Fourier Analysis

Fourier analysis is the branch of mathematics that decomposes functions into sums or integrals of simple oscillations, the sines and cosines. Its central insight, due to Joseph Fourier, is that a periodic function can be represented as a sum of sinusoids whose frequencies are whole-number multiples of a fundamental frequency. This representation, the Fourier series, expresses each component by a coefficient that records how much of that particular frequency the original function contains.

The individual sinusoids are called harmonics. Combining them in the right amplitudes and phases reconstructs even sharp-cornered shapes such as square or sawtooth waves, although near a discontinuity the approximation overshoots slightly, an effect known as the Gibbs phenomenon. Because each harmonic corresponds to a definite frequency, the Fourier series effectively separates a signal into its frequency content, revealing structure that is not obvious in the original time-domain form.

For functions that are not periodic, the Fourier transform generalises the idea, replacing the discrete set of harmonics with a continuous range of frequencies and producing a spectrum that shows how energy is distributed across them. The transform and its inverse let a signal be moved freely between its time and frequency descriptions. These techniques are foundational in physics and engineering, including acoustics, optics, image processing, communications and the numerical analysis of differential equations.

Frequently asked questions

What is a Fourier series?

It is a representation of a periodic function as a sum of sine and cosine waves whose frequencies are integer multiples of a fundamental frequency.

How does the Fourier transform differ from a Fourier series?

A Fourier series applies to periodic functions and uses a discrete set of frequencies, while the Fourier transform handles non-periodic functions using a continuous range of frequencies.

What are harmonics?

Harmonics are the sinusoidal components of a signal whose frequencies are whole-number multiples of the fundamental; their combination shapes the overall waveform.