Convergence of Parametric Fourier Series Approximations

Author: Stone Allen

This project consists primarily of a MATLAB function and complementary Java application. Its objective is to demonstrate the ability of Fourier Series to approximate an arbitrary data set, with increasing precision, and to show the convergence of the Fourier approximation onto the original data. The result is an interactive set of programs which illustrate this convergence geometrically with discrete parametric equations. Users can select a set of two signals, (x(t), y(t)), which are steadily reconstructed using more and more terms of their Fourier Series and displayed graphically via MATLAB.

Full video: Hummingbird.avi

The MATLAB function performs the following:

• Computes the discrete Fourier Series coefficients (a₁ through a_N) for X(t) and Y(t)
• Begins reconstructing both X(t) and Y(t) using only a₀, then adds the a_-1 and a₁ complex exponential terms as a pair
• a_-i and a_i terms are added in pairs for 1 ≤ i ≤ floor((N - 1) / 2).
• if N even, a_{N / 2} term is added at the end
• approximations are periodically graphed, and graphs are compiled into a video, which is displayed onscreen (and saved)

The Java Application allows the user to create graphical parametric datasets and save them for use with the MATLAB script. Its use is fairly straightforward.

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