Fourier analysis

Advanced

Any periodic wave is a sum of sines.

Any periodic wave can be written as a sum of sinusoids at multiples of the fundamental. A square wave contains odd harmonics with amplitudes 1/n.

Worked example

Audio spectrum analysers show these components.

Common mistake

Non-periodic signals need the Fourier transform.

In more detail

Fourier analysis breaks any periodic signal into a sum of sinusoids at harmonic frequencies. It reveals the frequency content of a sound or signal.

Audio equalisers and spectrum analysers use Fourier methods.
Medical MRI scans reconstruct images using Fourier transforms.
Engineers analyse vibration data by decomposing signals into sine waves, which shows which frequencies dominate.

Can any periodic function be decomposed?

Yes, under mild conditions, into a sum of sines and cosines.

Why is it useful?

It turns a complicated waveform into simple frequency components.

What is Fourier analysis used for?

Separating a complex signal into its frequency components, which is used in audio, imaging and communications.