Mathematics
Fourier Series for Representing Periodic Functions Beyond Sines and Cosines
Quick fact
The trigonometric Fourier series is just one of infinitely many ways to decompose a periodic function; functions can be represented using any complete orthonormal set, such as complex exponentials, Walsh functions, or Haar wavelets, each highlighting different features of the signal.
Why this is interesting
You've heard that any repeating wave can be built from smooth sine waves—but what if we used square waves or even random-looking blips instead? The music might sound different, but the information would be exactly the same.
Read the full explanation
Understanding Fourier Series for Representing Periodic Functions Beyond Sines and Cosines
When we say 'Fourier series,' we often picture adding up sine and cosine waves to build a square wave or a sawtooth. This works because sines and cosines are like the 'primary colors' of periodic signals. But these are not the only such set. A periodic function can be thought of as a vector in an infinite-dimensional space. Any set of mutually perpendicular (orthogonal) functions that spans that space can serve as a coordinate system. For example, instead of sines, we could use complex exponentials (which are equivalent but often cleaner), or we could use rectangular pulses like Walsh functions. Just as you can describe a point using Cartesian or polar coordinates, you can describe a periodic function using different bases. The key is that the basis functions must be orthogonal - they must overlap with each other in a particular way - so that we can compute each coefficient independently, and they must be complete, meaning they can represent any function in that space.
A deeper explanation
The mechanism behind any generalized Fourier series is the inner product of functions. For real functions on a period, the inner product is the integral of their product over one period. Two functions are orthogonal if this integral is zero. Given an orthonormal set {φn}, the coefficients are simply the projections: cn = ∫ f(x)φn(x) dx. Trig functions are one such set, but so are complex exponentials e^{i n x}, which diagonalize derivatives and are fundamental in quantum mechanics. Walsh functions are piecewise constant and form a complete orthonormal set on [0,1]; they are used in digital signal processing because they are simple and fast. Haar wavelets are orthogonal and can localize in both time and frequency, making them basis for wavelet analysis. The trigonometric series is not unique; it is a choice that trades off features like smoothness and locality. This insight reveals that the power of Fourier series lies not in sines themselves but in the principle of orthogonal expansions, which unifies many areas of mathematics and engineering.