Characterization of periodic variations in the GPS satellite clocks

Abstract

The clock products of the International Global Navigation Satellite Systems (GNSS) Service (IGS) are used to characterize the timing performance of the GPS satellites. Using 5-min and 30-s observational samples and focusing only on the sub-daily regime, approximate power-law stochastic processes are found. The Block IIA Rb and Cs clocks obey predominantly random walk phase (or white frequency) noise processes. The Rb clocks are up to nearly an order of magnitude more stable and show a flicker phase noise component over intervals shorter than about 100 s. Due to the onboard Time Keeping System in the newer Block IIR and IIR-M satellites, their Rb clocks behave in a more complex way: as an apparent random walk phase process up to about 100 s and then changing to flicker phase up to a few thousand seconds. Superposed on this random background, periodic signals have been detected in all clock types at four harmonic frequencies, n × (2.0029 ± 0.0005) cycles per day (24 h coordinated universal time or UTC), for n = 1, 2, 3, and 4. The equivalent fundamental period is 11.9826 ± 0.0030 h, which surprisingly differs from the reported mean GPS orbital period of 11.9659 ± 0.0007 h by 60 ± 11 s. We cannot account for this apparent discrepancy but note that a clear relationship between the periodic signals and the orbital dynamics is evidenced for some satellites by modulations of the spectral amplitudes with eclipse season. All four harmonics are much smaller for the IIR and IIR-M satellites than for the older blocks. Awareness of the periodic variations can be used to improve the clock modeling, including for interpolation of tabulated IGS products for higher-rate GPS positioning and for predictions in real-time applications. This is especially true for high-accuracy uses, but could also benefit the standard GPS operational products. The observed stochastic properties of each satellite clock type are used to estimate the growth of interpolation and prediction errors with time interval.

Appendix: Allan deviation

We now give a very brief introduction to the Allan deviation as a measure of frequency instability in clocks, or oscillators. In order to remain brief, several simplifications have been made which the detailed reader will likely want to avoid. For further reading on clock measurement and modeling, we suggest the collection of papers that make up NIST (1990). For a concise mathematical description of the the Allan and Modified Allan deviations consider also Walter (1994).

The voltage output of an oscillator/clock is usually represented as

$$ V(t)=V_0\sin\left(2\pi\nu_0 t + \phi(t)\right) $$

where V ₀ is its nominal peak voltage amplitude, ν ₀ is the nominal fundamental frequency, and ϕ denotes the deviations in phase from the nominal frequency; fluctuations in amplitude do not contribute appreciably to the frequency instability of most oscillators and so the amplitude has been considered constant here. The instantaneous frequency may be defined as the derivative (divided by 2π) of the total phase of the sine,

$$ \nu(t)=\nu_0 + \frac{1}{2\pi}\frac{{\rm d}\phi}{{\rm d}t}. $$

But, because the magnitude of this frequency is generally quite small for most oscillators and in order to avoid the dependence on nominal frequency, we generally define instead the normalized (or fractional) frequency,

$$ y(t) = \frac{\nu(t)-\nu_0}{\nu_0} = \frac{1} {2\pi\nu_0}\frac{{\rm d}\phi}{{\rm d}t}=\frac{{\rm d}x}{{\rm d}t}, $$

where

$$ x(t) = \frac{\phi(t)}{2\pi\nu_0} $$

is the phase expressed in time units.

The phase (or time) of a clock is generally subject to both systematic as well as random fluctuations. Combinations of five common power-law processes, that is processes whose power spectral density vary as powers of their Fourier frequency, are often used to describe the random behavior of most clocks. Table 2 summarizes these common power law processes.

Table 2 Power-law processes common in clocks

The IEEE has recommended (Barnes et al. 1971) the 2-sample, or Allan variance σ ² _y (τ) (or square-root for Allan deviation) as a time-domain measure of frequency instability, defined as

$$ \sigma_y^2(\tau)=\left\langle\frac{1} {2}\left({\bar{y}}_{k-1}-{\bar{y}}_k\right)^2\right\rangle, $$

where 〈·〉 represents infinite time average and

$$ {\bar{y}}_k = \frac{x(t_k+\tau)-x(t_k)}{\tau} $$

is the average fractional frequency over a specified interval of interest τ. This measure is desirable because it is convergent for the power-law processes shown in Table 2 and because the level and type of power law may be inferred easily by inspecting σ _y (τ) over a range of τ.

Because the Allan deviation has the same τ relationship for white-x and flicker-x noises, a Modified Allan deviation statistic was developed which essentially introduces a measurement bandwidth dependence such that these two noises may be differentiated from one another. We neither show the definition of the Modified Allan nor the Hadamard deviation here, but Table 2 shows the relevant τ relationships.