedia page:



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tiguous measurements (see also Rau & Schieder 1984). expectati

One has to consider a signal-function s(t), which is the In other

Optimizatiniostnanotafnehoeusteoruotpduyt nsiegnoalbosfearvspaetcitoronmsetuersicnhgannel or of spectro

AoinfltlaeagncroantvetadinrufioaurmnacdteiemtemectieonratesfrouvrarleexTmamreeppnlrete.ssTenhteinoguatpnuetstisimnaotwe

equivalen If we

of the mRe.aSnchsieidgenraalnwd hCi.cKhraims esrtored as spectrometer data in the computer:

(r2(T ) =

I. Physikalisches Institut, Universit?at zu Ko?ln, Zu?lpitcher Stra?e 77, 50937 Ko?ln, Germany

Received 11 January 2001 / Axc(cTep,tetd) 2=6 A1p/riTl 2001 s(t)dt.

(1)

t-T

Aofbstthreacqtu. aSlittaybioliftyratdeisot-sabstarsoeTndoohmneicthaleeAixnlpslatrenucvmtaearniattnaioctienomn.evtThaholeduyheaavreeobvfeecroxym(seTima,pstlte)anadniasdrdstipmhroueclaretdeeufrotehrfeeorsittihudeaeteivnoantluicwatahioelnn detecting weak signals buriedwinitlahrgethnoeiseeflxupcteucattiaontsi.oFnorothfessp(etc)i.alFcoonrdittihones doubrisnegrovbasetrvioatnionosfanwoeuatlikne

omfetahsuerbemaseicntpsraorpeerptrieesseonftetdh.seBiAgalnsleaadnlosvna,riaaanrcaetchieesrrgtsiiavmeinpn,leanmnduatsmhoemmbeaetgriucaidlNetlirneeaostmfheonwdtitcffoleeainrrteernruplcerseetsfotrhoeofbrseestruwvltastoioonf osthfine "Position-Switch", "Beam-" tohr e"sFereqdueantcay-,Swaitc"hs"i,g"nOanl-T-mhee-Falys-u"raenmd e"Rnats"terx-Ms apapnindg"am"odreefaererednercivee-d.

Also, a simple analysis leads

t"orualecoonf ctlhuesivtmheusemtarbas"teufgoryrehamnoweensttotim"paltaxenro,rfaadthiroee-aossptutriombnoutmmraictcaimlteoindbgseffrorvroattmihoenseo.baPsceahrrvtaicotuitolahnrselyirs:foforuanidr-.

The and

supseadcew-bitohrnme aoxbismeruvmatoeffiriecsieintciyds.

Tv=ehreyxaismna-plyosrxitsarnshtotuolddehteelrpmtionein, chroewasethteheexsctrieenmteifilyc

precious observing yield in such cases

stiigmneificcaa(nn2t)blye.

A2 (T ) =

Accordin smaller th as there i measurem butions fr the simpl signal to technique

xn correspond to clock ticks, which

Key words. instrumentation: smoisctehllaanteotuhs e? mdeetshiordes:ddastiagannaallysaisl,oonbseervbaeticonoaml ?esspavciesviebhlieclews: hinestnrumaevn-ts in radio-a ? techniques: spectroscopic ?etrelaesgcionpegs. Typically, each of the two measurements are done laborator

integrate (count) an oscillator signal

at different times, after the telescope has moved between switching

1. Introduction

two positions on skyd.ent. Thus, it is always necessary to verify the similatriiotyn as is

? ?

