PTTI-tutorial

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Clock Statistics: A TutorialDon PercivalApplied Physics LaboratoryUniversity of Washington, SeattleMotivating Example: I• consider following measurements:30-40-11010 1.160 0-10 -1.160 256 512t (days)− top: X =time (phase) difference between clock 55 andtUSNO time scale at day t (adjusted for systematic drift)(1)− bottom: X = X −X ∝ fractional frequency deviatet t−1taveraged over one day1X - X (ns) X (ns)t t-1 t13parts in 10Motivating Example: II• clock statistics used to summarize performance(1)− if X constant, clock 55 agrees with time scale (essentially)t(1)−X has stochastic (noise-like) fluctuationst− statistics used to quantify fluctuations• sample statistics(1)N−11− mean: µˆ = Xtt=0N(here N =512=# of measurements)(1)N−12 1 2− variance: σˆ = (X −µˆ)t=0 tN−σˆ (standard deviation) is measure of spread• easiest to interpret µˆ &ˆ σ if data taken to be independentsamples from Gaussian (i.e., normal) distribution2Motivating Example: III• Q: is Gaussian assumption reasonable?• comparison of histogram to probability density function:0.20.10.0-10 -5 0 5 10x (ns)− Gaussian assumption seems reasonable3PDFsMotivating Example: IV• Q: is independent assumption reasonable?• under Gaussianity, uncorrelatedness implies independence• sample autocorrelation sequence measures uncorrelatedness:(1) (1)N−τ−1(X −µˆ)(X −µˆ)t=0 t t+τρˆ = ,τ =1, 2,...,N− 1τ(1)N−12(X −µˆ)tt=0• can interpret ρˆ as correlation ...

Informations

• Publié par: Phaen
• Nombre de lectures: 22
• Langue: English

Extrait

Clock Statistics: A Tutorial

Don Percival

Applied Physics Laboratory University of Washington, Seattle

Motivating Example: I

•consider following measurements: 30

-40

-110 10

0

-10

0

256 t (days)

512

1.16

0

-1.16

−top:X t =time (phase) difference between clock 55 and USNO time scale at dayt(adjusted for systematic drift) (1) −bottom:X t =X t −X t−1 ∝fractional frequency deviate averaged over one day

1

Motivating Example: II

•clock statistics used to summarize performance −ifX t (1) constant, clock 55 agrees with time scale (essentially) −X t (1) has stochastic (noise-like) fluctuations −statistics used to quantify fluctuations •sample statistics −mean:µˆ = N 1  t N = −1 0 X t (1) (hereN=512 =# of measurements) −airav:ecnσˆ 2 = N 1  t N = −1 0 (X t (1) −ˆ) 2 µ −σ(standard deviation) is measure of spread ˆ • ˆeasiest to interpretµ& ˆσif data taken to be independent samples from Gaussian (i.e., normal) distribution

2

Motivating Example: III

•Q: is Gaussian assumption reasonable?

•comparison of histogram to probability density function:

0.2

0.1

0.0

-10 -5

0 x (ns)

5

−Gaussian assumption seems reasonable

3

10

Motivating Example: IV

•Q: is independent assumption reasonable? •under Gaussianity, uncorrelatedness implies independence •sample autocorrelation sequence measures uncorrelatedness: ρˆ τ =  t N =0 −τ  −1 t N ( = X 0 − t 1 (1 ( ) X − (1) µˆ)(X µˆ t ( ) 1) + 2 τ −µˆ)  ,, . N1 τ= 1,2, . .− − t •can interpretρˆ τ as correlation coefficient: 1

0

-1

0

τ (days)

32

−sinceρˆ τ ≈0, uncorrelatedness seems reasonable

4

Conclusions from Motivating Example

•X t (1) well-modeled as uncorrelated Gaussian deviates (sometimes called Gaussian white noise) •theory saysµ&σˆ 2 are sufficient statistics for summarizing ˆ statistical information about clock 55

•implies ‘random walk’ model for time difference dataX t •seems we need little more than what is taught in ‘Statistics 101’

5

Reality Bites!

•alas, other clocks do not have such simple statistical properties

30

-40

-110 -110

-135

-160 10

0

1

0

-1 1

0

-1 1

0

-10 -1 0 256 512 0 32 t (days) τ (days) •µˆ & ˆσ 2 not sufficient summaries for clock in middle plot 6

Overview of Remainder of Tutorial

•discussion of models for interpreting clock statistics

−models specified via spectrum (spectral density function) −while white noise & random walk models depend onµ&σ 2 , more comprehensive models depend onµand spectrum −in simplest case, spectrum itself depends on 2 parameters ∗σ  2 , a parameter setting overall level of spectrum ∗α, a so-called ‘power law’ parameter

•clock statistics based upon 2 variance decompositionslook at

−spectral analysis −wavelet analysis

7

The Spectrum

•letX t be a stochastic process, i.e., collection of random vari- ables (RVs) indexed byt •suppose further thatX t is stationary •implies certain theoretical properties do not change with time •in particular, its varianceσ 2 =var{X t }is the same for allt •spectrumS X (·) decomposesσ 2 across frequenciesf: /2 var{X t }=  1 S X (f)df −1/2 hereffrequency with units of cycles per unit timeis a Fourier (e.g., cycles per day for process sampled once per day)

8

Physical Interpretation of Spectrum via Filtering

•leta u be a filter, and formY t =  u ∞ =−∞ a u X t−u •Y t has spectrumS Y (f) =A(f)S X (f), where A(f) =  ∞  a u e −i2πf u  2 is squared gain function u=−∞ •ifa u narrow-band of bandwidth ∆faboutf, i.e., 1 A(f  ) =  2 0, ∆f , f oth − er ∆ w 2 f ise ≤ , |f  | ≤f+ ∆ 2 f then have following interpretation forS X (f):  1/2 S Y (f  )df  =  −1 1/ / 2 2 A(f  )S var{Y t }= X (f  )df  ≈S X (f) −1/2

9

Spectrum for White Noise Process

•simplest stationary process is white noise • t is white noise process if −E{ t }=µ  for allt(usually takeµ  =0), whereE{ t }denotes expected value of RV t −var{ t }=σ  2 for allt − t and t  are uncorrelated for allt=t  •spectrum for white noise is justS  (f) =σ  2 •note that  

1/2  −1 1 / /2 2 σ  2 df=σ  2 =var{ t }, S  (f)df= −1/2 as required

10