Fourier transform

The Fourier transform (FT) of a function forms a new function of a particular type. The transform provides an alternate, equivalent function that models the same situation using different quantities. A variant on the transform (inverse Fourier transform or backward Fourier transform as opposed to the normal, or forward Fourier transform) "undoes" this. Despite being opposite transformations, the computation-work to carry them out is very nearly the same and software that does one can do the other after minor adjustments. The forward transformation is:

F(s) is

 ∞
∫  f(x)e-2πisxdx
-∞

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.

(Note that given the eⁱ base, this integral can be expressed as that of the sum of sine and cosine terms through use of Euler's formula.) The inverse transformation produces the original function, f(x), from this F(s).

A common use of the Fourier transform is to calculate the coefficients of a Fourier series, which is a series of periodic functions equivalent to some original periodic function (or function over a finite span) in that the some of the series of functions matches the original function, much as does a Taylor series associated with a function. This is commonly used for the analysis of periodic functions that represent waves, such as electromagnetic radiation or sound. It is typically used to determine what sine-like components make up a particular periodic function, in particular, to break a signal down into its harmonics.

The fast Fourier transform (FFT) algorithm, an efficient means of computing such a Fourier series, is ubiquitous in physics, engineering analysis, and modeling.