Fourier Series

Any repeating signal, even a sharp square wave, is a sum of smooth sine waves. Add more of them and the corners get sharper.

How it works

Infinite series of sines

In one sentence

A square wave of period $2\pi$ equals $\frac{4}{\pi}\sum_{k\ge 1}\frac{\sin((2k-1)x)}{2k-1}$ : the sum of odd sine harmonics with amplitudes falling like $1/n$ .

General formula

{\htmlData{sym=2}{f(x)}} = {\htmlData{sym=0}{\frac{4}{\pi}}}\sum_{{\htmlData{sym=3}{k}}=1}^{\infty}\frac{\sin\big(({\htmlData{sym=1}{2k-1}}){\htmlData{sym=4}{x}}\big)}{{\htmlData{sym=1}{2k-1}}}

Each sine adds a finer ripple, and together they build the corners; the $1/(2k-1)$ weights say higher harmonics matter less.

What the symbols mean

$\frac{4}{\pi}$
Scale factor
Sets the square wave's height to exactly ±1.
$2k-1$
Odd harmonics
Only odd frequencies (1, 3, 5, …) appear in a square wave.
$f(x)$
Square wave
The signal being built out of sine waves.
$k$
Term number
Which sine wave in the sum: k = 1 is the fundamental.
$x$
Position
Time or position along the wave.

A square wave contains only odd harmonics, with amplitudes falling like $1/k$ .
Each coefficient comes from an integral measuring how much of that frequency is present.
The overshoot next to each jump never fully disappears. That is the Gibbs phenomenon.

Learn the math

Trig Functions and EquationsCalc I · Lesson 2, Review: The Functions of Calculus Integration by PartsCalc II · Lesson 1, Integration Techniques Integrals Involving Trig FunctionsCalc II · Lesson 2, Integration Techniques Orthogonal SetsLin Alg · Lesson 32, Orthogonality and Least Squares Periodic and Orthogonal FunctionsDiff Eq · Lesson 43, Boundary Value Problems and Fourier Series Fourier Sine SeriesDiff Eq · Lesson 44, Boundary Value Problems and Fourier Series Fourier SeriesDiff Eq · Lesson 46, Boundary Value Problems and Fourier Series The Fourier Series ToolkitPDE · Lesson 3, Foundations From Signals to Bits: Sampling and QuantizationDiscrete · Lesson 36, The Math Inside Computers

Learn the math

Trig Functions and EquationsCalc I · Lesson 2, Review: The Functions of Calculus

Integration by PartsCalc II · Lesson 1, Integration Techniques

Integrals Involving Trig FunctionsCalc II · Lesson 2, Integration Techniques

Orthogonal SetsLin Alg · Lesson 32, Orthogonality and Least Squares

Periodic and Orthogonal FunctionsDiff Eq · Lesson 43, Boundary Value Problems and Fourier Series

Fourier Sine SeriesDiff Eq · Lesson 44, Boundary Value Problems and Fourier Series

Fourier SeriesDiff Eq · Lesson 46, Boundary Value Problems and Fourier Series

The Fourier Series ToolkitPDE · Lesson 3, Foundations

From Signals to Bits: Sampling and QuantizationDiscrete · Lesson 36, The Math Inside Computers