Does Allan Varianced Determine the Specrum
Explore the source record for details and available documents.
Allan Variance↗
Engineering topics
Greenhall, C. A.
Publications and source records attributed to Greenhall, C. A..
A Derivation of the Long-Term Degradation of a Pulsed Atomic Frequency Standard from a Control-Loop Model
The phase of a frequency standard that uses periodic interrogation and control of a local oscillator (LO) is degraded by a long-term random-walk component induced by downconversion of LO noise into the loop passband. The Dick formula for the noise level of this degradation is derived from an explicit solution of an LO control-loop model.
Dick formula local oscillator frequency phase freq↗
Spectral Ambiguity of Allan Variance
We study the extent to which knowledge of Allan variance and other finite-difference variances determines the spectrum of a random process. The variance of first differences is known to determine the spectrum. We show that, in general, the Allan variance does not. A complete description of the ambiguity is given.
Allan Variance variances finite-difference varianc↗
Digital signal processing in the radio science stability analyzer
The Telecommunications Division has built a stability analyzer for testing Deep Space Network installations during flight radio science experiments. The low-frequency part of the analyzer operates by digitizing wave signals with bandwidths between 80 Hz and 45 kHz. Processed outputs include spectra of signal, phase, amplitude, and differential phase; time series of the same quantities; and Allan deviation of phase and differential phase. This article documents the digital signal-processing methods programmed into the analyzer.
Greenhall, C. A.↗
The Third-Difference Approach to Modified Allan Variance
This study gives strategies for estimating the modified Allan variance (mvar) and formulas for computing the equivalent degrees of freedom (edf) of the estimators. A third-difference formulation of mvar leads to a tractable formula for edf in the presence of power-law phase noise. The effect of estimation stride on edf is tabulated. First-degree rational-function approximations for edf are derived.
Allan variance frequency instability statistical m↗
Orthogonal sets of data windows constructed from trigonometric polynomials
Suboptimal, easily computable substitutes for the discrete prolate-spheroidal windows used by Thomson for spectral estimation are given. Trigonometric coefficients and energy leakages of the window polynomials are tabulated.
Greenhall, C. A.↗
Interval Counter Measures Stability Of Frequency
Propagation of errors and effects of dead time suppressed. System includes precise timing-pulse generator with interval counter and suitably programmed computer determines relative stability or instability of frequency of two signals differing in frequency by about 1 Hz. Designed for use in frequency-standards laboratory.
Greenhall, C. A.↗
A method for using a time interval counter to measure frequency stability
It is shown how a commercial time interval counter can be used to measure the relative stability of two signals that are offset in frequency and mixed down to a beat note of about 1 Hz. To avoid the dead-time problem, the counter is set up to read the time interval between each beat note upcrossing and the next pulse of a 10 Hz reference pulse train. The actual upcrossing times are recovered by a simple algorithm whose outputs can be used for computing residuals and Allan variance. A noise floor-test yielded a delta f/f Allan deviation of 1.3 times 10 to the minus 9th power/tau relative to the beat frequency.
Greenhall, C. A.↗
Frequency stability review
Certain aspects of the description and measurement of oscillator stability are treated. Topics covered are time and frequency deviations, Allan variance, the zero-crossing counter measurement technique, frequency drift removal, and the three-cornered hat.
Greenhall, C. A.↗
Initializing a flicker-noise generator
The behavior of the stationary process, y(s), resulting from a particular Barnes-Jarvis (1971) flicker-noise generator filter initialization is analyzed along with that of the usual output, y(p), and the results are compared to those derived from the theory of true nonstationary flicker noise. A computer simulation and theoretical analysis indicates that though y(s) and y(p) two-sample variances are almost the same, they have significantly different time-interval errors (TIE). As time increases, up to the useful life of the generator output, more and more of the y(s) TIE is found to be due to the transient part.
Greenhall, C. A.↗
Removing Drift From Frequency-Stability Measurements
Method for eliminating frequency drifts from frequency stability measurements produces more accurate indication of stability of such frequency standard as hydrogen maser.
Greenhall, C. A.↗
Time Interval Errors of a Flicker-noise Generator
Time interval error (TIE) is the error of a clock at time t after it is synchronized and syntonized at time zero. Previous simulations of Flicker FM noise yielded a mean-square TIE proportional to sq t. It is shown that the order of growth is actually sq t log t. The earlier sq t result is explained and a modified version of the Barnes-Jarvis simulation algorithm is given.
Greenhall, C. A.↗
A structure function representation theorem with applications to frequency stability estimation
Random processes with stationary nth differences serve as models for oscillator phase noise. The theorem proved here allows one to obtain the structure function (covariances of the nth differences) of such a process in terms of the differences of a single function of one time variable. 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.
Greenhall, C. A.↗
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.
Greenhall, C. A.↗
A compact presentation of DSN array telemetry performance
The telemetry performance of an arrayed receiver system, including radio losses, is often given by a family of curves giving bit error rate vs bit SNR, with tracking loop SNR at one receiver held constant along each curve. This study shows how to process this information into a more compact, useful format in which the minimal total signal power and optimal carrier suppression, for a given fixed bit error rate, are plotted vs data rate. Examples for baseband-only combining are given. When appropriate dimensionless variables are used for plotting, receiver arrays with different numbers of antennas and different threshold tracking loop bandwidths look much alike, and a universal curve for optimal carrier suppression emerges.
Greenhall, C. A.↗
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.
Greenhall, C. A.↗
An error budget for the JPL hydrogen maser receiver
The hydrogen maser receiver is a synthesizer that converts the maser signal, at about 1420 MHz, to a set of output signals at 100, 20, 10, 5, 1, and 0.1 MHz. The contribution of the receiver and its component modules to the frequency instability and phase noise of each of its outputs is estimated, and these results are compared to published measurements of the 100-MHz output of the frequency standard, which consists of a maser plus its receiver. One can then assess how much the receiver degrades the performance of the frequency standard.
Greenhall, C. A.↗
Removal of drift from frequency stability measurements
A method of estimating frequency drift rate and removing its effect from Allan variance plots is given. When tried on a test of hydrogen masers, the methods gives consistent results. An error in the previous Allan variance computation algorithm is corrected.
Greenhall, C. A.↗