Fourier Series

Fourier Series encaptulates the idea that any periodic function can be expressed as a sum of sine and cosine waves. This simulation has been programmed to contain the visual framework that is featured in 3Blue1Brown's video on Fourier Transforms.

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Intuition

\[ F(\omega) = \int_{-\infty}^{\infty} f(x) \ e^{-i 2 \pi \omega x} dx = a + i b \]

\(f(x)\) is the function that we want to analyze. It consists of a bunch of sine (\( \sin 2 \pi f x \)) and cosine waves (\( \cos 2 \pi f x \)) of different frequencies added together. The above formula gives us the relative amplitude of the sines and cosines corresponding to the frequency \( \omega \). The real part \(a\) is the amplitude of the cosine wave and the imaginary part \(b\) is the amplitude of the sine wave. Both \(a\) and \(b\) are evaluated in the above visualization. The \( e^{-i 2 \pi \omega x} \) term depicts winding of the function around the origin, and \( \omega \) is the rate at which the function is sampled. The integral is akin to finding the "centre of mass" of the winded function.

Please watch 3Blue1Brown's video for a better explanation. Here are some instructions about using this visualization:

1. In the default configuration, sine waves of frequencies 3 and 7 are present. When the sampling frequencies are equal to these frequencies, a peak in the magnitude is observed. Moreover, the magnitude entirely comes from the imaginary part of the Fourier transform.
2. Now, remove the waves and add cosine waves of the two distinct frequencies. When the sampling frequencies are equal to these frequencies, a peak in the magnitude is observed. This time, the magnitude comes from the real part of the Fourier transform.

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Developed by ChanRT | Fork me at GitHub