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Another cool thing is that orthonormal bases are not unique - there are many other basis functions that you can choose beyond just sine and cosine to decompose a function (or digital signal). Though they are a natural choice if you are specifically looking to analyze periodicities.

One direction to go in for further study:


Yes! Same goes for pulling eigenvectors out of a data set e.g. for PCA (Principal Components Analysis) -- those form an orthonormal basis. You can even pick a set of random orthogonal vectors of the same number and dimension as your original data and re-represent the data with no loss of information.

I had a lot of fun demonstrating this concept for reconstructing images via a Processing sketch a few years ago and still use it for teaching from time to time. All source code for quick and dirty Haar, Eigen, Random, and Fourier-like methods included here: http://www.cc.gatech.edu/~phlosoft/transforms/

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