principles of time scales
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Principles of Time Scales. Judah Levine Time and Frequency Division NIST Boulder [email protected] 303 497-3903. Outline. Time scale principles Examples of special cases AT1 and EAL Large Drift or Long averaging Large measurement noise or near real-time The general problem - PowerPoint PPT PresentationTRANSCRIPT
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
Judah Levine, NIST, CENAM, Oct2012 2
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
Judah Levine, NIST, CENAM, Oct2012 3
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
Judah Levine, NIST, CENAM, Oct2012 4
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)
Judah Levine, NIST, CENAM, Oct2012 5
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
Judah Levine, NIST, CENAM, Oct2012 6
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
Judah Levine, NIST, CENAM, Oct2012 7
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
Judah Levine, NIST, CENAM, Oct2012 8
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
Judah Levine, NIST, CENAM, Oct2012 9
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
Judah Levine, NIST, CENAM, Oct2012 10
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
Judah Levine, NIST, CENAM, Oct2012 11
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
Judah Levine, NIST, CENAM, Oct2012 12
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
Judah Levine, NIST, CENAM, Oct2012 13
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
Judah Levine, NIST, CENAM, Oct2012 14
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
Judah Levine, NIST, CENAM, Oct2012 15
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
Judah Levine, NIST, CENAM, Oct2012 16
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
Judah Levine, NIST, CENAM, Oct2012 17
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
Judah Levine, NIST, CENAM, Oct2012 18
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
Judah Levine, NIST, CENAM, Oct2012 19
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
Judah Levine, NIST, CENAM, Oct2012 20
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)
Judah Levine, NIST, CENAM, Oct2012 21
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
Judah Levine, NIST, CENAM, Oct2012 22
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
Judah Levine, NIST, CENAM, Oct2012 23
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
Judah Levine, NIST, CENAM, Oct2012 24
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
Judah Levine, NIST, CENAM, Oct2012 25
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
Judah Levine, NIST, CENAM, Oct2012 26
Steered clockSteered clock
Measurement system
and time scale
computation
Datafrom clockensemble
Phasestepper
SteeringControl
Steered output
Clocksarenot steered
Judah Levine, NIST, CENAM, Oct2012 27
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
Judah Levine, NIST, CENAM, Oct2012 28
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
Judah Levine, NIST, CENAM, Oct2012 29
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
Judah Levine, NIST, CENAM, Oct2012 30
Judah Levine, NIST, CENAM, Oct2012 31
Judah Levine, NIST, CENAM, Oct2012 33
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