principles of time scales

Judah Levine, NIST, CENAM, Oct2012 1

Principles of Time ScalesPrinciples of Time Scales

Judah LevineJudah LevineTime and Frequency DivisionTime and Frequency Division

NIST BoulderNIST [email protected]

303 497-3903303 497-3903

mailto:[email protected]


OutlineOutline Time scale principlesTime scale principles

– Examples of special casesExamples of special cases• AT1 and EALAT1 and EAL• Large Drift or Long averagingLarge Drift or Long averaging• Large measurement noise or near real-timeLarge measurement noise or near real-time

– The general problemThe general problem• Kalman SolutionKalman Solution

Adding a steered clockAdding a steered clock Steering the time scaleSteering the time scale


What and why?What and why? A time scale is a procedure for A time scale is a procedure for

combining the data from several combining the data from several clocksclocks

Inputs:Inputs:– (Initial estimates of the statistical (Initial estimates of the statistical

characteristics of each member)characteristics of each member)– Measurements of times or frequencies of Measurements of times or frequencies of

all members with respect to a reference all members with respect to a reference devicedevice• Reference device need not be specialReference device need not be special


What and why?What and why? A time scale is a procedure for A time scale is a procedure for

combining the data from several combining the data from several clocksclocks

Outputs:Outputs:– ensemble time and frequencyensemble time and frequency– Statistical performance of each memberStatistical performance of each member– Update to model for each clockUpdate to model for each clock– (Physical realization of ensemble time)(Physical realization of ensemble time)


What and why?What and why? Advantages:Advantages:

– Minimize single points of failureMinimize single points of failure– Output does not depend on a single Output does not depend on a single

devicedevice– Ensemble provides error detectionEnsemble provides error detection– Get the best of each contributorGet the best of each contributor

• Nominally identical clocks may not be Nominally identical clocks may not be equalequal

• Combine clocks with different propertiesCombine clocks with different properties


Partition of input time differences

Noise of the measurement process– Time noise with no frequency aspect

Deterministic model of each clock Stochastic contribution of each

clock Non-statistical glitches for each

clock


TDEV of measurement systems in seconds, common clock into two channels

1.00E-14

1.00E-13

1.00E-12

1.00E+02 1.00E+03 1.00E+04 1.00E+05 1.00E+06

Averaging time, s

sec


Time Scale Clock ModelEach clock in time scale has iterative model:

jkjkj

jkjkjkj

jkjkjkjkj

tdtd

ttdtyty

ttdttytxtx

)()(

)()()(

))((21)()()(

1

11

2111

AT1 Model: j=j=0 for all j

Measurement interval, clock model, and noise parametersare related and must be considered together


Variance in AT1 clock model

)()(

)()()(

))((21)()()(

1

11

2111

kjkj

jkjkjkj

kjkjkjkj

tdtd

ttdtyty

ttdttytxtx

In AT1 model, variance of time differences Is due to pure white frequency noise

Frequency drift is constant parameter


AT1 Algorithm, continued Measured time differences represent

differences of time states of clocks Frequency estimate has deterministic and

white noise contributions– Averaging statistically appropriate

• Time constant determined by flicker frequency floor– Frequency estimate (x/t) freq. state y(tk)

Drift parameter determined outside of algorithm– Treated as a constant by AT1


Ensemble Time Computed as weighted average of each clock

– Weight derived from prediction error on previous cycles

– Sum of weights is 1

222

21

2

22

21

2211

...

/1

1......

n

jj

jj

n

nne

w

if

wwwxwxwxwx

Statisticallyoptimumweights


Ensemble Frequency and Drift AT1 algorithm does not explicitly

calculate these parameters– Ensemble frequency is time evolution of

ensemble time– Ensemble frequency drift is time evolution of

ensemble frequency

ttyty

td

ttxtx

ty

kjkjkj

kjkjkj

)()()(

)()()(

1

1 Statistically ok over WFMnoise domain

Statistically difficult, Estimate not robust


Clock Correlation Correction - 1 Every clock is a member of ensemble

used to evaluate its performance Prediction error is always too small

– Weight is biased too large– Error detection is degraded– Positive Feedback loop

2

22

1~)(

)()(~)(

jkj

kkekjkj

tw

txtxt


Clock correlation Correction - 2

Administrative weight limiting: NIST: 30%, EAL: 2.5/N

Weight limiting always degrades the time scaleMost serious in small ensemble with very different true weights

