Fourier Epicycles

The Discrete Fourier Transform

Approximating and drawing closed curves with epicycles

by Juan Carlos Ponce Campuzano, 20/August/2018

Introduction

The Discrete Fourier Transform is defined as \begin{aligned} X_{k}&={\dfrac{1}{N}\sum _{n=0}^{N-1}x_{n}\cdot e^{-i k {\frac {2\pi }{N}}n}}\\ &={ \dfrac{1}{N}\sum _{n=0}^{N-1}x_{n}\left[\cos \left(k{\frac {2\pi }{N}}n\right)-i\, \sin \left({k\frac {2\pi }{N}}n\right)\right],} \end{aligned} where the last expression follows from the first one by Euler's formula. Essentially, it transforms a sequence of $N$ complex numbers $$\displaystyle \mathbf {x} :=x_{0},x_{1},\ldots ,x_{N-1}$$ into another sequence of complex numbers, $$\displaystyle \mathbf {X} :=X_{0},X_{1},\ldots ,X_{N-1}.$$

A few important notes:

$N =$ number of time samples we have
$n =$ current sample we're considering $(0 \ldots N-1)$
$x_n =$ value of the signal at time $n$
$k =$ current frequency we're considering (0 Hertz up to $N-1$ Hertz)
$X_k =$ amount of frequency $k$ in the signal (amplitude and phase, a complex number)

Drawing a T-Rex with epicycles

The animation below with epicycles demonstrates how a simple sum of complex numbers in terms of phases/amplitudes can be nicely visualized as a set of concatenated circles in the complex plane. Each circle represents a Fourier coefficient $X_k$ sorted by amplitude.

Live Demo

Originally the T-Rex curve (signal) is defined with 203 points. After calculating the Fourier coefficient $X_k$, we can easily determine its phase and amplitude for every value of $k$.

Aproximating T-Rex with parametric equations

It is possible to approximate the T-Rex curve. In this case we can use the Fourier coefficients $X_k$ to define the parametric equations: \begin{eqnarray*} x & = \sum_{k=0}^{N-1} \big[ a_k \, \cos\left( k \,t \right) - b_k \, \sin\left( k \,t \right) \big], \\ y & = \sum_{k=0}^{N-1} \big[ b_k \, \cos\left( k \,t \right) + a_k \, \sin\left( k \,t \right) \big] \end{eqnarray*} with $t\in[0, 2\pi)$. The coefficients $a_k$ and $b_k$ are the real and imaginary parts of $X_k$. The expression is obtained using the inverse of the Discrete Fourier Transform: \begin{aligned} x_{n}&=\sum _{k=0}^{N-1}X_{k}\cdot e^{i k {\frac {2\pi }{N}}n}\\ &= \sum _{k=0}^{N-1}x_{n}\left[\cos \left(k{\frac {2\pi }{N}}n\right)+i\,\sin \left({k\frac {2\pi }{N}}n\right)\right]. \end{aligned}

The method to create the system of epicycles to draw a T-rex works also for any possible periodic signal.

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