Principles of Time Scales Judah Levine Time and Frequency Division NIST Boulder jlevine@boulder.nist.gov 303 497-3903

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

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

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)

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

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 Averaging time, s sec

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

Variance in AT1 clock model 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 Statistically optimum weights

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

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

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

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

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

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 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

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 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 ensemble Interaction between weighting algorithm and clock noise usually results in random walk at longer term Every ensemble requires external data for steering

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

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

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 ps @ 12 minutes Steering loop drives steering error to 0 Long-period performance defined by stability of the scale

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

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

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