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ivtfarrhepeaeaecdalrtiif(vocfoeenorrrareairmlengcscealpogeoocs)fnko=vdfpisxtethiiob+rei1iotlih-dtyexoi ver (barstocaacatptqAaio(dttKohShieihamrrrssfueeeyomlleeteecsavclreiOnatrSapehafaanFpenwhooKbopunulbiSlunlocifneteriopeesebfilMu?ossyrhemdeepohllevmrelllssdyaemnmastaAaerattcosrteai,hviedefarhntusyflfie)fuiereiislubioattcinnainptottltwhtmrcieOnrnthgide-ilrheahesltinaitdoceeaoeebattelriooisaeev.yxofithbflmedstffoAle1minaenoosrvetdlrmemo9ram.olahtrinre-eeloavtfi8ranrcrafetatmTsteipafseoe5yotioonhhnoauctlcosr;hctunemeoetsyoainprehrlKavmshu.deerliyoteeuitsatloviaemrchsoycopiusreferhprpamAtoedt-eimoemrlraarohietieisinlrunooaffchsnnpliecvetetalsmtfctoehecatstntisqu?panredtirreanaatauoeicrldreaqessnantrstavlmuilfnuln.Suapeytaeoaslcmncora2chrcouaereomrroeeobsettg0tdulaibetaTpotwoFvrTnisipeiaettfeses0vns(ts-hieotdarhnfbrnenaodAAea222yTlM0toethhhhbiarb,yabirecticerz)stsc(rtroacvnle((vatuaeei.,aiostiiwobhTeilaIetatilrcaoTTesshnalPtff,nlsowdspomalnetsuerarisn:hibeuetdvomn)oaomeyednayvec))iwhseawenrwtrrvrnrffhneecsAceeaotuseh=kytaioaethisnn==ihmtteit,es,edestatrrincahscoaesdialcbemeisy,oetgrptidleulrpdnfrstrsedyk.eothraua.e11arespilpamoigihbeh(Aretar1ffettccnnmenMnlTlr//selredbhrirlifu9(ettoiettvseocretscslnAcar22heeaec8atrnimsyetatentvorl:rL1at-rheecee9ncrOslqhsidmofenaknioer"as:y)iottzsd(orloutmsnent.Sret-isaunaeeacrwfdrieeafra2teot)kTtemndcooitaoefitnondbfaatttmieate-evrrei-fma"oehhhlcbvwdsriaIepno?reeelxlindynyyynaeeersstsl----,-e,)tiusalpec2taytsnstdeieiaogppsosbimuwbcpmdptabsoaoehntcneipeoafannnt"(lraensaree=oni)eesreonthonoeetvtAwrcaltnnast.td2stahia.caebasktcrIemlArhhnlatifkynirkreolmeTllu-elatiogosalaesloeslstsrflyelpdlmilraeefy.mpdhresaaeoyn=daucgorcsmty.sppsdt2ltdTeedeneeinrtoshs,nessahfiiatiameqtFnntnoayhctremoeaecs1aerhtuutitoesva1maoeacny-arivsrhdavhere/rlsabltsemytpiasm9ineat"iniesarsrntatahl2nieovcriiptolai6taceaiolaebdes,eoclhpaeocatrrr[fmeyrtmffenutdcode6.mroonabileabraeoelsi,eue1ewrbxedcii)ecgss.s.oculrbOrmngcdevtsueteehls,2y/tta2pBstdiThedterhreleonelhsoaoeay.yomiii2lvtnriuiemechnfauerisn.nfnnf-nfnetokevtdotcore-stnlcrtAodrecireIoikaytgarhdpthuoitmientaeicitrewtfnnlmrsnvhhtlaeaahlhoslueuaifieesedipoauoiaiaurnlarceimtseaapdsndfrwlnarsnanrssroaenooaooisluogddecelblsedocloflrtwtfnidvivdartteatmn2fhtpnnoeysehoneatdemtoliifimadoitciiet]pfnlohrhgedrnrgffatbocolos.iieeaorfiniegoeteemAvheztiraspsmen-eiectaftfinbainemefanantroaivfutsimsaorndyrlizanebdsnropacunelfnvraltvfsteaotroloaneuanpaehrarttleiiahtttotnmrttnecncfeaietioethhinenprrhbohliaddtleogskgochisaneesecos.lnreseetsonreisotebeoeexnimeimiasmhndeIaetnscaosrlsmnfispwaistoreon,eoavstrabeeoitotimiwseftfoanhoesarrstrstorasorcem1hraeftbhuientcie(epiontittvrcfaecreavolzAt9iromthwhrolmaldhcrllrhaiarvriepautdeiiiei(m6Otlcedcmymgteei,esoierhbshlr3yilanonaaw6hratfarSto,looiynaioao.evraep)tt---bftls.b)ablibpiveraprci.at,nOlirsslaesetaeetaenosypeteeechw.iuhtrnnhvsaorrexoamadeft"saihvivoeiottaTIneoaaFrcinrrtcathmennovPnnstnu[soieiehhiivthcnnemnbtfeonanothex2nehvoternumoilae(tsehiwedeadncalndngogeeerdcrto-t----sl-teeeyneea/n)IWudgo.tdnirsx"ncanatrw,ueefeTorratesirfioror"gtntfi-ltthovnrnh.ehirhaneemaoeinmmeu

