COMPRESSIVE LOGARITHMIC GENERATING ELEMENTS FOR INERTIALESS FILTERING OF SYMMETRIC NOISE

Abstract

We develop a polynomial inertialess filter for recovering a Gaussian signal from additive non-Gaussian noise within the Kunchenko decomposition framework, taking the generating element to be a compressive, sub-polynomial family: after the mandatory linear element , the odd, monotone elements . Because these elements grow more slowly than any power of , they compress rather than amplify the tails, and the normal system  is finite whenever the noise has finite variance alone – no moment above the second is ever required, so the body cannot «diverge» on heavy tails. The cost of this generality is numerical: successive powers of a slowly varying function are strongly collinear, so the body is severely ill-conditioned and is solved under Tikhonov regularisation with a cross-validated ridge. We show that this controlled cost buys a measurable advantage on the noise regime to which the compressive shape is matched. Monte-Carlo experiments with 95% bootstrap confidence intervals, benchmarked against the Wiener linear filter through the Kunchenko coefficient , show the logarithmic basis to be the better filter on platykurtic / multimodal noise, where its slowly growing nonlinearity resolves inter-mode structure that the linear filter does not use (  = 0.851 at  =0.5, widening to 0.729 – a 27% reduction in residual variance over the linear filter – at q = 0.25). A controlled experiment on a real CWRU bearing carrier with a platykurtic disturbance reproduces this finding (  = 0.853 on a bimodal disturbance at q = 0.5), whereas raw real noise with negative excess kurtosis alone gives no advantage – i.e. the gain is driven by the disturbance’s structure, not by its kurtosis. Because the elements are odd and sign-preserving, the mechanism is matched to symmetric disturbances as a class: it improves on the linear filter on all three symmetric classes examined, and is only marginal on the asymmetric one, where odd compression captures skew inefficiently. We therefore recommend the logarithmic basis for symmetric non-Gaussian disturbances, most strongly on their platykurtic and multimodal corner, and we note that negative excess kurtosis on its own is not a sufficient condition for an advantage.