Dynamic Mathematics

by Juan Carlos Ponce Campuzano, 20/August/2018

Introduction

The Discrete Fourier Transform is defined as $$\begin{array}{rl}{X}_k& ={\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}{x}_n\cdot e^{-ik\frac{2\pi }{N}n}\\ & ={\displaystyle \frac{1}{N}}\sum _{n=0}^{N-1}{x}_n[\cos \left(k\frac{2\pi }{N}n\right)-i\sin \left(k\frac{2\pi }{N}n\right)],\end{array}$$ where the last expression follows from the first one by Euler's formula. Essentially, it transforms a sequence of $N$ complex numbers $${\displaystyle \mathbf{x}:=x_0,x_1,\dots ,x_{N-1}}$$ into another sequence of complex numbers, $${\displaystyle \mathbf{X}:=X_0,X_1,\dots ,X_{N-1}.}$$

A few important notes:

• $N=$ number of time samples we have
• $n=$ current sample we're considering $(0\dots N-1)$
• $x_n=$ 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)

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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.

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: $$\begin{array}{rc}x& =\sum _{k=0}^{N-1}[a_k\cos \left(kt\right)-b_k\sin \left(kt\right)],\\ y& =\sum _{k=0}^{N-1}[b_k\cos \left(kt\right)+a_k\sin \left(kt\right)]\end{array}$$ 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: $$\begin{array}{rl}{x}_n& =\sum _{k=0}^{N-1}{X}_k\cdot e^{ik\frac{2\pi }{N}n}\\ & =\sum _{k=0}^{N-1}{x}_n[\cos \left(k\frac{2\pi }{N}n\right)+i\sin \left(k\frac{2\pi }{N}n\right)].\end{array}$$

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

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Further details

For more details about how this method works and how to implement it in p5js, see:

How it works?

• Tracing closed curves with epicycles: A fun application of the discrete Fourier transform.
• Trigonometric intepolation using the Discrete Fourier Transform.

p5.js implementation - Daniel Shiffman's Coding Challenge #125 and #130.

Finally, if you find this content useful, please consider supporting my work using the links below.

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More applets

Here are more fun interactive applets that you can play with. Just click on the image to open the live demo! Enjoy! 😃

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Further reading

1. Brigham, E. O. (1988). The fast Fourier transform and its applications. Prentice-Hall, Inc. USA.
2. Cooley, J. W. and Tukey, J. W. (1965). An algorithm for the machine calculation of complex fourierseries. Mathematics of Computation, 19:297-301.
3. Ginnobili, S. and Carman, C. C. (2008). Deferentes, epiciclos y adaptaciones. Associacao de Filosofiae Historia da Ciencia do Cone Sul (AFHIC), 5:399-408.
4. Hanson, N. R. (1965). The mathematical power of epicyclical astronomy. Isis, 51:150-158.
5. Ponce Campuzano, J. C. (2023). Tracing closed curves with epicycles: A fun application of the Discrete Fourier Transform. North American GeoGebra Journal. Vol 11. No. 1. pp. 1-14.
6. Ponce Campuzano, J. C. (2022). Trigonometric intepolation using the Discrete Fourier Transform. North American GeoGebra Journal. Vol. 10. No. 1. pp. 1-13.
7. Sundararajan, D. (2018). Fourier Analysis - A Signal Processing Approach. Springer Singapur.
8. Stein, E. M. and Shakarchi, R. (2003). Fourier Analysis: An Introduction. Princeton University PressOxford.