Theory of oscillator instability based upon structure functions

Structure functions are apparently unfamiliar to most engineers and the unifying role they play in oscillator instability theory has largely gone unrecognized. This paper introduces and places into perspective the role which Kolmogorov structure functions have in theory. It is demonstrated that the rms fractional frequency deviation (phase accumulation) introduced by Cutler and Searle is related to the first phase structure function; the two-sample Allan variance is related to the second phase structure function. In addition, it is shown how the two-sample Allan variance is related to the rms fractional frequency deviation under suitable conditions. The L-sample Allan variance is also identified in terms of the first phase structure function; it is shown to be an asymptotically unbiased estimator of the rms fractional frequency deviation squared if the latter existL The utility of higher order structure functions of frequency and phase in the theory of instability is also demonstrated; in particular, how the frequency drift and "flicker"-type noise convergence problems can be overcome.