First stage of LISA data processing: Clock synchronization and arm-length determination via a hybrid-extended Kalman filter

In this paper, we describe a hybrid-extended Kalman filter algorithm to synchronize the clocks and to precisely determine the interspacecraft distances for space-based gravitational wave detectors, such as (e)LISA. According to the simulation, the algorithm has significantly improved the ranging accuracy and synchronized the clocks, making the phase-meter raw measurements qualified for time-delay interferometry algorithms.

Figure 1

Schematic configuration of LISA S/C and the exchanged laser beams (by S. Barke [8]).

Figure 2

LISA data processing chain.

Figure 3

Schematic power spectral density plot of LISA carrier laser beam, clock-sideband modulation and the PRN modulation. Horizontal direction denotes frequency and vertical direction denotes power. In the middle, the two peaks are the two beating carriers. Around the carriers are the PRN modulations. On the sides of the figure are the clock sidebands modulation.

Figure 4

Scatter plot of clock measurements $C_{ij}$.

Figure 5

Scatter plot of Doppler measurements $D_{ij}$. Unlike clock measurements, scatter plots of Doppler measurements exhibit elliptical clouds.

Figure 6

Scatter plot of ranging measurements $R_{ij}$. The arm-length variation is much larger than the ranging measurement noise. Therefore, we see only lines in the off-diagonal plots, which mainly show the arm-length changes. The ranging measurement noise is too small compared to the arm-length change to be visible in the plot.

Figure 7

Scatter plot of different measurements $C_{ij}$, $D_{ij}$, $R_{ij}$. Ranging measurements are correlated with Doppler measurements, but neither of them are correlated with clock measurements.

Figure 8

A priori covariance matrices $P_k^{-}$ at different steps. The absolute value of each component of the covariance matrix is represented by a color. The color map indicates the magnitude of each component in logarithmic scale $\ln (|P_k^{-}|)$.

Figure 9

A posteriori matrices $P_k^{+}$ at different steps. The absolute value of each component of the covariance matrix is represented by a color. The color map indicates the magnitude of each component in logarithmic scale.

Figure 10

The estimation error of the measurements, $H_k{P}_k^{+}{H}_k^T$ at different steps. The absolute value of each component is represented by a color. The color map indicates the magnitude of each component in logarithmic scale.

Figure 11

Histograms of errors of raw arm-length measurements and Kalman filter estimates, where the deviations of both raw arm-length measurements (excluding the initial clock bias) and the Kalman filter estimates from the true arm lengths are shown.

Figure 12

Plots of relative clock jitter and biases. Figure (a) shows typical results of estimates of relative clock jitters and biases. Figure (b) shows the deviations of the raw measurements and the Kalman filter estimates from the true values in histograms. Notice that the standard deviations in the legend have been converted to equivalent lengths.

Figure 13

The raw measurements, Kalman filter estimates and the true values of frequency differences between the USO in S/C 1 and the USO in S/C 2. The Kalman filter estimates are so good that they overlap with the true values in the figure.

Figure 14

A zoomed-in plot of Fig. 13. The true USO frequency differences and the Kalman filter estimates can clearly be seen in this figure.

Figure 15

The histograms of the deviations of the raw measurements and the Kalman filter estimates from the true values.