Characterizing frequency stability: a continuous power-law model with discrete sampling

This paper examines several aspects of the Allan variance and the modified Allan variance. New expressions for these variances are derived for noise processes that produce power spectral densities with both integer and noninteger powers (/spl alpha/) in their functional dependence on f. A single expression, continuous over /spl alpha/, is presented for each of these variances. Also investigated are the effects of discrete sampling and finite data length. Discrete equations are developed and compared with more familiar continuous expressions. In addition, the uncertainty of the estimates for the Allan variance and the modified Allan variance for fully overlapping data usage is presented. The uncertainties can be calculated for arbitrary /spl alpha/. The results presented are compared with computer simulations and found to be in excellent agreement. >

[1]  D. B. Percival,et al.  Characterization of frequency stability: frequency-domain estimation of stability measures , 1991, Proc. IEEE.

[2]  I. S. Gradshteyn,et al.  Table of Integrals, Series, and Products , 1976 .

[3]  Todd Walter.  A Multi-Variance Analysis in the Time Domain , 1992 .

[4]  F. L. Walls,et al.  Characterization of frequency stability in precision frequency sources , 1991, Proc. IEEE.

[5]  F. Harris.  On the use of windows for harmonic analysis with the discrete Fourier transform , 1978, Proceedings of the IEEE.

[6]  J. Rutman.  Characterization of phase and frequency instabilities in precision frequency sources: Fifteen years of progress , 1978, Proceedings of the IEEE.

[7]  D. W. Allan,et al.  Statistics of atomic frequency standards , 1966 .

[8]  Todd Walter,et al.  DISCRETE SIMULATION OF POWER LAW NOISE , 1992 .

[9]  C. L. Searle,et al.  Some aspects of the theory and measurement of frequency fluctuations in frequency standards , 1965 .

[10]  D. W. Allan,et al.  A statistical model of flicker noise , 1966 .

[11]  C. Audoin,et al.  Characterization of Frequency Stability: Uncertainty due to the Finite Number of Measurements , 1973 .

[12]  N. Kasdin.  Discrete simulation of colored noise and stochastic processes and 1/fα power law noise generation , 1995, Proc. IEEE.

[13]  Kazuyuki Yoshimura,et al.  Characterization of Frequency Stability: Uncertainty Due to the Autocorrelation of the Frequency Fluctuations , 1978, IEEE Transactions on Instrumentation and Measurement.

[14]  J. Rutman,et al.  Characterization of Frequency Stability: A Transfer Function Approach and Its Application to Measurements via Filtering of Phase Noise , 1974 .

[15]  Todd Walter,et al.  Discrete simulation of power law noise (for oscillator stability evaluation) , 1992, Proceedings of the 1992 IEEE Frequency Control Symposium.

[16]  P. Lesage,et al.  Characterization of Frequency Stability: Analysis of the Modified Allan Variance and Properties of Its Estimate , 1984, IEEE Transactions on Instrumentation and Measurement.

[17]  David A. Howe,et al.  Properties of Signal Sources and Measurement Methods , 1981 .

[18]  D. W. Allan,et al.  A Modified "Allan Variance" with Increased Oscillator Characterization Ability , 1981 .