### Fourier transform

(systematic method of breaking down functions into periodic components)

The **Fourier transform** of a function forms a new function of
a particular type. The transform is useful because it can be made
to produce a series of periodic functions equivalent to the original
function (in some sense like a **Taylor series**), if the original
function is periodic (or limited to a finite domain). This is
commonly used for the analysis of periodic functions representing
waves such as electromagnetic radiation or sound. As such, it is typically used to
determine what sine-like components make up a particular periodic
function, i.e., break a signal down into its harmonics. A variant
on the transform (**inverse Fourier transform** or **backward Fourier transform**
as opposed to the normal, or **forward Fourier transform**)
can "undo" this. Despite being opposite transformations,
the computation work to carry them out is virtually the same and
software that does one can easily be used to do the other.

The forward transformation is:

F(s) is
∞
∫ f(x)e^{-2πisx}dx
-∞

Where

- x is a real number.
- f(x) is a function on x.
- s is a real number with the inverse unit of x, e.g., if x is time, s is a frequency.
- F(s) is the transformed function.
*i* is the square root of -1.

The inverse transformation produces the original function, f(x), from this F(s).

The fast Fourier transform algorithm, an efficient means of computing a type
of *Fourier transform* is ubiquitous in physics, engineering analysis,
and modeling.

(*mathematics*)
**Further reading:**

http://en.wikipedia.org/wiki/Fourier_transform

http://www.thefouriertransform.com/

**Referenced by pages:**

Canada-France-Hawaii Telescope (CFHT)

convolution

fast Fourier transform (FFT)

Fourier space

imaging Fourier transform spectroscopy (IFTS)

linear theory

phase dispersion minimization (PDM)

spectral correlator

spectral density

spectral method

Index