time m Send offprint requests to: R. Schieder,1

e-mail: schieder@ph1.uni-koeln.de

This

original

defins(OiittiTioonFn-)",mth"eBaroseuaurmgem-h"enothrtse"oFrdre"iqffRueaersnetcenyrc--MSewaoiptfpchisn"ag, m""Obpanlse-eTsdhoen-Fdtlahyr"ed

may be altered by using the ratio of contiguous data instead. 1/N

defin

N n=

Article published by EDP Sciences and available at or

References provided by Peter & Anh

Allan , 1987 ( IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT ) Land, Levick & Hand , 2007 (Meas. Sci. Technol. 18 (2007)

646

IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. IM-36, NO.2, JUNE 1987

Should the Classical Variance Be Used As a Basic Measure in Standards Metrology?

IOP PUBLISHING

DAVID W. ALLAN Meas. Sci. Technol. 18 (2007) 1917?1928

The use of the Allan deviatio Abstract-Since a measurement is no better than its uncertainty, appropriate places to use classical statistics within this

specifying the uncertainty is a very important part of metrology. One is inclined to believe that the fundamental constants in physics are invariant with time and that they are the foundation upon which to build internationl system (SI) standards and metrology. Therefore clearly specifying uncertainties for these physical invariants at state-of-the-art levels should be one of the principal goals of metrology. However, by

measurement of the noise an discipline, however, they are limited. Since the statistical methods developed for time and

performance of microwave ra frequency metrology are generally applicable for any

equispaced time series, opportunity was taken to apply these methods in some other areas of metrology, namely,

the very act of observing some physical quantity we may perturb the standard, thus introducing uncertainties. The random deviations in a series of observations may be caused by the measurement system, by environmental coupling or by intrinsic deviations in the standard. For

standard voltage cells. Gauge block data were also stud-

ied but itations

thtoDestehVedLaptaraonwcdee1dr,euArneo.Pt TeLhqeeuvsiiescpkwa2cilealdnbydeieJdlidWsicnugHsssaeodnmdein3litmhe-

these reasons and because correlated random noise is as commonly occurring in nature as uncorrelated random noise, the universal use of the classical variance, and the standard deviation of the mean may cloud rather than clarify questions regarding uncertainties; l.e., these measures are well behaved only for random uncorrelated deviations

text.

1 Department of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, UK

It is wo2 rTthhernmoatliMngetrtohloagty,iNf aotinoenahl aPshyasicvaliaLbableoramtooryd,eTledfdoirngaton TW11 0LW, UK

time serie3sD, ivaisisopneocftCralilnidcaelnSsciiteyncems,oIdmeplerpiarloCpoollretgieo,nHaalmtmoefrsam,ith Hospital, Du Cane Road,

and if (X L==ond-on1,Wt1h2e0nNtNh,eUKclassical variance and standard

f IOP PUBLISHING

(white noise), and white noise is typically a subset of the MsEpAeSUcRtErMuEmNT SoCfIENCEdANeDvTiEaCHtiNoOLnOGEYa-rmeaidl:ivd.elarngde@ntp.hyBsiecsc.agluas.aec.oukf, tAhnedruebwi.lqeuviictko@usnpnl.acotu.urkeand j.hand@imperial.ac.uk

Meas.

Sci.

Technol.

