MaxPol is a lowpass derivative kernel optimized to have "ideal" spectral properties. See literature: https://arxiv.org/abs/1709.08321, https://ieeexplore.ieee.org/document/7944698 for more details. Unfortunately there is no closed form for the kernel with general parameters. I present analytical (exact) solutions for the centralized MaxPol kernel, with s=0 (no shift). Implements the same algorithm as the original solver, but in Mathematica instead of Matlab. Shifting can be implemented trivially if required.
This repository contains the solutions for n=0..4, l=1..100 (i.e. zero'th to fourth derivatives with a filter length up to 201) and all possible P,Q values (degree of lowpass filtering vs. tangency to exact derivative). The n=0 setting corresponds to filtering only.
maxpol_calc contains numerical values, truncated at 18 significant digits (more than f64 precision). Each value is printed on a newline, starting from the leftmost filter coefficient. Each solution is named maxpol_n<n>_l<l>_P<P>.
maxpol_calc_exact contains all the exact fractions formatted in the same way.
It is recommended to make use of an intermediate storage format, such as the Sled database as implemented in maxpol_rust.
The approximate maxima of the spectra are stored in maxpol_calc_cutoff, to be used as an indication of the cutoff frequency.
Everything is released under the GPL-3.0 license. If you use my work in academic context, a citation would be appreciated:
@misc{maxpol_pvdb,
author = {Pim van den Berg},
title = {MaxPol analytical solutions up to l=100},
howpublished = {https://github.com/PvdBerg1998/MaxPol/}
year = {2022}
}