Fourier Transforms

Overview

Fourier Transforms are ubiquitous in mathematical areas which require the decomposition of a function into its constituent frequencies.

This work attempts to capture the beautiful piece of mathematics that governs Fourier Transforms and is inspired by 3Blue1Brown's video. As explained in the video, wrapping input signals around a circle allows for varying patterns and when the centre of mass shifts from the origin (centre of the image), we see a pure frequency.

Some of the key features of this work include:

Wrapping input signals in a circular fashion to capture the patterns that emerge for varying wrapping frequencies.
Visualizing the aligned patterns that emerge when the signal frequency matches the wrapping frequency, thus indicating the pure frequencies of the signal.
Visualization of the square wave and sawtooth wave approximated by their Fourier series.

All images were generated using Python, by using packages like Matplotlib, NumPy and SciPy.

Project Page

More details can be found at the project page - Fourier Transforms.

Code Usage

Square and Sawtooth Wave Animations

Modify the following lines in sawtooth_animation.py and square_wave_animation.py to generate your own animations:

f = n

for series in range (min_terms, max_terms, step):

Here, f denotes the frequency of the wave, assigned using n and series denotes the number of fourier terms used, with variation from min_terms to max_terms in increments of step.

Signal Wrapping

Vary the following parameters in signal_wrap.py to generate your own wrappings for signals:


f_wave = n

signal = np.cos(k1*omega*t)+np.sin(k2*omega*t)

f_list = np.arange(min_freq, max_freq, step_size)

Here, f_wave denotes the frequency of the wave, signal denotes the actual signal to be wrapped and (min_freq, max_freq, step_size) denote the minimum and maximum frequency values to be used for the animation, with increments of step_size.