Fourier transformThe Fourier transform, named for Jean Baptiste Joseph Fourier, is a frequency transform that decomposes a function into its sine and/or cosine parts (basis functions) for several frequencies. The result is a function in the frequency domain, representing the frequency spectrum of the original function.
The Fourier transform taking functions with domain A into functions with domain B may be:
- a Fourier series, in which A is an interval, such as [-π, &pi] and B is the set of all integers)
- a discrete Fourier transform in which A and B are segments of natural numbers - usually 0, ..., N − 1)
- a Continuous Fourier transform, in which A and B are the whole real line.
The Fourier transform can be viewed as a special case of the Z-transform: the Fourier transform is the Z-transform evaluated at the unit circle in the complex space.
See the Fourier transform in action on the SETI at home project.
Actual implementations of Fourier transforms of arbitrary signals are computationally intensive. Such transforms are used in some types of RF modulation.