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Concept
Sine and cosine transforms
Summary
In mathematics, the Fourier sine and cosine transforms are forms of the Fourier transform that do not use complex numbers or require negative frequency. They are the forms originally used by Joseph Fourier and are still preferred in some applications, such as signal processing or statistics.
The Fourier sine transform of f(t), sometimes denoted by either or , is
If t means time, then ξ is frequency in cycles per unit time, but in the abstract, they can be any pair of variables which are dual to each other.
This transform is necessarily an odd function of frequency, i.e. for all ξ:
The numerical factors in the Fourier transforms are defined uniquely only by their product. Here, in order that the Fourier inversion formula not have any numerical factor, the factor of 2 appears because the sine function has L2 norm of
The Fourier cosine transform of f(t), sometimes denoted by either or , is
It is necessarily an even function of frequency, i.e. for all ξ:
Since positive frequencies can fully express the transform, the non-trivial concept of negative frequency needed in the regular Fourier transform can be avoided.
Some authors only define the cosine transform for even functions of t, in which case its sine transform is zero. Since cosine is also even, a simpler formula can be used,
Similarly, if f is an odd function, then the cosine transform is zero and the sine transform can be simplified to
Just like the Fourier transform takes the form of different equations with different constant factors (see ), other authors also define the cosine transform as
and sine as
or, the cosine transform as and the sine transform as using as the transformation variable. And while t is typically used to represent the time domain, x is often used alternatively, particularly when representing frequencies in a spatial domain.
The original function f can be recovered from its transform under the usual hypotheses, that f and both of its transforms should be absolutely integrable. For more details on the different hypotheses, see Fourier inversion theorem.
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Even and odd functions
In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. They are important in many areas of mathematical analysis, especially the theory of power series and Fourier series. They are named for the parity of the powers of the power functions which satisfy each condition: the function is an even function if n is an even integer, and it is an odd function if n is an odd integer.
Fourier series
A Fourier series (ˈfʊrieɪ,_-iər) is an expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series, but not all trigonometric series are Fourier series. By expressing a function as a sum of sines and cosines, many problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Joseph Fourier to find solutions to the heat equation.
Fourier transform
In physics and mathematics, the Fourier transform (FT) is a transform that converts a function into a form that describes the frequencies present in the original function. The output of the transform is a complex-valued function of frequency. The term Fourier transform refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made the Fourier transform is sometimes called the frequency domain representation of the original function.