Problems on fourier transform
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Answer:
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"). There are many closely-related variations of this transform, summarized below, depending upon the type of function being transformed. See also: List of Fourier-related transforms.
Contents
1 Applications
2 Variants of the Fourier transform
2.1 Continuous Fourier transform
2.2 Fourier series
2.3 Discrete Fourier transform
2.4 Other variants
2.5 Family of Fourier transforms
3 Interpretation in terms of time and frequency
4 References
5 See also
6 External links
Applications
Fourier transforms have many scientific applications — in physics, number theory, combinatorics, signal processing, probability theory, statistics, cryptography, acoustics, oceanography, optics, geometry, and other areas. (In signal processing and related fields, the Fourier transform is typically thought of as decomposing a signal into its component frequencies and their amplitudes.) This wide applicability stems from several useful properties of the transforms:
The transforms are linear operators and, with proper normalization, are unitary as well (a property known as Parseval's theorem or, more generally, as the Plancherel theorem, and most generally via Pontryagin duality).
The transforms are invertible, and in fact the inverse transform has almost the same form as the forward transform.
The sinusoidal basis functions are eigenfunctions of differentiation, which means that this representation transforms linear differential equations with constant coefficients into ordinary algebraic ones. (For example, in a linear time-invariant physical system, frequency is a conserved quantity, so the behavior at each frequency can be solved independently.)
By the convolution theorem, Fourier transforms turn the complicated convolution operation into simple multiplication, which means that they provide an efficient way to compute convolution-based operations such as polynomial multiplication and multiplying large numbers.
The discrete version of the Fourier transform (see below) can be evaluated quickly on computers using fast Fourier transform (FFT) algorithms.
Variants of the Fourier transform
Continuous Fourier transform
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):
{\displaystyle f(t)={\mathcal {F}}^{-1}(F)(t)={\frac {1}{\sqrt {2\pi }}}\int \limits _{-\infty }^{\infty }F(\omega )e^{i\omega t}\,d\omega .} {\displaystyle f(t)={\mathcal {F}}^{-1}(F)(t)={\frac {1}{\sqrt {2\pi }}}\int \limits _{-\infty }^{\infty }F(\omega )e^{i\omega t}\,d\omega .}
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. See continuous Fourier transform for more information, including a table of transforms, discussion of the transform properties, and the various conventions. A generalization of this transform is the fractional Fourier transform, by which the transform can be raised to any real "power".
When f(t) is an even or odd function, the sine or cosine terms disappear and one is left with the cosine transform or sine transform, respectively. Another important case is where f(t) is purely real, where it follows that F(−ω) = F(ω)*. (Similar special cases appear for all other variants of the Fourier transform as well.)
Fourier series
The continuous transform is itself actually a generalization of an earlier concept, a Fourier series, which was specific to periodic (or finite-domain) functions f(x) (with period 2π), and represents these functions as a series of sinusoids: