Fourier transform

The Fourier transform, named for Jean Baptiste Joseph Fourier, is a frequency transform that decomposes a function into its sine and cosine parts (basis functions) for several frequencies. The result is a function in the frequency domain, representing the frequency spectrum of the original function.

It is based on the theorem that any continuous periodic function can be represented as the sum of a series of sine waves:

f(x) = ∑ A₁ sin ω + A₂ sin ω/2 + A₃ sin ω/3 + ...

The first sine wave is the principal and the subsequent waves are the 2nd, 3rd, nth harmonics.

Alternatively, one can use as base functions the exponential function with imaginary arguments (i.e. exp(iωt) or e^iωt) because in this case the real part of the exponential function equals the cosine and the imaginary part equals the sine. This form is also called complex Fourier transform because the transformed function takes complex values and the original function can take complex values.

The Fourier transform is a linear operator which transforms functions with domain A into functions with domain B. Depending on A and B, we distinguish:

• Fourier series (A: interval - usually [-π, π], B: integers)

• Discrete Fourier transform (A and B: segment of natural numbers - usually 0, ..., N-1)

• Continuous Fourier transform (A and B: the set R of real numbers)

All the above are generalized by the Fourier transform on locally compact topological groups, which is studied in harmonic analysis; here, A is the group and B is its dual group. This treatment also allows a general formulation of the convolution theorem, which relates Fourier transforms and convolutions.

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 compute intensive. Such transforms are used in some types of RF modulation.