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

12 transits (2017

onds. Afterwards it starts increasing again in response to

kHz bandwidth)

Formula

of the

dishfor

paper

(Figpart

19) of

rotation

of in

thesection

sky. The6.6

Allan

variance

the real

declination aroun

the visibility is calculated as follows:

degrees in right a

arcmin resolution

1

The visibilit

? 2 (m¡¤0 ) =

?

2

2(m¡¤0 ) (N ¡Ù 2m)

array, where 720

N ¡Ù2m

of observation, ea

?

2

(Re [Vn+2m ¡Ù 2Vn+m + Vn ])

(3)

interval, and 121

n=1

cross-correlations

and 1. A comple

where ? 2 (m¡¤0 ) is the overlapping Allan deviation at an avcomparing the o

eraging time of ¡¤ = m¡¤0 , Vn is the time-series of the visifor the Tianlai a

bility, spaced by measurement interval ¡¤0 = 1 s with length

with only one po

N ? 438 ? 103 seconds (9 nights), depending on the baseUsing the s

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

construct a clean

variance between chunks of data of equal size ¡¤ .

size, the four hou

parts, each cover

map tiles, each co

7 MAPS AROUND SOURCES

guard area have

xi

to obtain the fu

The 16 dishes of the array provide 16 auto-correlations

and

in figure 20 (bot

120 cross correlations visibilities for each of the two linmap quality

ear polarisations (HH or VV), as well as 256

po-corresponds

to the is th

yi crossThis

visible.

OnZithe o

larisation (HV) visibilities. To illustrate the array perforvariable which

I call

duces low amplit

mance, we have reconstructed sky maps around few bright

Gaussian beam i

point sources by combining single linear polarisation

HH or

y

i is simply the successive

We also ma

VV signals. The sky maps shown here have been obtained

xFigure

i+1 - xi 21 and 2

through several algorithms which characteristics aredifferences

briefly

using 1 hour of d

outlined in Appendix C.

frequency chann

From wikipedia page:



xn correspond to clock ticks, which

integrate (count) an oscillator signal

?

?

If a

A&A 373, 746¨C756 (2001)

DOI: 10.1051/0004-6361:20010611any

test procedure is defined for use at any time

and at

Astronomy

¦Òs2 (T ) =

location, it needs to be as simple and unique

as

pos&

c ESO 2001

?

Astrophysics

sible. Therefore, we understand the Allan variance

as the gsr (T ) is

ordinary statistical variance of the di?erence of two con- two data

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

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

instantaneous

output signal

of a spectrometer

channel or of spectro

Optimization

of heterodyne

observations

using

of a continuum

detector

for example. The output is now equivalen

Allan

variance

measurements

integrated for a time interval T representing an estimate

If we

of the mean

signalandwhich

is stored as spectrometer data (¦Ò2 (T ) =

R. Schieder

C. Kramer

r

in the computer:

I. Physikalisches Institut, Universita?t zu Ko?ln, Zu?lpicher

2

! t Stra?e 77, 50937 Ko?ln, Germany

¦ÒA

(T ) =

¡ä

¡ä

x(T,

t)

=

1/T

s(t

)dt

.

(1)

Received 11 January 2001 / Accepted 26 April 2001

t?T

Accordin

Abstract. Stability tests based on the Allan variance method have become a standard procedure for the evaluation

The expectation

x(T,

t) and

is simulate

therefore

identical

of the quality of radio-astronomical

instrumentation. value

They areof

very

simple

the situation

when smaller th

detecting weak signals buriedwith

in largethe

noiseexpectation

fluctuations. For of

the s(t).

special For

conditions

observations

outline as there i

the during

observation

ofanweak

of the basic properties of the Allan variance is given, and some guidelines how to interpret the results of the

certain

number Ntreatment

of di?erences

two ofin measurem

measurements are presented.signals,

Based on aarather

simple mathematical

clear rules for of

observations

¡°Position-Switch¡±, ¡°Beam-¡± these

or ¡°Frequency-Switch¡±,

¡°On-The-Fly-¡± and ¡°Raster-Mapping¡±

are derived. butions fr

data, a ¡°signal-measurement¡±

xs and amode

¡°referenceAlso, a simple ¡°rule of the thumb¡± for an estimate of the optimum timing for the observations is found. The

measurement¡± x , are subtracted from each other:

the simpl

analysis leads to a conclusive strategy how to planr radio-astronomical observations. Particularly for air- and

space-borne observatories it is very important to determine, how the extremely precious observing time can be signal to

d The

= xanalysis

(2)

s ? xrshould help to increase the scientific yield in such cases significantly.

used with maximum e?ciency.

technique

that the¨C methods:

desireddata

signal

alone

becomes

visible

when

av- in radio-a

Key words. instrumentation: so

miscellaneous

analysis,

observational

¨C space

vehicles:

instruments

¨C techniques: spectroscopic ¨C eraging.

