Fourier Series

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A Fourier transform is a way of breaking down a complex function into (infinite) sums of sine and cosine waves. In short, given a smoothie, it finds its recipe. Fourier series is the Fourier transform of a periodic function and it aims to represent the periodic function as a sum of sinusoidal waves. It is analogous to the Taylor series, which represents functions as an infinite sum of monomial terms.

$S_{n}(x) = \frac{a_{o}}{2}+\sum_{n=1}^{N}(a_{n}\cos(\frac{2\pi nx}{P}) + b_{n}\sin(\frac{2\pi nx}{P}))$

This sum of sine and cosine waves can also be thought of as a list of phasors. Hence, the Fourier series generates a list of phasors which, when summed together, reproduces the original signal. The below animation shows how this can look like: ¹²

Simulate approximates the following waves:

1. Sawtooth function
2. Square function

We have the flexibility to fiddle with the following parameters:

• N: The number of phasors (sine and cosine terms) generated

• Amplitude: The coefficients ‘a_n’ and ‘b_n’

• Frequency: The value of ‘n’ to adjust the frequency of sine and cosine terms

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1. https://upload.wikimedia.org/wikipedia/commons/b/bd/Fourier_series_square_wave_circles_animation.svg ↩

2. https://upload.wikimedia.org/wikipedia/commons/1/1e/Fourier_series_sawtooth_wave_circles_animation.svg ↩

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Last update: December 25, 2021