FourierBasis

The Fourier basis is a fundamental concept in signal processing and mathematics, forming the bedrock of Fourier analysis. It represents any signal as a sum of simple sinusoidal waves of different frequencies and amplitudes. These sinusoidal waves, known as basis functions, are the cosine and sine functions. Specifically, the Fourier basis for a continuous-time signal is composed of the set of functions {cos(nω₀t), sin(nω₀t)} for integer values of n and a fundamental frequency ω₀. For discrete-time signals, the basis functions are {cos(2πkn/N), sin(2πkn/N)} where N is the number of samples and k is an integer.

The core idea behind the Fourier basis is that any sufficiently well-behaved periodic signal can be uniquely decomposed into a sum of these orthogonal basis functions. This decomposition is achieved through the Fourier series. For a non-periodic signal, the Fourier transform extends this concept, representing the signal as an integral over a continuous spectrum of frequencies.

The utility of the Fourier basis lies in its ability to transform a signal from its time-domain representation to its frequency-domain representation. This shift often simplifies analysis, revealing hidden periodicities, noise characteristics, or essential frequency components that are not apparent in the time domain. Applications are widespread, including audio and image compression, filtering, spectral analysis, and solving differential equations. The Fourier basis provides a powerful lens through which to understand and manipulate signals by dissecting them into their constituent frequencies.