An Interactive Introduction to Fourier Transforms
Even after taking a college level signal processing course, I didn't really grok why we use sines and cosines. Any explanation of Fourier theory just starts with the idea to break up a signal into sine and cosine components. Perhaps they'd say those are the simplest, purest instantiation of a frequency. But the idea still just falls out of the sky.<p>I believe I know now: it's because of the convolution theorem. When performing a Fourier transform, we represent the signal in a new basis, where each component will independently get transformed by linear shift invariant systems.<p>Basically the Fourier basis diagonalizes the convolution operator. And the even deeper reason for that is that a complex exponential function can be shifted by simply multiplying it with a constant.<p>In simpler words, it comes down to the barber pole illusion. If you rotate a spring-shaped 3d curve, it looks as if it was traveling upwards. And the 2d projection of the spring are sines and cosines.<p>And it turns out that linear, shift-invariant systems are really common (or are at least a good approximation of many common natural phenomena), so it's very helpful to break up a signal into pieces that each get independently transformed, without any interaction effects.