This course on Fourier Analysis aims to teach students the fundamental concepts and applications of Fourier series, Fourier transform, and related topics. By the end of the course, learners will be able to understand and apply complex Fourier series, compute the Discrete Fourier Transform (DFT) and Fast Fourier Transform (FFT), solve partial differential equations (PDEs) using FFT, analyze signals using spectrograms and Gabor transform, and explore image compression techniques with FFT and wavelets. The course utilizes a combination of theoretical explanations, practical examples in Matlab and Python, and hands-on exercises to enhance learning. This course is designed for students and professionals interested in signal processing, data analysis, image processing, and mathematical modeling.
Overview
Syllabus
Fourier Analysis: Overview.
Fourier Series: Part 1.
Fourier Series: Part 2.
Inner Products in Hilbert Space.
Complex Fourier Series.
Fourier Series [Matlab].
Fourier Series [Python].
Fourier Series and Gibbs Phenomena [Matlab].
Fourier Series and Gibbs Phenomena [Python].
The Fourier Transform.
The Fourier Transform and Derivatives.
The Fourier Transform and Convolution Integrals.
Parseval's Theorem.
Solving the Heat Equation with the Fourier Transform.
The Discrete Fourier Transform (DFT).
Computing the DFT Matrix.
The Fast Fourier Transform (FFT).
The Fast Fourier Transform Algorithm.
Denoising Data with FFT [Matlab].
Denoising Data with FFT [Python].
Computing Derivatives with FFT [Matlab].
Computing Derivatives with FFT [Python].
Solving PDEs with the FFT [Matlab].
Solving PDEs with the FFT [Python].
Solving PDEs with the FFT, Part 2 [Matlab].
Solving PDEs with the FFT, Part 2 [Python].
The Spectrogram and the Gabor Transform.
Spectrogram Examples [Matlab].
Spectrogram Examples [Python].
Uncertainty Principles and the Fourier Transform.
Wavelets and Multiresolution Analysis.
Image Compression and the FFT.
Image Compression with Wavelets (Examples in Python).
Image Compression with the FFT (Examples in Matlab).
Image Compression and Wavelets (Examples in Matlab).
Image Compression and the FFT (Examples in Python).
The Laplace Transform: A Generalized Fourier Transform.
Laplace Transforms and Differential Equations.
Laplace Transform Examples.
Taught by
Steve Brunton