Statistical Weight Adjustment (Tavella, EFTF):

)(11

86400)(

calcwtusedw

jj


Error detection and clock resets

)()()( 1 kjkekj tktxtx

)()4()( kjkj twktw

Assume clock error if:

NIST model:

K < 3: no error

3<k<4:

k>4: 0)( kj tw

Error is modeled as a single time step with no change in frequency or drift parameters


The frequency drift problem

jkjkj

jkjkjkj

jkjkjkjkj

tdtd

ttdtyty

ttdttytxtx

)()(

)()()(

))((21)()()(

1

11

2111

Suppose: jj t

Frequency variance no longer white frequency noiseAT1-type algorithm no longer statistically robust

AT1-type algorithms cannot be used when t too large and frequency drift has significant variance


Frequency Drift Solutions Short measurement interval

– Frequency variance approximately wfm

Mixed ensemble computed iteratively– Separate computation for clocks with

negligible drift Full Kalman algorithm

– Complex and difficult to handle errors


The measurement noise problem

jkjkj

jkjkjkj

jkjkjkjkj

tdtd

ttdtyty

ttdttytxtx

)()(

)()()(

))((21)()()(

1

11

2111

Suppose: 211 ))((

21)( ttdtty kjkjj

Measured time differences due to two sourcesTime state differences no longer time differencesFrequency estimator no longer statistically robust

AT1-type algorithms cannot be used at sufficiently short averaging times


Significant Measurement Noise Problem important when time

differences are noisy or as t 0– AT1 algorithm cannot be used for

near real-time systems Measured time differences must

be partitioned into measurement noise and clock noise– Measurement noise must not degrade

clock parameter estimates


Kalman Solution Partition variance of

measurements based on initial estimates of noise parameters and covariance matrix– Jones and Tryon, TA(NBS)– GPS Composite clock (Brown)– KAS2 (Sam Stein, Symmetricom)


Summary - 1 AT1-type algorithms assign

variance to frequency noise– Measurement noise very small– Frequency drift constant (or 0)

Errors are modeled as simple time steps with no change in parameters


Summary - 2 AT1-type algorithms are appropriate only

over a range of averaging times determined from the clock statistics– Lower limit from measurement noise– Upper limit from frequency variance

Kalman-type algorithms can handle more complex noise types– More sophisticated partition of measured

variance– Reset/Error detection more difficult to handle

• Reset machinery is outside of statistical considerations


Correlations among clocks Time scale algorithms assume

variance of clocks is not cross-correlated

Common-mode effects are a serious problem– Common time step in high-weight

clocks• Wrong clocks are reset


Clock steeringClock steering

Time and frequency of the scale are paper parameters

Scale algorithm defines offset of each member relative to the ensemble average

No member clock realizes the ensemble-average values


Statistics of a real-time ensembleStatistics of a real-time ensemble

Interaction between weighting algorithmand clock noise usually results in random

walk at longer term

Every ensemble requires external data for steering


Steered clockSteered clock

Measurement system

and time scale

computation

Datafrom clockensemble

Phasestepper

SteeringControl

Steered output

Clocksarenot steered


Steered Clock Error Signal Steered clock usually steered

based on time:– Simple steering drives xs 0

• Steered clock realizes ensemble time– More complex steering

• Steered clock is UTC(lab) steered to UTC– Error signal is UTC(lab)-UTC from Circular T

• xsx0+y(t-t0)+0.5*d*(t-t0)2


Statistics of the steered outputStatistics of the steered outputFree-running performance defined by statistics of steered clock reference

Time Noise in the reference clock for the phase stepper: 510-131/2 = 13 ps @ 12 minutes

Steering loop drives steering error to 0

Long-period performance defined bystability of the scale


Types of steering algorithmsTypes of steering algorithms

Time-driven: Minimize time error

Frequency driven: Minimize frequency excursions

Bang-bang Drift: Frequency and time continuous

Steering algorithm set by administrative considerations and by needs of users

No Universal “perfect” solution


References Realizing UTC(NIST) at a Remote

Location– Metrologia, Vol. 45, page S23, 2008

Other papers in this volume of Metrologia

The Statistical Model of Atomic Clocks and the Design of Time Scales– Review of Scientific Instruments, Feb.

2012

principles of time scales

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