The fundamental structure function of oscillator noise models

Continuous-time models of oscillator phase noise x(t) usually have stationary nth differences, for some n. The covariance structure of such a model can be characterized in the time domain by the structure function: D sub n (t;gamma sub 1, gamma sub 2) = E delta (n) sub gamma sub 1 x(s+t) delta(n) sub gamma sub 2 x (s). Although formulas for the special case D sub 2 (0;gamma,gamma) (the Allan variance times 2 gamma(2)) exist for power-law spectral models, certain estimation problems require a more complete knowledge of (0). Exhibited is a much simpler function of one time variable, D(t), from which (0) can easily be obtained from the spectral density by uncomplicated integrations. Believing that D(t) is the simplest function of time that holds the same information as (0), D(t) is called the fundamental structure function. D(t) is computed for several power-law spectral models. Two examples are D(t) = K/t/(3) for random walk FM, D(t) = Kt(2) 1n/t/ for flicker FM. Then, to demonstrate its use, a BASIC program is given that computes means and variances of two Allan variance estimators, one of which incorporates a method of frequency drift estimation and removal

The Parabolic variance (PVAR), a wavelet variance based on least-square fit

This article introduces the Parabolic Variance (PVAR), a wavelet variance similar to the Allan variance, based on the Linear Regression (LR) of phase data. The companion article arXiv:1506.05009 [physics.ins-det] details the $\Omega$ frequency counter, which implements the LR estimate. The PVAR combines the advantages of AVAR and MVAR. PVAR is good for long-term analysis because the wavelet spans over $2 \tau$, the same of the AVAR wavelet; and good for short-term analysis because the response to white and flicker PM is $1/\tau^3$ and $1/\tau^2$, same as the MVAR. After setting the theoretical framework, we study the degrees of freedom and the confidence interval for the most common noise types. Then, we focus on the detection of a weak noise process at the transition - or corner - where a faster process rolls off. This new perspective raises the question of which variance detects the weak process with the shortest data record. Our simulations show that PVAR is a fortunate tradeoff. PVAR is superior to MVAR in all cases, exhibits the best ability to divide between fast noise phenomena (up to flicker FM), and is almost as good as AVAR for the detection of random walk and drift

A sound card based multi-channel frequency measurement system

For physical processes which express themselves as a frequency, for example magnetic field measurements using optically-pumped alkali-vapor magnetometers, the precise extraction of the frequency from the noisy signal is a classical problem. We describe herein a frequency measurement system based on an inexpensive commercially available computer sound card coupled with a software single-tone estimator which reaches Cram\'er--Rao limited performance, a feature which commercial frequency counters often lack. Characterization of the system and examples of its successful application to magnetometry are presented.Comment: 4 pages, 3 figures, 1 tabl

The Omega Counter, a Frequency Counter Based on the Linear Regression

This article introduces the {\Omega} counter, a frequency counter -- or a frequency-to-digital converter, in a different jargon -- based on the Linear Regression (LR) algorithm on time stamps. We discuss the noise of the electronics. We derive the statistical properties of the {\Omega} counter on rigorous mathematical basis, including the weighted measure and the frequency response. We describe an implementation based on a SoC, under test in our laboratory, and we compare the {\Omega} counter to the traditional {\Pi} and {\Lambda} counters. The LR exhibits optimum rejection of white phase noise, superior to that of the {\Pi} and {\Lambda} counters. White noise is the major practical problem of wideband digital electronics, both in the instrument internal circuits and in the fast processes which we may want to measure. The {\Omega} counter finds a natural application in the measurement of the Parabolic Variance, described in the companion article arXiv:1506.00687 [physics.data-an].Comment: 8 pages, 6 figure, 2 table

An analytic technique for statistically modeling random atomic clock errors in estimation

Minimum variance estimation requires that the statistics of random observation errors be modeled properly. If measurements are derived through the use of atomic frequency standards, then one source of error affecting the observable is random fluctuation in frequency. This is the case, for example, with range and integrated Doppler measurements from satellites of the Global Positioning and baseline determination for geodynamic applications. An analytic method is presented which approximates the statistics of this random process. The procedure starts with a model of the Allan variance for a particular oscillator and develops the statistics of range and integrated Doppler measurements. A series of five first order Markov processes is used to approximate the power spectral density obtained from the Allan variance

Application of the Allan Variance to Time Series Analysis in Astrometry and Geodesy: A Review

The Allan variance (AVAR) was introduced 50 years ago as a statistical tool for assessing of the frequency standards deviations. For the past decades, AVAR has increasingly being used in geodesy and astrometry to assess the noise characteristics in geodetic and astrometric time series. A specific feature of astrometric and geodetic measurements, as compared with the clock measurements, is that they are generally associated with uncertainties; thus, an appropriate weighting should be applied during data analysis. Besides, some physically connected scalar time series naturally form series of multi-dimensional vectors. For example, three station coordinates time series $X$, $Y$, and $Z$ can be combined to analyze 3D station position variations. The classical AVAR is not intended for processing unevenly weighted and/or multi-dimensional data. Therefore, AVAR modifications, namely weighted AVAR (WAVAR), multi-dimensional AVAR (MAVAR), and weighted multi-dimensional AVAR (WMAVAR), were introduced to overcome these deficiencies. In this paper, a brief review is given of the experience of using AVAR and its modifications in processing astro-geodetic time series

On the measurement of frequency and of its sample variance with high-resolution counters

A frequency counter measures the input frequency $\bar{\nu}$ averaged over a suitable time $\tau$, versus the reference clock. High resolution is achieved by interpolating the clock signal. Further increased resolution is obtained by averaging multiple frequency measurements highly overlapped. In the presence of additive white noise or white phase noise, the square uncertainty improves from $\smash{\sigma^2_\nu\propto1/\tau^2}$ to $\smash{\sigma^2_\nu\propto1/\tau^3}$. Surprisingly, when a file of contiguous data is fed into the formula of the two-sample (Allan) variance $\smash{\sigma^2_y(\tau)=\mathbb{E}\{\frac12(\bar{y}_{k+1}-\bar{y}_k) ^2\}}$ of the fractional frequency fluctuation $y$, the result is the \emph{modified} Allan variance mod $\sigma^2_y(\tau)$. But if a sufficient number of contiguous measures are averaged in order to get a longer $\tau$ and the data are fed into the same formula, the results is the (non-modified) Allan variance. Of course interpretation mistakes are around the corner if the counter internal process is not well understood.Comment: 14 pages, 5 figures, 1 table, 18 reference

A structure function representation theorem with applications to frequency stability estimation

Random processes with stationary nth differences serve as models for oscillator phase noise. A theorem which obtains the structure function (covariance of the nth differences) of such a process in terms of the differences of a single function of one time variable is proven. In turn, this function can easily be obtained from the spectral density of the process. The theorem is used for computing the variance of two estimators of frequency stability