telescopes Typically, each of the two measurements are done

laborator

at di?erent times, after the telescope has moved between switching

tion as is

two positions on sky.dent. Thus, it is always necessary to verify the similarity

1. Introduction

of

all

frequency

channels

of

the

spectrometer

by

investiIn order

to obtain a plausible estimate of the error of

We h

Allan variance measurements have been

demonstrated

gating the baseline noise of measured spectra for examas a useful tool for the characterization

of the stabilthe di?erence

we useple.the

standard

of the

about th

Typical

problemdefinition

areas for instance

arevarilight scatter

ity of radio-astronomical equipment

such

as

Millimeter

ance:

For our a

problems in acousto-optical spectrometers (AOS), where

or Submillimeter-receivers or large bandwidth back-ends

speckles may a?ect individual channels more heavily than

rived from

(Schieder et al. 1985; Kooi et al. 2000).

2 Particularly for the 2 others. 2The same 2is true for filterbanks which have occa¦Ò

(T

)

=

?(d

?

?d?)

?

=

?d

?

?

?d?

.

d

development of acousto-optical spectrometers (AOS) at

channel.

sionally same peculiar channels even in a well maintained

the Ko?lner Observatorium fu?r Sub-Millimeter Astronomy

integratio

back-end system. But in all normal cases of well behaved

brackets

(KOSMA) the method has playedThe

a very

important¡±??¡±

role,stand for the expectation value. In cominstrumentation, the Allan variance plot is a most useful

other in

because it provides clear evidenceparison,

that the spectrometers

this definition

is similar

to the

original

method

to precisely

characterize

thedefinition

instrumentation in

are well suited for the use at an observatory

by means

of a

of the Allan

variance

(Allan 1966), if one considers a situa- the two m

use.

reliable test laboratory procedure (Tolls et al. 1989). The

the instru

In general,

observations

at an observatory

tion,makes

where

theeasy

expectation

value

of the di?erence

is zeroare done

simple definition of the Allan variance

it very

with the available instrumentation as is, and it can and

not ¡°r¡± m

to apply such measurements also which

for the characterization

is practically be¡°normal¡±

radio-astronomical

modified orduring

even improved

by the observer. On the

of the stability of other instruments,

a very elementary

observations:

contrary, the observer has to find the correct observing

The corre

The formulae in section 6.6 of the paper correspond to the

formulae above with xn replaced by real (or imag) of the

visibility - The yn correspond to the fractional clock period

change (yn = 1/¦Ó ( xn+2 - 2 xn+1 + xn) )

¦Ò (T ) = 1/2?d ?¡¤

For further treatment we use the standard definition of the

NO : yn correspond to successive differences

= xi+1 - xi

variance, but leave the factor of 1/2 in place for historical

since it was already introduced by Allan in 1966.

We need to replace the visibility by the reasons,

average

of visibility over

Thus we use :

¦Ò (T ) = 1/2?(d ? ?d?) ? = 1/2[?d ? ? ?d? ].

(3)

time m ¦Ó

case is the definition of the quality of a simple Lock-In

2

2

amplifier for example.

A as used in radioFor a real time spectrometer,

astronomy with many simultaneously operating frequency

channels, it is a very important condition that all channels

are behaving identically in a statistical sense. Therefore,

the use of the Allan variance for the investigation of the

1

performance of the spectrometer is based on the assumption that there are no di?erences between di?erent frequency channels. That this is not 2always correct is evi-

parameters in order to use the available hardware in a

?[x /x ?

most economic way. It is the purpose of this paper tos r

In case the

develop a strategy for an optimization of the observing

?s(t)?, one

process. For this the knowledge of the stability parameters is decisive. Once this information is available from

new defini

an Allan variance measurement for example, it should

even at va

be a rather straightforward matter to determine the es2

sential parameters like length of integration per position In gen

only a fin

on sky et cetera. The following mathematical treatment

2

2

2

analyses

the

commonly

used

observing

methods,

i.e.

¡°Poance. The

A

sition-¡±, ¡°Beam-¡± or ¡°Frequency-Switch¡±, ¡°On-The-Fly¡±

Send o?print requests to: R. Schieder,1

dard defin

This original definition

through theordi?erence

of samples

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

(OTF) measurements

¡°Raster-Mapping¡±

based on the "

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

Article published by EDP Sciences and available at or

1/N

N

n=

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?

