# Fourier Analysis

A mathematical analysis of waves, discovered by the French mathematician Fourier (1768-1830). Fourier proved that any periodic sound, or any non-periodic sound of limited duration, could be represented (Fourier analysis) or created out of (Fourier synthesis) the sum of a set of pure tones with different frequencies, amplitudes and phases.

- For the function f(x) defined over the interval c ≤ x ≤ c+2L

- Where

### Special Fourier Series

**Square wave**given by the function

- This can be expressed as the fourier series:

- As the number of terms, n in the series is increased so the accuracy of the approximation improves.

**Sawtooth**given by the function

- This can be expressed as the fourier series:

**Triangular wave**given by:

- This can be expressed as the fourier series:

**Half wave rectified sine wave**given by:

- This can be expressed as the fourier series:

**The full wave rectified sine wave**given by

- This can be expressed as the fourier series:

### Acknowledgement

We would like to thank B. Kanmani (Department of Telecommunication Engineering, BMS College of Engineering, Bangalore, India) for corrections to this page.

**See also: **Discrete Fourier Transform, Fourier, Baron Jean Baptiste Joseph.

**Subjects: ** Analysis Signal Processing

- Fourier Transform and Its Applications, , McGraw-Hill Book Company Ltd
- Fast Fourier Transform and Its Applications, , Prentice-Hall

**Further reading:**

- Acoustics and Vibrations Animations A collection of animations produced by Daniel A. Russell, Ph.D., Associate Professor of Applied Physics at Kettering University.

**Weblinks:**