18

(2007)

191o7b?1s9e2r8ved deviations. The assumption that each is independent because the measurements are

measurement taken at diffe

ridenonia:t1s0tei.1mr0iee88ss/0957or-e0f2a31s3o//1n8b/7nl/0eo1Ri8tsoeecaefsoikvretldhoe2w9q-ufJreaesnqtuiuoaernnycp2yo0s0ceo7dm, iinpnotfihnneeantlittfslo,er.pmAer2sh3iadpMesfarirotcmhis

2007

The use of the Allan deviation for the should be called into question if, in fact, the series is not random and uncorrelated, i.e., does not have a white spectrum. In this paper, studies of frequency standards, standard-volt cells, and gauge blocks pro-

frequencyPusbtalinsdhaerdds151M/ fayno2i0s0e7has been observed in im-

portant sOynsltienmesa:t sttarcaknss.iisotpo.rorgju/MncStTio/1n8s,/19s1e7miconductor

measurement of the noise and drift vide examples of long-term random-correlated time series which indi- diodes, reAsbissttorrasc,t thermistors, carbon microphones, thin cate behavior that is not "white" (not random and uncorrelated). This films, ligThthesouusrecoefs,thRe FAlplaronpdaegvaitaitoionnfflourcttuhaetaionnasly, saisndofisnignal noise and drift

performance of microwave radiometers paper outlines and illustrates a straightforward time-domain statistical approach, which for power-law spectra yields an alternative estimation method for most of the important random power-law processes encountered. Knowing the spectrum provides for clearer uncertainty as-

a surprisicnogmnpuomnebnetrs oisfcootnhseidr eprreodcienssthees c[o1n],te[x2t]o. fImf incorosiweave radiometry. The with ex no- is1e ibsefhoauvniodutroofbetwaorteyapseosnoafblmeicmroowdealveforradmioema-eter is modelled and surementcdoemvipaatrieodnswiinthbmaseiacssutraenmdeanrdtss oinf tgheenpeerraflo, rtmheanncteheof these radiometers

D V Land1, A P Levicskes2samnedntJinWthHe apnreds3ence of correlated random deviations, the statis-

tical approach outlined also provides a simple test for a white spectrum,

classical avnaarliyasnecde uasnindgstthaendAalrladnddeevviiaattiioonn mmeatyhohda.ve lim-

ited usefulness. The problem becomes significant when

1 Department of Physics and Atshturosnoamllyo,wUinnivgerasitmy oeftrGollaosggoiwst, GtolaskgnoowwG1w2h8eQtQhe, Ur Kuse of the classical vari- very longK-teeyrwmoradvs:erAagllianngdiesviuastieodn., Tmhiicsroiws apvreecraisdeiloymwethrya,t noise signal analysis

2 Thermal Metrology, Nationaal nPcheysiicsalsLuaibtoarbaltoeryo,rTewddhinegtthoenrTtWo11in0cLoWr,pUoKrate better uncertainty assess3 Division of Clinical Sciencesm, IemnpterpiarloCcoeldleugree, sH,ame.mge.,rsamsithouHtolsipnietadl, iDnutCheanepRapoaedr,.

London W12 0NN, UK

is required for maintenance of fixed standards which form the "invariant" building blocks of our measurement sys-

E-mail: d.land@physics.gla.ac.uk, Andrew.levick@npl.co.uk and j.hand@imperial.ac.uk I. INTRODUCTION

T Received 29 January 2007,

Published 15 May 2007

inIfiMnaEl foArmN2D3

MFaRrcEhQ20U0E7

N

CY

metrology

provides

some

Online at stacks.MST/o1f8/1th9e17 most accurate measurements known to man.

tem. 1. Introduction

In maintaining a set of standards and deriving calibra-

tions of oAtlhlemr esatsaunrdemarednst fsryosmtemtshehasveet,a smeveearsaulremqueensttiroensoslution arise. Twthoatimultpimorattaenlyt mounsetsbearleim: it1e)d btoy twhehramtaldlyeginredeucdedoerasndom

The Allan devia applied to the measu application to noise limited (Allan 1987, H

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