58380809017 http://1ucasvb.tumblr.com/post/58380809017/the-additive-synthesis-through-fourier-series-of 416 The additive synthesis, through Fourier series, of...

The additive synthesis, through Fourier series, of square, sawtooth and triangle waves

I just updated these three animations. The old ones I had on Wikipedia were among the first I’ve ever done, and were ugly, tiny and outdated.

What you’re seeing here is how a periodic function (in blue) can be approximated by a Fourier series (in red). The number N shows how many terms are being used.

A Fourier series is just a sum (usually infinite) of sines and cosines of different frequencies and amplitudes that approximates a desired function. These frequencies and amplitudes constitute what we call the frequency domain of the function, though this is more useful when we consider a continuous spectrum of frequencies.

The function doesn’t need to be continuous, as you can see in the case of the square and the sawtooth wave.

However, the Fourier series approximation does get a bit wacky around the discontinuities. The wavyness near those points never really goes away, but it usually stays within a certain limit. This is known as Gibbs phenomenon, and it’s a familiar problem in signal processing.

Due to the simplicity of sines and cosines, Fourier series are a great tool when studying the behavior of more complicated periodic functions, a common problem in differential equations with extremely wide applications in physics and engineering.

An important aspect of all of this that gets brushed over in most classes is that you don’t really need both sines and cosines in a Fourier series, since the sum of a sine and a cosine of the same frequency is just a senoidal function with a different phase and amplitude. In other words, you only use sine and cosines in order to encode a phase.

This is why Fourier series are much more elegantly handled using complex numbers, as the complex exponential can handle both phase and amplitude very succintly.

In the future I’m hoping to make a post explaining, in terms most people with a basic understanding of math can understand, why all of this Fourier analysis stuff works in the first place, and why you should think it is awesome. Because it is very awesome.

Previous posts on Fourier analysis

 
  1. solidstatesounds reblogged this from alyxandros
  2. alyxandros reblogged this from 1ucasvb
  3. archonsandtaters reblogged this from 1ucasvb
  4. quantumvyp3r reblogged this from 1ucasvb
  5. claireever reblogged this from 1ucasvb
  6. rycallahan reblogged this from 1ucasvb and added:
    The best kind of waves…
  7. 6analysis reblogged this from 1ucasvb
  8. yellowtypophile reblogged this from visualizingmath
  9. ahmed-el3azab reblogged this from 1ucasvb
  10. verylittleprivacy reblogged this from 1ucasvb
  11. vietdanny reblogged this from bennbatt
  12. spaghettiplot reblogged this from 1ucasvb
  13. jania-marie reblogged this from visualizingmath
  14. andreuinyu reblogged this from 1ucasvb and added:
    quieres Fourier? Dos tazas
  15. javierid reblogged this from physicsshiny
  16. can-i-leave-this-blank reblogged this from psychomath
  17. physicsshiny reblogged this from psychomath
  18. psychomath reblogged this from recursiverecursion
  19. darkoyster reblogged this from sleepytimejesse and added:
    Math is so crazy.
  20. sleepytimejesse reblogged this from 1ucasvb