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Applications of Fourier transform

Define a Fourier Transform. What are its properties and application areas ? Describe its application in signal processing.

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The Fourier transform, named after Joseph Fourier , is an integral transform that re-expresses a function in terms of sinusoidal basis functions , i.e. as a sum or integral of sinusoidal functions multiplied by some coefficients ("amplitudes").

Most often, the unqualified term "Fourier transform" refers to the continuous Fourier Transform, representing any square-integrable function f(t) as a sum of complex exponentials with angular frequencies ω and complex amplitudes F(ω):

This is actually the inverse continuous Fourier transform, whereas the Fourier transform expresses F(ω) in terms of f(t); the original function and its transform are sometimes called a transform pair.
Fourier transforms have many scientific applications — in physics , number theory , signal processing , probability theory , statistics , cryptography , acoustics , oceanography , optics , geometry , and other areas. In signal processing and related fields, the ...

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This solution is comprised of a detailed explanation to define a Fourier Transform.