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Fourier Transform

The Fourier transform is a mathematical transformation that re-expresses a function in terms of sinusoidal functions, and vice versa. Although there are several different specific definitions and conventions, the form of the Fourier transform with unitary normalization constants is

where ƒ(t) is a function of time, i is the square root of -1, and the Fourier transform F(ω) is the Fourier transform of ƒ(t) and is a function of (time)^-1, or frequency. If the original function ƒ is a function of distance, its Fourier transform is a function of (distance)^-1, or wavenumber.

For real, even functions, the above definition becomes

which is also called the cosine transform.

The function ƒ(t) is related to F(v) via a Fourier transform; that is,

In 1892, Lord Rayleigh recognized that a spectrum was related to its interferogram via a Fourier transform. The conversion of an interferogram into a spectrum was first accomplished by Fellgett in 1949. However, it was not until the introduction of spectrometer-dedicated computers and the development of the Cooley-Tukey algorithm, called the fast Fourier transform, in 1965 that Fourier transformbased spectroscopy began to grow in popularity.