Fourier transform

The Fourier transform (FT) 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 (that sum to it, 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) "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).

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