Continuous Wavelet Transform (CWT) of TWSTT and other Data.

Generally, "stationarity" of timekeeping and time transfer data is assumed, i.e. there are no systematic changes n their statistical properties with time. However, real world data are often affected by perturbations from environmental, hardware failure, control by steering, and other similar types of sources which introduce small-amplitude variability. A key concept is that nothing in our universe can be assumed to be stationary except as an approximation over a short interval of time. Therefore, time-frequency analysis is a useful tool in analysis of measured data.

Simple 1-day averages of TWSTT time differences between USNO's Master Clock #2 and the AMC's Master Clock #1 were generated from the nominal hourly measurements. Several times during the approximately 1000 days of data analyzed, small perturbations were introduced from various sources.

Figure 1 shows the time-frequency representation (TFR) of the signal and was produced using the software in Reference 1. The original signal is shown at the bottom as a standard x,y plot.

Figure 1. This plot shows the CWT time-frequency representation of the one-day simple averages of the USNO (MC #2) - USNO AMC (MC #1) TWSTT data.

The top of the figure shows the CWT TFR. The x-axis is still time, but the y-axis now becomes the "scale of the signal" from which the period of the signal may be extracted. A period of 0 length would be at the top and a period equal to the full data length (~1000 days) would be at the bottom. The left side color bar shows the reciprocal of the wavelet coefficient amplitude, which is similar to the power of the signal at the indicated period and instant of time. Impulses from a perturbation would stretch up towards the top of the image. The extent (width) of a structure in the x-axis is an indication of the length of time the impulse lasted.

Most of the structure seen in the TFR is normal structure inherent in the physics of atomic clocks and our steering towards UTC of this system. However, a real impulse caused by an environmental perturbation is located near the numeric one located at the bottom of the figure.

Monitoring, analysis, and identification of the source(s) of the subtle impulse structures in TFRs makes it possible to focus resources where improvements may be made in environmental control, device failure, device sensitivity indentification and resolution, measurement system improvements, etc.

CWT TFRs of Synthetic White Phase-Modulation (PM) and Synthetic White Frequency-Modulation (FM) in the Time Domain

For comparison purposes, CWT TFRs of synthetically produced white PM and white FM follow. The white PM dataset is in the time domain and has the characteristic of randomness with a Gaussian distribution, and is made up of 5000 points. The white FM data are a white FM noise with a Gaussian distribution integrated to time also with 5000 points.

Figure 2 (above) displays the CWT TFR of the white PM data. As expected, random disturbances at all times and scales are evident. Since the data are a Gaussian rather than a uniform distribution, there is a slight ramp from top to bottom.

Figure 3 (below) displays the CWT TFR of the white FM integrated to time case. Here, the disturbances in general are still showing up at almost all scales, but the distribution across the plot shows a stronger ramp. This signal is clearly more "deterministic," i.e. attributable to a particular cause rather than random noise.

Final review and comparison of Figure 1 shows that overall the noise characteristic of the CWT TFR is similar to the synthetic white FM integrated to time shown in Figure 3. The environmentally induced disturbances (impulses) in the TWSTT data produce signal that is in general larger than that produced by the simple integration of white FM to time, however.

  1. Chan, A. K., and Liu, S. J., 1998, "Wavelet Toolware," software, Academic Press, Boston, Massachusetts, USA.

    (07 May 1999)