1 Abstract

This report investigates the Fourier Transform, starting with the mathematics of wave superposition, explaining how a Fourier Transform is computed, and deriving a version reproducible in Desmos using Euler's Formula. After this, the report explores how computers turn real-world audio signals using ADC conversion into digital signals, using the Nyquist-Shannon Sampling Theorem, and what happens when this rule is broken, in the form of aliasing. After this is a derivation of a Discrete Fourier Transform, which is then further optimised into the Cooley-Tukey Radix 2 FFT algorithm, reducing computational complexity to the point where modern computers can run them with very

little computing power. Finally, the report examines a real-world application of these FFT algorithms, in the form of the wave simulation system used in the video games, Sea of Thieves.

1.1 Introduction

The Fourier Transform is a way of decomposing an input non-periodic wave into multiple frequency components. This can be useful for determining the strongest frequency in a signal or isolating certain frequencies in an audio recording. In this paper, I will go through an example wave to deconstruct, how this can be reproduced in Desmos, how to optimise the Fourier Transform to enable it to be run with minimal processing power, and also the real-world application of this technology. As I have been producing music since the COVID-19 pandemic, I know that a core idea of sound synthesis and even composing chords is stacking different frequency waves on top of each other to create new waves and more complex sounds. But then I wondered: how would you go backward from a mixed-frequency wave to find the fundamental frequencies again? How does AI extract chords from an audio file without the original sheet music? How does an EQ work, displaying multiple different frequency levels? My research led me to this concept, and here I will explain what I have learned and apply the transform to a specific wave function to isolate its composite frequencies, before talking about how computers can perform this transform as fast as possible and how video games use this technology to simulate in-game aspects such as ocean waves.