1. Principles of Time Scales
Judah Levine
Time and Frequency Division
NIST Boulder

2. Outline Time scale principles Adding a steered clock
Examples of special cases
AT1 and EAL
Large Drift or Long averaging
Large measurement noise or near real-time
The general problem
Kalman Solution
Adding a steered clock
Steering the time scale

3. What and why?
A time scale is a procedure for combining the data from several clocks
Inputs:
(Initial estimates of the statistical characteristics of each member)
Measurements of times or frequencies of all members with respect to a reference device
Reference device need not be special

4. What and why?
A time scale is a procedure for combining the data from several clocks
Outputs:
ensemble time and frequency
Statistical performance of each member
Update to model for each clock
(Physical realization of ensemble time)

5. What and why? Advantages: Minimize single points of failure
Output does not depend on a single device
Ensemble provides error detection
Get the best of each contributor
Nominally identical clocks may not be equal
Combine clocks with different properties

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

7. TDEV of measurement systems in seconds, common clock into two channels
Averaging time, s
sec

8. Time Scale Clock Model Each clock in time scale has iterative model:
AT1 Model: j=j=0 for all j
Measurement interval, clock model, and noise parameters
are related and must be considered together

9. Variance in AT1 clock model
In AT1 model, variance of time differences
Is due to pure white frequency noise
Frequency drift is constant parameter

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

11. Ensemble Time Computed as weighted average of each clock
Weight derived from prediction error on previous cycles
Sum of weights is 1
Statistically
optimum
weights

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
Statistically ok over WFM
noise domain
Statistically difficult,
Estimate not robust

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

14. Clock correlation Correction - 2
Statistical Weight Adjustment (Tavella, EFTF):
Administrative weight limiting: NIST: 30%, EAL: 2.5/N
Weight limiting always degrades the time scale
Most serious in small ensemble with very different true weights

15. Error detection and clock resets
Assume clock error if:
NIST model:
K < 3: no error
3<k<4:
k>4:
Error is modeled as a single time step with
no change in frequency or drift parameters

16. The frequency drift problem
Suppose:
Frequency variance no longer white frequency noise
AT1-type algorithm no longer statistically robust
AT1-type algorithms cannot be used when t too
large and frequency drift has significant variance

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

18. The measurement noise problem
Suppose:
Measured time differences due to two sources
Time state differences no longer time differences
Frequency estimator no longer statistically robust
AT1-type algorithms cannot be used
at sufficiently short averaging times

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

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)

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

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

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

24. Clock 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

25. Statistics of a real-time ensemble
Interaction between weighting algorithm
and clock noise usually results in random
walk at longer term
Every ensemble requires
external data for steering

26. Steered clock Measurement system and Data Clocks time scale from are
computation
Data
from
clock
ensemble
Clocks
are
not
steered
Steering
Control
Phase
stepper
Steered output

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
xsx0+y(t-t0)+0.5*d*(t-t0)2

28. Statistics of the steered output
Free-running performance defined by
statistics of steered clock reference
Time Noise in the reference clock
for the phase stepper: 510-131/2 =
13 12 minutes
Steering loop drives steering error to 0
Long-period performance defined by
stability of the scale

29. Types 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

32. Summary Advantages of time scale algorithms:
More robust – no single point of failure
Provides statistical evaluation of members
Provides a natural platform for steering
Problems with time scale algorithms
Real clock behavior may not conform to model
High-weight clocks are difficult to handle
Error handling/Reset algorithm is arbitrary and not always statistically robust
Variable frequency drift hard to model
Real-time time scales have special problems

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