Kalman Filtering¶
- class ptp.kalman.KalmanFilter(data, T, N=1, obs_model='scalar', s_0=None, P_0=None, R=None, Q=None)[source]¶
Kalman Filter for Time/Frequency Offset Processing
The Kalman state vector is:
s[n] = [ x[n] ] [ y[n] ]
that is, composed by the time offset (\(x[n]\)) and frequency offset (\(y[n]\)).
The recursive model for the time offset is:
\[x[n] = x[n-1] + y[n-1]*T + w_x[n],\]where \(w_x[n]\) is the time offset state noise.
The recursive model for the frequency offset is:
\[y[n] = y[n-1] + w_y[n],\]where \(w_y[n]\) is the frequency offset state noise.
Thus, the state transition matrix becomes:
A = [ 1 T ], [ 0 1 ]
such that:
\[s[n+1] = A*s[n] + w[n],\]where \(w[n] \sim N(0, Q)\) is the state noise vector and Q is the so-called state noise covariance matrix.
The state noise covariance matrix takes the expected variability of the true state into account. The true frequency offset changes over time due to several oscillator noise sources. Similarly, the time offset uncertainty comes from phase noise and other effects. Both of these uncertainties are small in magnitude, so that the transition covariance matrix is expected to have small values.
The model in [1] considers random-walk in both time and frequency. These are given in terms of normalized variances, which must be scaled by the observation period (Sync message period). Here, in contrast, we consider non-normalized variances, “var_y” and “var_x”.
Q = [ var_x, 0, ] [ 0, var_y ]
In addition to the state model, the Kalman filter also uses a measurement model.
The measurement model can rely on scalar observations, as presented in [2], where only the time offset is observed, or vector observations, as presented in [1], where both time and frequency offsets are observed.
In the scalar-observation model, the observation is given by:
\[z[n] = x[n] + v_x[n],\]where \(z[n]\), \(x[n]\) (time offset), and \(v_x[n]\) are all scalars.
In the vector-observation model, it is given by:
z[n] = [ x[n] ] + [ v_x[n] ] [ y[n] ] [ v_y[n] ],
where \(z[n]\) is (2x1).
The observed values come directly from the raw time and frequency offset measurements that are taken after each delay request-response exchange. That is, \(z[n] = [\tilde{x}[n], \tilde{y}[n]]^T\).
The observation covariance matrix reflects the confidence on the observations. Both time and frequency offset observations are expected to be very noisy. Thus, it is important to define sufficiently high variances/covariances in this covariance matrix.
When ignoring the timestamping noise (e.g., due to the timestamp granularity) and assuming that the PDV is the predominant noise, the observation noise covariance matrix can be shown to be given by:
R = [ (Var{d_ms} + Var{d_sm})/4 ]
when the scalar-observation model is used, or:
R = [ (Var{d_ms} + Var{d_sm})/4 , Var{d_ms} / (2*N*T) ] [ Var{d_ms} / (2*N*T) , 2*Var{d_ms} / (N*T)**2 ]
when the vector-observation model is used.
The former (scalar-observation) is easier to compute in practice because it can be computed by “Var{d_est}”, namely the variance of the delay estimates taken from PTP two-way exchanges. In contrast, the latter (vector-observation) requires the knowledge of the individual m-to-s and s-to-m variances, which a practical slave does not know. This formulation also assumes that m-to-s and s-to-m delays are independent (distinct) WSS and white discrete-time random processes.
References:
G. Giorgi and C. Narduzzi, “Performance Analysis of Kalman-Filter-Based Clock Synchronization in IEEE 1588 Networks,” in IEEE Transactions on Instrumentation and Measurement, vol. 60, no. 8, pp. 2902-2909, Aug. 2011.
G. Giorgi, “An Event-Based Kalman Filter for Clock Synchronization,” in IEEE Transactions on Instrumentation and Measurement, vol. 64, no. 2, pp. 449-457, Feb. 2015.
Brown, R. and Hwang, P., Introduction To Random Signals And Applied Kalman Filtering With Matlab Exercises, 4Th Edition. John Wiley & Sons, 2012.
- Parameters
data – Array of objects with simulation data
T – Nominal Sync period in sec
N – Interval used for frequency offset estimations
s_0 – Initial state
P_0 – Initial state’s covariance matrix
Q – Process noise covariance matrix
R – Measurement covariance matrix
- optimize(cache=None, error_metric='mse', early_stopping=True, force=False, skip=0.2)[source]¶
Optimize the process noise covariance matrix
Tries the filtering with several state noise covariance matrices and sets the internal matrix (Q) to the best evaluated value.
- Parameters
cache – Cache handler used to save the optimized configuration in a json file
error_metric – Chosen error metric: ‘max-te’ or ‘mse’
early_stopping – Whether to stop the search when min{erro} stalls
patience – Number of consecutive iterations without improvement to wait before signaling an early stop
skip – Fraction of the dataset to skip during the error assessment. Should be set to an acceptable transient for analysis.
Returns: