# Continuous Fourier Transform: A practical approach for truncated signals and suggestions for improvements in thermography

## TL;DR

This paper introduces a practical Continuous Fourier Transform approach for truncated signals, benchmarking it against FFT, and applies it to thermography defect analysis, showing improved consistency and artifact reduction.

## Contribution

It derives analytical solutions for CFT of truncated signals and demonstrates its advantages over FFT in thermography applications for defect detection.

## Key findings

- CFT provides artifact-free amplitude and phase analysis.
- FFT results depend heavily on sampling rates and introduce artifacts.
- Proposed CFT method yields consistent results with less artifact presence.

## Abstract

The fundamentals of Fourier Transform are presented, with analytical solutions derived for Continuous Fourier Transform (CFT) of truncated signals, to benchmark against Fast Fourier Transform (FFT). Certain artifacts from FFT were identified for decay curves. An existing method for Infrared Thermography, Pulse Phase Thermography (PPT), was benchmarked against a proposed method using polynomial fitting with CFT, to analyse cooling curves for defect identification in Non-Destructive Testing (NDT). Existing FFT methods used in PPT were shown to be dependent on sampling rates, with inherent artifacts and inconsistencies in both amplitude and phase. It was shown that the proposed method produced consistent amplitude and phase, with no artifacts, as long as the start of the cooling curves are sufficiently represented. It is hoped that a collaborative approach will be adopted to unify data in Thermography for machine learning models to thrive, in order to facilitate automated geometry and defect recognition and move the field forward.