Spectral Derivatives

TL;DR

This paper explains the mathematical foundation of the spectral-derivatives Python package, enabling accurate computation of derivatives in the Fourier domain with careful handling of signal processing principles.

Contribution

It provides a detailed mathematical explanation and implementation guidance for spectral derivatives, enhancing computational accuracy in signal processing.

Findings

Clarifies the mathematical basis of spectral derivatives

Provides implementation details for the spectral-derivatives package

Highlights considerations for accurate Fourier-based differentiation

Abstract

One of the happiest accidents in all math is the ease of transforming a function to and taking derivatives in the Fourier frequency domain. But in order to exploit this extraordinary fact without serious artefacting, and in order to be able to use a computer, we need quite a bit of extra knowledge and care. This document sets out the math behind the spectral-derivatives Python package. I touch on fundamental signal processing and calculus concepts as necessary and build upwards.