This Python project explores various applications of the Fourier transform, including the spectral derivative and convolutions. The Fourier transform is a standard tool in signal processing, image processing, and many other fields, where signals are decomposed into their frequency components. Often a difficult calculation in a time or space domain can be made simpler in the frequency domain.
This project contains applications and visualizations that make use of Fourier-based methods. The python scripts and jupyter notebooks in the applications directory include overviews of the following:
- Discrete Fourier Transform Animation: This module animates the discrete Fourier transform (DFT) of a complex sequence as described in this 3 Blue 1 Brown video.
applications/complex_dft/lissajous.py: animation the DFT of a parametrically defined function: Lissajous Curve
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Spectral Derivative: Taking a numerical derivative of a function
$f(x)$ can sometimes be more stable when operating in the frequency domain. If$\mathcal{F}(f)(x) = \hat{f}(\omega)$ is the Fourier transform of a differentiable function$f(x)$ , then$\mathcal{F}(f')(x) = i \omega \hat{f}(\omega)$ . -
Convolution: The convolution theorem says that under suitable conditions, convolution of two functions in the time domain is equivalent to pointwise multiplication in the frequency domain.
- Clone the repository
git clone https://github.com/gregory-bopp/fourier-transform-python.git- Install the project dependencies
pip install -r requirements.txt- Install project library
python -m pip install --upgrade setuptools
pip install fourierThis project is licensed under the MIT License. See the LICENSE file for details.