# Fourier Transform

The Fourier transform, named after Joseph Fourier, is a mathematical transform with many applications in physics and engineering. Very commonly it transforms a mathematical function of time, into a new function, sometimes denoted by or whose argument is frequency with units of cycles or radians per second. The new function is then known as the Fourier transform and/or the frequency spectrum of the function The Fourier transform is also a reversible operation. Thus, given the function one can determine the original function, (See Fourier inversion theorem.) and are also respectively known as time domain and frequency domain representations of the same "event". Most often perhaps, is a real-valued function, and is complex valued, where a complex number describes both the amplitude and phase of a corresponding frequency component. In general, is also complex, such as the analytic representation of a real-valued function. The term "Fourier transform" refers to both the transform operation and to the complex-valued function it produces.

In the case of a periodic function (for example, a continuous but not necessarily sinusoidal musical sound), the Fourier transform can be simplified to the calculation of a discrete set of complex amplitudes, called Fourier series coefficients. Also, when a time-domain function is sampled to facilitate storage or computer-processing, it is still possible to recreate a version of the original Fourier transform according to the Poisson summation formula, also known as discrete-time Fourier transform. These topics are addressed in separate articles. For an overview of those and other related operations, refer to Fourier analysis or List of Fourier-related transforms.

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