Calculating Spectra by Sequential High-Pass Filtering

TL;DR

This paper introduces an improved sequential high-pass filtering method for calculating inhomogeneous field spectra, enabling extraction of steeper spectra than previous low-pass filtering techniques, and discusses limitations of second-order structure functions.

Contribution

It demonstrates that sequential high-pass filtering can extract steeper spectra with fewer kernel vanishing moments, expanding spectral analysis capabilities.

Findings

High-pass filtering allows extraction of spectra with <2p+3.

Any spectrum shallower than k^{-5} can be extracted with a 1st order kernel.

Second-order structure functions cannot detect shallow spectral peaks.

Abstract

We expand on the method of sequential filtering for calculating spectra of inhomogeneous fields. Sadek & Aluie [Phys. Rev. Fluids, 3, 124610 (2018)] showed that the kernel has to have at least $p$ vanishing moments to extract a power-law spectrum $k^{-\alpha}$ with $\alpha<p+2$ by low-pass filtering. Here, we show that sequential high-pass filtering allows for extracting steeper spectra with $\alpha<2p+3$ using the same $p$-th order kernel. For example, any spectrum of a field that is shallower than $k^{-5}$ can be extracted by sequential high-pass filtering using any 1st order kernel such as a Gaussian or top-hat. Finally, we demonstrate how second-order structure functions fail to capture spectral peaks because they cannot detect scaling that is too shallow.