A Physical Interpretation of Difference Variances

Most variances of the time error x(t) used in industry can be written as delta² _X,M (tau) M^th order difference variances of x(t) 2 over the interval tau or their finite sample statistics, delta² _X,M is normalized so all M-orders are equivalent when x(t) consists solely of white-x noise. However, all M-orders are not equivalent when negative power law (neg-beta) noise is present, and the 2 question arises as to which order of delta² _X,M is most appropriate to 2 use in a given problem. The paper shows that delta² _X,M can be 2 interpreted as an approximate measure of delta² _tau,M the variance of the residual error after an M-order polynomial X_a,M (t, A) with adjustable coefficients A is removed from x(t) by least squares fitting X_a,M (t, A) to samples of x(t) over the interval T = Mtau . This interpretation of delta² _X,M then provides an objective rationale for choosing the appropriate M-order of delta² _X,M based on the order of the X_a,M (t,A) removed from x(t) in the problem addressed. It is further noted that X_a,M (t, A) can represent either a causal aging function or more generally a polynomial representation of any causal information extracted from x(t). The aging interpretation explains the sensitivity of Allan based variances (delta² _X,M) to frequency drift and the insensitivity of Hadamard based variances (delta² _X,M) to such drift. The information extraction interpretation leads to an even more important conclusion, that the process of extracting information from data highpass filters the residual noise with increasing efficiency as the complexity of the extracted data (as measured by M) increases. Another consequence of interpreting delta² _X,M as a measure of delta² _tau,M is that the M-order of delta² _X,M is not a free parameter that one can arbitrarily change to avoid a divergence problem caused by the presence of neg-beta noise. The paper shows that such divergences are indications of real problems in the design, specification, or analysis of a system, and examples are given of how to properly mitigate such divergences without arbitrarily changing M It is finally noted that the results of this paper can be applied to variances of other variables such as phi(t)and y(t).