DAVID W. ALLAN

IOP PUBLISHING

Meas. Sci. Technol. 18 (2007) 1917¨C1928

The use of the Allan deviatio

measurement of the noise an

performance of microwave ra

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

discipline, however, they are limited.

is inclined to believe that the fundamental constants in physics are inSince the statistical methods developed for time and

variant with time and that they are the foundation upon which to build

frequency metrology are generally applicable for any

internationl system (SI) standards and metrology. Therefore clearly

equispaced time series, opportunity was taken to apply

specifying uncertainties for these physical invariants at state-of-the-art

levels should be one of the principal goals of metrology. However, by

these methods in some other areas of metrology, namely,

the very act of observing some physical quantity we may perturb the

standard voltage cells. Gauge block data were also studstandard, thus introducing uncertainties. The random deviations in a

ied but these data were not equispaced yielding some limseries of observations may be caused by the measurement system, by

1, A P Levick2 and J W Hand3

V Land

procedure.

These will be discussed in the

itations toDthe

environmental coupling or by intrinsic deviations in the standard. For

text.

these reasons and because correlated random noise is as commonly oc1

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

curring in nature as uncorrelated random noise, the universal use of

noting

that ifNational

one has

a viable

model

for a TW11 0LW, UK

It is worth

2

Thermal

Metrology,

Physical

Laboratory,

Teddington

the classical variance, and the standard deviation of the mean may

3

time series,Division

a spectral

density

model

proportional

to fa, Hospital, Du Cane Road,

of Clinical

Sciences,

Imperial

College, Hammersmith

cloud rather than clarify questions regarding uncertainties; l.e., these

0NN,

and if (X London

== - 1,W12

then

theUK

classical variance and standard

measures are well behaved only for random uncorrelated deviations

are divergent.

Because of the

ubiquitous natureand j.hand@imperial.ac.uk

(white noise), and white noise is typically a subset of the Mspectrum

ofCIENCE deviation

E-mail:

d.land@physics.gla.ac.uk,

Andrew.levick@npl.co.uk

IOP PUBLISHING

EASUREMENT S

AND TECHNOLOGY

of 1/ f noise for low-frequency components, perhaps it is

observed deviations. The assumption that each measurement in a series

Meas. Sci. Technol. 18 (2007) 1917¨C1928

doi:10.1088/0957-0233/18/7/018

is independent because the measurements are taken at different times

Januaryposed

2007, in

in the

finaltitle.

formAside

23 March

reasonbleReceived

to ask the29question

from 2007

should be called into question if, in fact, the series is not random and

2007has been observed in imstandards151/May

f noise

frequencyPublished

uncorrelated, i.e., does not have a white spectrum. In this paper, studOnline

at

stacks.MST/18/1917

portant systems: transistor junctions, semiconductor

ies of frequency standards, standard-volt cells, and gauge blocks prodiodes, resistors, thermistors, carbon microphones, thin

vide examples of long-term random-correlated time series which indiAbstract

films, light sources, RF propagation fluctuations, and in

cate behavior that is not "white" (not random and uncorrelated). This

The use of the Allan deviation for the analysis of signal noise and drift

paper outlines and illustrates a straightforward time-domain statistical

a surprising number of other processes [1], [2]. If nosie

components is considered in the context of microwave radiometry. The

approach, which for power-law spectra yields an alternative estimation

with ex noise

- 1 is found to be a reasonable model for meabehaviour of two types of microwave radiometer is modelled and

method for most of the important random power-law processes ensurement

deviations

in basic

standards of

in the

general,

then theof these radiometers

compared with

measurements

performance

countered. Knowing the spectrum provides for clearer uncertainty asclassical analysed

variance using

and standard

may have limsessment

presence

of correlated random deviations, the statisthe Allandeviation

deviation method.

2 and JinWthe

3

D V Land1, A P Levick

Hand

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

ited usefulness. The problem becomes significant when

1

thus

allowing

a

metrologist

to

know

whether

use

of

the

classical

variKeywords:

Allan deviation,

microwave

radiometry,

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

very long-term

averaging

is used. This

is precisely

what noise signal analysis

2

Thermal Metrology, Nationalance

Physical

Laboratory,

0LW, UK

is suitable

orTeddington

whetherTW11

to incorporate

better uncertainty assessis required for maintenance of fixed standards which form

3

Division of Clinical Sciences,ment

Imperial

College, Hammersmith

Hospital, in

Duthe

Canepaper.

Road,

procedures,

e.g., as outlined

The use of the Allan deviation for the

measurement of the noise and drift

performance of microwave radiometers

London W12 0NN, UK

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, inIME

final form

23 March

2007

AND

FREQUENCY

metrology provides some

Published 15 May 2007

of the most accurate measurements known to man.

Online at stacks.MST/18/1917

the "invariant" building blocks of our measurement system.

1. Introduction

In maintaining a set of standards and deriving calibrations of other

standards from

thehave

set,a several

questions

All measurement

systems

measurement

resolution

important

ones

are:

1)

to

what

degree

does

arise. Two

that ultimately must be limited by thermally induced random

The Allan devia

applied to the measu

application to noise

limited (Allan 1987, H

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