How Extracting Information from Data Highpass Filters Its Additive Noise
Abstract
We examines characteristics of three types of random error measures in presence of negative power law (neg-p) noise: (a) observable residual error after removing an estimate of an information containing causal function from data, (b) jitter, the residual error with additional highpass (HP) filtering, and (c) Mth-order difference (delta) variances, such as the Allan variance (1st-order delta-variance of the fractional frequency error y(t)) and the Hadamard-Picinbono variance (2nd-order delta-variance of y(t)). Measures (b) and (c) are used to mitigate perceived divergence problems in the mean square (MS) of Measure (a) due to the presence of neg-p noise. This paper proves that this perception is wrong; it shows that the MS of Measure (a) converges in the presence of neg-p noise by demonstrating that extracting a statistically optimal estimate of the causal behavior from data HP filters the noise in the measure. It is further shown that the order of this noise HP filtering increases with the complexity of the model function used to estimate the causal behavior in the data. Thus, if one is free to choose the complexity of the model function, the MS observable residual error is guaranteed to converge for any negative power in the noise PSD. Because of this, it is shown that jitter can be defined simply as observable residual error without additional HP filtering, making jitter and residual error the same error measure. The paper finally shows that an Mth-order delta-variance is also a measure of the MS of observable residual error for any number of data samples when the model function is an (M-1)th-order polynomial. This completes the equivalence, showing Measures (a), (b), and (c) all measure the same kind of error when the model function for causal behavior is a polynomial. The consequences of this equivalence then explored. Among these is a physical explanation for the fact that Allan variance is sensitive to frequency drift, while Hadamard-Picinbono variance is not.
Document Details
- Document Type
- Technical Report
- Publication Date
- Nov 01, 2007
- Accession Number
- ADA483372
Entities
People
- Victor S. Reinhardt
Organizations
- RTX