Theo1 is the first new species of variance that addresses a particularly difficult measurement problem, namely, obtaining reliable estimation of frequency stability for sample periods that are long compared to the length of a data run. Theo1 has statistical properties that are like the Allan variance (Avar), but Theo1 also has two advantages over other estimators of frequency stability: (1) it can evaluate frequency stability at a sample period (τ) of 3/4 the length of a data run, and (2) it presently attains the highest equivalent degrees of freedom (edf) of any estimator of frequency stability including Total-var and overlapping-Avar. Theo1 is unbiased relative to Avar for WHFM noise. Theo1 is biased slightly low with FLFM and RWFM, and we present a formula for a hybrid statistic (TheoH) made up of a combination of Theo1 and Avar in which bias is automatically removed. We explain the sampling function used in Theo1 and show that its frequency response is nearly ideal for extracting power-law noise processes of the types encountered with precision oscillators and clocks. We present results which, for a given data run, show how Theo1 anticipates the levels of frequency stability that are determined by Avar when given a longer data run from the same set of clocks.
-"Jitter" is the noise modulation due to random time shifts on an otherwise ideal, or perfectly on-time, signal transition. In the absence of ultra-high-speed jitter analyzers, spectrum analysis is an alternate noise measurement for timing jitter. Conventionally, jitter has been defined as a the integral of the phase noise. This paper presents a modified way of calculating timing jitter using phasemodulation (PM) noise measurements of high-speed digital clocks, which considers the frequency response of the jitter analyzer, providing a more accurate map. Measurements of phase noise are typically much more sensitive to phase (or time) fluctuations than a jitter analyzer. A summary table is provided for mapping the results of these measurements in the Fourier frequency domain to jitter in the τ domain for various random (specifically, power-law) noise types, spurs, vibration, and power-supply ripple. In general, one cannot unambiguously map back, that is, translate from jitter measurements to phase noise.
Theoretical variance #1 ( Theo1) has been developed at NIST to improve the estimation of longterm frequency stability. Its square-root ( Theo1-dev) has two significant improvements over the Allan deviation σ y (τ ) (Adev) in estimating long-term frequency stability, in that (1) it can evaluate frequency stability at averaging times 50 % longer than those of Adev, and (2) it can estimate frequency stability with greater confidence than any other estimator. We discuss a method for determining the exact confidence intervals of Theo1, particularly useful for small sample sizes, using analytic techniques. The confidence intervals of Theo1 are narrower and less skewed (more symmetric) than confidence intervals based on chi-square.
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