⊕

Fourier Series

Undergraduate

Definition

Fourier series represents periodic functions as infinite sums of sines and cosines. Any periodic function can be decomposed into oscillatory components.

Formulas

f(x) = (a₀)/(2) + ∑_n=1^∈fty (aₙ cos nx + bₙ sin nx)

Fourier series

aₙ = (1)/(π) ∈t_-π^π f(x) cos(nx) dx

Fourier coefficient (cosine)

bₙ = (1)/(π) ∈t_-π^π f(x) sin(nx) dx

Fourier coefficient (sine)

Examples

Example 1

Find the Fourier series of f(x) = x for -π < x < π.

History

Discovered by: Joseph Fourier (1822)

Fourier developed this to solve the heat equation.

Applications

Signal Processing

Frequency analysis

Acoustics

Sound analysis, speech recognition

Image Processing

JPEG compression